id	sid	tid	token	lemma	pos
ejpam-6275	1	1	european	european	PROPN
ejpam-6275	1	2	journal	journal	PROPN
ejpam-6275	1	3	of	of	ADP
ejpam-6275	1	4	pure	pure	ADJ
ejpam-6275	1	5	and	and	CCONJ
ejpam-6275	1	6	applied	applied	ADJ
ejpam-6275	1	7	mathematics	mathematic	NOUN
ejpam-6275	1	8	2025	2025	NUM
ejpam-6275	1	9	,	,	PUNCT
ejpam-6275	1	10	vol	vol	NOUN
ejpam-6275	1	11	.	.	PROPN
ejpam-6275	1	12	18	18	NUM
ejpam-6275	1	13	,	,	PUNCT
ejpam-6275	1	14	issue	issue	NOUN
ejpam-6275	1	15	3	3	NUM
ejpam-6275	1	16	,	,	PUNCT
ejpam-6275	1	17	article	article	NOUN
ejpam-6275	1	18	number	number	NOUN
ejpam-6275	1	19	6275	6275	NUM
ejpam-6275	1	20	issn	issn	PROPN
ejpam-6275	1	21	1307	1307	NUM
ejpam-6275	1	22	-	-	SYM
ejpam-6275	1	23	5543	5543	NUM
ejpam-6275	1	24	–	–	PUNCT
ejpam-6275	1	25	ejpam.com	ejpam.com	X
ejpam-6275	1	26	published	publish	VERB
ejpam-6275	1	27	by	by	ADP
ejpam-6275	1	28	new	new	PROPN
ejpam-6275	1	29	york	york	PROPN
ejpam-6275	1	30	business	business	PROPN
ejpam-6275	1	31	global	global	ADJ
ejpam-6275	1	32	design	design	NOUN
ejpam-6275	1	33	and	and	CCONJ
ejpam-6275	1	34	analysis	analysis	NOUN
ejpam-6275	1	35	of	of	ADP
ejpam-6275	1	36	shortened	shorten	VERB
ejpam-6275	1	37	bose	bose	NOUN
ejpam-6275	1	38	-	-	PUNCT
ejpam-6275	1	39	chaudhuri	chaudhuri	PROPN
ejpam-6275	1	40	-	-	PUNCT
ejpam-6275	1	41	hocquenghem	hocquenghem	PROPN
ejpam-6275	1	42	codes	code	NOUN
ejpam-6275	1	43	for	for	ADP
ejpam-6275	1	44	efficient	efficient	ADJ
ejpam-6275	1	45	data	datum	NOUN
ejpam-6275	1	46	transmission	transmission	NOUN
ejpam-6275	1	47	over	over	ADP
ejpam-6275	1	48	the	the	DET
ejpam-6275	1	49	eisenstein	eisenstein	PROPN
ejpam-6275	1	50	fields	field	VERB
ejpam-6275	1	51	muhammad	muhammad	PROPN
ejpam-6275	1	52	sajjad1,∗	sajjad1,∗	PROPN
ejpam-6275	1	53	,	,	PUNCT
ejpam-6275	1	54	rida	rida	PROPN
ejpam-6275	1	55	asghar2	asghar2	PROPN
ejpam-6275	1	56	,	,	PUNCT
ejpam-6275	2	1	mushtaq	mushtaq	PROPN
ejpam-6275	2	2	k.	k.	PROPN
ejpam-6275	2	3	abdalrahem3	abdalrahem3	PROPN
ejpam-6275	2	4	,	,	PUNCT
ejpam-6275	2	5	adnan	adnan	PROPN
ejpam-6275	2	6	burhan	burhan	PROPN
ejpam-6275	2	7	rajab4	rajab4	PROPN
ejpam-6275	2	8	,	,	PUNCT
ejpam-6275	2	9	mahesha	mahesha	PROPN
ejpam-6275	2	10	narayana5	narayana5	PROPN
ejpam-6275	2	11	,	,	PUNCT
ejpam-6275	2	12	moin	moin	X
ejpam-6275	2	13	-	-	PUNCT
ejpam-6275	2	14	ud	ud	ADP
ejpam-6275	2	15	-	-	PUNCT
ejpam-6275	2	16	din	din	NOUN
ejpam-6275	2	17	junjua6,∗	junjua6,∗	NOUN
ejpam-6275	2	18	,	,	PUNCT
ejpam-6275	2	19	salman	salman	PROPN
ejpam-6275	2	20	a.	a.	PROPN
ejpam-6275	2	21	alqahtani7	alqahtani7	PROPN
ejpam-6275	2	22	1	1	NUM
ejpam-6275	2	23	nutech	nutech	NOUN
ejpam-6275	2	24	school	school	NOUN
ejpam-6275	2	25	of	of	ADP
ejpam-6275	2	26	applied	apply	VERB
ejpam-6275	2	27	science	science	NOUN
ejpam-6275	2	28	and	and	CCONJ
ejpam-6275	2	29	humanities	humanity	NOUN
ejpam-6275	2	30	,	,	PUNCT
ejpam-6275	2	31	national	national	ADJ
ejpam-6275	2	32	university	university	PROPN
ejpam-6275	2	33	of	of	ADP
ejpam-6275	2	34	technology	technology	NOUN
ejpam-6275	2	35	,	,	PUNCT
ejpam-6275	2	36	islamabad	islamabad	PROPN
ejpam-6275	2	37	,	,	PUNCT
ejpam-6275	2	38	44000	44000	NUM
ejpam-6275	2	39	,	,	PUNCT
ejpam-6275	2	40	pakistan	pakistan	PROPN
ejpam-6275	2	41	2	2	NUM
ejpam-6275	2	42	department	department	NOUN
ejpam-6275	2	43	of	of	ADP
ejpam-6275	2	44	mathematics	mathematic	NOUN
ejpam-6275	2	45	,	,	PUNCT
ejpam-6275	2	46	quaid	quaid	PROPN
ejpam-6275	2	47	-	-	PUNCT
ejpam-6275	2	48	i	i	PROPN
ejpam-6275	2	49	-	-	PUNCT
ejpam-6275	2	50	azam	azam	PROPN
ejpam-6275	2	51	university	university	PROPN
ejpam-6275	2	52	,	,	PUNCT
ejpam-6275	2	53	islamabad	islamabad	PROPN
ejpam-6275	2	54	,	,	PUNCT
ejpam-6275	2	55	pakistan	pakistan	PROPN
ejpam-6275	2	56	3	3	NUM
ejpam-6275	2	57	college	college	NOUN
ejpam-6275	2	58	of	of	ADP
ejpam-6275	2	59	pharmacy	pharmacy	NOUN
ejpam-6275	2	60	,	,	PUNCT
ejpam-6275	2	61	university	university	NOUN
ejpam-6275	2	62	of	of	ADP
ejpam-6275	2	63	al	al	PROPN
ejpam-6275	2	64	-	-	PUNCT
ejpam-6275	2	65	ameed	ameed	PROPN
ejpam-6275	2	66	4	4	NUM
ejpam-6275	2	67	department	department	NOUN
ejpam-6275	2	68	of	of	ADP
ejpam-6275	2	69	computer	computer	NOUN
ejpam-6275	2	70	engineering	engineering	NOUN
ejpam-6275	2	71	,	,	PUNCT
ejpam-6275	2	72	college	college	NOUN
ejpam-6275	2	73	of	of	ADP
ejpam-6275	2	74	engineering	engineering	PROPN
ejpam-6275	2	75	,	,	PUNCT
ejpam-6275	2	76	knowledge	knowledge	PROPN
ejpam-6275	2	77	university	university	PROPN
ejpam-6275	2	78	,	,	PUNCT
ejpam-6275	2	79	erbil	erbil	PROPN
ejpam-6275	2	80	44001	44001	NUM
ejpam-6275	2	81	,	,	PUNCT
ejpam-6275	2	82	iraq	iraq	PROPN
ejpam-6275	2	83	5	5	NUM
ejpam-6275	2	84	department	department	NOUN
ejpam-6275	2	85	of	of	ADP
ejpam-6275	2	86	mathematics	mathematic	NOUN
ejpam-6275	2	87	,	,	PUNCT
ejpam-6275	2	88	the	the	DET
ejpam-6275	2	89	university	university	NOUN
ejpam-6275	2	90	of	of	ADP
ejpam-6275	2	91	the	the	DET
ejpam-6275	2	92	west	west	PROPN
ejpam-6275	2	93	indies	indies	PROPN
ejpam-6275	2	94	,	,	PUNCT
ejpam-6275	2	95	kingston	kingston	PROPN
ejpam-6275	2	96	7	7	NUM
ejpam-6275	2	97	,	,	PUNCT
ejpam-6275	2	98	jamaica	jamaica	PROPN
ejpam-6275	2	99	6	6	NUM
ejpam-6275	2	100	school	school	NOUN
ejpam-6275	2	101	of	of	ADP
ejpam-6275	2	102	mathematical	mathematical	ADJ
ejpam-6275	2	103	sciences	sciences	PROPN
ejpam-6275	2	104	,	,	PUNCT
ejpam-6275	2	105	zhejiang	zhejiang	PROPN
ejpam-6275	2	106	normal	normal	PROPN
ejpam-6275	2	107	university	university	PROPN
ejpam-6275	2	108	,	,	PUNCT
ejpam-6275	2	109	jinhua	jinhua	PROPN
ejpam-6275	2	110	321004	321004	NUM
ejpam-6275	2	111	,	,	PUNCT
ejpam-6275	2	112	china	china	PROPN
ejpam-6275	2	113	7	7	NUM
ejpam-6275	2	114	new	new	ADJ
ejpam-6275	2	115	emerging	emerge	VERB
ejpam-6275	2	116	technologies	technology	NOUN
ejpam-6275	2	117	and	and	CCONJ
ejpam-6275	2	118	5	5	NUM
ejpam-6275	2	119	g	g	NOUN
ejpam-6275	2	120	network	network	NOUN
ejpam-6275	2	121	and	and	CCONJ
ejpam-6275	2	122	beyond	beyond	ADP
ejpam-6275	2	123	research	research	NOUN
ejpam-6275	2	124	chair	chair	NOUN
ejpam-6275	2	125	,	,	PUNCT
ejpam-6275	2	126	department	department	NOUN
ejpam-6275	2	127	of	of	ADP
ejpam-6275	2	128	computer	computer	NOUN
ejpam-6275	2	129	engineering	engineering	NOUN
ejpam-6275	2	130	,	,	PUNCT
ejpam-6275	2	131	college	college	NOUN
ejpam-6275	2	132	of	of	ADP
ejpam-6275	2	133	computer	computer	NOUN
ejpam-6275	2	134	and	and	CCONJ
ejpam-6275	2	135	information	information	NOUN
ejpam-6275	2	136	sciences	science	NOUN
ejpam-6275	2	137	,	,	PUNCT
ejpam-6275	2	138	king	king	NOUN
ejpam-6275	2	139	saud	saud	PROPN
ejpam-6275	2	140	university	university	PROPN
ejpam-6275	2	141	,	,	PUNCT
ejpam-6275	2	142	riyadh	riyadh	PROPN
ejpam-6275	2	143	,	,	PUNCT
ejpam-6275	2	144	saudi	saudi	PROPN
ejpam-6275	2	145	arabia	arabia	PROPN
ejpam-6275	2	146	abstract	abstract	NOUN
ejpam-6275	2	147	.	.	PUNCT
ejpam-6275	3	1	this	this	DET
ejpam-6275	3	2	article	article	NOUN
ejpam-6275	3	3	focuses	focus	VERB
ejpam-6275	3	4	on	on	ADP
ejpam-6275	3	5	the	the	DET
ejpam-6275	3	6	constructing	construct	VERB
ejpam-6275	3	7	and	and	CCONJ
ejpam-6275	3	8	decoding	decode	VERB
ejpam-6275	3	9	of	of	ADP
ejpam-6275	3	10	shortened	shorten	VERB
ejpam-6275	3	11	bose	bose	NOUN
ejpam-6275	3	12	-	-	PUNCT
ejpam-6275	3	13	chaudhurihocquenghem	chaudhurihocquenghem	PROPN
ejpam-6275	3	14	(	(	PUNCT
ejpam-6275	3	15	bch	bch	PROPN
ejpam-6275	3	16	)	)	PUNCT
ejpam-6275	3	17	codes	code	NOUN
ejpam-6275	3	18	over	over	ADP
ejpam-6275	3	19	the	the	DET
ejpam-6275	3	20	eisenstein	eisenstein	NOUN
ejpam-6275	3	21	fields	field	NOUN
ejpam-6275	3	22	,	,	PUNCT
ejpam-6275	3	23	based	base	VERB
ejpam-6275	3	24	on	on	ADP
ejpam-6275	3	25	the	the	DET
ejpam-6275	3	26	berlekamp	berlekamp	NOUN
ejpam-6275	3	27	-	-	PUNCT
ejpam-6275	3	28	massey	massey	PROPN
ejpam-6275	3	29	algorithm	algorithm	NOUN
ejpam-6275	3	30	(	(	PUNCT
ejpam-6275	3	31	bma	bma	PROPN
ejpam-6275	3	32	)	)	PUNCT
ejpam-6275	3	33	with	with	ADP
ejpam-6275	3	34	improved	improved	ADJ
ejpam-6275	3	35	decoding	decode	VERB
ejpam-6275	3	36	performance	performance	NOUN
ejpam-6275	3	37	.	.	PUNCT
ejpam-6275	4	1	thus	thus	ADV
ejpam-6275	4	2	,	,	PUNCT
ejpam-6275	4	3	eisenstein	eisenstein	NOUN
ejpam-6275	4	4	fields	field	NOUN
ejpam-6275	4	5	,	,	PUNCT
ejpam-6275	4	6	being	be	AUX
ejpam-6275	4	7	a	a	DET
ejpam-6275	4	8	natural	natural	ADJ
ejpam-6275	4	9	generalization	generalization	NOUN
ejpam-6275	4	10	of	of	ADP
ejpam-6275	4	11	gaussian	gaussian	ADJ
ejpam-6275	4	12	fields	field	NOUN
ejpam-6275	4	13	,	,	PUNCT
ejpam-6275	4	14	form	form	VERB
ejpam-6275	4	15	a	a	DET
ejpam-6275	4	16	well	well	ADV
ejpam-6275	4	17	-	-	PUNCT
ejpam-6275	4	18	devised	devise	VERB
ejpam-6275	4	19	algebraic	algebraic	ADJ
ejpam-6275	4	20	structure	structure	NOUN
ejpam-6275	4	21	for	for	ADP
ejpam-6275	4	22	error	error	NOUN
ejpam-6275	4	23	-	-	PUNCT
ejpam-6275	4	24	correcting	correct	VERB
ejpam-6275	4	25	codes	code	NOUN
ejpam-6275	4	26	and	and	CCONJ
ejpam-6275	4	27	have	have	VERB
ejpam-6275	4	28	better	well	ADJ
ejpam-6275	4	29	parameters	parameter	NOUN
ejpam-6275	4	30	for	for	ADP
ejpam-6275	4	31	transmission	transmission	NOUN
ejpam-6275	4	32	of	of	ADP
ejpam-6275	4	33	information	information	NOUN
ejpam-6275	4	34	in	in	ADP
ejpam-6275	4	35	noisy	noisy	ADJ
ejpam-6275	4	36	channels	channel	NOUN
ejpam-6275	4	37	.	.	PUNCT
ejpam-6275	5	1	the	the	DET
ejpam-6275	5	2	work	work	NOUN
ejpam-6275	5	3	starts	start	VERB
ejpam-6275	5	4	with	with	ADP
ejpam-6275	5	5	the	the	DET
ejpam-6275	5	6	definition	definition	NOUN
ejpam-6275	5	7	of	of	ADP
ejpam-6275	5	8	shortened	shorten	VERB
ejpam-6275	5	9	bch	bch	PROPN
ejpam-6275	5	10	codes	code	NOUN
ejpam-6275	5	11	over	over	ADP
ejpam-6275	5	12	the	the	DET
ejpam-6275	5	13	eisenstein	eisenstein	NOUN
ejpam-6275	5	14	fields	field	NOUN
ejpam-6275	5	15	with	with	ADP
ejpam-6275	5	16	the	the	DET
ejpam-6275	5	17	mention	mention	NOUN
ejpam-6275	5	18	of	of	ADP
ejpam-6275	5	19	generator	generator	NOUN
ejpam-6275	5	20	polynomials	polynomial	NOUN
ejpam-6275	5	21	and	and	CCONJ
ejpam-6275	5	22	the	the	DET
ejpam-6275	5	23	study	study	NOUN
ejpam-6275	5	24	of	of	ADP
ejpam-6275	5	25	certain	certain	ADJ
ejpam-6275	5	26	algebraic	algebraic	ADJ
ejpam-6275	5	27	characteristics	characteristic	NOUN
ejpam-6275	5	28	.	.	PUNCT
ejpam-6275	6	1	these	these	DET
ejpam-6275	6	2	codes	code	NOUN
ejpam-6275	6	3	are	be	AUX
ejpam-6275	6	4	designed	design	VERB
ejpam-6275	6	5	to	to	PART
ejpam-6275	6	6	provide	provide	VERB
ejpam-6275	6	7	the	the	DET
ejpam-6275	6	8	best	good	ADJ
ejpam-6275	6	9	balance	balance	NOUN
ejpam-6275	6	10	between	between	ADP
ejpam-6275	6	11	the	the	DET
ejpam-6275	6	12	word	word	NOUN
ejpam-6275	6	13	’s	’s	PART
ejpam-6275	6	14	length	length	NOUN
ejpam-6275	6	15	and	and	CCONJ
ejpam-6275	6	16	the	the	DET
ejpam-6275	6	17	ability	ability	NOUN
ejpam-6275	6	18	to	to	PART
ejpam-6275	6	19	cope	cope	VERB
ejpam-6275	6	20	with	with	ADP
ejpam-6275	6	21	error	error	NOUN
ejpam-6275	6	22	detection	detection	NOUN
ejpam-6275	6	23	and	and	CCONJ
ejpam-6275	6	24	correction	correction	NOUN
ejpam-6275	6	25	in	in	ADP
ejpam-6275	6	26	a	a	DET
ejpam-6275	6	27	variety	variety	NOUN
ejpam-6275	6	28	of	of	ADP
ejpam-6275	6	29	high	high	ADJ
ejpam-6275	6	30	-	-	PUNCT
ejpam-6275	6	31	efficiency	efficiency	NOUN
ejpam-6275	6	32	applications	application	NOUN
ejpam-6275	6	33	.	.	PUNCT
ejpam-6275	7	1	the	the	DET
ejpam-6275	7	2	modified	modified	PROPN
ejpam-6275	7	3	bma	bma	PROPN
ejpam-6275	7	4	is	be	AUX
ejpam-6275	7	5	presented	present	VERB
ejpam-6275	7	6	as	as	ADP
ejpam-6275	7	7	a	a	DET
ejpam-6275	7	8	decoding	decode	VERB
ejpam-6275	7	9	mechanism	mechanism	NOUN
ejpam-6275	7	10	that	that	PRON
ejpam-6275	7	11	is	be	AUX
ejpam-6275	7	12	equipped	equip	VERB
ejpam-6275	7	13	for	for	ADP
ejpam-6275	7	14	the	the	DET
ejpam-6275	7	15	translation	translation	NOUN
ejpam-6275	7	16	of	of	ADP
ejpam-6275	7	17	eisenstein	eisenstein	PROPN
ejpam-6275	7	18	field	field	PROPN
ejpam-6275	7	19	arithmetic	arithmetic	NOUN
ejpam-6275	7	20	,	,	PUNCT
ejpam-6275	7	21	which	which	PRON
ejpam-6275	7	22	is	be	AUX
ejpam-6275	7	23	different	different	ADJ
ejpam-6275	7	24	in	in	ADP
ejpam-6275	7	25	nature	nature	NOUN
ejpam-6275	7	26	.	.	PUNCT
ejpam-6275	8	1	the	the	DET
ejpam-6275	8	2	modification	modification	NOUN
ejpam-6275	8	3	concerns	concern	VERB
ejpam-6275	8	4	an	an	DET
ejpam-6275	8	5	improved	improved	ADJ
ejpam-6275	8	6	approach	approach	NOUN
ejpam-6275	8	7	to	to	ADP
ejpam-6275	8	8	the	the	DET
ejpam-6275	8	9	residual	residual	ADJ
ejpam-6275	8	10	terms	term	NOUN
ejpam-6275	8	11	defined	define	VERB
ejpam-6275	8	12	by	by	ADP
ejpam-6275	8	13	the	the	DET
ejpam-6275	8	14	classes	class	NOUN
ejpam-6275	8	15	of	of	ADP
ejpam-6275	8	16	higher	high	ADJ
ejpam-6275	8	17	level	level	NOUN
ejpam-6275	8	18	,	,	PUNCT
ejpam-6275	8	19	which	which	PRON
ejpam-6275	8	20	leads	lead	VERB
ejpam-6275	8	21	to	to	ADP
ejpam-6275	8	22	the	the	DET
ejpam-6275	8	23	efficiency	efficiency	NOUN
ejpam-6275	8	24	of	of	ADP
ejpam-6275	8	25	convergence	convergence	NOUN
ejpam-6275	8	26	and	and	CCONJ
ejpam-6275	8	27	the	the	DET
ejpam-6275	8	28	reduction	reduction	NOUN
ejpam-6275	8	29	of	of	ADP
ejpam-6275	8	30	numerical	numerical	ADJ
ejpam-6275	8	31	complexity	complexity	NOUN
ejpam-6275	8	32	in	in	ADP
ejpam-6275	8	33	contrast	contrast	NOUN
ejpam-6275	8	34	to	to	ADP
ejpam-6275	8	35	the	the	DET
ejpam-6275	8	36	bma	bma	PROPN
ejpam-6275	8	37	.	.	PUNCT
ejpam-6275	9	1	the	the	DET
ejpam-6275	9	2	imitations	imitation	NOUN
ejpam-6275	9	3	show	show	VERB
ejpam-6275	9	4	that	that	SCONJ
ejpam-6275	9	5	the	the	DET
ejpam-6275	9	6	proposed	propose	VERB
ejpam-6275	9	7	codes	code	NOUN
ejpam-6275	9	8	offer	offer	VERB
ejpam-6275	9	9	a	a	DET
ejpam-6275	9	10	higher	high	ADJ
ejpam-6275	9	11	error	error	NOUN
ejpam-6275	9	12	-	-	PUNCT
ejpam-6275	9	13	correcting	correct	VERB
ejpam-6275	9	14	capability	capability	NOUN
ejpam-6275	9	15	and	and	CCONJ
ejpam-6275	9	16	faster	fast	ADJ
ejpam-6275	9	17	computation	computation	NOUN
ejpam-6275	9	18	compared	compare	VERB
ejpam-6275	9	19	to	to	ADP
ejpam-6275	9	20	the	the	DET
ejpam-6275	9	21	traditional	traditional	ADJ
ejpam-6275	9	22	bch	bch	PROPN
ejpam-6275	9	23	codes	code	NOUN
ejpam-6275	9	24	over	over	ADP
ejpam-6275	9	25	finite	finite	ADJ
ejpam-6275	9	26	fields	field	NOUN
ejpam-6275	9	27	when	when	SCONJ
ejpam-6275	9	28	reliability	reliability	NOUN
ejpam-6275	9	29	and	and	CCONJ
ejpam-6275	9	30	low	low	ADJ
ejpam-6275	9	31	latency	latency	NOUN
ejpam-6275	9	32	are	be	AUX
ejpam-6275	9	33	desired	desire	VERB
ejpam-6275	9	34	.	.	PUNCT
ejpam-6275	10	1	2020	2020	NUM
ejpam-6275	10	2	mathematics	mathematic	NOUN
ejpam-6275	10	3	subject	subject	NOUN
ejpam-6275	10	4	classifications	classification	NOUN
ejpam-6275	10	5	:	:	PUNCT
ejpam-6275	10	6	94b75	94b75	NUM
ejpam-6275	10	7	,	,	PUNCT
ejpam-6275	10	8	11t71	11t71	NUM
ejpam-6275	10	9	,	,	PUNCT
ejpam-6275	10	10	94a24	94a24	NUM
ejpam-6275	10	11	,	,	PUNCT
ejpam-6275	10	12	68p30	68p30	NUM
ejpam-6275	10	13	,	,	PUNCT
ejpam-6275	10	14	14g50	14g50	NUM
ejpam-6275	10	15	,	,	PUNCT
ejpam-6275	10	16	94a05	94a05	NUM
ejpam-6275	10	17	key	key	ADJ
ejpam-6275	10	18	words	word	NOUN
ejpam-6275	10	19	and	and	CCONJ
ejpam-6275	10	20	phrases	phrase	NOUN
ejpam-6275	10	21	:	:	PUNCT
ejpam-6275	10	22	eisenstein	eisenstein	NOUN
ejpam-6275	10	23	field	field	NOUN
ejpam-6275	10	24	,	,	PUNCT
ejpam-6275	10	25	shortened	shorten	VERB
ejpam-6275	10	26	bch	bch	PROPN
ejpam-6275	10	27	codes	code	NOUN
ejpam-6275	10	28	,	,	PUNCT
ejpam-6275	10	29	modified	modified	ADJ
ejpam-6275	10	30	bma	bma	PROPN
ejpam-6275	10	31	∗corresponding	∗corresponding	NOUN
ejpam-6275	10	32	author	author	NOUN
ejpam-6275	10	33	.	.	PUNCT
ejpam-6275	11	1	∗corresponding	∗corresponde	VERB
ejpam-6275	11	2	author	author	NOUN
ejpam-6275	11	3	.	.	PUNCT
ejpam-6275	12	1	doi	doi	NOUN
ejpam-6275	12	2	:	:	PUNCT
ejpam-6275	12	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6275	https://doi.org/10.29020/nybg.ejpam.v18i3.6275	ADJ
ejpam-6275	12	4	email	email	NOUN
ejpam-6275	12	5	addresses	address	VERB
ejpam-6275	12	6	:	:	PUNCT
ejpam-6275	12	7	muhammad.sajjad@nutech.edu.pk	muhammad.sajjad@nutech.edu.pk	PROPN
ejpam-6275	12	8	(	(	PUNCT
ejpam-6275	12	9	m.	m.	PROPN
ejpam-6275	12	10	sajjad	sajjad	PROPN
ejpam-6275	12	11	)	)	PUNCT
ejpam-6275	12	12	,	,	PUNCT
ejpam-6275	12	13	moinuddin@zjnu.edu.cn	moinuddin@zjnu.edu.cn	NOUN
ejpam-6275	12	14	(	(	PUNCT
ejpam-6275	12	15	m.	m.	NOUN
ejpam-6275	12	16	junjua	junjua	PROPN
ejpam-6275	12	17	)	)	PUNCT
ejpam-6275	12	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6275	13	1	1	1	NUM
ejpam-6275	13	2	copyright	copyright	NOUN
ejpam-6275	13	3	:	:	PUNCT
ejpam-6275	13	4	©	©	PROPN
ejpam-6275	13	5	2025	2025	NUM
ejpam-6275	13	6	the	the	DET
ejpam-6275	13	7	author(s	author(s	NOUN
ejpam-6275	13	8	)	)	PUNCT
ejpam-6275	13	9	.	.	PUNCT
ejpam-6275	14	1	(	(	PUNCT
ejpam-6275	14	2	cc	cc	NOUN
ejpam-6275	14	3	by	by	ADP
ejpam-6275	14	4	-	-	PUNCT
ejpam-6275	14	5	nc	nc	PROPN
ejpam-6275	14	6	4.0	4.0	NUM
ejpam-6275	14	7	)	)	PUNCT
ejpam-6275	14	8	m.	m.	NOUN
ejpam-6275	14	9	sajjad	sajjad	PROPN
ejpam-6275	14	10	et	et	PROPN
ejpam-6275	14	11	al	al	PROPN
ejpam-6275	14	12	.	.	PUNCT
ejpam-6275	14	13	/	/	SYM
ejpam-6275	14	14	eur	eur	PROPN
ejpam-6275	14	15	.	.	PUNCT
ejpam-6275	15	1	j.	j.	PROPN
ejpam-6275	15	2	pure	pure	PROPN
ejpam-6275	15	3	appl	appl	PROPN
ejpam-6275	15	4	.	.	PROPN
ejpam-6275	15	5	math	math	PROPN
ejpam-6275	15	6	,	,	PUNCT
ejpam-6275	15	7	18	18	NUM
ejpam-6275	15	8	(	(	PUNCT
ejpam-6275	15	9	3	3	NUM
ejpam-6275	15	10	)	)	PUNCT
ejpam-6275	15	11	(	(	PUNCT
ejpam-6275	15	12	2025	2025	NUM
ejpam-6275	15	13	)	)	PUNCT
ejpam-6275	15	14	,	,	PUNCT
ejpam-6275	15	15	6275	6275	NUM
ejpam-6275	15	16	2	2	NUM
ejpam-6275	15	17	of	of	ADP
ejpam-6275	15	18	36	36	NUM
ejpam-6275	15	19	1	1	NUM
ejpam-6275	15	20	.	.	PUNCT
ejpam-6275	16	1	introduction	introduction	NOUN
ejpam-6275	16	2	nowadays	nowadays	ADP
ejpam-6275	16	3	transmission	transmission	NOUN
ejpam-6275	16	4	of	of	ADP
ejpam-6275	16	5	data	datum	NOUN
ejpam-6275	16	6	is	be	AUX
ejpam-6275	16	7	necessary	necessary	ADJ
ejpam-6275	16	8	,	,	PUNCT
ejpam-6275	16	9	and	and	CCONJ
ejpam-6275	16	10	error	error	NOUN
ejpam-6275	16	11	-	-	PUNCT
ejpam-6275	16	12	correcting	correct	VERB
ejpam-6275	16	13	hyphenate	hyphenate	ADJ
ejpam-6275	16	14	codes	code	NOUN
ejpam-6275	16	15	are	be	AUX
ejpam-6275	16	16	critically	critically	ADV
ejpam-6275	16	17	important	important	ADJ
ejpam-6275	16	18	for	for	ADP
ejpam-6275	16	19	realizing	realize	VERB
ejpam-6275	16	20	reliable	reliable	ADJ
ejpam-6275	16	21	data	data	NOUN
ejpam-6275	16	22	transmission	transmission	NOUN
ejpam-6275	16	23	via	via	ADP
ejpam-6275	16	24	noisy	noisy	ADJ
ejpam-6275	16	25	channels	channel	NOUN
ejpam-6275	16	26	.	.	PUNCT
ejpam-6275	17	1	of	of	ADP
ejpam-6275	17	2	them	they	PRON
ejpam-6275	17	3	,	,	PUNCT
ejpam-6275	17	4	bch	bch	PROPN
ejpam-6275	17	5	codes	code	NOUN
ejpam-6275	17	6	are	be	AUX
ejpam-6275	17	7	being	be	AUX
ejpam-6275	17	8	widely	widely	ADV
ejpam-6275	17	9	studied	study	VERB
ejpam-6275	17	10	because	because	SCONJ
ejpam-6275	17	11	of	of	ADP
ejpam-6275	17	12	their	their	PRON
ejpam-6275	17	13	algebraic	algebraic	ADJ
ejpam-6275	17	14	nature	nature	NOUN
ejpam-6275	17	15	and	and	CCONJ
ejpam-6275	17	16	superlative	superlative	NOUN
ejpam-6275	17	17	error	error	NOUN
ejpam-6275	17	18	correction	correction	NOUN
ejpam-6275	17	19	capability	capability	NOUN
ejpam-6275	17	20	.	.	PUNCT
ejpam-6275	18	1	originally	originally	ADV
ejpam-6275	18	2	built	build	VERB
ejpam-6275	18	3	over	over	ADP
ejpam-6275	18	4	finite	finite	ADJ
ejpam-6275	18	5	fields	field	NOUN
ejpam-6275	18	6	,	,	PUNCT
ejpam-6275	18	7	bch	bch	PROPN
ejpam-6275	18	8	codes	code	NOUN
ejpam-6275	18	9	are	be	AUX
ejpam-6275	18	10	widely	widely	ADV
ejpam-6275	18	11	used	use	VERB
ejpam-6275	18	12	in	in	ADP
ejpam-6275	18	13	storage	storage	NOUN
ejpam-6275	18	14	equipment	equipment	NOUN
ejpam-6275	18	15	,	,	PUNCT
ejpam-6275	18	16	wireless	wireless	ADJ
ejpam-6275	18	17	communication	communication	NOUN
ejpam-6275	18	18	,	,	PUNCT
ejpam-6275	18	19	and	and	CCONJ
ejpam-6275	18	20	other	other	ADJ
ejpam-6275	18	21	data	data	NOUN
ejpam-6275	18	22	-	-	PUNCT
ejpam-6275	18	23	applying	apply	VERB
ejpam-6275	18	24	fields	field	NOUN
ejpam-6275	18	25	.	.	PUNCT
ejpam-6275	19	1	however	however	ADV
ejpam-6275	19	2	,	,	PUNCT
ejpam-6275	19	3	as	as	ADP
ejpam-6275	19	4	the	the	DET
ejpam-6275	19	5	requirement	requirement	NOUN
ejpam-6275	19	6	for	for	ADP
ejpam-6275	19	7	higher	high	ADJ
ejpam-6275	19	8	reliability	reliability	NOUN
ejpam-6275	19	9	and	and	CCONJ
ejpam-6275	19	10	system	system	NOUN
ejpam-6275	19	11	efficiency	efficiency	NOUN
ejpam-6275	19	12	increases	increase	NOUN
ejpam-6275	19	13	in	in	ADP
ejpam-6275	19	14	the	the	DET
ejpam-6275	19	15	next	next	ADJ
ejpam-6275	19	16	generation	generation	NOUN
ejpam-6275	19	17	of	of	ADP
ejpam-6275	19	18	communication	communication	NOUN
ejpam-6275	19	19	systems	system	NOUN
ejpam-6275	19	20	,	,	PUNCT
ejpam-6275	19	21	researchers	researcher	NOUN
ejpam-6275	19	22	are	be	AUX
ejpam-6275	19	23	considering	consider	VERB
ejpam-6275	19	24	different	different	ADJ
ejpam-6275	19	25	algebraic	algebraic	ADJ
ejpam-6275	19	26	contexts	contexts	NOUN
ejpam-6275	19	27	that	that	PRON
ejpam-6275	19	28	will	will	AUX
ejpam-6275	19	29	help	help	VERB
ejpam-6275	19	30	improve	improve	VERB
ejpam-6275	19	31	these	these	DET
ejpam-6275	19	32	codes	code	NOUN
ejpam-6275	19	33	.	.	PUNCT
ejpam-6275	20	1	such	such	ADJ
ejpam-6275	20	2	field	field	NOUN
ejpam-6275	20	3	-	-	PUNCT
ejpam-6275	20	4	based	base	VERB
ejpam-6275	20	5	approaches	approach	NOUN
ejpam-6275	20	6	are	be	AUX
ejpam-6275	20	7	also	also	ADV
ejpam-6275	20	8	possible	possible	ADJ
ejpam-6275	20	9	,	,	PUNCT
ejpam-6275	20	10	and	and	CCONJ
ejpam-6275	20	11	eisenstein	eisenstein	PROPN
ejpam-6275	20	12	fields	field	NOUN
ejpam-6275	20	13	appear	appear	VERB
ejpam-6275	20	14	to	to	PART
ejpam-6275	20	15	be	be	AUX
ejpam-6275	20	16	logical	logical	ADJ
ejpam-6275	20	17	extensions	extension	NOUN
ejpam-6275	20	18	of	of	ADP
ejpam-6275	20	19	gaussian	gaussian	ADJ
ejpam-6275	20	20	fields	field	NOUN
ejpam-6275	20	21	.	.	PUNCT
ejpam-6275	21	1	several	several	ADJ
ejpam-6275	21	2	algebraic	algebraic	ADJ
ejpam-6275	21	3	characteristics	characteristic	NOUN
ejpam-6275	21	4	of	of	ADP
ejpam-6275	21	5	eisenstein	eisenstein	PROPN
ejpam-6275	21	6	fields	field	NOUN
ejpam-6275	21	7	,	,	PUNCT
ejpam-6275	21	8	including	include	VERB
ejpam-6275	21	9	unique	unique	ADJ
ejpam-6275	21	10	factorization	factorization	NOUN
ejpam-6275	21	11	and	and	CCONJ
ejpam-6275	21	12	extended	extended	ADJ
ejpam-6275	21	13	residue	residue	NOUN
ejpam-6275	21	14	classes	class	NOUN
ejpam-6275	21	15	,	,	PUNCT
ejpam-6275	21	16	can	can	AUX
ejpam-6275	21	17	be	be	AUX
ejpam-6275	21	18	critically	critically	ADV
ejpam-6275	21	19	exploited	exploit	VERB
ejpam-6275	21	20	for	for	ADP
ejpam-6275	21	21	efficient	efficient	ADJ
ejpam-6275	21	22	codes	code	NOUN
ejpam-6275	21	23	as	as	ADV
ejpam-6275	21	24	well	well	ADV
ejpam-6275	21	25	as	as	ADP
ejpam-6275	21	26	being	be	AUX
ejpam-6275	21	27	resilient	resilient	ADJ
ejpam-6275	21	28	to	to	ADP
ejpam-6275	21	29	errors	error	NOUN
ejpam-6275	21	30	.	.	PUNCT
ejpam-6275	22	1	nonetheless	nonetheless	ADV
ejpam-6275	22	2	,	,	PUNCT
ejpam-6275	22	3	the	the	DET
ejpam-6275	22	4	use	use	NOUN
ejpam-6275	22	5	of	of	ADP
ejpam-6275	22	6	the	the	DET
ejpam-6275	22	7	eisenstein	eisenstein	PROPN
ejpam-6275	22	8	fields	field	NOUN
ejpam-6275	22	9	in	in	ADP
ejpam-6275	22	10	coding	code	VERB
ejpam-6275	22	11	theory	theory	NOUN
ejpam-6275	22	12	has	have	AUX
ejpam-6275	22	13	not	not	PART
ejpam-6275	22	14	been	be	AUX
ejpam-6275	22	15	fully	fully	ADV
ejpam-6275	22	16	researched	research	VERB
ejpam-6275	22	17	until	until	ADP
ejpam-6275	22	18	now	now	ADV
ejpam-6275	22	19	,	,	PUNCT
ejpam-6275	22	20	especially	especially	ADV
ejpam-6275	22	21	in	in	ADP
ejpam-6275	22	22	the	the	DET
ejpam-6275	22	23	construction	construction	NOUN
ejpam-6275	22	24	and	and	CCONJ
ejpam-6275	22	25	decoding	decoding	NOUN
ejpam-6275	22	26	of	of	ADP
ejpam-6275	22	27	shortened	shorten	VERB
ejpam-6275	22	28	bch	bch	PROPN
ejpam-6275	22	29	codes	code	NOUN
ejpam-6275	22	30	.	.	PUNCT
ejpam-6275	23	1	this	this	PRON
ejpam-6275	23	2	is	be	AUX
ejpam-6275	23	3	especially	especially	ADV
ejpam-6275	23	4	due	due	ADJ
ejpam-6275	23	5	to	to	ADP
ejpam-6275	23	6	the	the	DET
ejpam-6275	23	7	fact	fact	NOUN
ejpam-6275	23	8	that	that	SCONJ
ejpam-6275	23	9	shortened	shorten	VERB
ejpam-6275	23	10	codes	code	NOUN
ejpam-6275	23	11	are	be	AUX
ejpam-6275	23	12	renowned	renowned	ADJ
ejpam-6275	23	13	for	for	ADP
ejpam-6275	23	14	their	their	PRON
ejpam-6275	23	15	capability	capability	NOUN
ejpam-6275	23	16	to	to	PART
ejpam-6275	23	17	sustain	sustain	VERB
ejpam-6275	23	18	relatively	relatively	ADV
ejpam-6275	23	19	short	short	ADJ
ejpam-6275	23	20	codes	code	NOUN
ejpam-6275	23	21	with	with	ADP
ejpam-6275	23	22	good	good	ADJ
ejpam-6275	23	23	error	error	NOUN
ejpam-6275	23	24	correction	correction	NOUN
ejpam-6275	23	25	capabilities	capability	NOUN
ejpam-6275	23	26	.	.	PUNCT
ejpam-6275	24	1	shortened	shorten	VERB
ejpam-6275	24	2	codes	code	NOUN
ejpam-6275	24	3	are	be	AUX
ejpam-6275	24	4	well	well	ADV
ejpam-6275	24	5	suited	suited	ADJ
ejpam-6275	24	6	for	for	ADP
ejpam-6275	24	7	any	any	DET
ejpam-6275	24	8	application	application	NOUN
ejpam-6275	24	9	that	that	PRON
ejpam-6275	24	10	requires	require	VERB
ejpam-6275	24	11	concise	concise	ADJ
ejpam-6275	24	12	and	and	CCONJ
ejpam-6275	24	13	efficient	efficient	ADJ
ejpam-6275	24	14	communication	communication	NOUN
ejpam-6275	24	15	code	code	NOUN
ejpam-6275	24	16	.	.	PUNCT
ejpam-6275	25	1	other	other	ADJ
ejpam-6275	25	2	components	component	NOUN
ejpam-6275	25	3	of	of	ADP
ejpam-6275	25	4	error	error	NOUN
ejpam-6275	25	5	-	-	PUNCT
ejpam-6275	25	6	correcting	correct	VERB
ejpam-6275	25	7	codes	code	NOUN
ejpam-6275	25	8	as	as	ADP
ejpam-6275	25	9	a	a	DET
ejpam-6275	25	10	standard	standard	NOUN
ejpam-6275	25	11	are	be	AUX
ejpam-6275	25	12	decoding	decode	VERB
ejpam-6275	25	13	algorithms	algorithm	NOUN
ejpam-6275	25	14	.	.	PUNCT
ejpam-6275	26	1	in	in	ADP
ejpam-6275	26	2	a	a	DET
ejpam-6275	26	3	number	number	NOUN
ejpam-6275	26	4	of	of	ADP
ejpam-6275	26	5	situations	situation	NOUN
ejpam-6275	26	6	,	,	PUNCT
ejpam-6275	26	7	the	the	DET
ejpam-6275	26	8	berlekamp	berlekamp	NOUN
ejpam-6275	26	9	-	-	PUNCT
ejpam-6275	26	10	massey	massey	PROPN
ejpam-6275	26	11	algorithm	algorithm	NOUN
ejpam-6275	26	12	(	(	PUNCT
ejpam-6275	26	13	bma	bma	PROPN
ejpam-6275	26	14	)	)	PUNCT
ejpam-6275	26	15	has	have	AUX
ejpam-6275	26	16	been	be	AUX
ejpam-6275	26	17	a	a	DET
ejpam-6275	26	18	standard	standard	ADJ
ejpam-6275	26	19	tool	tool	NOUN
ejpam-6275	26	20	in	in	ADP
ejpam-6275	26	21	the	the	DET
ejpam-6275	26	22	decoding	decoding	NOUN
ejpam-6275	26	23	of	of	ADP
ejpam-6275	26	24	bch	bch	PROPN
ejpam-6275	26	25	codes	code	NOUN
ejpam-6275	26	26	.	.	PUNCT
ejpam-6275	27	1	nevertheless	nevertheless	ADV
ejpam-6275	27	2	,	,	PUNCT
ejpam-6275	27	3	its	its	PRON
ejpam-6275	27	4	use	use	NOUN
ejpam-6275	27	5	in	in	ADP
ejpam-6275	27	6	eisenstein	eisenstein	NOUN
ejpam-6275	27	7	fields	field	NOUN
ejpam-6275	27	8	may	may	AUX
ejpam-6275	27	9	be	be	AUX
ejpam-6275	27	10	problematic	problematic	ADJ
ejpam-6275	27	11	due	due	ADP
ejpam-6275	27	12	to	to	ADP
ejpam-6275	27	13	the	the	DET
ejpam-6275	27	14	fact	fact	NOUN
ejpam-6275	27	15	that	that	SCONJ
ejpam-6275	27	16	the	the	DET
ejpam-6275	27	17	residue	residue	NOUN
ejpam-6275	27	18	classes	class	NOUN
ejpam-6275	27	19	are	be	AUX
ejpam-6275	27	20	extended	extend	VERB
ejpam-6275	27	21	and	and	CCONJ
ejpam-6275	27	22	the	the	DET
ejpam-6275	27	23	computations	computation	NOUN
ejpam-6275	27	24	are	be	AUX
ejpam-6275	27	25	more	more	ADV
ejpam-6275	27	26	cumbersome	cumbersome	ADJ
ejpam-6275	27	27	.	.	PUNCT
ejpam-6275	28	1	overcoming	overcome	VERB
ejpam-6275	28	2	these	these	DET
ejpam-6275	28	3	challenges	challenge	NOUN
ejpam-6275	28	4	calls	call	VERB
ejpam-6275	28	5	for	for	ADP
ejpam-6275	28	6	propositions	proposition	NOUN
ejpam-6275	28	7	on	on	ADP
ejpam-6275	28	8	the	the	DET
ejpam-6275	28	9	classical	classical	ADJ
ejpam-6275	28	10	bma	bma	PROPN
ejpam-6275	28	11	for	for	ADP
ejpam-6275	28	12	faster	fast	ADJ
ejpam-6275	28	13	and	and	CCONJ
ejpam-6275	28	14	more	more	ADV
ejpam-6275	28	15	accurate	accurate	ADJ
ejpam-6275	28	16	decoding	decode	VERB
ejpam-6275	28	17	for	for	ADP
ejpam-6275	28	18	codes	code	NOUN
ejpam-6275	28	19	over	over	ADP
ejpam-6275	28	20	eisenstein	eisenstein	NOUN
ejpam-6275	28	21	fields	field	NOUN
ejpam-6275	28	22	[	[	X
ejpam-6275	28	23	1–6	1–6	NUM
ejpam-6275	28	24	]	]	X
ejpam-6275	28	25	.	.	PUNCT
ejpam-6275	29	1	among	among	ADP
ejpam-6275	29	2	the	the	DET
ejpam-6275	29	3	coding	code	VERB
ejpam-6275	29	4	techniques	technique	NOUN
ejpam-6275	29	5	,	,	PUNCT
ejpam-6275	29	6	the	the	DET
ejpam-6275	29	7	error	error	NOUN
ejpam-6275	29	8	-	-	PUNCT
ejpam-6275	29	9	correcting	correct	VERB
ejpam-6275	29	10	code	code	NOUN
ejpam-6275	29	11	,	,	PUNCT
ejpam-6275	29	12	especially	especially	ADV
ejpam-6275	29	13	the	the	DET
ejpam-6275	29	14	shortened	shorten	VERB
ejpam-6275	29	15	bch	bch	PROPN
ejpam-6275	29	16	code	code	NOUN
ejpam-6275	29	17	of	of	ADP
ejpam-6275	29	18	a	a	DET
ejpam-6275	29	19	given	give	VERB
ejpam-6275	29	20	order	order	NOUN
ejpam-6275	29	21	,	,	PUNCT
ejpam-6275	29	22	has	have	AUX
ejpam-6275	29	23	been	be	AUX
ejpam-6275	29	24	an	an	DET
ejpam-6275	29	25	important	important	ADJ
ejpam-6275	29	26	topic	topic	NOUN
ejpam-6275	29	27	in	in	ADP
ejpam-6275	29	28	coding	code	VERB
ejpam-6275	29	29	theory	theory	NOUN
ejpam-6275	29	30	because	because	SCONJ
ejpam-6275	29	31	of	of	ADP
ejpam-6275	29	32	its	its	PRON
ejpam-6275	29	33	random	random	ADJ
ejpam-6275	29	34	error	error	NOUN
ejpam-6275	29	35	-	-	PUNCT
ejpam-6275	29	36	correcting	correct	VERB
ejpam-6275	29	37	capability	capability	NOUN
ejpam-6275	29	38	and	and	CCONJ
ejpam-6275	29	39	use	use	NOUN
ejpam-6275	29	40	in	in	ADP
ejpam-6275	29	41	high	high	ADJ
ejpam-6275	29	42	-	-	PUNCT
ejpam-6275	29	43	reliability	reliability	NOUN
ejpam-6275	29	44	communication	communication	NOUN
ejpam-6275	29	45	systems	system	NOUN
ejpam-6275	29	46	.	.	PUNCT
ejpam-6275	30	1	the	the	DET
ejpam-6275	30	2	theoretical	theoretical	ADJ
ejpam-6275	30	3	characteristics	characteristic	NOUN
ejpam-6275	30	4	of	of	ADP
ejpam-6275	30	5	shortened	shorten	VERB
ejpam-6275	30	6	cyclic	cyclic	ADJ
ejpam-6275	30	7	codes	code	NOUN
ejpam-6275	30	8	and	and	CCONJ
ejpam-6275	30	9	their	their	PRON
ejpam-6275	30	10	uses	use	NOUN
ejpam-6275	30	11	,	,	PUNCT
ejpam-6275	30	12	especially	especially	ADV
ejpam-6275	30	13	for	for	ADP
ejpam-6275	30	14	burst	burst	NOUN
ejpam-6275	30	15	-	-	PUNCT
ejpam-6275	30	16	error	error	NOUN
ejpam-6275	30	17	correction	correction	NOUN
ejpam-6275	30	18	,	,	PUNCT
ejpam-6275	30	19	were	be	AUX
ejpam-6275	30	20	investigated	investigate	VERB
ejpam-6275	30	21	by	by	ADP
ejpam-6275	30	22	kasami	kasami	PROPN
ejpam-6275	30	23	and	and	CCONJ
ejpam-6275	30	24	hsu	hsu	PROPN
ejpam-6275	30	25	.	.	PUNCT
ejpam-6275	31	1	further	far	ADV
ejpam-6275	31	2	,	,	PUNCT
ejpam-6275	31	3	they	they	PRON
ejpam-6275	31	4	pointed	point	VERB
ejpam-6275	31	5	out	out	ADP
ejpam-6275	31	6	that	that	SCONJ
ejpam-6275	31	7	,	,	PUNCT
ejpam-6275	31	8	while	while	SCONJ
ejpam-6275	31	9	optimizing	optimize	VERB
ejpam-6275	31	10	code	code	NOUN
ejpam-6275	31	11	length	length	NOUN
ejpam-6275	31	12	,	,	PUNCT
ejpam-6275	31	13	it	it	PRON
ejpam-6275	31	14	is	be	AUX
ejpam-6275	31	15	necessary	necessary	ADJ
ejpam-6275	31	16	to	to	PART
ejpam-6275	31	17	balance	balance	VERB
ejpam-6275	31	18	the	the	DET
ejpam-6275	31	19	amount	amount	NOUN
ejpam-6275	31	20	of	of	ADP
ejpam-6275	31	21	correction	correction	NOUN
ejpam-6275	31	22	capacity	capacity	NOUN
ejpam-6275	31	23	[	[	X
ejpam-6275	31	24	7	7	NUM
ejpam-6275	31	25	,	,	PUNCT
ejpam-6275	31	26	8	8	NUM
ejpam-6275	31	27	]	]	PUNCT
ejpam-6275	31	28	.	.	PUNCT
ejpam-6275	32	1	helgert	helgert	PROPN
ejpam-6275	32	2	and	and	CCONJ
ejpam-6275	32	3	stinaff	stinaff	PROPN
ejpam-6275	32	4	extended	extend	VERB
ejpam-6275	32	5	the	the	DET
ejpam-6275	32	6	construction	construction	NOUN
ejpam-6275	32	7	and	and	CCONJ
ejpam-6275	32	8	decoding	decoding	NOUN
ejpam-6275	32	9	of	of	ADP
ejpam-6275	32	10	shortened	shorten	VERB
ejpam-6275	32	11	bch	bch	PROPN
ejpam-6275	32	12	codes	code	NOUN
ejpam-6275	32	13	and	and	CCONJ
ejpam-6275	32	14	the	the	DET
ejpam-6275	32	15	fact	fact	NOUN
ejpam-6275	32	16	that	that	SCONJ
ejpam-6275	32	17	these	these	DET
ejpam-6275	32	18	codes	code	NOUN
ejpam-6275	32	19	are	be	AUX
ejpam-6275	32	20	well	well	ADV
ejpam-6275	32	21	suited	suited	ADJ
ejpam-6275	32	22	to	to	ADP
ejpam-6275	32	23	any	any	DET
ejpam-6275	32	24	narrow	narrow	ADJ
ejpam-6275	32	25	bandwidth	bandwidth	ADJ
ejpam-6275	32	26	systems	system	NOUN
ejpam-6275	32	27	[	[	X
ejpam-6275	32	28	9	9	NUM
ejpam-6275	32	29	]	]	PUNCT
ejpam-6275	32	30	.	.	PUNCT
ejpam-6275	33	1	traditionally	traditionally	ADV
ejpam-6275	33	2	linear	linear	ADJ
ejpam-6275	33	3	codes	code	NOUN
ejpam-6275	33	4	have	have	AUX
ejpam-6275	33	5	been	be	AUX
ejpam-6275	33	6	discussed	discuss	VERB
ejpam-6275	33	7	in	in	ADP
ejpam-6275	33	8	numerous	numerous	ADJ
ejpam-6275	33	9	algebraic	algebraic	ADJ
ejpam-6275	33	10	contexts	contexts	NOUN
ejpam-6275	33	11	,	,	PUNCT
ejpam-6275	33	12	for	for	ADP
ejpam-6275	33	13	example	example	NOUN
ejpam-6275	33	14	,	,	PUNCT
ejpam-6275	33	15	bch	bch	PROPN
ejpam-6275	33	16	codes	code	NOUN
ejpam-6275	33	17	.	.	PUNCT
ejpam-6275	34	1	cyclotomic	cyclotomic	ADJ
ejpam-6275	34	2	linear	linear	PROPN
ejpam-6275	34	3	codes	code	NOUN
ejpam-6275	34	4	of	of	ADP
ejpam-6275	34	5	order	order	NOUN
ejpam-6275	34	6	three	three	NUM
ejpam-6275	34	7	were	be	AUX
ejpam-6275	34	8	studied	study	VERB
ejpam-6275	34	9	by	by	ADP
ejpam-6275	34	10	ding	ding	NOUN
ejpam-6275	34	11	and	and	CCONJ
ejpam-6275	34	12	niederreiter	niederreiter	VERB
ejpam-6275	34	13	,	,	PUNCT
ejpam-6275	34	14	and	and	CCONJ
ejpam-6275	34	15	cyclic	cyclic	ADJ
ejpam-6275	34	16	codes	code	NOUN
ejpam-6275	34	17	with	with	ADP
ejpam-6275	34	18	certain	certain	ADJ
ejpam-6275	34	19	specified	specify	VERB
ejpam-6275	34	20	weight	weight	NOUN
ejpam-6275	34	21	distributions	distribution	NOUN
ejpam-6275	34	22	were	be	AUX
ejpam-6275	34	23	investigated	investigate	VERB
ejpam-6275	34	24	by	by	ADP
ejpam-6275	34	25	ding	ding	NOUN
ejpam-6275	34	26	et	et	PROPN
ejpam-6275	34	27	al	al	PROPN
ejpam-6275	34	28	.	.	PUNCT
ejpam-6275	35	1	[	[	X
ejpam-6275	35	2	10	10	NUM
ejpam-6275	35	3	,	,	PUNCT
ejpam-6275	35	4	11	11	NUM
ejpam-6275	35	5	]	]	PUNCT
ejpam-6275	35	6	,	,	PUNCT
ejpam-6275	35	7	because	because	SCONJ
ejpam-6275	35	8	such	such	ADJ
ejpam-6275	35	9	codes	code	NOUN
ejpam-6275	35	10	have	have	AUX
ejpam-6275	35	11	been	be	AUX
ejpam-6275	35	12	deemed	deem	VERB
ejpam-6275	35	13	useful	useful	ADJ
ejpam-6275	35	14	in	in	ADP
ejpam-6275	35	15	application	application	NOUN
ejpam-6275	35	16	,	,	PUNCT
ejpam-6275	35	17	particularly	particularly	ADV
ejpam-6275	35	18	in	in	ADP
ejpam-6275	35	19	efficient	efficient	ADJ
ejpam-6275	35	20	error	error	NOUN
ejpam-6275	35	21	correction	correction	NOUN
ejpam-6275	35	22	.	.	PUNCT
ejpam-6275	36	1	shah	shah	PROPN
ejpam-6275	36	2	et	et	PROPN
ejpam-6275	36	3	al	al	PROPN
ejpam-6275	36	4	.	.	PROPN
ejpam-6275	36	5	introduced	introduce	VERB
ejpam-6275	36	6	codes	code	NOUN
ejpam-6275	36	7	and	and	CCONJ
ejpam-6275	36	8	decoding	decode	VERB
ejpam-6275	36	9	using	use	VERB
ejpam-6275	36	10	generalized	generalized	ADJ
ejpam-6275	36	11	polynomials	polynomial	NOUN
ejpam-6275	36	12	and	and	CCONJ
ejpam-6275	36	13	sequences	sequence	NOUN
ejpam-6275	36	14	in	in	ADP
ejpam-6275	36	15	[	[	X
ejpam-6275	36	16	12	12	NUM
ejpam-6275	36	17	,	,	PUNCT
ejpam-6275	36	18	13	13	NUM
ejpam-6275	36	19	]	]	PUNCT
ejpam-6275	36	20	.	.	PUNCT
ejpam-6275	37	1	linear	linear	PROPN
ejpam-6275	37	2	codes	code	NOUN
ejpam-6275	37	3	from	from	ADP
ejpam-6275	37	4	2	2	NUM
ejpam-6275	37	5	-	-	PUNCT
ejpam-6275	37	6	designs	design	NOUN
ejpam-6275	37	7	have	have	AUX
ejpam-6275	37	8	also	also	ADV
ejpam-6275	37	9	been	be	AUX
ejpam-6275	37	10	designed	design	VERB
ejpam-6275	37	11	by	by	ADP
ejpam-6275	37	12	ding	ding	NOUN
ejpam-6275	37	13	,	,	PUNCT
ejpam-6275	37	14	who	who	PRON
ejpam-6275	37	15	offered	offer	VERB
ejpam-6275	37	16	a	a	DET
ejpam-6275	37	17	new	new	ADJ
ejpam-6275	37	18	way	way	NOUN
ejpam-6275	37	19	to	to	PART
ejpam-6275	37	20	define	define	VERB
ejpam-6275	37	21	codes	code	NOUN
ejpam-6275	37	22	with	with	ADP
ejpam-6275	37	23	performance	performance	NOUN
ejpam-6275	37	24	characteristics	characteristic	NOUN
ejpam-6275	37	25	that	that	PRON
ejpam-6275	37	26	would	would	AUX
ejpam-6275	37	27	be	be	AUX
ejpam-6275	37	28	suitable	suitable	ADJ
ejpam-6275	37	29	for	for	ADP
ejpam-6275	37	30	various	various	ADJ
ejpam-6275	37	31	practical	practical	ADJ
ejpam-6275	37	32	uses	use	NOUN
ejpam-6275	37	33	[	[	X
ejpam-6275	37	34	14	14	NUM
ejpam-6275	37	35	]	]	PUNCT
ejpam-6275	37	36	.	.	PUNCT
ejpam-6275	38	1	an	an	DET
ejpam-6275	38	2	important	important	ADJ
ejpam-6275	38	3	part	part	NOUN
ejpam-6275	38	4	of	of	ADP
ejpam-6275	38	5	the	the	DET
ejpam-6275	38	6	development	development	NOUN
ejpam-6275	38	7	of	of	ADP
ejpam-6275	38	8	coding	code	VERB
ejpam-6275	38	9	theory	theory	NOUN
ejpam-6275	38	10	demands	demand	VERB
ejpam-6275	38	11	the	the	DET
ejpam-6275	38	12	consideration	consideration	NOUN
ejpam-6275	38	13	of	of	ADP
ejpam-6275	38	14	shortened	shorten	VERB
ejpam-6275	38	15	and	and	CCONJ
ejpam-6275	38	16	punctured	punctured	ADJ
ejpam-6275	38	17	codes	code	NOUN
ejpam-6275	38	18	.	.	PUNCT
ejpam-6275	39	1	goldwasser	goldwasser	PROPN
ejpam-6275	39	2	studied	study	VERB
ejpam-6275	39	3	the	the	DET
ejpam-6275	39	4	relation	relation	NOUN
ejpam-6275	39	5	between	between	ADP
ejpam-6275	39	6	code	code	NOUN
ejpam-6275	39	7	shortening	shortening	NOUN
ejpam-6275	39	8	and	and	CCONJ
ejpam-6275	39	9	macwilliams	macwilliam	NOUN
ejpam-6275	39	10	identities	identity	NOUN
ejpam-6275	39	11	by	by	ADP
ejpam-6275	39	12	giving	give	VERB
ejpam-6275	39	13	significant	significant	ADJ
ejpam-6275	39	14	information	information	NOUN
ejpam-6275	39	15	about	about	ADP
ejpam-6275	39	16	the	the	DET
ejpam-6275	39	17	structure	structure	NOUN
ejpam-6275	39	18	of	of	ADP
ejpam-6275	39	19	these	these	DET
ejpam-6275	39	20	codes	code	NOUN
ejpam-6275	39	21	[	[	X
ejpam-6275	39	22	15	15	NUM
ejpam-6275	39	23	]	]	PUNCT
ejpam-6275	39	24	.	.	PUNCT
ejpam-6275	40	1	yardi	yardi	PROPN
ejpam-6275	40	2	and	and	CCONJ
ejpam-6275	40	3	pellikaan	pellikaan	PROPN
ejpam-6275	40	4	also	also	ADV
ejpam-6275	40	5	investigated	investigate	VERB
ejpam-6275	40	6	m.	m.	NOUN
ejpam-6275	40	7	sajjad	sajjad	PROPN
ejpam-6275	40	8	et	et	PROPN
ejpam-6275	40	9	al	al	PROPN
ejpam-6275	40	10	.	.	PUNCT
ejpam-6275	40	11	/	/	SYM
ejpam-6275	40	12	eur	eur	PROPN
ejpam-6275	40	13	.	.	PUNCT
ejpam-6275	41	1	j.	j.	PROPN
ejpam-6275	41	2	pure	pure	PROPN
ejpam-6275	41	3	appl	appl	PROPN
ejpam-6275	41	4	.	.	PROPN
ejpam-6275	41	5	math	math	PROPN
ejpam-6275	41	6	,	,	PUNCT
ejpam-6275	41	7	18	18	NUM
ejpam-6275	41	8	(	(	PUNCT
ejpam-6275	41	9	3	3	NUM
ejpam-6275	41	10	)	)	PUNCT
ejpam-6275	41	11	(	(	PUNCT
ejpam-6275	41	12	2025	2025	NUM
ejpam-6275	41	13	)	)	PUNCT
ejpam-6275	41	14	,	,	PUNCT
ejpam-6275	41	15	6275	6275	NUM
ejpam-6275	41	16	3	3	NUM
ejpam-6275	41	17	of	of	ADP
ejpam-6275	41	18	36	36	NUM
ejpam-6275	41	19	the	the	DET
ejpam-6275	41	20	behaviour	behaviour	NOUN
ejpam-6275	41	21	of	of	ADP
ejpam-6275	41	22	shortened	shorten	VERB
ejpam-6275	41	23	cyclic	cyclic	ADJ
ejpam-6275	41	24	codes	code	NOUN
ejpam-6275	41	25	in	in	ADP
ejpam-6275	41	26	more	more	ADJ
ejpam-6275	41	27	detail	detail	NOUN
ejpam-6275	41	28	regarding	regard	VERB
ejpam-6275	41	29	that	that	PRON
ejpam-6275	41	30	being	be	AUX
ejpam-6275	41	31	useful	useful	ADJ
ejpam-6275	41	32	for	for	ADP
ejpam-6275	41	33	solving	solve	VERB
ejpam-6275	41	34	error	error	NOUN
ejpam-6275	41	35	correction	correction	NOUN
ejpam-6275	41	36	for	for	ADP
ejpam-6275	41	37	a	a	DET
ejpam-6275	41	38	constraint	constraint	NOUN
ejpam-6275	42	1	[	[	X
ejpam-6275	42	2	16	16	NUM
ejpam-6275	42	3	]	]	PUNCT
ejpam-6275	42	4	.	.	PUNCT
ejpam-6275	43	1	gaussian	gaussian	ADJ
ejpam-6275	43	2	fields	field	NOUN
ejpam-6275	43	3	are	be	AUX
ejpam-6275	43	4	extended	extend	VERB
ejpam-6275	43	5	to	to	ADP
ejpam-6275	43	6	eisenstein	eisenstein	NOUN
ejpam-6275	43	7	fields	field	NOUN
ejpam-6275	43	8	,	,	PUNCT
ejpam-6275	43	9	which	which	PRON
ejpam-6275	43	10	have	have	AUX
ejpam-6275	43	11	attracted	attract	VERB
ejpam-6275	43	12	attention	attention	NOUN
ejpam-6275	43	13	as	as	ADP
ejpam-6275	43	14	a	a	DET
ejpam-6275	43	15	sound	sound	ADJ
ejpam-6275	43	16	algebraic	algebraic	ADJ
ejpam-6275	43	17	structure	structure	NOUN
ejpam-6275	43	18	for	for	ADP
ejpam-6275	43	19	implementing	implement	VERB
ejpam-6275	43	20	richer	rich	ADJ
ejpam-6275	43	21	error	error	NOUN
ejpam-6275	43	22	-	-	PUNCT
ejpam-6275	43	23	correcting	correct	VERB
ejpam-6275	43	24	codes	code	NOUN
ejpam-6275	43	25	.	.	PUNCT
ejpam-6275	44	1	huber	huber	PROPN
ejpam-6275	44	2	extended	extend	VERB
ejpam-6275	44	3	codes	code	NOUN
ejpam-6275	44	4	over	over	ADP
ejpam-6275	44	5	eisenstein	eisenstein	PROPN
ejpam-6275	44	6	-	-	PUNCT
ejpam-6275	44	7	jacobi	jacobi	PROPN
ejpam-6275	44	8	integers	integer	NOUN
ejpam-6275	44	9	,	,	PUNCT
ejpam-6275	44	10	which	which	PRON
ejpam-6275	44	11	laid	lay	VERB
ejpam-6275	44	12	down	down	ADP
ejpam-6275	44	13	the	the	DET
ejpam-6275	44	14	groundwork	groundwork	NOUN
ejpam-6275	44	15	for	for	ADP
ejpam-6275	44	16	approaching	approach	VERB
ejpam-6275	44	17	these	these	DET
ejpam-6275	44	18	fields	field	NOUN
ejpam-6275	44	19	in	in	ADP
ejpam-6275	44	20	coding	code	VERB
ejpam-6275	44	21	theory	theory	NOUN
ejpam-6275	44	22	[	[	X
ejpam-6275	44	23	17	17	NUM
ejpam-6275	44	24	]	]	PUNCT
ejpam-6275	44	25	.	.	PUNCT
ejpam-6275	45	1	later	later	ADV
ejpam-6275	45	2	on	on	ADV
ejpam-6275	45	3	,	,	PUNCT
ejpam-6275	45	4	sajjad	sajjad	PROPN
ejpam-6275	45	5	et	et	PROPN
ejpam-6275	45	6	al	al	PROPN
ejpam-6275	45	7	.	.	PROPN
ejpam-6275	45	8	did	do	VERB
ejpam-6275	45	9	more	more	ADV
ejpam-6275	45	10	advanced	advanced	ADJ
ejpam-6275	45	11	work	work	NOUN
ejpam-6275	45	12	on	on	ADP
ejpam-6275	45	13	eisenstein	eisenstein	NOUN
ejpam-6275	45	14	and	and	CCONJ
ejpam-6275	45	15	gaussian	gaussian	ADJ
ejpam-6275	45	16	fields	field	NOUN
ejpam-6275	45	17	using	use	VERB
ejpam-6275	45	18	cryptology	cryptology	NOUN
ejpam-6275	45	19	and	and	CCONJ
ejpam-6275	45	20	error	error	NOUN
ejpam-6275	45	21	correction	correction	NOUN
ejpam-6275	45	22	codes	code	NOUN
ejpam-6275	45	23	,	,	PUNCT
ejpam-6275	45	24	showing	show	VERB
ejpam-6275	45	25	more	more	ADJ
ejpam-6275	45	26	efficiency	efficiency	NOUN
ejpam-6275	45	27	in	in	ADP
ejpam-6275	45	28	noisy	noisy	ADJ
ejpam-6275	45	29	channel	channel	NOUN
ejpam-6275	45	30	communication	communication	NOUN
ejpam-6275	45	31	systems	system	NOUN
ejpam-6275	45	32	[	[	X
ejpam-6275	45	33	18	18	NUM
ejpam-6275	45	34	,	,	PUNCT
ejpam-6275	45	35	19	19	NUM
ejpam-6275	45	36	]	]	PUNCT
ejpam-6275	45	37	.	.	PUNCT
ejpam-6275	46	1	the	the	DET
ejpam-6275	46	2	berlekamp	berlekamp	PROPN
ejpam-6275	46	3	-	-	PUNCT
ejpam-6275	46	4	massey	massey	PROPN
ejpam-6275	46	5	algorithm	algorithm	NOUN
ejpam-6275	46	6	(	(	PUNCT
ejpam-6275	46	7	bma	bma	PROPN
ejpam-6275	46	8	)	)	PUNCT
ejpam-6275	46	9	has	have	AUX
ejpam-6275	46	10	played	play	VERB
ejpam-6275	46	11	an	an	DET
ejpam-6275	46	12	important	important	ADJ
ejpam-6275	46	13	role	role	NOUN
ejpam-6275	46	14	in	in	ADP
ejpam-6275	46	15	decoding	decode	VERB
ejpam-6275	46	16	schemes	scheme	NOUN
ejpam-6275	46	17	of	of	ADP
ejpam-6275	46	18	linear	linear	ADJ
ejpam-6275	46	19	and	and	CCONJ
ejpam-6275	46	20	cyclic	cyclic	ADJ
ejpam-6275	46	21	codes	code	NOUN
ejpam-6275	46	22	,	,	PUNCT
ejpam-6275	46	23	bch	bch	PROPN
ejpam-6275	46	24	codes	code	NOUN
ejpam-6275	46	25	.	.	PUNCT
ejpam-6275	47	1	appropriate	appropriate	ADJ
ejpam-6275	47	2	overviews	overview	NOUN
ejpam-6275	47	3	of	of	ADP
ejpam-6275	47	4	the	the	DET
ejpam-6275	47	5	overall	overall	ADJ
ejpam-6275	47	6	algorithm	algorithm	NOUN
ejpam-6275	47	7	and	and	CCONJ
ejpam-6275	47	8	explanations	explanation	NOUN
ejpam-6275	47	9	of	of	ADP
ejpam-6275	47	10	its	its	PRON
ejpam-6275	47	11	uses	use	NOUN
ejpam-6275	47	12	were	be	AUX
ejpam-6275	47	13	given	give	VERB
ejpam-6275	47	14	by	by	ADP
ejpam-6275	47	15	lin	lin	PROPN
ejpam-6275	47	16	and	and	CCONJ
ejpam-6275	47	17	huffman	huffman	PROPN
ejpam-6275	47	18	and	and	CCONJ
ejpam-6275	47	19	pless	pless	PROPN
ejpam-6275	47	20	for	for	ADP
ejpam-6275	47	21	decoding	decode	VERB
ejpam-6275	47	22	finite	finite	ADJ
ejpam-6275	47	23	geometry	geometry	NOUN
ejpam-6275	47	24	and	and	CCONJ
ejpam-6275	47	25	cyclic	cyclic	ADJ
ejpam-6275	47	26	codes	code	NOUN
ejpam-6275	47	27	[	[	X
ejpam-6275	47	28	20	20	NUM
ejpam-6275	47	29	,	,	PUNCT
ejpam-6275	47	30	21	21	NUM
ejpam-6275	47	31	]	]	PUNCT
ejpam-6275	47	32	.	.	PUNCT
ejpam-6275	48	1	nonetheless	nonetheless	ADV
ejpam-6275	48	2	,	,	PUNCT
ejpam-6275	48	3	the	the	DET
ejpam-6275	48	4	computational	computational	ADJ
ejpam-6275	48	5	complexity	complexity	NOUN
ejpam-6275	48	6	of	of	ADP
ejpam-6275	48	7	the	the	DET
ejpam-6275	48	8	classical	classical	ADJ
ejpam-6275	48	9	bma	bma	PROPN
ejpam-6275	48	10	remains	remain	VERB
ejpam-6275	48	11	fairly	fairly	ADV
ejpam-6275	48	12	high	high	ADJ
ejpam-6275	48	13	,	,	PUNCT
ejpam-6275	48	14	and	and	CCONJ
ejpam-6275	48	15	innovative	innovative	ADJ
ejpam-6275	48	16	modified	modify	VERB
ejpam-6275	48	17	algorithms	algorithm	NOUN
ejpam-6275	48	18	have	have	AUX
ejpam-6275	48	19	been	be	AUX
ejpam-6275	48	20	adapted	adapt	VERB
ejpam-6275	48	21	to	to	PART
ejpam-6275	48	22	solve	solve	VERB
ejpam-6275	48	23	concrete	concrete	ADJ
ejpam-6275	48	24	algebraic	algebraic	ADJ
ejpam-6275	48	25	structures	structure	NOUN
ejpam-6275	48	26	.	.	PUNCT
ejpam-6275	49	1	the	the	DET
ejpam-6275	49	2	adoption	adoption	NOUN
ejpam-6275	49	3	of	of	ADP
ejpam-6275	49	4	such	such	ADJ
ejpam-6275	49	5	modifications	modification	NOUN
ejpam-6275	49	6	,	,	PUNCT
ejpam-6275	49	7	mainly	mainly	ADV
ejpam-6275	49	8	for	for	ADP
ejpam-6275	49	9	eisenstein	eisenstein	NOUN
ejpam-6275	49	10	fields	field	NOUN
ejpam-6275	49	11	,	,	PUNCT
ejpam-6275	49	12	is	be	AUX
ejpam-6275	49	13	a	a	DET
ejpam-6275	49	14	useful	useful	ADJ
ejpam-6275	49	15	step	step	NOUN
ejpam-6275	49	16	forward	forward	ADV
ejpam-6275	49	17	in	in	ADP
ejpam-6275	49	18	terms	term	NOUN
ejpam-6275	49	19	of	of	ADP
ejpam-6275	49	20	simplification	simplification	NOUN
ejpam-6275	49	21	of	of	ADP
ejpam-6275	49	22	decoding	decode	VERB
ejpam-6275	49	23	and	and	CCONJ
ejpam-6275	49	24	increasing	increase	VERB
ejpam-6275	49	25	the	the	DET
ejpam-6275	49	26	convergence	convergence	NOUN
ejpam-6275	49	27	rate	rate	NOUN
ejpam-6275	49	28	[	[	X
ejpam-6275	49	29	19	19	NUM
ejpam-6275	49	30	]	]	PUNCT
ejpam-6275	49	31	.	.	PUNCT
ejpam-6275	50	1	other	other	ADJ
ejpam-6275	50	2	important	important	ADJ
ejpam-6275	50	3	works	work	NOUN
ejpam-6275	50	4	have	have	AUX
ejpam-6275	50	5	brought	bring	VERB
ejpam-6275	50	6	to	to	ADP
ejpam-6275	50	7	the	the	DET
ejpam-6275	50	8	knowledge	knowledge	NOUN
ejpam-6275	50	9	of	of	ADP
ejpam-6275	50	10	issues	issue	NOUN
ejpam-6275	50	11	on	on	ADP
ejpam-6275	50	12	the	the	DET
ejpam-6275	50	13	algebraic	algebraic	ADJ
ejpam-6275	50	14	structures	structure	NOUN
ejpam-6275	50	15	for	for	ADP
ejpam-6275	50	16	the	the	DET
ejpam-6275	50	17	coding	code	VERB
ejpam-6275	50	18	theory	theory	NOUN
ejpam-6275	50	19	.	.	PUNCT
ejpam-6275	51	1	the	the	DET
ejpam-6275	51	2	combinatorial	combinatorial	ADJ
ejpam-6275	51	3	characteristics	characteristic	NOUN
ejpam-6275	51	4	of	of	ADP
ejpam-6275	51	5	linear	linear	PROPN
ejpam-6275	51	6	codes	code	NOUN
ejpam-6275	51	7	were	be	AUX
ejpam-6275	51	8	examined	examine	VERB
ejpam-6275	51	9	using	use	VERB
ejpam-6275	51	10	t	t	NOUN
ejpam-6275	51	11	-	-	PUNCT
ejpam-6275	51	12	designs	design	NOUN
ejpam-6275	51	13	by	by	ADP
ejpam-6275	51	14	assmus	assmus	NOUN
ejpam-6275	51	15	and	and	CCONJ
ejpam-6275	51	16	mattson	mattson	NOUN
ejpam-6275	51	17	,	,	PUNCT
ejpam-6275	51	18	while	while	SCONJ
ejpam-6275	51	19	t	t	NOUN
ejpam-6275	51	20	-	-	PUNCT
ejpam-6275	51	21	covers	cover	NOUN
ejpam-6275	51	22	of	of	ADP
ejpam-6275	51	23	the	the	DET
ejpam-6275	51	24	finite	finite	PROPN
ejpam-6275	51	25	projective	projective	ADJ
ejpam-6275	51	26	spaces	space	NOUN
ejpam-6275	51	27	were	be	AUX
ejpam-6275	51	28	studied	study	VERB
ejpam-6275	51	29	by	by	ADP
ejpam-6275	51	30	beutelspacher	beutelspacher	NOUN
ejpam-6275	51	31	,	,	PUNCT
ejpam-6275	51	32	which	which	PRON
ejpam-6275	51	33	also	also	ADV
ejpam-6275	51	34	focused	focus	VERB
ejpam-6275	51	35	on	on	ADP
ejpam-6275	51	36	the	the	DET
ejpam-6275	51	37	geometry	geometry	NOUN
ejpam-6275	51	38	[	[	X
ejpam-6275	51	39	20	20	NUM
ejpam-6275	51	40	,	,	PUNCT
ejpam-6275	51	41	22	22	NUM
ejpam-6275	51	42	]	]	PUNCT
ejpam-6275	51	43	.	.	PUNCT
ejpam-6275	52	1	these	these	DET
ejpam-6275	52	2	concepts	concept	NOUN
ejpam-6275	52	3	have	have	AUX
ejpam-6275	52	4	been	be	AUX
ejpam-6275	52	5	further	far	ADV
ejpam-6275	52	6	generalized	generalize	VERB
ejpam-6275	52	7	to	to	ADP
ejpam-6275	52	8	mixed	mixed	ADJ
ejpam-6275	52	9	binary	binary	ADJ
ejpam-6275	52	10	/	/	SYM
ejpam-6275	52	11	ternary	ternary	ADJ
ejpam-6275	52	12	codes	code	NOUN
ejpam-6275	52	13	by	by	ADP
ejpam-6275	52	14	brower	brower	PROPN
ejpam-6275	52	15	et	et	PROPN
ejpam-6275	52	16	al	al	PROPN
ejpam-6275	52	17	.	.	PUNCT
ejpam-6275	53	1	[	[	X
ejpam-6275	53	2	16	16	NUM
ejpam-6275	53	3	]	]	PUNCT
ejpam-6275	53	4	and	and	CCONJ
ejpam-6275	53	5	to	to	ADP
ejpam-6275	53	6	codes	code	NOUN
ejpam-6275	53	7	obtained	obtain	VERB
ejpam-6275	53	8	from	from	ADP
ejpam-6275	53	9	boolean	boolean	ADJ
ejpam-6275	53	10	functions	function	NOUN
ejpam-6275	53	11	by	by	ADP
ejpam-6275	53	12	tang	tang	PROPN
ejpam-6275	53	13	et	et	PROPN
ejpam-6275	53	14	al	al	PROPN
ejpam-6275	53	15	.	.	PUNCT
ejpam-6275	54	1	[	[	X
ejpam-6275	54	2	23	23	NUM
ejpam-6275	54	3	]	]	PUNCT
ejpam-6275	54	4	,	,	PUNCT
ejpam-6275	54	5	thus	thus	ADV
ejpam-6275	54	6	proving	prove	VERB
ejpam-6275	54	7	the	the	DET
ejpam-6275	54	8	versatility	versatility	NOUN
ejpam-6275	54	9	of	of	ADP
ejpam-6275	54	10	algebraic	algebraic	ADJ
ejpam-6275	54	11	methods	method	NOUN
ejpam-6275	54	12	in	in	ADP
ejpam-6275	54	13	terms	term	NOUN
ejpam-6275	54	14	of	of	ADP
ejpam-6275	54	15	reliable	reliable	ADJ
ejpam-6275	54	16	code	code	NOUN
ejpam-6275	54	17	construction	construction	NOUN
ejpam-6275	54	18	.	.	PUNCT
ejpam-6275	55	1	this	this	DET
ejpam-6275	55	2	paper	paper	NOUN
ejpam-6275	55	3	extends	extend	VERB
ejpam-6275	55	4	the	the	DET
ejpam-6275	55	5	literature	literature	NOUN
ejpam-6275	55	6	presented	present	VERB
ejpam-6275	55	7	above	above	ADV
ejpam-6275	55	8	by	by	ADP
ejpam-6275	55	9	developing	develop	VERB
ejpam-6275	55	10	abbreviated	abbreviate	VERB
ejpam-6275	55	11	bch	bch	NOUN
ejpam-6275	55	12	codes	code	NOUN
ejpam-6275	55	13	over	over	ADP
ejpam-6275	55	14	eisenstein	eisenstein	NOUN
ejpam-6275	55	15	fields	field	NOUN
ejpam-6275	55	16	and	and	CCONJ
ejpam-6275	55	17	presenting	present	VERB
ejpam-6275	55	18	an	an	DET
ejpam-6275	55	19	adapted	adapted	ADJ
ejpam-6275	55	20	bma	bma	NOUN
ejpam-6275	55	21	appropriate	appropriate	ADJ
ejpam-6275	55	22	to	to	ADP
ejpam-6275	55	23	the	the	DET
ejpam-6275	55	24	arithmetic	arithmetic	NOUN
ejpam-6275	55	25	of	of	ADP
ejpam-6275	55	26	the	the	DET
ejpam-6275	55	27	shortened	shorten	VERB
ejpam-6275	55	28	bch	bch	PROPN
ejpam-6275	55	29	codes	code	NOUN
ejpam-6275	55	30	.	.	PUNCT
ejpam-6275	56	1	this	this	DET
ejpam-6275	56	2	work	work	NOUN
ejpam-6275	56	3	fills	fill	VERB
ejpam-6275	56	4	the	the	DET
ejpam-6275	56	5	gap	gap	NOUN
ejpam-6275	56	6	of	of	ADP
ejpam-6275	56	7	existing	exist	VERB
ejpam-6275	56	8	eisenstein	eisenstein	NOUN
ejpam-6275	56	9	field	field	NOUN
ejpam-6275	56	10	theoretical	theoretical	ADJ
ejpam-6275	56	11	developments	development	NOUN
ejpam-6275	56	12	and	and	CCONJ
ejpam-6275	56	13	applies	apply	VERB
ejpam-6275	56	14	them	they	PRON
ejpam-6275	56	15	to	to	ADP
ejpam-6275	56	16	error	error	NOUN
ejpam-6275	56	17	correction	correction	NOUN
ejpam-6275	56	18	and	and	CCONJ
ejpam-6275	56	19	amplification	amplification	NOUN
ejpam-6275	56	20	of	of	ADP
ejpam-6275	56	21	several	several	ADJ
ejpam-6275	56	22	features	feature	NOUN
ejpam-6275	56	23	,	,	PUNCT
ejpam-6275	56	24	promoting	promote	VERB
ejpam-6275	56	25	the	the	DET
ejpam-6275	56	26	reliability	reliability	NOUN
ejpam-6275	56	27	and	and	CCONJ
ejpam-6275	56	28	performance	performance	NOUN
ejpam-6275	56	29	of	of	ADP
ejpam-6275	56	30	communication	communication	NOUN
ejpam-6275	56	31	systems	system	NOUN
ejpam-6275	56	32	.	.	PUNCT
ejpam-6275	57	1	thus	thus	ADV
ejpam-6275	57	2	,	,	PUNCT
ejpam-6275	57	3	this	this	DET
ejpam-6275	57	4	thesis	thesis	NOUN
ejpam-6275	57	5	supplements	supplement	NOUN
ejpam-6275	57	6	prior	prior	ADJ
ejpam-6275	57	7	research	research	NOUN
ejpam-6275	57	8	and	and	CCONJ
ejpam-6275	57	9	helps	helps	AUX
ejpam-6275	57	10	further	far	ADV
ejpam-6275	57	11	develop	develop	VERB
ejpam-6275	57	12	the	the	DET
ejpam-6275	57	13	theory	theory	NOUN
ejpam-6275	57	14	of	of	ADP
ejpam-6275	57	15	coding	code	VERB
ejpam-6275	57	16	for	for	ADP
ejpam-6275	57	17	the	the	DET
ejpam-6275	57	18	next	next	ADJ
ejpam-6275	57	19	generations	generation	NOUN
ejpam-6275	57	20	by	by	ADP
ejpam-6275	57	21	utilizing	utilize	VERB
ejpam-6275	57	22	the	the	DET
ejpam-6275	57	23	algebraic	algebraic	ADJ
ejpam-6275	57	24	characteristics	characteristic	NOUN
ejpam-6275	57	25	of	of	ADP
ejpam-6275	57	26	eisenstein	eisenstein	PROPN
ejpam-6275	57	27	fields	field	NOUN
ejpam-6275	57	28	and	and	CCONJ
ejpam-6275	57	29	enhancing	enhance	VERB
ejpam-6275	57	30	the	the	DET
ejpam-6275	57	31	decoding	decode	VERB
ejpam-6275	57	32	approach	approach	NOUN
ejpam-6275	57	33	[	[	X
ejpam-6275	57	34	7	7	NUM
ejpam-6275	57	35	,	,	PUNCT
ejpam-6275	57	36	9	9	NUM
ejpam-6275	57	37	,	,	PUNCT
ejpam-6275	57	38	17–19	17–19	NUM
ejpam-6275	57	39	,	,	PUNCT
ejpam-6275	57	40	24	24	NUM
ejpam-6275	57	41	]	]	PUNCT
ejpam-6275	57	42	.	.	PUNCT
ejpam-6275	58	1	shortened	shorten	VERB
ejpam-6275	58	2	bose	bose	NOUN
ejpam-6275	58	3	-	-	PUNCT
ejpam-6275	58	4	chaudhuri	chaudhuri	PROPN
ejpam-6275	58	5	-	-	PUNCT
ejpam-6275	58	6	hocquenghem	hocquenghem	PROPN
ejpam-6275	58	7	(	(	PUNCT
ejpam-6275	58	8	bch	bch	PROPN
ejpam-6275	58	9	)	)	PUNCT
ejpam-6275	58	10	codes	code	NOUN
ejpam-6275	58	11	have	have	AUX
ejpam-6275	58	12	been	be	AUX
ejpam-6275	58	13	found	find	VERB
ejpam-6275	58	14	to	to	PART
ejpam-6275	58	15	be	be	AUX
ejpam-6275	58	16	very	very	ADV
ejpam-6275	58	17	useful	useful	ADJ
ejpam-6275	58	18	in	in	ADP
ejpam-6275	58	19	correcting	correct	VERB
ejpam-6275	58	20	errors	error	NOUN
ejpam-6275	58	21	,	,	PUNCT
ejpam-6275	58	22	thereby	thereby	ADV
ejpam-6275	58	23	facilitating	facilitate	VERB
ejpam-6275	58	24	smooth	smooth	ADJ
ejpam-6275	58	25	data	datum	NOUN
ejpam-6275	58	26	communication	communication	NOUN
ejpam-6275	58	27	in	in	ADP
ejpam-6275	58	28	noisy	noisy	ADJ
ejpam-6275	58	29	media	medium	NOUN
ejpam-6275	58	30	.	.	PUNCT
ejpam-6275	59	1	their	their	PRON
ejpam-6275	59	2	flexibility	flexibility	NOUN
ejpam-6275	59	3	and	and	CCONJ
ejpam-6275	59	4	speed	speed	NOUN
ejpam-6275	59	5	are	be	AUX
ejpam-6275	59	6	ideal	ideal	ADJ
ejpam-6275	59	7	for	for	ADP
ejpam-6275	59	8	the	the	DET
ejpam-6275	59	9	current	current	ADJ
ejpam-6275	59	10	data	data	NOUN
ejpam-6275	59	11	transmission	transmission	NOUN
ejpam-6275	59	12	networks	network	NOUN
ejpam-6275	59	13	where	where	SCONJ
ejpam-6275	59	14	the	the	DET
ejpam-6275	59	15	probability	probability	NOUN
ejpam-6275	59	16	of	of	ADP
ejpam-6275	59	17	error	error	NOUN
ejpam-6275	59	18	and	and	CCONJ
ejpam-6275	59	19	computational	computational	ADJ
ejpam-6275	59	20	time	time	NOUN
ejpam-6275	59	21	are	be	AUX
ejpam-6275	59	22	critical	critical	ADJ
ejpam-6275	59	23	concerns	concern	NOUN
ejpam-6275	59	24	[	[	X
ejpam-6275	59	25	9	9	NUM
ejpam-6275	59	26	,	,	PUNCT
ejpam-6275	59	27	24	24	NUM
ejpam-6275	59	28	]	]	PUNCT
ejpam-6275	59	29	.	.	PUNCT
ejpam-6275	60	1	while	while	SCONJ
ejpam-6275	60	2	the	the	DET
ejpam-6275	60	3	classical	classical	ADJ
ejpam-6275	60	4	bch	bch	PROPN
ejpam-6275	60	5	codes	code	NOUN
ejpam-6275	60	6	over	over	ADP
ejpam-6275	60	7	finite	finite	ADJ
ejpam-6275	60	8	fields	field	NOUN
ejpam-6275	60	9	are	be	AUX
ejpam-6275	60	10	highly	highly	ADV
ejpam-6275	60	11	efficient	efficient	ADJ
ejpam-6275	60	12	,	,	PUNCT
ejpam-6275	60	13	they	they	PRON
ejpam-6275	60	14	seem	seem	VERB
ejpam-6275	60	15	to	to	PART
ejpam-6275	60	16	have	have	VERB
ejpam-6275	60	17	drawbacks	drawback	NOUN
ejpam-6275	60	18	that	that	PRON
ejpam-6275	60	19	cropped	crop	VERB
ejpam-6275	60	20	up	up	ADP
ejpam-6275	60	21	in	in	ADP
ejpam-6275	60	22	some	some	DET
ejpam-6275	60	23	high	high	ADJ
ejpam-6275	60	24	-	-	PUNCT
ejpam-6275	60	25	reliability	reliability	NOUN
ejpam-6275	60	26	applications	application	NOUN
ejpam-6275	60	27	,	,	PUNCT
ejpam-6275	60	28	such	such	ADJ
ejpam-6275	60	29	as	as	ADP
ejpam-6275	60	30	the	the	DET
ejpam-6275	60	31	trade	trade	NOUN
ejpam-6275	60	32	-	-	PUNCT
ejpam-6275	60	33	off	off	NOUN
ejpam-6275	60	34	between	between	ADP
ejpam-6275	60	35	the	the	DET
ejpam-6275	60	36	code	code	NOUN
ejpam-6275	60	37	length	length	NOUN
ejpam-6275	60	38	and	and	CCONJ
ejpam-6275	60	39	error	error	NOUN
ejpam-6275	60	40	-	-	PUNCT
ejpam-6275	60	41	correcting	correct	VERB
ejpam-6275	60	42	capability	capability	NOUN
ejpam-6275	60	43	[	[	X
ejpam-6275	60	44	7	7	NUM
ejpam-6275	60	45	,	,	PUNCT
ejpam-6275	60	46	8	8	NUM
ejpam-6275	60	47	]	]	PUNCT
ejpam-6275	60	48	.	.	PUNCT
ejpam-6275	61	1	the	the	DET
ejpam-6275	61	2	proposed	propose	VERB
ejpam-6275	61	3	concept	concept	NOUN
ejpam-6275	61	4	of	of	ADP
ejpam-6275	61	5	eisenstein	eisenstein	NOUN
ejpam-6275	61	6	fields	field	NOUN
ejpam-6275	61	7	as	as	ADP
ejpam-6275	61	8	an	an	DET
ejpam-6275	61	9	algebraic	algebraic	ADJ
ejpam-6275	61	10	structure	structure	NOUN
ejpam-6275	61	11	is	be	AUX
ejpam-6275	61	12	prospective	prospective	ADJ
ejpam-6275	61	13	to	to	PART
ejpam-6275	61	14	open	open	VERB
ejpam-6275	61	15	a	a	DET
ejpam-6275	61	16	new	new	ADJ
ejpam-6275	61	17	direction	direction	NOUN
ejpam-6275	61	18	toward	toward	ADP
ejpam-6275	61	19	research	research	NOUN
ejpam-6275	61	20	on	on	ADP
ejpam-6275	61	21	the	the	DET
ejpam-6275	61	22	construction	construction	NOUN
ejpam-6275	61	23	of	of	ADP
ejpam-6275	61	24	higher	high	ADJ
ejpam-6275	61	25	coding	code	VERB
ejpam-6275	61	26	schemes	scheme	NOUN
ejpam-6275	61	27	.	.	PUNCT
ejpam-6275	62	1	eisenstein	eisenstein	PROPN
ejpam-6275	62	2	fields	field	NOUN
ejpam-6275	62	3	are	be	AUX
ejpam-6275	62	4	a	a	DET
ejpam-6275	62	5	natural	natural	ADJ
ejpam-6275	62	6	generalization	generalization	NOUN
ejpam-6275	62	7	of	of	ADP
ejpam-6275	62	8	gaussian	gaussian	ADJ
ejpam-6275	62	9	fields	field	NOUN
ejpam-6275	62	10	and	and	CCONJ
ejpam-6275	62	11	offer	offer	VERB
ejpam-6275	62	12	a	a	DET
ejpam-6275	62	13	higher	high	ADJ
ejpam-6275	62	14	level	level	NOUN
ejpam-6275	62	15	of	of	ADP
ejpam-6275	62	16	structural	structural	ADJ
ejpam-6275	62	17	complexity	complexity	NOUN
ejpam-6275	62	18	,	,	PUNCT
ejpam-6275	62	19	which	which	PRON
ejpam-6275	62	20	can	can	AUX
ejpam-6275	62	21	contribute	contribute	VERB
ejpam-6275	62	22	to	to	ADP
ejpam-6275	62	23	the	the	DET
ejpam-6275	62	24	improvement	improvement	NOUN
ejpam-6275	62	25	of	of	ADP
ejpam-6275	62	26	the	the	DET
ejpam-6275	62	27	error	error	NOUN
ejpam-6275	62	28	-	-	PUNCT
ejpam-6275	62	29	correcting	correct	VERB
ejpam-6275	62	30	code	code	NOUN
ejpam-6275	62	31	.	.	PUNCT
ejpam-6275	63	1	a	a	DET
ejpam-6275	63	2	number	number	NOUN
ejpam-6275	63	3	of	of	ADP
ejpam-6275	63	4	previous	previous	ADJ
ejpam-6275	63	5	works	work	NOUN
ejpam-6275	63	6	,	,	PUNCT
ejpam-6275	63	7	including	include	VERB
ejpam-6275	63	8	huber	huber	PROPN
ejpam-6275	63	9	and	and	CCONJ
ejpam-6275	63	10	sajjad	sajjad	PROPN
ejpam-6275	63	11	et	et	PROPN
ejpam-6275	63	12	al	al	PROPN
ejpam-6275	63	13	.	.	PROPN
ejpam-6275	63	14	,	,	PUNCT
ejpam-6275	63	15	have	have	AUX
ejpam-6275	63	16	shown	show	VERB
ejpam-6275	63	17	that	that	SCONJ
ejpam-6275	63	18	eisenstein	eisenstein	NOUN
ejpam-6275	63	19	and	and	CCONJ
ejpam-6275	63	20	gaussian	gaussian	ADJ
ejpam-6275	63	21	fields	field	NOUN
ejpam-6275	63	22	could	could	AUX
ejpam-6275	63	23	be	be	AUX
ejpam-6275	63	24	applied	apply	VERB
ejpam-6275	63	25	in	in	ADP
ejpam-6275	63	26	cryptographic	cryptographic	ADJ
ejpam-6275	63	27	m.	m.	NOUN
ejpam-6275	63	28	sajjad	sajjad	PROPN
ejpam-6275	63	29	et	et	PROPN
ejpam-6275	63	30	al	al	PROPN
ejpam-6275	63	31	.	.	PUNCT
ejpam-6275	63	32	/	/	SYM
ejpam-6275	63	33	eur	eur	PROPN
ejpam-6275	63	34	.	.	PUNCT
ejpam-6275	64	1	j.	j.	PROPN
ejpam-6275	64	2	pure	pure	PROPN
ejpam-6275	64	3	appl	appl	PROPN
ejpam-6275	64	4	.	.	PROPN
ejpam-6275	64	5	math	math	PROPN
ejpam-6275	64	6	,	,	PUNCT
ejpam-6275	64	7	18	18	NUM
ejpam-6275	64	8	(	(	PUNCT
ejpam-6275	64	9	3	3	NUM
ejpam-6275	64	10	)	)	PUNCT
ejpam-6275	64	11	(	(	PUNCT
ejpam-6275	64	12	2025	2025	NUM
ejpam-6275	64	13	)	)	PUNCT
ejpam-6275	64	14	,	,	PUNCT
ejpam-6275	64	15	6275	6275	NUM
ejpam-6275	64	16	4	4	NUM
ejpam-6275	64	17	of	of	ADP
ejpam-6275	64	18	36	36	NUM
ejpam-6275	64	19	applications	application	NOUN
ejpam-6275	64	20	and	and	CCONJ
ejpam-6275	64	21	coding	code	VERB
ejpam-6275	64	22	theory	theory	NOUN
ejpam-6275	64	23	;	;	PUNCT
ejpam-6275	64	24	hence	hence	ADV
ejpam-6275	64	25	,	,	PUNCT
ejpam-6275	64	26	more	more	ADJ
ejpam-6275	64	27	investigation	investigation	NOUN
ejpam-6275	64	28	needs	need	VERB
ejpam-6275	64	29	to	to	PART
ejpam-6275	64	30	be	be	AUX
ejpam-6275	64	31	made	make	VERB
ejpam-6275	64	32	on	on	ADP
ejpam-6275	64	33	the	the	DET
ejpam-6275	64	34	use	use	NOUN
ejpam-6275	64	35	of	of	ADP
ejpam-6275	64	36	eisenstein	eisenstein	NOUN
ejpam-6275	64	37	and	and	CCONJ
ejpam-6275	64	38	gaussian	gaussian	ADJ
ejpam-6275	64	39	fields	field	NOUN
ejpam-6275	64	40	in	in	ADP
ejpam-6275	64	41	the	the	DET
ejpam-6275	64	42	construction	construction	NOUN
ejpam-6275	64	43	of	of	ADP
ejpam-6275	64	44	bch	bch	PROPN
ejpam-6275	64	45	code	code	PROPN
ejpam-6275	65	1	[	[	X
ejpam-6275	65	2	17	17	NUM
ejpam-6275	65	3	,	,	PUNCT
ejpam-6275	65	4	18	18	NUM
ejpam-6275	65	5	]	]	PUNCT
ejpam-6275	65	6	.	.	PUNCT
ejpam-6275	66	1	moreover	moreover	ADV
ejpam-6275	66	2	,	,	PUNCT
ejpam-6275	66	3	current	current	ADJ
ejpam-6275	66	4	improvements	improvement	NOUN
ejpam-6275	66	5	in	in	ADP
ejpam-6275	66	6	decoding	decode	VERB
ejpam-6275	66	7	techniques	technique	NOUN
ejpam-6275	66	8	like	like	ADP
ejpam-6275	66	9	the	the	DET
ejpam-6275	66	10	berlekamp	berlekamp	NOUN
ejpam-6275	66	11	-	-	PUNCT
ejpam-6275	66	12	massey	massey	PROPN
ejpam-6275	66	13	algorithm	algorithm	NOUN
ejpam-6275	66	14	(	(	PUNCT
ejpam-6275	66	15	bma	bma	NOUN
ejpam-6275	66	16	)	)	PUNCT
ejpam-6275	66	17	offer	offer	VERB
ejpam-6275	66	18	potentialities	potentiality	NOUN
ejpam-6275	66	19	to	to	PART
ejpam-6275	66	20	make	make	VERB
ejpam-6275	66	21	moderate	moderate	ADJ
ejpam-6275	66	22	decoding	decoding	NOUN
ejpam-6275	66	23	procedures	procedure	NOUN
ejpam-6275	66	24	and	and	CCONJ
ejpam-6275	66	25	lessen	lessen	VERB
ejpam-6275	66	26	the	the	DET
ejpam-6275	66	27	computational	computational	ADJ
ejpam-6275	66	28	intricacy	intricacy	NOUN
ejpam-6275	66	29	required	require	VERB
ejpam-6275	66	30	to	to	PART
ejpam-6275	66	31	defend	defend	VERB
ejpam-6275	66	32	contemporary	contemporary	ADJ
ejpam-6275	66	33	communication	communication	NOUN
ejpam-6275	66	34	systems	system	NOUN
ejpam-6275	66	35	[	[	X
ejpam-6275	66	36	19	19	NUM
ejpam-6275	66	37	,	,	PUNCT
ejpam-6275	66	38	20	20	NUM
ejpam-6275	66	39	]	]	PUNCT
ejpam-6275	66	40	.	.	PUNCT
ejpam-6275	67	1	this	this	DET
ejpam-6275	67	2	research	research	NOUN
ejpam-6275	67	3	is	be	AUX
ejpam-6275	67	4	motivated	motivate	VERB
ejpam-6275	67	5	by	by	ADP
ejpam-6275	67	6	developing	develop	VERB
ejpam-6275	67	7	error	error	NOUN
ejpam-6275	67	8	-	-	PUNCT
ejpam-6275	67	9	correcting	correct	VERB
ejpam-6275	67	10	codes	code	NOUN
ejpam-6275	67	11	with	with	ADP
ejpam-6275	67	12	enhanced	enhanced	ADJ
ejpam-6275	67	13	performance	performance	NOUN
ejpam-6275	67	14	measures	measure	NOUN
ejpam-6275	67	15	and	and	CCONJ
ejpam-6275	67	16	channel	channel	NOUN
ejpam-6275	67	17	decoding	decode	VERB
ejpam-6275	67	18	techniques	technique	NOUN
ejpam-6275	67	19	using	use	VERB
ejpam-6275	67	20	eisenstein	eisenstein	NOUN
ejpam-6275	67	21	field	field	NOUN
ejpam-6275	67	22	advantages	advantage	NOUN
ejpam-6275	67	23	.	.	PUNCT
ejpam-6275	68	1	the	the	DET
ejpam-6275	68	2	proposed	propose	VERB
ejpam-6275	68	3	modification	modification	NOUN
ejpam-6275	68	4	of	of	ADP
ejpam-6275	68	5	bma	bma	PROPN
ejpam-6275	68	6	and	and	CCONJ
ejpam-6275	68	7	integration	integration	NOUN
ejpam-6275	68	8	with	with	ADP
ejpam-6275	68	9	these	these	DET
ejpam-6275	68	10	three	three	NUM
ejpam-6275	68	11	fields	field	NOUN
ejpam-6275	68	12	is	be	AUX
ejpam-6275	68	13	designed	design	VERB
ejpam-6275	68	14	to	to	PART
ejpam-6275	68	15	contribute	contribute	VERB
ejpam-6275	68	16	to	to	ADP
ejpam-6275	68	17	the	the	DET
ejpam-6275	68	18	further	further	ADJ
ejpam-6275	68	19	development	development	NOUN
ejpam-6275	68	20	of	of	ADP
ejpam-6275	68	21	coding	code	VERB
ejpam-6275	68	22	theory	theory	NOUN
ejpam-6275	68	23	and	and	CCONJ
ejpam-6275	68	24	real	real	ADJ
ejpam-6275	68	25	communication	communication	NOUN
ejpam-6275	68	26	applications	application	NOUN
ejpam-6275	68	27	.	.	PUNCT
ejpam-6275	69	1	despite	despite	SCONJ
ejpam-6275	69	2	significant	significant	ADJ
ejpam-6275	69	3	advancements	advancement	NOUN
ejpam-6275	69	4	in	in	ADP
ejpam-6275	69	5	coding	code	VERB
ejpam-6275	69	6	theory	theory	NOUN
ejpam-6275	69	7	,	,	PUNCT
ejpam-6275	69	8	several	several	ADJ
ejpam-6275	69	9	research	research	NOUN
ejpam-6275	69	10	gaps	gap	NOUN
ejpam-6275	69	11	persist	persist	VERB
ejpam-6275	69	12	in	in	ADP
ejpam-6275	69	13	the	the	DET
ejpam-6275	69	14	development	development	NOUN
ejpam-6275	69	15	of	of	ADP
ejpam-6275	69	16	shortened	shorten	VERB
ejpam-6275	69	17	bch	bch	PROPN
ejpam-6275	69	18	codes	code	NOUN
ejpam-6275	69	19	and	and	CCONJ
ejpam-6275	69	20	their	their	PRON
ejpam-6275	69	21	decoding	decode	VERB
ejpam-6275	69	22	techniques	technique	NOUN
ejpam-6275	69	23	.	.	PUNCT
ejpam-6275	70	1	although	although	SCONJ
ejpam-6275	70	2	the	the	DET
ejpam-6275	70	3	fields	field	NOUN
ejpam-6275	70	4	of	of	ADP
ejpam-6275	70	5	eisenstein	eisenstein	PROPN
ejpam-6275	70	6	have	have	AUX
ejpam-6275	70	7	been	be	AUX
ejpam-6275	70	8	discussed	discuss	VERB
ejpam-6275	70	9	in	in	ADP
ejpam-6275	70	10	the	the	DET
ejpam-6275	70	11	sphere	sphere	NOUN
ejpam-6275	70	12	of	of	ADP
ejpam-6275	70	13	cryptography	cryptography	NOUN
ejpam-6275	70	14	and	and	CCONJ
ejpam-6275	70	15	the	the	DET
ejpam-6275	70	16	theory	theory	NOUN
ejpam-6275	70	17	of	of	ADP
ejpam-6275	70	18	codes	code	NOUN
ejpam-6275	70	19	,	,	PUNCT
ejpam-6275	70	20	the	the	DET
ejpam-6275	70	21	application	application	NOUN
ejpam-6275	70	22	of	of	ADP
ejpam-6275	70	23	such	such	ADJ
ejpam-6275	70	24	fields	field	NOUN
ejpam-6275	70	25	to	to	PART
ejpam-6275	70	26	construct	construct	VERB
ejpam-6275	70	27	shortened	shorten	VERB
ejpam-6275	70	28	bch	bch	PROPN
ejpam-6275	70	29	codes	code	NOUN
ejpam-6275	70	30	has	have	AUX
ejpam-6275	70	31	not	not	PART
ejpam-6275	70	32	been	be	AUX
ejpam-6275	70	33	thoroughly	thoroughly	ADV
ejpam-6275	70	34	investigated	investigate	VERB
ejpam-6275	70	35	.	.	PUNCT
ejpam-6275	71	1	in	in	ADP
ejpam-6275	71	2	finite	finite	NOUN
ejpam-6275	71	3	and	and	CCONJ
ejpam-6275	71	4	gaussian	gaussian	ADJ
ejpam-6275	71	5	fields	field	NOUN
ejpam-6275	71	6	,	,	PUNCT
ejpam-6275	71	7	most	most	ADJ
ejpam-6275	71	8	of	of	ADP
ejpam-6275	71	9	the	the	DET
ejpam-6275	71	10	previous	previous	ADJ
ejpam-6275	71	11	studies	study	NOUN
ejpam-6275	71	12	have	have	AUX
ejpam-6275	71	13	been	be	AUX
ejpam-6275	71	14	performed	perform	VERB
ejpam-6275	71	15	on	on	ADP
ejpam-6275	71	16	them	they	PRON
ejpam-6275	71	17	[	[	X
ejpam-6275	71	18	17–19	17–19	NUM
ejpam-6275	71	19	]	]	PUNCT
ejpam-6275	71	20	.	.	PUNCT
ejpam-6275	72	1	the	the	DET
ejpam-6275	72	2	classical	classical	ADJ
ejpam-6275	72	3	bma	bma	PROPN
ejpam-6275	72	4	has	have	AUX
ejpam-6275	72	5	been	be	AUX
ejpam-6275	72	6	studied	study	VERB
ejpam-6275	72	7	in	in	ADP
ejpam-6275	72	8	detail	detail	NOUN
ejpam-6275	72	9	in	in	ADP
ejpam-6275	72	10	literature	literature	NOUN
ejpam-6275	72	11	,	,	PUNCT
ejpam-6275	72	12	but	but	CCONJ
ejpam-6275	72	13	its	its	PRON
ejpam-6275	72	14	complexity	complexity	NOUN
ejpam-6275	72	15	is	be	AUX
ejpam-6275	72	16	high	high	ADJ
ejpam-6275	72	17	and	and	CCONJ
ejpam-6275	72	18	thus	thus	ADV
ejpam-6275	72	19	becomes	become	VERB
ejpam-6275	72	20	a	a	DET
ejpam-6275	72	21	problem	problem	NOUN
ejpam-6275	72	22	when	when	SCONJ
ejpam-6275	72	23	one	one	PRON
ejpam-6275	72	24	needs	need	VERB
ejpam-6275	72	25	to	to	PART
ejpam-6275	72	26	use	use	VERB
ejpam-6275	72	27	it	it	PRON
ejpam-6275	72	28	in	in	ADP
ejpam-6275	72	29	the	the	DET
ejpam-6275	72	30	realtime	realtime	NOUN
ejpam-6275	72	31	application	application	NOUN
ejpam-6275	72	32	.	.	PUNCT
ejpam-6275	73	1	extensions	extension	NOUN
ejpam-6275	73	2	of	of	ADP
ejpam-6275	73	3	the	the	DET
ejpam-6275	73	4	above	above	ADV
ejpam-6275	73	5	-	-	PUNCT
ejpam-6275	73	6	described	describe	VERB
ejpam-6275	73	7	bma	bma	PROPN
ejpam-6275	73	8	,	,	PUNCT
ejpam-6275	73	9	especially	especially	ADV
ejpam-6275	73	10	those	those	PRON
ejpam-6275	73	11	that	that	PRON
ejpam-6275	73	12	take	take	VERB
ejpam-6275	73	13	into	into	ADP
ejpam-6275	73	14	account	account	NOUN
ejpam-6275	73	15	arithmetic	arithmetic	ADJ
ejpam-6275	73	16	properties	property	NOUN
ejpam-6275	73	17	of	of	ADP
ejpam-6275	73	18	eisenstein	eisenstein	NOUN
ejpam-6275	73	19	fields	field	NOUN
ejpam-6275	73	20	,	,	PUNCT
ejpam-6275	73	21	are	be	AUX
ejpam-6275	73	22	few	few	ADJ
ejpam-6275	73	23	and	and	CCONJ
ejpam-6275	73	24	far	far	ADV
ejpam-6275	73	25	between	between	ADP
ejpam-6275	73	26	in	in	ADP
ejpam-6275	73	27	the	the	DET
ejpam-6275	73	28	literature	literature	NOUN
ejpam-6275	73	29	[	[	X
ejpam-6275	73	30	19	19	NUM
ejpam-6275	73	31	,	,	PUNCT
ejpam-6275	73	32	21	21	NUM
ejpam-6275	73	33	,	,	PUNCT
ejpam-6275	73	34	25	25	NUM
ejpam-6275	73	35	]	]	PUNCT
ejpam-6275	73	36	.	.	PUNCT
ejpam-6275	74	1	previous	previous	ADJ
ejpam-6275	74	2	research	research	NOUN
ejpam-6275	74	3	works	work	NOUN
ejpam-6275	74	4	that	that	PRON
ejpam-6275	74	5	analyze	analyze	VERB
ejpam-6275	74	6	the	the	DET
ejpam-6275	74	7	shortened	shorten	VERB
ejpam-6275	74	8	bch	bch	PROPN
ejpam-6275	74	9	codes	code	NOUN
ejpam-6275	74	10	mostly	mostly	ADV
ejpam-6275	74	11	focus	focus	VERB
ejpam-6275	74	12	on	on	ADP
ejpam-6275	74	13	either	either	CCONJ
ejpam-6275	74	14	the	the	DET
ejpam-6275	74	15	length	length	NOUN
ejpam-6275	74	16	reduction	reduction	NOUN
ejpam-6275	74	17	aspect	aspect	NOUN
ejpam-6275	74	18	or	or	CCONJ
ejpam-6275	74	19	the	the	DET
ejpam-6275	74	20	error	error	NOUN
ejpam-6275	74	21	-	-	PUNCT
ejpam-6275	74	22	correcting	correct	VERB
ejpam-6275	74	23	capability	capability	NOUN
ejpam-6275	74	24	dimension	dimension	NOUN
ejpam-6275	74	25	without	without	ADP
ejpam-6275	74	26	providing	provide	VERB
ejpam-6275	74	27	a	a	DET
ejpam-6275	74	28	balance	balance	NOUN
ejpam-6275	74	29	between	between	ADP
ejpam-6275	74	30	the	the	DET
ejpam-6275	74	31	two	two	NUM
ejpam-6275	74	32	factors	factor	NOUN
ejpam-6275	74	33	.	.	PUNCT
ejpam-6275	75	1	such	such	ADJ
ejpam-6275	75	2	balance	balance	NOUN
ejpam-6275	75	3	is	be	AUX
ejpam-6275	75	4	envisaged	envisage	VERB
ejpam-6275	75	5	for	for	ADP
ejpam-6275	75	6	applications	application	NOUN
ejpam-6275	75	7	that	that	PRON
ejpam-6275	75	8	demand	demand	VERB
ejpam-6275	75	9	both	both	PRON
ejpam-6275	75	10	high	high	ADJ
ejpam-6275	75	11	reliability	reliability	NOUN
ejpam-6275	75	12	and	and	CCONJ
ejpam-6275	75	13	low	low	ADJ
ejpam-6275	75	14	latency	latency	NOUN
ejpam-6275	75	15	[	[	X
ejpam-6275	75	16	7–9	7–9	NOUN
ejpam-6275	75	17	]	]	X
ejpam-6275	75	18	.	.	PUNCT
ejpam-6275	76	1	closing	close	VERB
ejpam-6275	76	2	these	these	DET
ejpam-6275	76	3	gaps	gap	NOUN
ejpam-6275	76	4	,	,	PUNCT
ejpam-6275	76	5	this	this	DET
ejpam-6275	76	6	research	research	NOUN
ejpam-6275	76	7	is	be	AUX
ejpam-6275	76	8	aimed	aim	VERB
ejpam-6275	76	9	at	at	ADP
ejpam-6275	76	10	developing	develop	VERB
ejpam-6275	76	11	shortened	shorten	VERB
ejpam-6275	76	12	bch	bch	PROPN
ejpam-6275	76	13	codes	code	NOUN
ejpam-6275	76	14	over	over	ADP
ejpam-6275	76	15	the	the	DET
ejpam-6275	76	16	eisenstein	eisenstein	NOUN
ejpam-6275	76	17	fields	field	NOUN
ejpam-6275	76	18	,	,	PUNCT
ejpam-6275	76	19	deriving	derive	VERB
ejpam-6275	76	20	the	the	DET
ejpam-6275	76	21	generator	generator	NOUN
ejpam-6275	76	22	polynomials	polynomial	NOUN
ejpam-6275	76	23	for	for	ADP
ejpam-6275	76	24	them	they	PRON
ejpam-6275	76	25	,	,	PUNCT
ejpam-6275	76	26	and	and	CCONJ
ejpam-6275	76	27	presenting	present	VERB
ejpam-6275	76	28	the	the	DET
ejpam-6275	76	29	modified	modified	ADJ
ejpam-6275	76	30	bma	bma	PROPN
ejpam-6275	76	31	for	for	ADP
ejpam-6275	76	32	decoding	decode	VERB
ejpam-6275	76	33	.	.	PUNCT
ejpam-6275	77	1	therefore	therefore	ADV
ejpam-6275	77	2	,	,	PUNCT
ejpam-6275	77	3	this	this	DET
ejpam-6275	77	4	study	study	NOUN
ejpam-6275	77	5	seeks	seek	VERB
ejpam-6275	77	6	to	to	PART
ejpam-6275	77	7	solve	solve	VERB
ejpam-6275	77	8	the	the	DET
ejpam-6275	77	9	problem	problem	NOUN
ejpam-6275	77	10	of	of	ADP
ejpam-6275	77	11	developing	develop	VERB
ejpam-6275	77	12	an	an	DET
ejpam-6275	77	13	understanding	understanding	NOUN
ejpam-6275	77	14	of	of	ADP
ejpam-6275	77	15	the	the	DET
ejpam-6275	77	16	trade	trade	NOUN
ejpam-6275	77	17	-	-	PUNCT
ejpam-6275	77	18	offs	off	NOUN
ejpam-6275	77	19	and	and	CCONJ
ejpam-6275	77	20	performance	performance	NOUN
ejpam-6275	77	21	measures	measure	NOUN
ejpam-6275	77	22	to	to	PART
ejpam-6275	77	23	close	close	VERB
ejpam-6275	77	24	the	the	DET
ejpam-6275	77	25	gap	gap	NOUN
ejpam-6275	77	26	that	that	PRON
ejpam-6275	77	27	exists	exist	VERB
ejpam-6275	77	28	between	between	ADP
ejpam-6275	77	29	theoretical	theoretical	ADJ
ejpam-6275	77	30	enhancements	enhancement	NOUN
ejpam-6275	77	31	and	and	CCONJ
ejpam-6275	77	32	application	application	NOUN
ejpam-6275	77	33	in	in	ADP
ejpam-6275	77	34	coding	code	VERB
ejpam-6275	77	35	theory	theory	NOUN
ejpam-6275	77	36	.	.	PUNCT
ejpam-6275	78	1	this	this	DET
ejpam-6275	78	2	work	work	NOUN
ejpam-6275	78	3	presents	present	VERB
ejpam-6275	78	4	new	new	ADJ
ejpam-6275	78	5	approaches	approach	NOUN
ejpam-6275	78	6	toward	toward	ADP
ejpam-6275	78	7	the	the	DET
ejpam-6275	78	8	construction	construction	NOUN
ejpam-6275	78	9	and	and	CCONJ
ejpam-6275	78	10	decoding	decoding	NOUN
ejpam-6275	78	11	of	of	ADP
ejpam-6275	78	12	errorcorrecting	errorcorrecte	VERB
ejpam-6275	78	13	codes	code	NOUN
ejpam-6275	78	14	using	use	VERB
ejpam-6275	78	15	shortened	shorten	VERB
ejpam-6275	78	16	bch	bch	PROPN
ejpam-6275	78	17	codes	code	NOUN
ejpam-6275	78	18	over	over	ADP
ejpam-6275	78	19	the	the	DET
ejpam-6275	78	20	eisenstein	eisenstein	NOUN
ejpam-6275	78	21	fields	field	NOUN
ejpam-6275	78	22	.	.	PUNCT
ejpam-6275	79	1	the	the	DET
ejpam-6275	79	2	study	study	NOUN
ejpam-6275	79	3	constructs	construct	VERB
ejpam-6275	79	4	these	these	DET
ejpam-6275	79	5	codes	code	NOUN
ejpam-6275	79	6	from	from	ADP
ejpam-6275	79	7	a	a	DET
ejpam-6275	79	8	fresh	fresh	ADJ
ejpam-6275	79	9	perspective	perspective	NOUN
ejpam-6275	79	10	with	with	ADP
ejpam-6275	79	11	reference	reference	NOUN
ejpam-6275	79	12	to	to	ADP
ejpam-6275	79	13	the	the	DET
ejpam-6275	79	14	properties	property	NOUN
ejpam-6275	79	15	of	of	ADP
ejpam-6275	79	16	eisenstein	eisenstein	NOUN
ejpam-6275	79	17	fields	field	NOUN
ejpam-6275	79	18	as	as	ADP
ejpam-6275	79	19	an	an	DET
ejpam-6275	79	20	augmentation	augmentation	NOUN
ejpam-6275	79	21	of	of	ADP
ejpam-6275	79	22	gaussian	gaussian	ADJ
ejpam-6275	79	23	fields	field	NOUN
ejpam-6275	79	24	to	to	PART
ejpam-6275	79	25	attain	attain	VERB
ejpam-6275	79	26	optimal	optimal	ADJ
ejpam-6275	79	27	generator	generator	NOUN
ejpam-6275	79	28	polynomials	polynomial	NOUN
ejpam-6275	79	29	and	and	CCONJ
ejpam-6275	79	30	algebraic	algebraic	ADJ
ejpam-6275	79	31	architecture	architecture	NOUN
ejpam-6275	79	32	.	.	PUNCT
ejpam-6275	80	1	the	the	DET
ejpam-6275	80	2	obtained	obtain	VERB
ejpam-6275	80	3	codes	code	NOUN
ejpam-6275	80	4	strike	strike	VERB
ejpam-6275	80	5	a	a	DET
ejpam-6275	80	6	perfect	perfect	ADJ
ejpam-6275	80	7	balance	balance	NOUN
ejpam-6275	80	8	between	between	ADP
ejpam-6275	80	9	possible	possible	ADJ
ejpam-6275	80	10	code	code	NOUN
ejpam-6275	80	11	length	length	NOUN
ejpam-6275	80	12	and	and	CCONJ
ejpam-6275	80	13	the	the	DET
ejpam-6275	80	14	code	code	NOUN
ejpam-6275	80	15	’s	’s	PART
ejpam-6275	80	16	ability	ability	NOUN
ejpam-6275	80	17	to	to	PART
ejpam-6275	80	18	correct	correct	VERB
ejpam-6275	80	19	errors	error	NOUN
ejpam-6275	80	20	,	,	PUNCT
ejpam-6275	80	21	making	make	VERB
ejpam-6275	80	22	it	it	PRON
ejpam-6275	80	23	perfect	perfect	ADJ
ejpam-6275	80	24	for	for	ADP
ejpam-6275	80	25	applications	application	NOUN
ejpam-6275	80	26	where	where	SCONJ
ejpam-6275	80	27	reliability	reliability	NOUN
ejpam-6275	80	28	is	be	AUX
ejpam-6275	80	29	paramount	paramount	ADJ
ejpam-6275	80	30	and	and	CCONJ
ejpam-6275	80	31	latency	latency	NOUN
ejpam-6275	80	32	is	be	AUX
ejpam-6275	80	33	low	low	ADJ
ejpam-6275	80	34	.	.	PUNCT
ejpam-6275	81	1	an	an	DET
ejpam-6275	81	2	original	original	ADJ
ejpam-6275	81	3	approach	approach	NOUN
ejpam-6275	81	4	proposed	propose	VERB
ejpam-6275	81	5	in	in	ADP
ejpam-6275	81	6	this	this	DET
ejpam-6275	81	7	work	work	NOUN
ejpam-6275	81	8	is	be	AUX
ejpam-6275	81	9	the	the	DET
ejpam-6275	81	10	modified	modify	VERB
ejpam-6275	81	11	berlekamp	berlekamp	NOUN
ejpam-6275	81	12	-	-	PUNCT
ejpam-6275	81	13	massey	massey	NOUN
ejpam-6275	81	14	algorithm	algorithm	NOUN
ejpam-6275	81	15	(	(	PUNCT
ejpam-6275	81	16	bma	bma	PROPN
ejpam-6275	81	17	)	)	PUNCT
ejpam-6275	81	18	suitable	suitable	ADJ
ejpam-6275	81	19	for	for	ADP
ejpam-6275	81	20	the	the	DET
ejpam-6275	81	21	eisenstein	eisenstein	PROPN
ejpam-6275	81	22	fields	field	NOUN
ejpam-6275	81	23	.	.	PUNCT
ejpam-6275	82	1	the	the	DET
ejpam-6275	82	2	approach	approach	NOUN
ejpam-6275	82	3	outlined	outline	VERB
ejpam-6275	82	4	in	in	ADP
ejpam-6275	82	5	this	this	DET
ejpam-6275	82	6	modification	modification	NOUN
ejpam-6275	82	7	is	be	AUX
ejpam-6275	82	8	capable	capable	ADJ
ejpam-6275	82	9	of	of	ADP
ejpam-6275	82	10	utilizing	utilize	VERB
ejpam-6275	82	11	more	more	ADJ
ejpam-6275	82	12	complex	complex	ADJ
ejpam-6275	82	13	methods	method	NOUN
ejpam-6275	82	14	to	to	PART
ejpam-6275	82	15	deal	deal	VERB
ejpam-6275	82	16	with	with	ADP
ejpam-6275	82	17	wider	wide	ADJ
ejpam-6275	82	18	residue	residue	NOUN
ejpam-6275	82	19	classes	class	NOUN
ejpam-6275	82	20	in	in	ADP
ejpam-6275	82	21	order	order	NOUN
ejpam-6275	82	22	to	to	PART
ejpam-6275	82	23	offer	offer	VERB
ejpam-6275	82	24	better	well	ADJ
ejpam-6275	82	25	convergence	convergence	NOUN
ejpam-6275	82	26	and	and	CCONJ
ejpam-6275	82	27	lower	low	ADJ
ejpam-6275	82	28	computational	computational	ADJ
ejpam-6275	82	29	demands	demand	NOUN
ejpam-6275	82	30	than	than	SCONJ
ejpam-6275	82	31	needed	need	VERB
ejpam-6275	82	32	for	for	ADP
ejpam-6275	82	33	the	the	DET
ejpam-6275	82	34	classical	classical	ADJ
ejpam-6275	82	35	bma	bma	PROPN
ejpam-6275	82	36	.	.	PUNCT
ejpam-6275	83	1	in	in	ADP
ejpam-6275	83	2	the	the	DET
ejpam-6275	83	3	analysis	analysis	NOUN
ejpam-6275	83	4	of	of	ADP
ejpam-6275	83	5	performance	performance	NOUN
ejpam-6275	83	6	,	,	PUNCT
ejpam-6275	83	7	the	the	DET
ejpam-6275	83	8	work	work	NOUN
ejpam-6275	83	9	reveals	reveal	VERB
ejpam-6275	83	10	that	that	SCONJ
ejpam-6275	83	11	the	the	DET
ejpam-6275	83	12	proposed	propose	VERB
ejpam-6275	83	13	codes	code	NOUN
ejpam-6275	83	14	exhibit	exhibit	VERB
ejpam-6275	83	15	higher	high	ADJ
ejpam-6275	83	16	error	error	NOUN
ejpam-6275	83	17	correction	correction	NOUN
ejpam-6275	83	18	and	and	CCONJ
ejpam-6275	83	19	better	well	ADJ
ejpam-6275	83	20	computational	computational	ADJ
ejpam-6275	83	21	complexity	complexity	NOUN
ejpam-6275	83	22	compared	compare	VERB
ejpam-6275	83	23	to	to	ADP
ejpam-6275	83	24	conventional	conventional	ADJ
ejpam-6275	83	25	bch	bch	PROPN
ejpam-6275	83	26	over	over	ADP
ejpam-6275	83	27	finite	finite	ADJ
ejpam-6275	83	28	fields	field	NOUN
ejpam-6275	83	29	.	.	PUNCT
ejpam-6275	84	1	in	in	ADP
ejpam-6275	84	2	addition	addition	NOUN
ejpam-6275	84	3	,	,	PUNCT
ejpam-6275	84	4	this	this	DET
ejpam-6275	84	5	research	research	NOUN
ejpam-6275	84	6	expands	expand	VERB
ejpam-6275	84	7	on	on	ADP
ejpam-6275	84	8	the	the	DET
ejpam-6275	84	9	cyclic	cyclic	ADJ
ejpam-6275	84	10	subgroups	subgroup	NOUN
ejpam-6275	84	11	of	of	ADP
ejpam-6275	84	12	fields	field	NOUN
ejpam-6275	84	13	of	of	ADP
ejpam-6275	84	14	eisenstein	eisenstein	NOUN
ejpam-6275	84	15	for	for	ADP
ejpam-6275	84	16	coding	code	VERB
ejpam-6275	84	17	theory	theory	NOUN
ejpam-6275	84	18	,	,	PUNCT
ejpam-6275	84	19	as	as	SCONJ
ejpam-6275	84	20	it	it	PRON
ejpam-6275	84	21	now	now	ADV
ejpam-6275	84	22	incorporates	incorporate	VERB
ejpam-6275	84	23	settings	setting	NOUN
ejpam-6275	84	24	outside	outside	ADP
ejpam-6275	84	25	of	of	ADP
ejpam-6275	84	26	cryptographic	cryptographic	ADJ
ejpam-6275	84	27	applications	application	NOUN
ejpam-6275	84	28	,	,	PUNCT
ejpam-6275	84	29	thereby	thereby	ADV
ejpam-6275	84	30	finding	find	VERB
ejpam-6275	84	31	its	its	PRON
ejpam-6275	84	32	usefulness	usefulness	NOUN
ejpam-6275	84	33	in	in	ADP
ejpam-6275	84	34	next	next	ADJ
ejpam-6275	84	35	-	-	PUNCT
ejpam-6275	84	36	generation	generation	NOUN
ejpam-6275	84	37	communication	communication	NOUN
ejpam-6275	84	38	systems	system	NOUN
ejpam-6275	84	39	.	.	PUNCT
ejpam-6275	85	1	these	these	DET
ejpam-6275	85	2	contributions	contribution	NOUN
ejpam-6275	85	3	are	be	AUX
ejpam-6275	85	4	firm	firm	ADJ
ejpam-6275	85	5	to	to	PART
ejpam-6275	85	6	fill	fill	VERB
ejpam-6275	85	7	the	the	DET
ejpam-6275	85	8	existing	exist	VERB
ejpam-6275	85	9	m.	m.	NOUN
ejpam-6275	85	10	sajjad	sajjad	PROPN
ejpam-6275	85	11	et	et	PROPN
ejpam-6275	85	12	al	al	PROPN
ejpam-6275	85	13	.	.	PUNCT
ejpam-6275	85	14	/	/	SYM
ejpam-6275	85	15	eur	eur	PROPN
ejpam-6275	85	16	.	.	PUNCT
ejpam-6275	86	1	j.	j.	PROPN
ejpam-6275	86	2	pure	pure	PROPN
ejpam-6275	86	3	appl	appl	PROPN
ejpam-6275	86	4	.	.	PROPN
ejpam-6275	86	5	math	math	PROPN
ejpam-6275	86	6	,	,	PUNCT
ejpam-6275	86	7	18	18	NUM
ejpam-6275	86	8	(	(	PUNCT
ejpam-6275	86	9	3	3	NUM
ejpam-6275	86	10	)	)	PUNCT
ejpam-6275	86	11	(	(	PUNCT
ejpam-6275	86	12	2025	2025	NUM
ejpam-6275	86	13	)	)	PUNCT
ejpam-6275	86	14	,	,	PUNCT
ejpam-6275	86	15	6275	6275	NUM
ejpam-6275	86	16	5	5	NUM
ejpam-6275	86	17	of	of	ADP
ejpam-6275	86	18	36	36	NUM
ejpam-6275	86	19	gaps	gap	NOUN
ejpam-6275	86	20	in	in	ADP
ejpam-6275	86	21	coding	code	VERB
ejpam-6275	86	22	theory	theory	NOUN
ejpam-6275	86	23	and	and	CCONJ
ejpam-6275	86	24	also	also	ADV
ejpam-6275	86	25	plan	plan	VERB
ejpam-6275	86	26	a	a	DET
ejpam-6275	86	27	strong	strong	ADJ
ejpam-6275	86	28	base	base	NOUN
ejpam-6275	86	29	for	for	ADP
ejpam-6275	86	30	many	many	ADJ
ejpam-6275	86	31	more	more	ADJ
ejpam-6275	86	32	inventions	invention	NOUN
ejpam-6275	86	33	in	in	ADP
ejpam-6275	86	34	the	the	DET
ejpam-6275	86	35	field	field	NOUN
ejpam-6275	86	36	.	.	PUNCT
ejpam-6275	87	1	2	2	X
ejpam-6275	87	2	.	.	X
ejpam-6275	87	3	eisenstein	eisenstein	PROPN
ejpam-6275	87	4	field	field	NOUN
ejpam-6275	87	5	from	from	ADP
ejpam-6275	87	6	[	[	X
ejpam-6275	87	7	26	26	NUM
ejpam-6275	87	8	]	]	PUNCT
ejpam-6275	87	9	,	,	PUNCT
ejpam-6275	87	10	assume	assume	VERB
ejpam-6275	87	11	ω	ω	NUM
ejpam-6275	87	12	being	be	AUX
ejpam-6275	87	13	the	the	DET
ejpam-6275	87	14	cubic	cubic	ADJ
ejpam-6275	87	15	root	root	NOUN
ejpam-6275	87	16	of	of	ADP
ejpam-6275	87	17	unity	unity	NOUN
ejpam-6275	87	18	correspondingly	correspondingly	ADV
ejpam-6275	87	19	1	1	NUM
ejpam-6275	87	20	+	+	NUM
ejpam-6275	87	21	ω	ω	NUM
ejpam-6275	87	22	+	+	CCONJ
ejpam-6275	88	1	ω2	ω2	ADJ
ejpam-6275	88	2	=	=	SYM
ejpam-6275	88	3	0	0	X
ejpam-6275	88	4	.	.	PUNCT
ejpam-6275	88	5	assume	assume	VERB
ejpam-6275	88	6	that	that	SCONJ
ejpam-6275	88	7	z[ω	z[ω	PROPN
ejpam-6275	88	8	]	]	X
ejpam-6275	88	9	=	=	SYM
ejpam-6275	88	10	{	{	PUNCT
ejpam-6275	88	11	a+bω	a+bω	INTJ
ejpam-6275	88	12	:	:	PUNCT
ejpam-6275	88	13	a	a	PRON
ejpam-6275	88	14	,	,	PUNCT
ejpam-6275	88	15	b	b	X
ejpam-6275	88	16	∈	∈	PROPN
ejpam-6275	88	17	z	z	AUX
ejpam-6275	88	18	}	}	PUNCT
ejpam-6275	88	19	be	be	AUX
ejpam-6275	88	20	the	the	DET
ejpam-6275	88	21	euclidean	euclidean	ADJ
ejpam-6275	88	22	domain	domain	NOUN
ejpam-6275	88	23	of	of	ADP
ejpam-6275	88	24	the	the	DET
ejpam-6275	88	25	eisenstein	eisenstein	NOUN
ejpam-6275	88	26	integers	integer	NOUN
ejpam-6275	88	27	.	.	PUNCT
ejpam-6275	89	1	consequently	consequently	ADV
ejpam-6275	89	2	,	,	PUNCT
ejpam-6275	89	3	it	it	PRON
ejpam-6275	89	4	zp[ω	zp[ω	X
ejpam-6275	89	5	]	]	X
ejpam-6275	90	1	=	=	X
ejpam-6275	90	2	{	{	PUNCT
ejpam-6275	91	1	a+bω	a+bω	INTJ
ejpam-6275	91	2	:	:	PUNCT
ejpam-6275	91	3	a	a	PRON
ejpam-6275	91	4	,	,	PUNCT
ejpam-6275	91	5	b	b	PROPN
ejpam-6275	91	6	∈	∈	PROPN
ejpam-6275	91	7	zp	zp	PROPN
ejpam-6275	91	8	}	}	PUNCT
ejpam-6275	91	9	is	be	AUX
ejpam-6275	91	10	a	a	DET
ejpam-6275	91	11	commutative	commutative	ADJ
ejpam-6275	91	12	ring	ring	NOUN
ejpam-6275	91	13	with	with	ADP
ejpam-6275	91	14	identity	identity	NOUN
ejpam-6275	91	15	and	and	CCONJ
ejpam-6275	91	16	zp[ω	zp[ω	PROPN
ejpam-6275	91	17	]	]	X
ejpam-6275	91	18	is	be	AUX
ejpam-6275	91	19	an	an	DET
ejpam-6275	91	20	eisenstein	eisenstein	NOUN
ejpam-6275	91	21	field	field	NOUN
ejpam-6275	91	22	(	(	PUNCT
ejpam-6275	91	23	ef	ef	PROPN
ejpam-6275	91	24	)	)	PUNCT
ejpam-6275	91	25	if	if	SCONJ
ejpam-6275	91	26	p	p	PRON
ejpam-6275	91	27	≡	≡	PROPN
ejpam-6275	91	28	2	2	NUM
ejpam-6275	91	29	(	(	PUNCT
ejpam-6275	91	30	mod	mod	NOUN
ejpam-6275	91	31	3	3	NUM
ejpam-6275	91	32	)	)	PUNCT
ejpam-6275	91	33	.	.	PUNCT
ejpam-6275	92	1	illustration	illustration	NOUN
ejpam-6275	92	2	2.1	2.1	NUM
ejpam-6275	92	3	:	:	PUNCT
ejpam-6275	92	4	from	from	ADP
ejpam-6275	92	5	[	[	X
ejpam-6275	92	6	24	24	NUM
ejpam-6275	92	7	,	,	PUNCT
ejpam-6275	92	8	26	26	NUM
ejpam-6275	92	9	]	]	PUNCT
ejpam-6275	92	10	,	,	PUNCT
ejpam-6275	92	11	let	let	VERB
ejpam-6275	92	12	z2[ω	z2[ω	PRON
ejpam-6275	92	13	]	]	X
ejpam-6275	92	14	=	=	PUNCT
ejpam-6275	92	15	{	{	PUNCT
ejpam-6275	92	16	0	0	NUM
ejpam-6275	92	17	,	,	PUNCT
ejpam-6275	92	18	1	1	NUM
ejpam-6275	92	19	,	,	PUNCT
ejpam-6275	92	20	ω	ω	PROPN
ejpam-6275	92	21	,	,	PUNCT
ejpam-6275	92	22	1+ω	1+ω	NUM
ejpam-6275	92	23	}	}	PUNCT
ejpam-6275	92	24	be	be	AUX
ejpam-6275	92	25	the	the	DET
ejpam-6275	92	26	ef	ef	PROPN
ejpam-6275	92	27	,	,	PUNCT
ejpam-6275	92	28	as	as	SCONJ
ejpam-6275	92	29	every	every	DET
ejpam-6275	92	30	nonzero	nonzero	NOUN
ejpam-6275	92	31	element	element	NOUN
ejpam-6275	92	32	of	of	ADP
ejpam-6275	92	33	z2[ω	z2[ω	NOUN
ejpam-6275	92	34	]	]	X
ejpam-6275	92	35	is	be	AUX
ejpam-6275	92	36	a	a	DET
ejpam-6275	92	37	unit	unit	NOUN
ejpam-6275	92	38	element	element	NOUN
ejpam-6275	92	39	and	and	CCONJ
ejpam-6275	92	40	the	the	DET
ejpam-6275	92	41	cardinality	cardinality	NOUN
ejpam-6275	92	42	of	of	ADP
ejpam-6275	92	43	z2[ω	z2[ω	NOUN
ejpam-6275	92	44	]	]	PUNCT
ejpam-6275	92	45	is	be	AUX
ejpam-6275	92	46	2	2	NUM
ejpam-6275	92	47	2	2	NUM
ejpam-6275	92	48	=	=	SYM
ejpam-6275	92	49	4	4	NUM
ejpam-6275	92	50	.	.	NOUN
ejpam-6275	92	51	remark	remark	VERB
ejpam-6275	92	52	2.1	2.1	NUM
ejpam-6275	93	1	[	[	X
ejpam-6275	93	2	26	26	NUM
ejpam-6275	93	3	]	]	SYM
ejpam-6275	93	4	:	:	PUNCT
ejpam-6275	93	5	zp[ω	zp[ω	PROPN
ejpam-6275	93	6	]	]	PUNCT
ejpam-6275	93	7	has	have	VERB
ejpam-6275	93	8	p	p	PROPN
ejpam-6275	93	9	2	2	NUM
ejpam-6275	93	10	elements	element	NOUN
ejpam-6275	93	11	if	if	SCONJ
ejpam-6275	93	12	p	p	PRON
ejpam-6275	93	13	≡	≡	PROPN
ejpam-6275	93	14	2	2	NUM
ejpam-6275	93	15	(	(	PUNCT
ejpam-6275	93	16	mod	mod	NOUN
ejpam-6275	93	17	3	3	NUM
ejpam-6275	93	18	)	)	PUNCT
ejpam-6275	93	19	.	.	PUNCT
ejpam-6275	94	1	2.1	2.1	NUM
ejpam-6275	94	2	.	.	PUNCT
ejpam-6275	95	1	eisenstein	eisenstein	PROPN
ejpam-6275	95	2	field	field	NOUN
ejpam-6275	95	3	extension	extension	NOUN
ejpam-6275	95	4	from	from	ADP
ejpam-6275	95	5	[	[	X
ejpam-6275	95	6	26	26	NUM
ejpam-6275	95	7	]	]	PUNCT
ejpam-6275	95	8	,	,	PUNCT
ejpam-6275	95	9	assume	assume	VERB
ejpam-6275	95	10	zp[ω	zp[ω	PROPN
ejpam-6275	95	11	]	]	PUNCT
ejpam-6275	95	12	with	with	ADP
ejpam-6275	95	13	p	p	PROPN
ejpam-6275	95	14	≡	≡	PROPN
ejpam-6275	95	15	2	2	NUM
ejpam-6275	95	16	(	(	PUNCT
ejpam-6275	95	17	mod	mod	NOUN
ejpam-6275	95	18	3	3	NUM
ejpam-6275	95	19	)	)	PUNCT
ejpam-6275	95	20	is	be	AUX
ejpam-6275	95	21	an	an	DET
ejpam-6275	95	22	ef	ef	PROPN
ejpam-6275	95	23	,	,	PUNCT
ejpam-6275	95	24	then	then	ADV
ejpam-6275	95	25	zp[ω][x	zp[ω][x	NOUN
ejpam-6275	95	26	]	]	PUNCT
ejpam-6275	95	27	is	be	AUX
ejpam-6275	95	28	a	a	DET
ejpam-6275	95	29	euclidean	euclidean	ADJ
ejpam-6275	95	30	domain	domain	NOUN
ejpam-6275	95	31	.	.	PUNCT
ejpam-6275	96	1	for	for	ADP
ejpam-6275	96	2	the	the	DET
ejpam-6275	96	3	extension	extension	NOUN
ejpam-6275	96	4	of	of	ADP
ejpam-6275	96	5	ef	ef	PROPN
ejpam-6275	96	6	,	,	PUNCT
ejpam-6275	96	7	zp[ω	zp[ω	PROPN
ejpam-6275	96	8	]	]	X
ejpam-6275	96	9	m	m	PROPN
ejpam-6275	96	10	,	,	PUNCT
ejpam-6275	96	11	we	we	PRON
ejpam-6275	96	12	have	have	VERB
ejpam-6275	96	13	:	:	PUNCT
ejpam-6275	96	14	zp[ω	zp[ω	PROPN
ejpam-6275	96	15	]	]	X
ejpam-6275	96	16	m	m	VERB
ejpam-6275	96	17	=	=	SYM
ejpam-6275	96	18	zp[ω][x	zp[ω][x	NOUN
ejpam-6275	96	19	]	]	PUNCT
ejpam-6275	96	20	⟨h(x)⟩	⟨h(x)⟩	PART
ejpam-6275	96	21	∼=	∼=	PROPN
ejpam-6275	96	22	gf(p2	gf(p2	PROPN
ejpam-6275	96	23	m	m	NOUN
ejpam-6275	96	24	)	)	PUNCT
ejpam-6275	96	25	,	,	PUNCT
ejpam-6275	96	26	where	where	SCONJ
ejpam-6275	96	27	⟨h(x)⟩	⟨h(x)⟩	PRON
ejpam-6275	96	28	is	be	AUX
ejpam-6275	96	29	the	the	DET
ejpam-6275	96	30	maximal	maximal	ADJ
ejpam-6275	96	31	ideal	ideal	NOUN
ejpam-6275	96	32	generated	generate	VERB
ejpam-6275	96	33	by	by	ADP
ejpam-6275	96	34	an	an	DET
ejpam-6275	96	35	irreducible	irreducible	ADJ
ejpam-6275	96	36	polynomial	polynomial	ADJ
ejpam-6275	96	37	h(x	h(x	PROPN
ejpam-6275	96	38	)	)	PUNCT
ejpam-6275	96	39	of	of	ADP
ejpam-6275	96	40	degree	degree	NOUN
ejpam-6275	96	41	k	k	PROPN
ejpam-6275	96	42	in	in	ADP
ejpam-6275	96	43	zp[ω][x	zp[ω][x	NOUN
ejpam-6275	96	44	]	]	PUNCT
ejpam-6275	96	45	.	.	PUNCT
ejpam-6275	97	1	let	let	VERB
ejpam-6275	97	2	β	β	X
ejpam-6275	97	3	be	be	AUX
ejpam-6275	97	4	the	the	DET
ejpam-6275	97	5	coset	coset	NOUN
ejpam-6275	97	6	x+	x+	ADJ
ejpam-6275	97	7	⟨h(x)⟩	⟨h(x)⟩	X
ejpam-6275	97	8	,	,	PUNCT
ejpam-6275	97	9	so	so	SCONJ
ejpam-6275	97	10	that	that	SCONJ
ejpam-6275	97	11	h(β	h(β	VERB
ejpam-6275	97	12	)	)	PUNCT
ejpam-6275	97	13	=	=	SYM
ejpam-6275	97	14	0	0	X
ejpam-6275	97	15	.	.	PUNCT
ejpam-6275	98	1	moreover	moreover	ADV
ejpam-6275	98	2	,	,	PUNCT
ejpam-6275	98	3	zp[ω	zp[ω	PROPN
ejpam-6275	98	4	]	]	X
ejpam-6275	98	5	m	m	PROPN
ejpam-6275	98	6	=	=	SYM
ejpam-6275	98	7	{	{	PUNCT
ejpam-6275	98	8	a0	a0	PROPN
ejpam-6275	98	9	+	+	CCONJ
ejpam-6275	98	10	a1β	a1β	PROPN
ejpam-6275	99	1	+	+	CCONJ
ejpam-6275	99	2	a2β	a2β	SYM
ejpam-6275	99	3	2	2	NUM
ejpam-6275	99	4	+	+	CCONJ
ejpam-6275	99	5	·	·	PUNCT
ejpam-6275	99	6	·	·	PUNCT
ejpam-6275	99	7	·	·	PUNCT
ejpam-6275	99	8	+	+	CCONJ
ejpam-6275	99	9	am−1β	am−1β	PROPN
ejpam-6275	99	10	m−1	m−1	PROPN
ejpam-6275	99	11	:	:	PUNCT
ejpam-6275	99	12	∀a0	∀a0	ADJ
ejpam-6275	99	13	,	,	PUNCT
ejpam-6275	99	14	a1	a1	NOUN
ejpam-6275	99	15	,	,	PUNCT
ejpam-6275	99	16	.	.	PUNCT
ejpam-6275	99	17	.	.	PUNCT
ejpam-6275	99	18	.	.	PUNCT
ejpam-6275	100	1	,	,	PUNCT
ejpam-6275	100	2	am−1	am−1	PROPN
ejpam-6275	100	3	∈	∈	PROPN
ejpam-6275	100	4	zp[ω	zp[ω	PROPN
ejpam-6275	100	5	]	]	PUNCT
ejpam-6275	100	6	}	}	PUNCT
ejpam-6275	100	7	.	.	PUNCT
ejpam-6275	101	1	zp[ω	zp[ω	PROPN
ejpam-6275	101	2	]	]	X
ejpam-6275	102	1	m	m	VERB
ejpam-6275	102	2	is	be	AUX
ejpam-6275	102	3	an	an	DET
ejpam-6275	102	4	m	m	NOUN
ejpam-6275	102	5	-	-	PUNCT
ejpam-6275	102	6	degree	degree	NOUN
ejpam-6275	102	7	extension	extension	NOUN
ejpam-6275	102	8	field	field	NOUN
ejpam-6275	102	9	of	of	ADP
ejpam-6275	102	10	zp[ω	zp[ω	PROPN
ejpam-6275	102	11	]	]	PUNCT
ejpam-6275	102	12	.	.	PUNCT
ejpam-6275	103	1	the	the	DET
ejpam-6275	103	2	multiplicative	multiplicative	ADJ
ejpam-6275	103	3	group	group	NOUN
ejpam-6275	103	4	zp[ω	zp[ω	PROPN
ejpam-6275	103	5	]	]	X
ejpam-6275	103	6	∗m	∗m	PROPN
ejpam-6275	103	7	=	=	PUNCT
ejpam-6275	103	8	zp[ω	zp[ω	PROPN
ejpam-6275	103	9	]	]	X
ejpam-6275	103	10	m	m	VERB
ejpam-6275	103	11	\	\	X
ejpam-6275	103	12	{	{	PUNCT
ejpam-6275	103	13	0	0	NUM
ejpam-6275	103	14	}	}	PUNCT
ejpam-6275	103	15	is	be	AUX
ejpam-6275	103	16	a	a	DET
ejpam-6275	103	17	cyclic	cyclic	ADJ
ejpam-6275	103	18	group	group	NOUN
ejpam-6275	103	19	(	(	PUNCT
ejpam-6275	103	20	cg	cg	NOUN
ejpam-6275	103	21	)	)	PUNCT
ejpam-6275	103	22	of	of	ADP
ejpam-6275	103	23	order	order	NOUN
ejpam-6275	103	24	p2	p2	NOUN
ejpam-6275	103	25	m	m	NOUN
ejpam-6275	103	26	−	−	NOUN
ejpam-6275	103	27	1	1	NUM
ejpam-6275	103	28	.	.	PUNCT
ejpam-6275	103	29	remark	remark	VERB
ejpam-6275	103	30	2.2	2.2	NUM
ejpam-6275	103	31	[	[	SYM
ejpam-6275	103	32	26	26	NUM
ejpam-6275	103	33	,	,	PUNCT
ejpam-6275	103	34	remark	remark	NOUN
ejpam-6275	103	35	3	3	NUM
ejpam-6275	103	36	]	]	PUNCT
ejpam-6275	103	37	:	:	PUNCT
ejpam-6275	103	38	the	the	DET
ejpam-6275	103	39	cardinality	cardinality	NOUN
ejpam-6275	103	40	of	of	ADP
ejpam-6275	103	41	zp[ω	zp[ω	PROPN
ejpam-6275	103	42	]	]	X
ejpam-6275	103	43	m	m	VERB
ejpam-6275	103	44	is	be	AUX
ejpam-6275	103	45	p2	p2	NOUN
ejpam-6275	103	46	m.	m.	NOUN
ejpam-6275	103	47	illustration	illustration	NOUN
ejpam-6275	103	48	2.2	2.2	NUM
ejpam-6275	103	49	:	:	PUNCT
ejpam-6275	103	50	assume	assume	VERB
ejpam-6275	103	51	that	that	SCONJ
ejpam-6275	103	52	the	the	DET
ejpam-6275	103	53	quotient	quotient	NOUN
ejpam-6275	103	54	ring	ring	NOUN
ejpam-6275	103	55	z2[ω][x	z2[ω][x	PROPN
ejpam-6275	103	56	]	]	X
ejpam-6275	103	57	⟨x2	⟨x2	PROPN
ejpam-6275	103	58	+	+	CCONJ
ejpam-6275	103	59	ωx+	ωx+	NOUN
ejpam-6275	103	60	ω⟩	ω⟩	NOUN
ejpam-6275	103	61	=	=	PRON
ejpam-6275	103	62	{	{	PUNCT
ejpam-6275	103	63	a0	a0	NOUN
ejpam-6275	103	64	+	+	CCONJ
ejpam-6275	103	65	a1x	a1x	NOUN
ejpam-6275	103	66	:	:	PUNCT
ejpam-6275	103	67	∀a0	∀a0	ADJ
ejpam-6275	103	68	,	,	PUNCT
ejpam-6275	103	69	a1	a1	PROPN
ejpam-6275	103	70	∈	∈	PROPN
ejpam-6275	103	71	z2[ω	z2[ω	NOUN
ejpam-6275	103	72	]	]	X
ejpam-6275	103	73	}	}	PUNCT
ejpam-6275	103	74	=	=	SYM
ejpam-6275	103	75	z2[ω	z2[ω	NOUN
ejpam-6275	103	76	]	]	X
ejpam-6275	103	77	2	2	NUM
ejpam-6275	103	78	,	,	PUNCT
ejpam-6275	103	79	is	be	AUX
ejpam-6275	103	80	generated	generate	VERB
ejpam-6275	103	81	by	by	ADP
ejpam-6275	103	82	the	the	DET
ejpam-6275	103	83	primitive	primitive	ADJ
ejpam-6275	103	84	irreducible	irreducible	ADJ
ejpam-6275	103	85	polynomial	polynomial	ADJ
ejpam-6275	103	86	h(x	h(x	PROPN
ejpam-6275	103	87	)	)	PUNCT
ejpam-6275	104	1	=	=	SYM
ejpam-6275	105	1	x2	x2	NOUN
ejpam-6275	106	1	+	+	CCONJ
ejpam-6275	106	2	ωx+	ωx+	PROPN
ejpam-6275	106	3	ω	ω	PROPN
ejpam-6275	106	4	over	over	ADP
ejpam-6275	106	5	z2[ω	z2[ω	NOUN
ejpam-6275	106	6	]	]	PUNCT
ejpam-6275	106	7	.	.	PUNCT
ejpam-6275	107	1	let	let	VERB
ejpam-6275	107	2	α	α	PRON
ejpam-6275	107	3	be	be	AUX
ejpam-6275	107	4	a	a	DET
ejpam-6275	107	5	root	root	NOUN
ejpam-6275	107	6	of	of	ADP
ejpam-6275	107	7	h(x	h(x	PROPN
ejpam-6275	107	8	)	)	PUNCT
ejpam-6275	107	9	in	in	ADP
ejpam-6275	107	10	the	the	DET
ejpam-6275	107	11	extension	extension	NOUN
ejpam-6275	107	12	field	field	NOUN
ejpam-6275	107	13	z2[ω][x	z2[ω][x	NOUN
ejpam-6275	107	14	]	]	PUNCT
ejpam-6275	107	15	,	,	PUNCT
ejpam-6275	107	16	then	then	ADV
ejpam-6275	107	17	h(α	h(α	ADJ
ejpam-6275	107	18	)	)	PUNCT
ejpam-6275	107	19	=	=	SYM
ejpam-6275	107	20	0	0	PUNCT
ejpam-6275	107	21	as	as	ADP
ejpam-6275	107	22	α2	α2	ADJ
ejpam-6275	107	23	+	+	CCONJ
ejpam-6275	107	24	ωα	ωα	X
ejpam-6275	107	25	+	+	NUM
ejpam-6275	107	26	ω	ω	NUM
ejpam-6275	107	27	=	=	SYM
ejpam-6275	107	28	0	0	NUM
ejpam-6275	107	29	,	,	PUNCT
ejpam-6275	107	30	and	and	CCONJ
ejpam-6275	107	31	z2[ω	z2[ω	NOUN
ejpam-6275	107	32	]	]	X
ejpam-6275	107	33	∗2	∗2	PUNCT
ejpam-6275	107	34	=	=	SYM
ejpam-6275	107	35	z2[ω	z2[ω	NOUN
ejpam-6275	107	36	]	]	X
ejpam-6275	107	37	2	2	NUM
ejpam-6275	107	38	\	\	NOUN
ejpam-6275	107	39	{	{	PUNCT
ejpam-6275	107	40	0	0	NUM
ejpam-6275	107	41	}	}	PUNCT
ejpam-6275	107	42	is	be	AUX
ejpam-6275	107	43	a	a	DET
ejpam-6275	107	44	cyclic	cyclic	ADJ
ejpam-6275	107	45	group	group	NOUN
ejpam-6275	107	46	(	(	PUNCT
ejpam-6275	107	47	cg	cg	NOUN
ejpam-6275	107	48	)	)	PUNCT
ejpam-6275	107	49	of	of	ADP
ejpam-6275	107	50	order	order	NOUN
ejpam-6275	108	1	22·2	22·2	NUM
ejpam-6275	108	2	−	−	NOUN
ejpam-6275	108	3	1	1	NUM
ejpam-6275	108	4	=	=	SYM
ejpam-6275	108	5	15	15	NUM
ejpam-6275	108	6	as	as	SCONJ
ejpam-6275	108	7	shown	show	VERB
ejpam-6275	108	8	in	in	ADP
ejpam-6275	108	9	table	table	NOUN
ejpam-6275	108	10	1	1	NUM
ejpam-6275	108	11	.	.	PUNCT
ejpam-6275	109	1	now	now	ADV
ejpam-6275	109	2	,	,	PUNCT
ejpam-6275	109	3	for	for	ADP
ejpam-6275	109	4	finding	find	VERB
ejpam-6275	109	5	the	the	DET
ejpam-6275	109	6	total	total	ADJ
ejpam-6275	109	7	number	number	NOUN
ejpam-6275	109	8	of	of	ADP
ejpam-6275	109	9	possible	possible	ADJ
ejpam-6275	109	10	polynomials	polynomial	NOUN
ejpam-6275	109	11	,	,	PUNCT
ejpam-6275	109	12	the	the	DET
ejpam-6275	109	13	possible	possible	ADJ
ejpam-6275	109	14	irreducible	irreducible	ADJ
ejpam-6275	109	15	polynomials	polynomial	NOUN
ejpam-6275	109	16	,	,	PUNCT
ejpam-6275	109	17	and	and	CCONJ
ejpam-6275	109	18	the	the	DET
ejpam-6275	109	19	number	number	NOUN
ejpam-6275	109	20	of	of	ADP
ejpam-6275	109	21	primitive	primitive	ADJ
ejpam-6275	109	22	irreducible	irreducible	ADJ
ejpam-6275	109	23	polynomials	polynomial	NOUN
ejpam-6275	109	24	over	over	ADP
ejpam-6275	109	25	the	the	DET
ejpam-6275	109	26	eisenstein	eisenstein	NOUN
ejpam-6275	109	27	field	field	NOUN
ejpam-6275	109	28	(	(	PUNCT
ejpam-6275	109	29	ef	ef	PROPN
ejpam-6275	109	30	)	)	PUNCT
ejpam-6275	109	31	z2[ω][x	z2[ω][x	PROPN
ejpam-6275	109	32	]	]	PUNCT
ejpam-6275	109	33	⟨f(x)⟩	⟨f(x)⟩	X
ejpam-6275	109	34	for	for	ADP
ejpam-6275	109	35	f(x	f(x	PROPN
ejpam-6275	109	36	)	)	PUNCT
ejpam-6275	109	37	being	be	AUX
ejpam-6275	109	38	a	a	DET
ejpam-6275	109	39	polynomial	polynomial	NOUN
ejpam-6275	109	40	of	of	ADP
ejpam-6275	109	41	degree	degree	NOUN
ejpam-6275	109	42	2	2	NUM
ejpam-6275	109	43	,	,	PUNCT
ejpam-6275	109	44	consider	consider	VERB
ejpam-6275	109	45	table	table	NOUN
ejpam-6275	109	46	2	2	NUM
ejpam-6275	109	47	.	.	PUNCT
ejpam-6275	109	48	m.	m.	PROPN
ejpam-6275	109	49	sajjad	sajjad	PROPN
ejpam-6275	109	50	et	et	PROPN
ejpam-6275	109	51	al	al	PROPN
ejpam-6275	109	52	.	.	PUNCT
ejpam-6275	109	53	/	/	SYM
ejpam-6275	109	54	eur	eur	PROPN
ejpam-6275	109	55	.	.	PUNCT
ejpam-6275	110	1	j.	j.	PROPN
ejpam-6275	110	2	pure	pure	PROPN
ejpam-6275	110	3	appl	appl	PROPN
ejpam-6275	110	4	.	.	PROPN
ejpam-6275	110	5	math	math	PROPN
ejpam-6275	110	6	,	,	PUNCT
ejpam-6275	110	7	18	18	NUM
ejpam-6275	110	8	(	(	PUNCT
ejpam-6275	110	9	3	3	NUM
ejpam-6275	110	10	)	)	PUNCT
ejpam-6275	110	11	(	(	PUNCT
ejpam-6275	110	12	2025	2025	NUM
ejpam-6275	110	13	)	)	PUNCT
ejpam-6275	110	14	,	,	PUNCT
ejpam-6275	110	15	6275	6275	NUM
ejpam-6275	110	16	6	6	NUM
ejpam-6275	110	17	of	of	ADP
ejpam-6275	110	18	36	36	NUM
ejpam-6275	110	19	table	table	NOUN
ejpam-6275	110	20	1	1	NUM
ejpam-6275	110	21	:	:	PUNCT
ejpam-6275	110	22	cyclic	cyclic	ADJ
ejpam-6275	110	23	group	group	NOUN
ejpam-6275	110	24	over	over	ADP
ejpam-6275	110	25	eisenstein	eisenstein	NOUN
ejpam-6275	110	26	integers	integer	NOUN
ejpam-6275	110	27	of	of	ADP
ejpam-6275	110	28	order	order	NOUN
ejpam-6275	111	1	15	15	NUM
ejpam-6275	111	2	s.	s.	PROPN
ejpam-6275	111	3	no	no	INTJ
ejpam-6275	111	4	.	.	PUNCT
ejpam-6275	112	1	power	power	NOUN
ejpam-6275	112	2	of	of	ADP
ejpam-6275	112	3	α	α	PROPN
ejpam-6275	112	4	’s	’s	PROPN
ejpam-6275	112	5	s.	s.	PROPN
ejpam-6275	112	6	no	no	PROPN
ejpam-6275	112	7	.	.	PUNCT
ejpam-6275	113	1	power	power	NOUN
ejpam-6275	113	2	of	of	ADP
ejpam-6275	113	3	α	α	PROPN
ejpam-6275	113	4	’s	’s	PART
ejpam-6275	113	5	1	1	NUM
ejpam-6275	113	6	α	α	NOUN
ejpam-6275	113	7	9	9	NUM
ejpam-6275	113	8	ω	ω	NUM
ejpam-6275	113	9	+	+	CCONJ
ejpam-6275	113	10	1	1	NUM
ejpam-6275	114	1	+	+	CCONJ
ejpam-6275	114	2	αω	αω	NUM
ejpam-6275	114	3	2	2	NUM
ejpam-6275	114	4	αω	αω	NUM
ejpam-6275	114	5	+	+	NOUN
ejpam-6275	114	6	ω	ω	NUM
ejpam-6275	114	7	10	10	NUM
ejpam-6275	114	8	1	1	NUM
ejpam-6275	114	9	+	+	NUM
ejpam-6275	114	10	ω	ω	NUM
ejpam-6275	114	11	3	3	NUM
ejpam-6275	114	12	α+	α+	NOUN
ejpam-6275	114	13	ω	ω	NOUN
ejpam-6275	114	14	+	+	CCONJ
ejpam-6275	114	15	1	1	NUM
ejpam-6275	114	16	11	11	NUM
ejpam-6275	114	17	α+	α+	SYM
ejpam-6275	114	18	αω	αω	NUM
ejpam-6275	114	19	4	4	NUM
ejpam-6275	114	20	α+	α+	NOUN
ejpam-6275	114	21	ω	ω	NUM
ejpam-6275	114	22	12	12	NUM
ejpam-6275	114	23	1	1	NUM
ejpam-6275	114	24	+	+	CCONJ
ejpam-6275	114	25	α	α	PROPN
ejpam-6275	114	26	5	5	NUM
ejpam-6275	114	27	ω	ω	NUM
ejpam-6275	114	28	13	13	NUM
ejpam-6275	114	29	ω	ω	NOUN
ejpam-6275	114	30	+	+	CCONJ
ejpam-6275	114	31	α(1	α(1	PROPN
ejpam-6275	114	32	+	+	CCONJ
ejpam-6275	114	33	ω	ω	NOUN
ejpam-6275	114	34	)	)	PUNCT
ejpam-6275	114	35	6	6	NUM
ejpam-6275	114	36	αω	αω	NUM
ejpam-6275	114	37	14	14	NUM
ejpam-6275	114	38	1	1	NUM
ejpam-6275	115	1	+	+	CCONJ
ejpam-6275	115	2	α+	α+	ADP
ejpam-6275	115	3	αω	αω	SYM
ejpam-6275	115	4	7	7	NUM
ejpam-6275	115	5	α(1	α(1	PROPN
ejpam-6275	115	6	+	+	PROPN
ejpam-6275	115	7	ω	ω	NUM
ejpam-6275	115	8	)	)	PUNCT
ejpam-6275	115	9	+	+	CCONJ
ejpam-6275	115	10	1	1	NUM
ejpam-6275	116	1	+	+	NUM
ejpam-6275	116	2	ω	ω	NUM
ejpam-6275	116	3	15	15	NUM
ejpam-6275	116	4	1	1	NUM
ejpam-6275	116	5	8	8	NUM
ejpam-6275	116	6	αω	αω	NUM
ejpam-6275	116	7	+	+	SYM
ejpam-6275	116	8	1	1	NUM
ejpam-6275	116	9	illustration	illustration	NOUN
ejpam-6275	116	10	2.3	2.3	NUM
ejpam-6275	116	11	:	:	PUNCT
ejpam-6275	116	12	let	let	VERB
ejpam-6275	116	13	the	the	DET
ejpam-6275	116	14	quotient	quotient	NOUN
ejpam-6275	116	15	ring	ring	NOUN
ejpam-6275	116	16	z2[ω][x	z2[ω][x	NOUN
ejpam-6275	116	17	]	]	X
ejpam-6275	117	1	⟨x3	⟨x3	PROPN
ejpam-6275	117	2	+	+	PROPN
ejpam-6275	117	3	x2	x2	PROPN
ejpam-6275	118	1	+	+	CCONJ
ejpam-6275	118	2	x+	x+	SYM
ejpam-6275	118	3	1	1	NUM
ejpam-6275	118	4	+	+	NUM
ejpam-6275	118	5	ω⟩	ω⟩	NOUN
ejpam-6275	118	6	=	=	SYM
ejpam-6275	118	7	{	{	PUNCT
ejpam-6275	118	8	a0	a0	NOUN
ejpam-6275	118	9	+	+	CCONJ
ejpam-6275	118	10	a1x+	a1x+	NOUN
ejpam-6275	118	11	a2x	a2x	PROPN
ejpam-6275	118	12	2	2	NUM
ejpam-6275	118	13	|	|	NOUN
ejpam-6275	118	14	a0	a0	NOUN
ejpam-6275	118	15	,	,	PUNCT
ejpam-6275	118	16	a1	a1	PROPN
ejpam-6275	118	17	,	,	PUNCT
ejpam-6275	118	18	a2	a2	PROPN
ejpam-6275	118	19	∈	∈	PROPN
ejpam-6275	118	20	z2[ω	z2[ω	NOUN
ejpam-6275	118	21	]	]	X
ejpam-6275	118	22	}	}	PUNCT
ejpam-6275	118	23	=	=	SYM
ejpam-6275	118	24	z2[ω][x	z2[ω][x	NOUN
ejpam-6275	118	25	]	]	X
ejpam-6275	118	26	3	3	NUM
ejpam-6275	118	27	,	,	PUNCT
ejpam-6275	118	28	be	be	AUX
ejpam-6275	118	29	generated	generate	VERB
ejpam-6275	118	30	by	by	ADP
ejpam-6275	118	31	the	the	DET
ejpam-6275	118	32	irreducible	irreducible	ADJ
ejpam-6275	118	33	primitive	primitive	ADJ
ejpam-6275	118	34	polynomial	polynomial	ADJ
ejpam-6275	118	35	h(x	h(x	PROPN
ejpam-6275	118	36	)	)	PUNCT
ejpam-6275	119	1	=	=	SYM
ejpam-6275	119	2	x3+x2+x+1+ω	x3+x2+x+1+ω	PROPN
ejpam-6275	119	3	over	over	ADP
ejpam-6275	119	4	z2[ω	z2[ω	NOUN
ejpam-6275	119	5	]	]	PUNCT
ejpam-6275	119	6	,	,	PUNCT
ejpam-6275	119	7	and	and	CCONJ
ejpam-6275	119	8	let	let	VERB
ejpam-6275	119	9	α	α	PRON
ejpam-6275	119	10	be	be	AUX
ejpam-6275	119	11	the	the	DET
ejpam-6275	119	12	root	root	NOUN
ejpam-6275	119	13	of	of	ADP
ejpam-6275	119	14	h(x	h(x	PROPN
ejpam-6275	119	15	)	)	PUNCT
ejpam-6275	119	16	in	in	ADP
ejpam-6275	119	17	the	the	DET
ejpam-6275	119	18	extension	extension	NOUN
ejpam-6275	119	19	field	field	NOUN
ejpam-6275	119	20	z2[ω][x	z2[ω][x	NOUN
ejpam-6275	119	21	]	]	PUNCT
ejpam-6275	119	22	.	.	PUNCT
ejpam-6275	120	1	then	then	ADV
ejpam-6275	120	2	h(α	h(α	ADV
ejpam-6275	120	3	)	)	PUNCT
ejpam-6275	120	4	=	=	SYM
ejpam-6275	120	5	0	0	NUM
ejpam-6275	120	6	implies	imply	VERB
ejpam-6275	120	7	α3	α3	PROPN
ejpam-6275	120	8	+	+	CCONJ
ejpam-6275	120	9	α2	α2	ADJ
ejpam-6275	120	10	+	+	CCONJ
ejpam-6275	120	11	α+	α+	PUNCT
ejpam-6275	120	12	1	1	NUM
ejpam-6275	120	13	+	+	NUM
ejpam-6275	120	14	ω	ω	NUM
ejpam-6275	120	15	=	=	SYM
ejpam-6275	120	16	0	0	NUM
ejpam-6275	120	17	⇒	⇒	NOUN
ejpam-6275	120	18	α3	α3	NOUN
ejpam-6275	120	19	=	=	SYM
ejpam-6275	120	20	α2	α2	PROPN
ejpam-6275	121	1	+	+	CCONJ
ejpam-6275	121	2	α+	α+	PUNCT
ejpam-6275	121	3	1	1	NUM
ejpam-6275	121	4	+	+	NUM
ejpam-6275	121	5	ω	ω	X
ejpam-6275	121	6	.	.	PUNCT
ejpam-6275	121	7	therefore	therefore	ADV
ejpam-6275	121	8	,	,	PUNCT
ejpam-6275	121	9	z2[ω	z2[ω	NOUN
ejpam-6275	121	10	]	]	X
ejpam-6275	121	11	∗	∗	NOUN
ejpam-6275	121	12	3	3	NUM
ejpam-6275	121	13	=	=	SYM
ejpam-6275	121	14	z2[ω	z2[ω	NOUN
ejpam-6275	121	15	]	]	X
ejpam-6275	121	16	3	3	NUM
ejpam-6275	121	17	\	\	NOUN
ejpam-6275	121	18	{	{	PUNCT
ejpam-6275	121	19	0	0	NUM
ejpam-6275	121	20	}	}	PUNCT
ejpam-6275	121	21	is	be	AUX
ejpam-6275	121	22	a	a	DET
ejpam-6275	121	23	cyclic	cyclic	ADJ
ejpam-6275	121	24	group	group	NOUN
ejpam-6275	121	25	of	of	ADP
ejpam-6275	121	26	order	order	NOUN
ejpam-6275	122	1	22·3	22·3	NUM
ejpam-6275	122	2	−	−	NOUN
ejpam-6275	122	3	1	1	NUM
ejpam-6275	122	4	=	=	SYM
ejpam-6275	122	5	64	64	NUM
ejpam-6275	122	6	−	−	NOUN
ejpam-6275	122	7	1	1	NUM
ejpam-6275	122	8	=	=	SYM
ejpam-6275	122	9	63	63	NUM
ejpam-6275	122	10	,	,	PUNCT
ejpam-6275	122	11	as	as	SCONJ
ejpam-6275	122	12	illustrated	illustrate	VERB
ejpam-6275	122	13	in	in	ADP
ejpam-6275	122	14	table	table	NOUN
ejpam-6275	122	15	3	3	NUM
ejpam-6275	122	16	.	.	PUNCT
ejpam-6275	122	17	remark	remark	NOUN
ejpam-6275	122	18	2.4	2.4	NUM
ejpam-6275	122	19	:	:	PUNCT
ejpam-6275	122	20	the	the	DET
ejpam-6275	122	21	order	order	NOUN
ejpam-6275	122	22	of	of	ADP
ejpam-6275	122	23	a	a	DET
ejpam-6275	122	24	subgroup	subgroup	NOUN
ejpam-6275	122	25	of	of	ADP
ejpam-6275	122	26	eisenstein	eisenstein	PROPN
ejpam-6275	122	27	integers	integer	NOUN
ejpam-6275	122	28	must	must	AUX
ejpam-6275	122	29	divide	divide	VERB
ejpam-6275	122	30	the	the	DET
ejpam-6275	122	31	order	order	NOUN
ejpam-6275	122	32	of	of	ADP
ejpam-6275	122	33	the	the	DET
ejpam-6275	122	34	corresponding	corresponding	ADJ
ejpam-6275	122	35	group	group	NOUN
ejpam-6275	122	36	of	of	ADP
ejpam-6275	122	37	eisenstein	eisenstein	PROPN
ejpam-6275	122	38	integers	integer	NOUN
ejpam-6275	122	39	.	.	PUNCT
ejpam-6275	123	1	illustration	illustration	NOUN
ejpam-6275	123	2	2.4	2.4	NUM
ejpam-6275	123	3	:	:	PUNCT
ejpam-6275	123	4	if	if	SCONJ
ejpam-6275	123	5	we	we	PRON
ejpam-6275	123	6	require	require	VERB
ejpam-6275	123	7	a	a	DET
ejpam-6275	123	8	subgroup	subgroup	NOUN
ejpam-6275	123	9	of	of	ADP
ejpam-6275	123	10	five	five	NUM
ejpam-6275	123	11	elements	element	NOUN
ejpam-6275	123	12	over	over	ADP
ejpam-6275	123	13	the	the	DET
ejpam-6275	123	14	extension	extension	NOUN
ejpam-6275	123	15	field	field	NOUN
ejpam-6275	123	16	z2[ω][x	z2[ω][x	NOUN
ejpam-6275	123	17	]	]	PUNCT
ejpam-6275	123	18	,	,	PUNCT
ejpam-6275	123	19	then	then	ADV
ejpam-6275	123	20	we	we	PRON
ejpam-6275	123	21	know	know	VERB
ejpam-6275	123	22	that	that	SCONJ
ejpam-6275	123	23	we	we	PRON
ejpam-6275	123	24	can	can	AUX
ejpam-6275	123	25	not	not	PART
ejpam-6275	123	26	directly	directly	ADV
ejpam-6275	123	27	obtain	obtain	VERB
ejpam-6275	123	28	it	it	PRON
ejpam-6275	123	29	from	from	ADP
ejpam-6275	123	30	any	any	DET
ejpam-6275	123	31	polynomial	polynomial	ADJ
ejpam-6275	123	32	f(x	f(x	PROPN
ejpam-6275	123	33	)	)	PUNCT
ejpam-6275	123	34	of	of	ADP
ejpam-6275	123	35	degree	degree	NOUN
ejpam-6275	123	36	2	2	NUM
ejpam-6275	123	37	over	over	ADP
ejpam-6275	123	38	z2[ω][x	z2[ω][x	NOUN
ejpam-6275	123	39	]	]	PUNCT
ejpam-6275	123	40	⟨f(x)⟩	⟨f(x)⟩	X
ejpam-6275	123	41	.	.	PUNCT
ejpam-6275	124	1	we	we	PRON
ejpam-6275	124	2	obtain	obtain	VERB
ejpam-6275	124	3	a	a	DET
ejpam-6275	124	4	cyclic	cyclic	ADJ
ejpam-6275	124	5	group	group	NOUN
ejpam-6275	124	6	of	of	ADP
ejpam-6275	124	7	order	order	NOUN
ejpam-6275	124	8	n	n	NOUN
ejpam-6275	124	9	=	=	SYM
ejpam-6275	124	10	(	(	PUNCT
ejpam-6275	124	11	(	(	PUNCT
ejpam-6275	124	12	2)2	2)2	NUM
ejpam-6275	124	13	)	)	SYM
ejpam-6275	124	14	2	2	NUM
ejpam-6275	124	15	−	−	NOUN
ejpam-6275	124	16	1	1	NUM
ejpam-6275	124	17	=	=	SYM
ejpam-6275	124	18	16−	16−	NUM
ejpam-6275	124	19	1	1	NUM
ejpam-6275	124	20	=	=	SYM
ejpam-6275	124	21	15	15	NUM
ejpam-6275	124	22	.	.	PUNCT
ejpam-6275	125	1	the	the	DET
ejpam-6275	125	2	desired	desire	VERB
ejpam-6275	125	3	subgroup	subgroup	NOUN
ejpam-6275	125	4	is	be	AUX
ejpam-6275	125	5	obtained	obtain	VERB
ejpam-6275	125	6	by	by	ADP
ejpam-6275	125	7	dividing	divide	VERB
ejpam-6275	125	8	n	n	PROPN
ejpam-6275	125	9	by	by	ADP
ejpam-6275	125	10	5	5	NUM
ejpam-6275	125	11	,	,	PUNCT
ejpam-6275	125	12	i.e.	i.e.	X
ejpam-6275	125	13	,	,	PUNCT
ejpam-6275	125	14	15	15	NUM
ejpam-6275	125	15	5	5	NUM
ejpam-6275	125	16	=	=	SYM
ejpam-6275	125	17	3	3	X
ejpam-6275	125	18	.	.	PUNCT
ejpam-6275	126	1	if	if	SCONJ
ejpam-6275	126	2	α	α	PRON
ejpam-6275	126	3	is	be	AUX
ejpam-6275	126	4	the	the	DET
ejpam-6275	126	5	root	root	NOUN
ejpam-6275	126	6	of	of	ADP
ejpam-6275	126	7	the	the	DET
ejpam-6275	126	8	extension	extension	NOUN
ejpam-6275	126	9	field	field	NOUN
ejpam-6275	126	10	z2[ω][x	z2[ω][x	NOUN
ejpam-6275	126	11	]	]	PUNCT
ejpam-6275	126	12	,	,	PUNCT
ejpam-6275	126	13	then	then	ADV
ejpam-6275	126	14	α3	α3	PROPN
ejpam-6275	126	15	is	be	AUX
ejpam-6275	126	16	the	the	DET
ejpam-6275	126	17	generator	generator	NOUN
ejpam-6275	126	18	of	of	ADP
ejpam-6275	126	19	the	the	DET
ejpam-6275	126	20	required	require	VERB
ejpam-6275	126	21	subgroup	subgroup	NOUN
ejpam-6275	126	22	of	of	ADP
ejpam-6275	126	23	z2[ω][x	z2[ω][x	NOUN
ejpam-6275	126	24	]	]	X
ejpam-6275	126	25	2	2	NUM
ejpam-6275	126	26	for	for	ADP
ejpam-6275	126	27	n	n	NOUN
ejpam-6275	126	28	=	=	SYM
ejpam-6275	126	29	5	5	NUM
ejpam-6275	126	30	.	.	X
ejpam-6275	126	31	similarly	similarly	ADV
ejpam-6275	126	32	,	,	PUNCT
ejpam-6275	126	33	we	we	PRON
ejpam-6275	126	34	can	can	AUX
ejpam-6275	126	35	find	find	VERB
ejpam-6275	126	36	a	a	DET
ejpam-6275	126	37	subgroup	subgroup	NOUN
ejpam-6275	126	38	for	for	ADP
ejpam-6275	126	39	n	n	NOUN
ejpam-6275	126	40	=	=	SYM
ejpam-6275	126	41	3	3	X
ejpam-6275	126	42	.	.	X
ejpam-6275	126	43	illustration	illustration	NOUN
ejpam-6275	126	44	2.5	2.5	NUM
ejpam-6275	126	45	:	:	PUNCT
ejpam-6275	126	46	if	if	SCONJ
ejpam-6275	126	47	we	we	PRON
ejpam-6275	126	48	want	want	VERB
ejpam-6275	126	49	a	a	DET
ejpam-6275	126	50	subgroup	subgroup	NOUN
ejpam-6275	126	51	of	of	ADP
ejpam-6275	126	52	twenty	twenty	NUM
ejpam-6275	126	53	-	-	PUNCT
ejpam-6275	126	54	one	one	NUM
ejpam-6275	126	55	elements	element	NOUN
ejpam-6275	126	56	over	over	ADP
ejpam-6275	126	57	z2[ω][x	z2[ω][x	NOUN
ejpam-6275	126	58	]	]	PUNCT
ejpam-6275	126	59	,	,	PUNCT
ejpam-6275	126	60	we	we	PRON
ejpam-6275	126	61	can	can	AUX
ejpam-6275	126	62	not	not	PART
ejpam-6275	126	63	directly	directly	ADV
ejpam-6275	126	64	obtain	obtain	VERB
ejpam-6275	126	65	it	it	PRON
ejpam-6275	126	66	from	from	ADP
ejpam-6275	126	67	any	any	DET
ejpam-6275	126	68	polynomial	polynomial	ADJ
ejpam-6275	126	69	f(x	f(x	PROPN
ejpam-6275	126	70	)	)	PUNCT
ejpam-6275	126	71	of	of	ADP
ejpam-6275	126	72	degree	degree	NOUN
ejpam-6275	126	73	3	3	NUM
ejpam-6275	126	74	over	over	ADP
ejpam-6275	126	75	z2[ω][x	z2[ω][x	NOUN
ejpam-6275	126	76	]	]	PUNCT
ejpam-6275	126	77	⟨f(x)⟩	⟨f(x)⟩	X
ejpam-6275	126	78	.	.	PUNCT
ejpam-6275	127	1	m.	m.	PROPN
ejpam-6275	127	2	sajjad	sajjad	PROPN
ejpam-6275	127	3	et	et	PROPN
ejpam-6275	127	4	al	al	PROPN
ejpam-6275	127	5	.	.	PUNCT
ejpam-6275	127	6	/	/	SYM
ejpam-6275	127	7	eur	eur	PROPN
ejpam-6275	127	8	.	.	PUNCT
ejpam-6275	128	1	j.	j.	PROPN
ejpam-6275	128	2	pure	pure	PROPN
ejpam-6275	128	3	appl	appl	PROPN
ejpam-6275	128	4	.	.	PROPN
ejpam-6275	128	5	math	math	PROPN
ejpam-6275	128	6	,	,	PUNCT
ejpam-6275	128	7	18	18	NUM
ejpam-6275	128	8	(	(	PUNCT
ejpam-6275	128	9	3	3	NUM
ejpam-6275	128	10	)	)	PUNCT
ejpam-6275	128	11	(	(	PUNCT
ejpam-6275	128	12	2025	2025	NUM
ejpam-6275	128	13	)	)	PUNCT
ejpam-6275	128	14	,	,	PUNCT
ejpam-6275	128	15	6275	6275	NUM
ejpam-6275	128	16	7	7	NUM
ejpam-6275	128	17	of	of	ADP
ejpam-6275	128	18	36	36	NUM
ejpam-6275	128	19	table	table	NOUN
ejpam-6275	128	20	2	2	NUM
ejpam-6275	128	21	:	:	PUNCT
ejpam-6275	128	22	details	detail	NOUN
ejpam-6275	128	23	of	of	ADP
ejpam-6275	128	24	irreducible	irreducible	ADJ
ejpam-6275	128	25	and	and	CCONJ
ejpam-6275	128	26	primitive	primitive	ADJ
ejpam-6275	128	27	irreducible	irreducible	ADJ
ejpam-6275	128	28	polynomials	polynomial	NOUN
ejpam-6275	128	29	polynomials	polynomial	NOUN
ejpam-6275	128	30	of	of	ADP
ejpam-6275	128	31	degree	degree	NOUN
ejpam-6275	128	32	two	two	NUM
ejpam-6275	128	33	over	over	ADP
ejpam-6275	128	34	z2[ω	z2[ω	NOUN
ejpam-6275	128	35	]	]	X
ejpam-6275	128	36	irreducible	irreducible	ADJ
ejpam-6275	128	37	primitive	primitive	ADJ
ejpam-6275	128	38	irreducible	irreducible	ADJ
ejpam-6275	129	1	x2	x2	INTJ
ejpam-6275	129	2	×	×	NOUN
ejpam-6275	129	3	×	×	NOUN
ejpam-6275	130	1	x2	x2	NOUN
ejpam-6275	131	1	+	+	CCONJ
ejpam-6275	131	2	1	1	NUM
ejpam-6275	131	3	×	×	NOUN
ejpam-6275	131	4	×	×	NOUN
ejpam-6275	131	5	x2	x2	PROPN
ejpam-6275	132	1	+	+	CCONJ
ejpam-6275	132	2	ω	ω	NUM
ejpam-6275	132	3	✓	✓	ADJ
ejpam-6275	132	4	×	×	NOUN
ejpam-6275	132	5	x2	x2	NOUN
ejpam-6275	133	1	+	+	CCONJ
ejpam-6275	133	2	1	1	NUM
ejpam-6275	133	3	+	+	CCONJ
ejpam-6275	133	4	ω	ω	NUM
ejpam-6275	133	5	✓	✓	ADJ
ejpam-6275	133	6	×	×	NOUN
ejpam-6275	133	7	x2	x2	NOUN
ejpam-6275	134	1	+	+	CCONJ
ejpam-6275	134	2	x	x	SYM
ejpam-6275	134	3	×	×	PROPN
ejpam-6275	134	4	×	×	NOUN
ejpam-6275	134	5	x2	x2	INTJ
ejpam-6275	135	1	+	+	CCONJ
ejpam-6275	135	2	ωx	ωx	PRON
ejpam-6275	135	3	×	×	NOUN
ejpam-6275	135	4	×	×	NOUN
ejpam-6275	135	5	x2	x2	INTJ
ejpam-6275	136	1	+	+	CCONJ
ejpam-6275	136	2	(	(	PUNCT
ejpam-6275	136	3	1	1	NUM
ejpam-6275	136	4	+	+	CCONJ
ejpam-6275	136	5	ω)x	ω)x	ADJ
ejpam-6275	136	6	×	×	ADJ
ejpam-6275	136	7	×	×	NOUN
ejpam-6275	136	8	x2	x2	NOUN
ejpam-6275	137	1	+	+	CCONJ
ejpam-6275	137	2	x+	x+	SYM
ejpam-6275	137	3	1	1	NUM
ejpam-6275	137	4	×	×	NOUN
ejpam-6275	137	5	×	×	NOUN
ejpam-6275	137	6	x2	x2	PROPN
ejpam-6275	138	1	+	+	CCONJ
ejpam-6275	139	1	x+	x+	PROPN
ejpam-6275	139	2	ω	ω	NUM
ejpam-6275	139	3	✓	✓	ADJ
ejpam-6275	139	4	✓	✓	ADJ
ejpam-6275	139	5	x2	x2	PROPN
ejpam-6275	140	1	+	+	CCONJ
ejpam-6275	140	2	x+	x+	SYM
ejpam-6275	140	3	1	1	NUM
ejpam-6275	140	4	+	+	NUM
ejpam-6275	140	5	ω	ω	NUM
ejpam-6275	140	6	✓	✓	ADJ
ejpam-6275	140	7	✓	✓	ADJ
ejpam-6275	140	8	x2	x2	NOUN
ejpam-6275	141	1	+	+	CCONJ
ejpam-6275	141	2	ωx+	ωx+	NOUN
ejpam-6275	141	3	1	1	NUM
ejpam-6275	141	4	✓	✓	ADJ
ejpam-6275	141	5	×	×	NOUN
ejpam-6275	141	6	x2	x2	NOUN
ejpam-6275	141	7	+	+	CCONJ
ejpam-6275	141	8	ωx+	ωx+	NOUN
ejpam-6275	141	9	ω	ω	NUM
ejpam-6275	141	10	✓	✓	ADJ
ejpam-6275	141	11	✓	✓	ADJ
ejpam-6275	141	12	x2	x2	NOUN
ejpam-6275	142	1	+	+	CCONJ
ejpam-6275	142	2	ωx+	ωx+	NOUN
ejpam-6275	142	3	1	1	NUM
ejpam-6275	143	1	+	+	NUM
ejpam-6275	143	2	ω	ω	NUM
ejpam-6275	143	3	×	×	NOUN
ejpam-6275	143	4	×	×	NOUN
ejpam-6275	143	5	x2	x2	NOUN
ejpam-6275	143	6	+	+	CCONJ
ejpam-6275	143	7	(	(	PUNCT
ejpam-6275	143	8	1	1	NUM
ejpam-6275	143	9	+	+	SYM
ejpam-6275	143	10	ω)x+	ω)x+	NUM
ejpam-6275	143	11	1	1	NUM
ejpam-6275	143	12	×	×	NOUN
ejpam-6275	143	13	×	×	NOUN
ejpam-6275	143	14	x2	x2	NOUN
ejpam-6275	144	1	+	+	CCONJ
ejpam-6275	144	2	(	(	PUNCT
ejpam-6275	144	3	1	1	NUM
ejpam-6275	144	4	+	+	SYM
ejpam-6275	144	5	ω)x+	ω)x+	NUM
ejpam-6275	144	6	ω	ω	NUM
ejpam-6275	144	7	×	×	NOUN
ejpam-6275	144	8	×	×	NOUN
ejpam-6275	144	9	x2	x2	NOUN
ejpam-6275	145	1	+	+	CCONJ
ejpam-6275	145	2	(	(	PUNCT
ejpam-6275	145	3	1	1	NUM
ejpam-6275	145	4	+	+	SYM
ejpam-6275	145	5	ω)x+	ω)x+	NUM
ejpam-6275	145	6	1	1	NUM
ejpam-6275	145	7	+	+	CCONJ
ejpam-6275	145	8	ω	ω	NUM
ejpam-6275	145	9	✓	✓	ADJ
ejpam-6275	145	10	×	×	NOUN
ejpam-6275	145	11	instead	instead	ADV
ejpam-6275	145	12	,	,	PUNCT
ejpam-6275	145	13	we	we	PRON
ejpam-6275	145	14	obtain	obtain	VERB
ejpam-6275	145	15	a	a	DET
ejpam-6275	145	16	cyclic	cyclic	ADJ
ejpam-6275	145	17	group	group	NOUN
ejpam-6275	145	18	of	of	ADP
ejpam-6275	145	19	order	order	NOUN
ejpam-6275	145	20	n	n	NOUN
ejpam-6275	145	21	=	=	SYM
ejpam-6275	145	22	(	(	PUNCT
ejpam-6275	145	23	(	(	PUNCT
ejpam-6275	145	24	2)2	2)2	NUM
ejpam-6275	145	25	)	)	PUNCT
ejpam-6275	145	26	3	3	NUM
ejpam-6275	145	27	−	−	NOUN
ejpam-6275	145	28	1	1	NUM
ejpam-6275	145	29	=	=	NOUN
ejpam-6275	146	1	64−	64−	NOUN
ejpam-6275	146	2	1	1	NUM
ejpam-6275	146	3	=	=	SYM
ejpam-6275	146	4	63	63	NUM
ejpam-6275	146	5	.	.	PUNCT
ejpam-6275	147	1	the	the	DET
ejpam-6275	147	2	desired	desire	VERB
ejpam-6275	147	3	subgroup	subgroup	NOUN
ejpam-6275	147	4	is	be	AUX
ejpam-6275	147	5	obtained	obtain	VERB
ejpam-6275	147	6	by	by	ADP
ejpam-6275	147	7	dividing	divide	VERB
ejpam-6275	147	8	n	n	PROPN
ejpam-6275	147	9	by	by	ADP
ejpam-6275	147	10	21	21	NUM
ejpam-6275	147	11	,	,	PUNCT
ejpam-6275	147	12	i.e.	i.e.	X
ejpam-6275	147	13	,	,	PUNCT
ejpam-6275	147	14	63	63	NUM
ejpam-6275	147	15	21	21	NUM
ejpam-6275	147	16	=	=	SYM
ejpam-6275	147	17	3	3	X
ejpam-6275	147	18	.	.	PUNCT
ejpam-6275	148	1	if	if	SCONJ
ejpam-6275	148	2	α	α	PRON
ejpam-6275	148	3	is	be	AUX
ejpam-6275	148	4	the	the	DET
ejpam-6275	148	5	root	root	NOUN
ejpam-6275	148	6	of	of	ADP
ejpam-6275	148	7	the	the	DET
ejpam-6275	148	8	extension	extension	NOUN
ejpam-6275	148	9	field	field	NOUN
ejpam-6275	148	10	z2[ω][x	z2[ω][x	NOUN
ejpam-6275	148	11	]	]	PUNCT
ejpam-6275	148	12	,	,	PUNCT
ejpam-6275	148	13	then	then	ADV
ejpam-6275	148	14	α3	α3	PROPN
ejpam-6275	148	15	is	be	AUX
ejpam-6275	148	16	the	the	DET
ejpam-6275	148	17	generator	generator	NOUN
ejpam-6275	148	18	of	of	ADP
ejpam-6275	148	19	the	the	DET
ejpam-6275	148	20	required	require	VERB
ejpam-6275	148	21	subgroup	subgroup	NOUN
ejpam-6275	148	22	of	of	ADP
ejpam-6275	148	23	z2[ω][x	z2[ω][x	NOUN
ejpam-6275	148	24	]	]	PUNCT
ejpam-6275	148	25	for	for	ADP
ejpam-6275	148	26	n	n	NOUN
ejpam-6275	148	27	=	=	SYM
ejpam-6275	148	28	21	21	NUM
ejpam-6275	148	29	.	.	PUNCT
ejpam-6275	149	1	similarly	similarly	ADV
ejpam-6275	149	2	,	,	PUNCT
ejpam-6275	149	3	subgroups	subgroup	NOUN
ejpam-6275	149	4	of	of	ADP
ejpam-6275	149	5	orders	order	NOUN
ejpam-6275	149	6	3	3	NUM
ejpam-6275	149	7	,	,	PUNCT
ejpam-6275	149	8	7	7	NUM
ejpam-6275	149	9	,	,	PUNCT
ejpam-6275	149	10	and	and	CCONJ
ejpam-6275	149	11	9	9	NUM
ejpam-6275	149	12	can	can	AUX
ejpam-6275	149	13	be	be	AUX
ejpam-6275	149	14	found	find	VERB
ejpam-6275	149	15	with	with	ADP
ejpam-6275	149	16	generators	generator	NOUN
ejpam-6275	149	17	α21	α21	PROPN
ejpam-6275	149	18	,	,	PUNCT
ejpam-6275	149	19	α9	α9	PROPN
ejpam-6275	149	20	,	,	PUNCT
ejpam-6275	149	21	and	and	CCONJ
ejpam-6275	149	22	α7	α7	NOUN
ejpam-6275	149	23	respectively	respectively	ADV
ejpam-6275	149	24	.	.	PUNCT
ejpam-6275	150	1	3	3	X
ejpam-6275	150	2	.	.	NUM
ejpam-6275	150	3	shortened	shorten	VERB
ejpam-6275	150	4	bch	bch	PROPN
ejpam-6275	150	5	codes	code	NOUN
ejpam-6275	150	6	over	over	ADP
ejpam-6275	150	7	the	the	DET
ejpam-6275	150	8	eisenstein	eisenstein	PROPN
ejpam-6275	150	9	field	field	NOUN
ejpam-6275	150	10	shortened	shorten	VERB
ejpam-6275	150	11	bch	bch	PROPN
ejpam-6275	150	12	codes	code	NOUN
ejpam-6275	150	13	over	over	ADP
ejpam-6275	150	14	the	the	DET
ejpam-6275	150	15	eisenstein	eisenstein	PROPN
ejpam-6275	150	16	fields	field	NOUN
ejpam-6275	150	17	involve	involve	VERB
ejpam-6275	150	18	taking	take	VERB
ejpam-6275	150	19	a	a	DET
ejpam-6275	150	20	longer	long	ADJ
ejpam-6275	150	21	bch	bch	PROPN
ejpam-6275	150	22	code	code	NOUN
ejpam-6275	150	23	,	,	PUNCT
ejpam-6275	150	24	perhaps	perhaps	ADV
ejpam-6275	150	25	over	over	ADP
ejpam-6275	150	26	a	a	DET
ejpam-6275	150	27	bigger	big	ADJ
ejpam-6275	150	28	eisenstein	eisenstein	NOUN
ejpam-6275	150	29	field	field	NOUN
ejpam-6275	150	30	,	,	PUNCT
ejpam-6275	150	31	and	and	CCONJ
ejpam-6275	150	32	shortening	shorten	VERB
ejpam-6275	150	33	it	it	PRON
ejpam-6275	150	34	to	to	PART
ejpam-6275	150	35	achieve	achieve	VERB
ejpam-6275	150	36	specific	specific	ADJ
ejpam-6275	150	37	design	design	NOUN
ejpam-6275	150	38	goals	goal	NOUN
ejpam-6275	150	39	,	,	PUNCT
ejpam-6275	150	40	such	such	ADJ
ejpam-6275	150	41	as	as	ADP
ejpam-6275	150	42	minimizing	minimize	VERB
ejpam-6275	150	43	complexity	complexity	NOUN
ejpam-6275	150	44	or	or	CCONJ
ejpam-6275	150	45	meeting	meet	VERB
ejpam-6275	150	46	certain	certain	ADJ
ejpam-6275	150	47	constraints	constraint	NOUN
ejpam-6275	150	48	.	.	PUNCT
ejpam-6275	151	1	it	it	PRON
ejpam-6275	151	2	involves	involve	VERB
ejpam-6275	151	3	nearly	nearly	ADV
ejpam-6275	151	4	the	the	DET
ejpam-6275	151	5	same	same	ADJ
ejpam-6275	151	6	steps	step	NOUN
ejpam-6275	151	7	as	as	ADP
ejpam-6275	151	8	shortened	shorten	VERB
ejpam-6275	151	9	bch	bch	PROPN
ejpam-6275	151	10	codes	code	NOUN
ejpam-6275	151	11	over	over	ADP
ejpam-6275	151	12	eisenstein	eisenstein	NOUN
ejpam-6275	151	13	fields	field	NOUN
ejpam-6275	151	14	[	[	X
ejpam-6275	151	15	13	13	NUM
ejpam-6275	151	16	,	,	PUNCT
ejpam-6275	151	17	17	17	NUM
ejpam-6275	151	18	,	,	PUNCT
ejpam-6275	151	19	26	26	NUM
ejpam-6275	151	20	]	]	PUNCT
ejpam-6275	151	21	.	.	PUNCT
ejpam-6275	152	1	3.1	3.1	NUM
ejpam-6275	152	2	.	.	X
ejpam-6275	153	1	encoding	encoding	NOUN
ejpam-6275	153	2	of	of	ADP
ejpam-6275	153	3	shortened	shorten	VERB
ejpam-6275	153	4	bch	bch	PROPN
ejpam-6275	153	5	code	code	NOUN
ejpam-6275	153	6	over	over	ADP
ejpam-6275	153	7	the	the	DET
ejpam-6275	153	8	eisenstein	eisenstein	NOUN
ejpam-6275	153	9	field	field	NOUN
ejpam-6275	153	10	first	first	ADV
ejpam-6275	153	11	of	of	ADP
ejpam-6275	153	12	all	all	PRON
ejpam-6275	153	13	,	,	PUNCT
ejpam-6275	153	14	obtain	obtain	VERB
ejpam-6275	153	15	the	the	DET
ejpam-6275	153	16	shortened	shorten	VERB
ejpam-6275	153	17	bch	bch	PROPN
ejpam-6275	153	18	code	code	NOUN
ejpam-6275	153	19	from	from	ADP
ejpam-6275	153	20	the	the	DET
ejpam-6275	153	21	full	full	ADJ
ejpam-6275	153	22	length	length	NOUN
ejpam-6275	153	23	bch	bch	PROPN
ejpam-6275	153	24	code	code	NOUN
ejpam-6275	153	25	over	over	ADP
ejpam-6275	153	26	the	the	DET
ejpam-6275	153	27	extension	extension	NOUN
ejpam-6275	153	28	field	field	NOUN
ejpam-6275	153	29	zp[ω	zp[ω	PROPN
ejpam-6275	153	30	]	]	X
ejpam-6275	153	31	,	,	PUNCT
ejpam-6275	153	32	where	where	SCONJ
ejpam-6275	153	33	p	p	PRON
ejpam-6275	153	34	≡	≡	PROPN
ejpam-6275	153	35	2	2	NUM
ejpam-6275	153	36	(	(	PUNCT
ejpam-6275	153	37	mod	mod	NOUN
ejpam-6275	153	38	3	3	NUM
ejpam-6275	153	39	)	)	PUNCT
ejpam-6275	153	40	.	.	PUNCT
ejpam-6275	154	1	assume	assume	VERB
ejpam-6275	154	2	that	that	SCONJ
ejpam-6275	154	3	the	the	DET
ejpam-6275	154	4	integers	integer	NOUN
ejpam-6275	154	5	c	c	VERB
ejpam-6275	154	6	,	,	PUNCT
ejpam-6275	154	7	n	n	CCONJ
ejpam-6275	154	8	,	,	PUNCT
ejpam-6275	154	9	k	k	PROPN
ejpam-6275	154	10	,	,	PUNCT
ejpam-6275	154	11	d	d	X
ejpam-6275	154	12	>	>	X
ejpam-6275	154	13	0	0	NUM
ejpam-6275	154	14	,	,	PUNCT
ejpam-6275	154	15	such	such	ADJ
ejpam-6275	154	16	m.	m.	NOUN
ejpam-6275	154	17	sajjad	sajjad	PROPN
ejpam-6275	154	18	et	et	PROPN
ejpam-6275	154	19	al	al	PROPN
ejpam-6275	154	20	.	.	PUNCT
ejpam-6275	154	21	/	/	SYM
ejpam-6275	154	22	eur	eur	PROPN
ejpam-6275	154	23	.	.	PUNCT
ejpam-6275	155	1	j.	j.	PROPN
ejpam-6275	155	2	pure	pure	PROPN
ejpam-6275	155	3	appl	appl	PROPN
ejpam-6275	155	4	.	.	PROPN
ejpam-6275	155	5	math	math	PROPN
ejpam-6275	155	6	,	,	PUNCT
ejpam-6275	155	7	18	18	NUM
ejpam-6275	155	8	(	(	PUNCT
ejpam-6275	155	9	3	3	NUM
ejpam-6275	155	10	)	)	PUNCT
ejpam-6275	155	11	(	(	PUNCT
ejpam-6275	155	12	2025	2025	NUM
ejpam-6275	155	13	)	)	PUNCT
ejpam-6275	155	14	,	,	PUNCT
ejpam-6275	155	15	6275	6275	NUM
ejpam-6275	155	16	8	8	NUM
ejpam-6275	155	17	of	of	ADP
ejpam-6275	155	18	36	36	NUM
ejpam-6275	155	19	table	table	NOUN
ejpam-6275	155	20	3	3	NUM
ejpam-6275	155	21	:	:	PUNCT
ejpam-6275	155	22	cyclic	cyclic	ADJ
ejpam-6275	155	23	group	group	NOUN
ejpam-6275	155	24	over	over	ADP
ejpam-6275	155	25	eisenstein	eisenstein	NOUN
ejpam-6275	155	26	integers	integer	NOUN
ejpam-6275	155	27	of	of	ADP
ejpam-6275	155	28	order	order	NOUN
ejpam-6275	155	29	63	63	NUM
ejpam-6275	155	30	s.	s.	PROPN
ejpam-6275	155	31	no	no	INTJ
ejpam-6275	155	32	.	.	PUNCT
ejpam-6275	156	1	power	power	NOUN
ejpam-6275	156	2	of	of	ADP
ejpam-6275	156	3	α	α	PROPN
ejpam-6275	156	4	’s	’s	PROPN
ejpam-6275	156	5	s.	s.	PROPN
ejpam-6275	156	6	no	no	PROPN
ejpam-6275	156	7	.	.	PUNCT
ejpam-6275	157	1	power	power	NOUN
ejpam-6275	157	2	of	of	ADP
ejpam-6275	157	3	α	α	PROPN
ejpam-6275	157	4	’s	’s	NOUN
ejpam-6275	157	5	1	1	NUM
ejpam-6275	157	6	α	α	NOUN
ejpam-6275	157	7	33	33	NUM
ejpam-6275	157	8	(	(	PUNCT
ejpam-6275	157	9	α+	α+	X
ejpam-6275	157	10	1	1	NUM
ejpam-6275	157	11	+	+	NUM
ejpam-6275	157	12	ωα)2	ωα)2	NOUN
ejpam-6275	157	13	2	2	NUM
ejpam-6275	157	14	α2	α2	ADJ
ejpam-6275	157	15	34	34	NUM
ejpam-6275	158	1	(	(	PUNCT
ejpam-6275	158	2	α(1	α(1	PROPN
ejpam-6275	158	3	+	+	PROPN
ejpam-6275	158	4	ω	ω	NUM
ejpam-6275	158	5	)	)	PUNCT
ejpam-6275	159	1	+	+	CCONJ
ejpam-6275	159	2	1	1	NUM
ejpam-6275	159	3	+	+	CCONJ
ejpam-6275	159	4	(	(	PUNCT
ejpam-6275	159	5	1	1	NUM
ejpam-6275	159	6	+	+	NUM
ejpam-6275	159	7	ω)α)2	ω)α)2	NUM
ejpam-6275	159	8	3	3	NUM
ejpam-6275	159	9	α2	α2	ADJ
ejpam-6275	159	10	+	+	CCONJ
ejpam-6275	159	11	α+	α+	PUNCT
ejpam-6275	159	12	1	1	NUM
ejpam-6275	159	13	+	+	NUM
ejpam-6275	159	14	ω	ω	NUM
ejpam-6275	159	15	35	35	NUM
ejpam-6275	159	16	ω	ω	NOUN
ejpam-6275	159	17	+	+	CCONJ
ejpam-6275	159	18	ωα	ωα	NUM
ejpam-6275	159	19	4	4	NUM
ejpam-6275	159	20	1	1	NUM
ejpam-6275	159	21	+	+	NUM
ejpam-6275	159	22	ω	ω	NUM
ejpam-6275	160	1	+	+	CCONJ
ejpam-6275	160	2	ωα	ωα	NUM
ejpam-6275	160	3	36	36	NUM
ejpam-6275	160	4	(	(	PUNCT
ejpam-6275	160	5	ωα+	ωα+	NOUN
ejpam-6275	160	6	ωα)2	ωα)2	NOUN
ejpam-6275	160	7	5	5	NUM
ejpam-6275	160	8	α+	α+	PRON
ejpam-6275	160	9	ωα+	ωα+	PROPN
ejpam-6275	160	10	α2ω	α2ω	PROPN
ejpam-6275	160	11	37	37	NUM
ejpam-6275	160	12	ωα+	ωα+	NOUN
ejpam-6275	160	13	1	1	NUM
ejpam-6275	160	14	6	6	NUM
ejpam-6275	160	15	α2	α2	ADJ
ejpam-6275	160	16	+	+	CCONJ
ejpam-6275	160	17	ωα+	ωα+	NOUN
ejpam-6275	160	18	1	1	NUM
ejpam-6275	160	19	38	38	NUM
ejpam-6275	160	20	α+	α+	NOUN
ejpam-6275	160	21	ωα2	ωα2	NOUN
ejpam-6275	160	22	7	7	NUM
ejpam-6275	160	23	α2(1	α2(1	NOUN
ejpam-6275	160	24	+	+	NOUN
ejpam-6275	160	25	ω	ω	NUM
ejpam-6275	160	26	)	)	PUNCT
ejpam-6275	161	1	+	+	CCONJ
ejpam-6275	161	2	1	1	NUM
ejpam-6275	162	1	+	+	NUM
ejpam-6275	162	2	ω	ω	NUM
ejpam-6275	162	3	39	39	NUM
ejpam-6275	162	4	(	(	PUNCT
ejpam-6275	162	5	1	1	NUM
ejpam-6275	162	6	+	+	X
ejpam-6275	162	7	ω)α2	ω)α2	NUM
ejpam-6275	162	8	+	+	CCONJ
ejpam-6275	162	9	1	1	NUM
ejpam-6275	162	10	+	+	CCONJ
ejpam-6275	162	11	ωα	ωα	NUM
ejpam-6275	162	12	8	8	NUM
ejpam-6275	162	13	α2(1	α2(1	NOUN
ejpam-6275	162	14	+	+	NOUN
ejpam-6275	162	15	ω	ω	NUM
ejpam-6275	162	16	)	)	PUNCT
ejpam-6275	162	17	+	+	CCONJ
ejpam-6275	162	18	ω	ω	NUM
ejpam-6275	162	19	40	40	NUM
ejpam-6275	162	20	α2	α2	ADJ
ejpam-6275	162	21	+	+	CCONJ
ejpam-6275	162	22	ω	ω	PROPN
ejpam-6275	163	1	+	+	CCONJ
ejpam-6275	163	2	ωα	ωα	NUM
ejpam-6275	163	3	9	9	NUM
ejpam-6275	163	4	α+	α+	DET
ejpam-6275	163	5	α2(1	α2(1	NOUN
ejpam-6275	163	6	+	+	CCONJ
ejpam-6275	163	7	ω	ω	NUM
ejpam-6275	163	8	)	)	PUNCT
ejpam-6275	163	9	+	+	CCONJ
ejpam-6275	164	1	ω	ω	NUM
ejpam-6275	164	2	41	41	NUM
ejpam-6275	164	3	1	1	NUM
ejpam-6275	164	4	+	+	NUM
ejpam-6275	164	5	ω	ω	NUM
ejpam-6275	164	6	+	+	CCONJ
ejpam-6275	164	7	(	(	PUNCT
ejpam-6275	164	8	1	1	NUM
ejpam-6275	164	9	+	+	NUM
ejpam-6275	164	10	ω)α2	ω)α2	NUM
ejpam-6275	164	11	+	+	CCONJ
ejpam-6275	164	12	(	(	PUNCT
ejpam-6275	164	13	1	1	NUM
ejpam-6275	164	14	+	+	CCONJ
ejpam-6275	164	15	ω)α	ω)α	X
ejpam-6275	164	16	10	10	NUM
ejpam-6275	164	17	α+	α+	SYM
ejpam-6275	164	18	ω	ω	NOUN
ejpam-6275	164	19	+	+	CCONJ
ejpam-6275	164	20	ωα2	ωα2	NOUN
ejpam-6275	164	21	42	42	NUM
ejpam-6275	164	22	ω	ω	NUM
ejpam-6275	164	23	11	11	NUM
ejpam-6275	164	24	α2(1	α2(1	NOUN
ejpam-6275	164	25	+	+	CCONJ
ejpam-6275	164	26	ω	ω	NUM
ejpam-6275	164	27	)	)	PUNCT
ejpam-6275	164	28	+	+	CCONJ
ejpam-6275	164	29	1	1	NUM
ejpam-6275	164	30	43	43	NUM
ejpam-6275	164	31	ωα	ωα	NUM
ejpam-6275	164	32	12	12	NUM
ejpam-6275	164	33	(	(	PUNCT
ejpam-6275	164	34	1	1	NUM
ejpam-6275	164	35	+	+	NUM
ejpam-6275	164	36	ω)α2	ω)α2	PROPN
ejpam-6275	164	37	+	+	NUM
ejpam-6275	164	38	ω	ω	PROPN
ejpam-6275	165	1	+	+	CCONJ
ejpam-6275	165	2	ωα	ωα	NUM
ejpam-6275	165	3	44	44	NUM
ejpam-6275	165	4	ωα2	ωα2	NOUN
ejpam-6275	165	5	13	13	NUM
ejpam-6275	165	6	α+	α+	NOUN
ejpam-6275	165	7	ω	ω	PROPN
ejpam-6275	165	8	+	+	CCONJ
ejpam-6275	165	9	α2	α2	ADJ
ejpam-6275	165	10	45	45	NUM
ejpam-6275	165	11	ωα2	ωα2	NOUN
ejpam-6275	165	12	+	+	CCONJ
ejpam-6275	165	13	1	1	NUM
ejpam-6275	166	1	+	+	CCONJ
ejpam-6275	166	2	ωα	ωα	NUM
ejpam-6275	166	3	14	14	NUM
ejpam-6275	166	4	1	1	NUM
ejpam-6275	166	5	+	+	NUM
ejpam-6275	166	6	ω	ω	NUM
ejpam-6275	166	7	+	+	CCONJ
ejpam-6275	166	8	(	(	PUNCT
ejpam-6275	166	9	1	1	NUM
ejpam-6275	166	10	+	+	CCONJ
ejpam-6275	166	11	ω)α	ω)α	ADJ
ejpam-6275	166	12	46	46	NUM
ejpam-6275	166	13	(	(	PUNCT
ejpam-6275	166	14	1	1	NUM
ejpam-6275	166	15	+	+	NUM
ejpam-6275	166	16	ω)α+	ω)α+	NUM
ejpam-6275	166	17	1	1	NUM
ejpam-6275	166	18	15	15	NUM
ejpam-6275	166	19	(	(	PUNCT
ejpam-6275	166	20	1	1	NUM
ejpam-6275	166	21	+	+	NUM
ejpam-6275	166	22	ω)α2	ω)α2	PROPN
ejpam-6275	167	1	+	+	CCONJ
ejpam-6275	167	2	ωα	ωα	NUM
ejpam-6275	167	3	47	47	NUM
ejpam-6275	167	4	α+	α+	PUNCT
ejpam-6275	167	5	(	(	PUNCT
ejpam-6275	167	6	1	1	NUM
ejpam-6275	167	7	+	+	NUM
ejpam-6275	167	8	ω)α2	ω)α2	PROPN
ejpam-6275	167	9	16	16	NUM
ejpam-6275	167	10	ω	ω	NOUN
ejpam-6275	167	11	+	+	CCONJ
ejpam-6275	167	12	(	(	PUNCT
ejpam-6275	167	13	1	1	NUM
ejpam-6275	167	14	+	+	CCONJ
ejpam-6275	167	15	ω)α	ω)α	X
ejpam-6275	167	16	48	48	NUM
ejpam-6275	167	17	ωα2	ωα2	NOUN
ejpam-6275	167	18	+	+	X
ejpam-6275	167	19	ω	ω	NUM
ejpam-6275	167	20	+	+	CCONJ
ejpam-6275	167	21	(	(	PUNCT
ejpam-6275	167	22	1	1	NUM
ejpam-6275	167	23	+	+	CCONJ
ejpam-6275	167	24	ω)α	ω)α	X
ejpam-6275	167	25	17	17	NUM
ejpam-6275	167	26	(	(	PUNCT
ejpam-6275	167	27	1	1	NUM
ejpam-6275	167	28	+	+	NUM
ejpam-6275	167	29	ω)α2	ω)α2	PROPN
ejpam-6275	167	30	+	+	CCONJ
ejpam-6275	167	31	ωα	ωα	NUM
ejpam-6275	167	32	49	49	NUM
ejpam-6275	167	33	α2	α2	ADJ
ejpam-6275	167	34	+	+	CCONJ
ejpam-6275	167	35	1	1	NUM
ejpam-6275	167	36	18	18	NUM
ejpam-6275	167	37	α2	α2	ADJ
ejpam-6275	167	38	+	+	CCONJ
ejpam-6275	167	39	ω	ω	NUM
ejpam-6275	167	40	+	+	CCONJ
ejpam-6275	167	41	(	(	PUNCT
ejpam-6275	167	42	1	1	NUM
ejpam-6275	167	43	+	+	CCONJ
ejpam-6275	167	44	ω)α	ω)α	ADJ
ejpam-6275	167	45	50	50	NUM
ejpam-6275	167	46	α2	α2	ADJ
ejpam-6275	167	47	+	+	CCONJ
ejpam-6275	167	48	1	1	NUM
ejpam-6275	167	49	+	+	NUM
ejpam-6275	167	50	ω	ω	NUM
ejpam-6275	167	51	19	19	NUM
ejpam-6275	167	52	ωα2	ωα2	NOUN
ejpam-6275	167	53	+	+	CCONJ
ejpam-6275	167	54	1	1	NUM
ejpam-6275	167	55	+	+	NUM
ejpam-6275	167	56	ω	ω	NUM
ejpam-6275	167	57	+	+	CCONJ
ejpam-6275	167	58	(	(	PUNCT
ejpam-6275	167	59	1	1	NUM
ejpam-6275	167	60	+	+	CCONJ
ejpam-6275	167	61	ω)α	ω)α	X
ejpam-6275	167	62	51	51	NUM
ejpam-6275	167	63	1	1	NUM
ejpam-6275	167	64	+	+	NUM
ejpam-6275	167	65	ω	ω	PROPN
ejpam-6275	167	66	+	+	CCONJ
ejpam-6275	167	67	α2	α2	ADJ
ejpam-6275	167	68	+	+	CCONJ
ejpam-6275	167	69	ωα	ωα	NUM
ejpam-6275	167	70	20	20	NUM
ejpam-6275	167	71	α+	α+	X
ejpam-6275	167	72	α2	α2	NOUN
ejpam-6275	167	73	+	+	CCONJ
ejpam-6275	167	74	1	1	NUM
ejpam-6275	167	75	52	52	NUM
ejpam-6275	167	76	1	1	NUM
ejpam-6275	167	77	+	+	NUM
ejpam-6275	167	78	ω	ω	NUM
ejpam-6275	167	79	+	+	CCONJ
ejpam-6275	167	80	(	(	PUNCT
ejpam-6275	167	81	1	1	NUM
ejpam-6275	167	82	+	+	NUM
ejpam-6275	167	83	ω)α2	ω)α2	PROPN
ejpam-6275	167	84	+	+	CCONJ
ejpam-6275	167	85	ωα	ωα	NUM
ejpam-6275	167	86	21	21	NUM
ejpam-6275	167	87	1	1	NUM
ejpam-6275	167	88	+	+	SYM
ejpam-6275	167	89	ω	ω	NUM
ejpam-6275	167	90	53	53	NUM
ejpam-6275	167	91	α2	α2	ADJ
ejpam-6275	167	92	+	+	CCONJ
ejpam-6275	167	93	ω	ω	NUM
ejpam-6275	167	94	22	22	NUM
ejpam-6275	167	95	(	(	PUNCT
ejpam-6275	167	96	1	1	NUM
ejpam-6275	167	97	+	+	CCONJ
ejpam-6275	167	98	ω)α	ω)α	X
ejpam-6275	167	99	54	54	NUM
ejpam-6275	167	100	α2	α2	ADJ
ejpam-6275	167	101	+	+	CCONJ
ejpam-6275	167	102	1	1	NUM
ejpam-6275	167	103	+	+	NUM
ejpam-6275	167	104	ω	ω	NUM
ejpam-6275	167	105	+	+	CCONJ
ejpam-6275	167	106	(	(	PUNCT
ejpam-6275	167	107	1	1	NUM
ejpam-6275	167	108	+	+	CCONJ
ejpam-6275	167	109	ω)α	ω)α	X
ejpam-6275	167	110	23	23	NUM
ejpam-6275	167	111	(	(	PUNCT
ejpam-6275	167	112	1	1	NUM
ejpam-6275	167	113	+	+	NUM
ejpam-6275	167	114	ω)α2	ω)α2	NUM
ejpam-6275	167	115	55	55	NUM
ejpam-6275	167	116	1	1	NUM
ejpam-6275	167	117	+	+	NUM
ejpam-6275	167	118	ω	ω	NUM
ejpam-6275	167	119	+	+	NUM
ejpam-6275	167	120	ωα2	ωα2	NOUN
ejpam-6275	167	121	+	+	CCONJ
ejpam-6275	167	122	ωα	ωα	NUM
ejpam-6275	167	123	24	24	NUM
ejpam-6275	167	124	(	(	PUNCT
ejpam-6275	167	125	1	1	NUM
ejpam-6275	167	126	+	+	NUM
ejpam-6275	167	127	ω)α2	ω)α2	PROPN
ejpam-6275	167	128	+	+	NUM
ejpam-6275	167	129	ω	ω	NUM
ejpam-6275	167	130	+	+	CCONJ
ejpam-6275	167	131	(	(	PUNCT
ejpam-6275	167	132	1	1	NUM
ejpam-6275	167	133	+	+	CCONJ
ejpam-6275	167	134	ω)α	ω)α	X
ejpam-6275	167	135	56	56	NUM
ejpam-6275	167	136	α+	α+	SYM
ejpam-6275	167	137	1	1	NUM
ejpam-6275	167	138	25	25	NUM
ejpam-6275	167	139	α+	α+	PUNCT
ejpam-6275	167	140	ω	ω	NUM
ejpam-6275	167	141	57	57	NUM
ejpam-6275	167	142	α+	α+	PUNCT
ejpam-6275	167	143	α2	α2	PROPN
ejpam-6275	167	144	26	26	NUM
ejpam-6275	167	145	α2	α2	ADJ
ejpam-6275	167	146	+	+	CCONJ
ejpam-6275	167	147	ωα	ωα	NUM
ejpam-6275	167	148	58	58	NUM
ejpam-6275	167	149	α+	α+	SYM
ejpam-6275	167	150	1	1	NUM
ejpam-6275	167	151	+	+	NUM
ejpam-6275	167	152	ω	ω	NUM
ejpam-6275	167	153	27	27	NUM
ejpam-6275	167	154	α+	α+	SYM
ejpam-6275	167	155	1	1	NUM
ejpam-6275	167	156	+	+	NUM
ejpam-6275	167	157	ω	ω	NUM
ejpam-6275	167	158	+	+	CCONJ
ejpam-6275	167	159	(	(	PUNCT
ejpam-6275	167	160	1	1	NUM
ejpam-6275	167	161	+	+	NUM
ejpam-6275	167	162	ω)α2	ω)α2	PROPN
ejpam-6275	167	163	59	59	NUM
ejpam-6275	167	164	α2	α2	ADJ
ejpam-6275	167	165	+	+	CCONJ
ejpam-6275	167	166	(	(	PUNCT
ejpam-6275	167	167	1	1	NUM
ejpam-6275	167	168	+	+	CCONJ
ejpam-6275	167	169	ω)α	ω)α	X
ejpam-6275	167	170	28	28	NUM
ejpam-6275	167	171	ωα2	ωα2	NOUN
ejpam-6275	167	172	+	+	X
ejpam-6275	167	173	ω	ω	NUM
ejpam-6275	167	174	60	60	NUM
ejpam-6275	167	175	α+	α+	NOUN
ejpam-6275	167	176	ω	ω	NOUN
ejpam-6275	167	177	+	+	CCONJ
ejpam-6275	167	178	1	1	NUM
ejpam-6275	167	179	+	+	CCONJ
ejpam-6275	167	180	ωα2	ωα2	NOUN
ejpam-6275	167	181	29	29	NUM
ejpam-6275	167	182	ωα2	ωα2	NOUN
ejpam-6275	167	183	+	+	CCONJ
ejpam-6275	167	184	1	1	NUM
ejpam-6275	167	185	61	61	NUM
ejpam-6275	167	186	α+	α+	NOUN
ejpam-6275	167	187	(	(	PUNCT
ejpam-6275	167	188	1	1	NUM
ejpam-6275	167	189	+	+	NUM
ejpam-6275	167	190	ω)α2	ω)α2	NUM
ejpam-6275	167	191	+	+	CCONJ
ejpam-6275	167	192	1	1	NUM
ejpam-6275	167	193	30	30	NUM
ejpam-6275	167	194	ωα2	ωα2	NOUN
ejpam-6275	167	195	+	+	CCONJ
ejpam-6275	167	196	(	(	PUNCT
ejpam-6275	167	197	1	1	NUM
ejpam-6275	167	198	+	+	NUM
ejpam-6275	167	199	ω)α+	ω)α+	NUM
ejpam-6275	167	200	1	1	NUM
ejpam-6275	167	201	62	62	NUM
ejpam-6275	167	202	ω	ω	NOUN
ejpam-6275	167	203	+	+	CCONJ
ejpam-6275	167	204	ωα+	ωα+	PROPN
ejpam-6275	167	205	ωα2	ωα2	NOUN
ejpam-6275	167	206	31	31	NUM
ejpam-6275	167	207	α2	α2	ADJ
ejpam-6275	167	208	+	+	CCONJ
ejpam-6275	167	209	(	(	PUNCT
ejpam-6275	167	210	1	1	NUM
ejpam-6275	167	211	+	+	NUM
ejpam-6275	167	212	ω)α+	ω)α+	NUM
ejpam-6275	167	213	1	1	NUM
ejpam-6275	167	214	63	63	NUM
ejpam-6275	167	215	1	1	NUM
ejpam-6275	167	216	32	32	NUM
ejpam-6275	167	217	ωα2	ωα2	NOUN
ejpam-6275	167	218	+	+	CCONJ
ejpam-6275	167	219	1	1	NUM
ejpam-6275	167	220	+	+	NUM
ejpam-6275	167	221	ω	ω	NOUN
ejpam-6275	167	222	that	that	SCONJ
ejpam-6275	167	223	k	k	PROPN
ejpam-6275	167	224	is	be	AUX
ejpam-6275	167	225	a	a	DET
ejpam-6275	167	226	prime	prime	ADJ
ejpam-6275	167	227	power	power	NOUN
ejpam-6275	167	228	,	,	PUNCT
ejpam-6275	167	229	2	2	NUM
ejpam-6275	167	230	≤	≤	NUM
ejpam-6275	167	231	d	d	PROPN
ejpam-6275	167	232	≤	≤	NOUN
ejpam-6275	167	233	n−1	n−1	PROPN
ejpam-6275	167	234	,	,	PUNCT
ejpam-6275	167	235	and	and	CCONJ
ejpam-6275	167	236	gcd(n	gcd(n	PROPN
ejpam-6275	167	237	,	,	PUNCT
ejpam-6275	167	238	k	k	NOUN
ejpam-6275	167	239	)	)	PUNCT
ejpam-6275	167	240	=	=	SYM
ejpam-6275	168	1	1	1	X
ejpam-6275	168	2	.	.	X
ejpam-6275	168	3	assume	assume	VERB
ejpam-6275	168	4	that	that	SCONJ
ejpam-6275	168	5	there	there	PRON
ejpam-6275	168	6	is	be	VERB
ejpam-6275	168	7	a	a	DET
ejpam-6275	168	8	smallest	small	ADJ
ejpam-6275	168	9	positive	positive	ADJ
ejpam-6275	168	10	integer	integer	NOUN
ejpam-6275	168	11	b	b	PROPN
ejpam-6275	168	12	such	such	ADJ
ejpam-6275	168	13	that	that	DET
ejpam-6275	168	14	p2b	p2b	PROPN
ejpam-6275	168	15	≡	≡	PROPN
ejpam-6275	168	16	1	1	NUM
ejpam-6275	168	17	(	(	PUNCT
ejpam-6275	168	18	mod	mod	NOUN
ejpam-6275	168	19	n	n	CCONJ
ejpam-6275	168	20	)	)	PUNCT
ejpam-6275	168	21	.	.	PUNCT
ejpam-6275	169	1	then	then	ADV
ejpam-6275	169	2	,	,	PUNCT
ejpam-6275	169	3	by	by	ADP
ejpam-6275	169	4	euler	euler	PROPN
ejpam-6275	169	5	’s	’s	PART
ejpam-6275	169	6	theorem	theorem	ADJ
ejpam-6275	169	7	,	,	PUNCT
ejpam-6275	169	8	if	if	SCONJ
ejpam-6275	169	9	p2φ(n	p2φ(n	NOUN
ejpam-6275	169	10	)	)	PUNCT
ejpam-6275	169	11	≡	≡	PROPN
ejpam-6275	169	12	1	1	NUM
ejpam-6275	169	13	(	(	PUNCT
ejpam-6275	169	14	mod	mod	NOUN
ejpam-6275	169	15	n	n	CCONJ
ejpam-6275	169	16	)	)	PUNCT
ejpam-6275	169	17	,	,	PUNCT
ejpam-6275	169	18	then	then	ADV
ejpam-6275	169	19	b	b	NOUN
ejpam-6275	169	20	divides	divide	VERB
ejpam-6275	169	21	φ(n	φ(n	ADJ
ejpam-6275	169	22	)	)	PUNCT
ejpam-6275	169	23	,	,	PUNCT
ejpam-6275	169	24	where	where	SCONJ
ejpam-6275	169	25	φ	φ	PROPN
ejpam-6275	169	26	is	be	AUX
ejpam-6275	169	27	the	the	DET
ejpam-6275	169	28	euler	euler	NOUN
ejpam-6275	169	29	phi	phi	NOUN
ejpam-6275	169	30	function	function	NOUN
ejpam-6275	169	31	.	.	PUNCT
ejpam-6275	170	1	thus	thus	ADV
ejpam-6275	170	2	,	,	PUNCT
ejpam-6275	170	3	n	n	CCONJ
ejpam-6275	170	4	|	|	ADV
ejpam-6275	170	5	(	(	PUNCT
ejpam-6275	170	6	p2b	p2b	NOUN
ejpam-6275	170	7	−	−	NOUN
ejpam-6275	170	8	1	1	NUM
ejpam-6275	170	9	)	)	PUNCT
ejpam-6275	170	10	.	.	PUNCT
ejpam-6275	171	1	let	let	VERB
ejpam-6275	171	2	β	β	PRON
ejpam-6275	171	3	be	be	AUX
ejpam-6275	171	4	an	an	DET
ejpam-6275	171	5	element	element	NOUN
ejpam-6275	171	6	of	of	ADP
ejpam-6275	171	7	the	the	DET
ejpam-6275	171	8	extension	extension	NOUN
ejpam-6275	171	9	field	field	NOUN
ejpam-6275	171	10	zp[ω	zp[ω	PROPN
ejpam-6275	171	11	]	]	X
ejpam-6275	171	12	b.	b.	PROPN
ejpam-6275	171	13	consider	consider	VERB
ejpam-6275	171	14	the	the	DET
ejpam-6275	171	15	minimal	minimal	ADJ
ejpam-6275	171	16	polynomials	polynomial	NOUN
ejpam-6275	171	17	bi(x	bi(x	NUM
ejpam-6275	171	18	)	)	PUNCT
ejpam-6275	171	19	∈	∈	PROPN
ejpam-6275	172	1	zp[ω][x	zp[ω][x	NOUN
ejpam-6275	172	2	]	]	PUNCT
ejpam-6275	172	3	of	of	ADP
ejpam-6275	172	4	β	β	PROPN
ejpam-6275	172	5	i.	i.	PROPN
ejpam-6275	172	6	the	the	DET
ejpam-6275	172	7	least	least	ADV
ejpam-6275	172	8	common	common	ADJ
ejpam-6275	172	9	multiple	multiple	ADJ
ejpam-6275	172	10	(	(	PUNCT
ejpam-6275	172	11	lcm	lcm	NOUN
ejpam-6275	172	12	)	)	PUNCT
ejpam-6275	172	13	of	of	ADP
ejpam-6275	172	14	all	all	DET
ejpam-6275	172	15	distinct	distinct	ADJ
ejpam-6275	172	16	polynomials	polynomial	NOUN
ejpam-6275	172	17	bi(x	bi(x	NUM
ejpam-6275	172	18	)	)	PUNCT
ejpam-6275	172	19	,	,	PUNCT
ejpam-6275	172	20	for	for	ADP
ejpam-6275	172	21	i	i	PRON
ejpam-6275	172	22	=	=	SYM
ejpam-6275	172	23	c	c	X
ejpam-6275	172	24	,	,	PUNCT
ejpam-6275	172	25	c+	c+	VERB
ejpam-6275	172	26	1	1	NUM
ejpam-6275	172	27	,	,	PUNCT
ejpam-6275	172	28	c+	c+	VERB
ejpam-6275	172	29	2	2	NUM
ejpam-6275	172	30	,	,	PUNCT
ejpam-6275	172	31	.	.	PUNCT
ejpam-6275	172	32	.	.	PUNCT
ejpam-6275	173	1	.	.	PUNCT
ejpam-6275	174	1	,	,	PUNCT
ejpam-6275	174	2	c+	c+	VERB
ejpam-6275	174	3	d−	d−	PROPN
ejpam-6275	174	4	2	2	NUM
ejpam-6275	174	5	,	,	PUNCT
ejpam-6275	174	6	is	be	AUX
ejpam-6275	174	7	known	know	VERB
ejpam-6275	174	8	as	as	ADP
ejpam-6275	174	9	the	the	DET
ejpam-6275	174	10	generator	generator	NOUN
ejpam-6275	174	11	polynomial	polynomial	ADJ
ejpam-6275	174	12	g(x	g(x	PROPN
ejpam-6275	174	13	)	)	PUNCT
ejpam-6275	174	14	,	,	PUNCT
ejpam-6275	174	15	i.e.	i.e.	X
ejpam-6275	174	16	,	,	PUNCT
ejpam-6275	174	17	g(x	g(x	NOUN
ejpam-6275	174	18	)	)	PUNCT
ejpam-6275	175	1	=	=	SYM
ejpam-6275	175	2	lcm{bi(x	lcm{bi(x	NOUN
ejpam-6275	175	3	)	)	PUNCT
ejpam-6275	176	1	|	|	ADV
ejpam-6275	176	2	i	i	PRON
ejpam-6275	176	3	=	=	SYM
ejpam-6275	176	4	c	c	X
ejpam-6275	176	5	,	,	PUNCT
ejpam-6275	176	6	c+	c+	VERB
ejpam-6275	176	7	1	1	NUM
ejpam-6275	176	8	,	,	PUNCT
ejpam-6275	176	9	c+	c+	VERB
ejpam-6275	176	10	2	2	NUM
ejpam-6275	176	11	,	,	PUNCT
ejpam-6275	176	12	.	.	PUNCT
ejpam-6275	176	13	.	.	PUNCT
ejpam-6275	177	1	.	.	PUNCT
ejpam-6275	178	1	,	,	PUNCT
ejpam-6275	178	2	c+	c+	VERB
ejpam-6275	178	3	d−	d−	PROPN
ejpam-6275	178	4	2	2	NUM
ejpam-6275	178	5	}	}	PUNCT
ejpam-6275	178	6	.	.	PUNCT
ejpam-6275	179	1	m.	m.	NOUN
ejpam-6275	179	2	sajjad	sajjad	PROPN
ejpam-6275	179	3	et	et	PROPN
ejpam-6275	179	4	al	al	PROPN
ejpam-6275	179	5	.	.	PUNCT
ejpam-6275	179	6	/	/	SYM
ejpam-6275	179	7	eur	eur	PROPN
ejpam-6275	179	8	.	.	PUNCT
ejpam-6275	180	1	j.	j.	PROPN
ejpam-6275	180	2	pure	pure	PROPN
ejpam-6275	180	3	appl	appl	PROPN
ejpam-6275	180	4	.	.	PROPN
ejpam-6275	180	5	math	math	PROPN
ejpam-6275	180	6	,	,	PUNCT
ejpam-6275	180	7	18	18	NUM
ejpam-6275	180	8	(	(	PUNCT
ejpam-6275	180	9	3	3	NUM
ejpam-6275	180	10	)	)	PUNCT
ejpam-6275	180	11	(	(	PUNCT
ejpam-6275	180	12	2025	2025	NUM
ejpam-6275	180	13	)	)	PUNCT
ejpam-6275	180	14	,	,	PUNCT
ejpam-6275	180	15	6275	6275	NUM
ejpam-6275	180	16	9	9	NUM
ejpam-6275	180	17	of	of	ADP
ejpam-6275	180	18	36	36	NUM
ejpam-6275	180	19	since	since	SCONJ
ejpam-6275	180	20	all	all	DET
ejpam-6275	180	21	minimal	minimal	ADJ
ejpam-6275	180	22	polynomials	polynomial	NOUN
ejpam-6275	180	23	divide	divide	VERB
ejpam-6275	180	24	xn−1	xn−1	PROPN
ejpam-6275	180	25	,	,	PUNCT
ejpam-6275	180	26	the	the	DET
ejpam-6275	180	27	generator	generator	NOUN
ejpam-6275	180	28	polynomial	polynomial	PROPN
ejpam-6275	180	29	g(x	g(x	PROPN
ejpam-6275	180	30	)	)	PUNCT
ejpam-6275	180	31	also	also	ADV
ejpam-6275	180	32	divides	divide	VERB
ejpam-6275	180	33	xn	xn	PROPN
ejpam-6275	180	34	−	−	PROPN
ejpam-6275	181	1	1	1	X
ejpam-6275	181	2	.	.	PUNCT
ejpam-6275	181	3	let	let	VERB
ejpam-6275	181	4	c	c	NOUN
ejpam-6275	181	5	be	be	AUX
ejpam-6275	181	6	the	the	DET
ejpam-6275	181	7	cyclic	cyclic	PROPN
ejpam-6275	181	8	code	code	NOUN
ejpam-6275	181	9	generated	generate	VERB
ejpam-6275	181	10	by	by	ADP
ejpam-6275	181	11	g(x	g(x	PROPN
ejpam-6275	181	12	)	)	PUNCT
ejpam-6275	181	13	in	in	ADP
ejpam-6275	181	14	the	the	DET
ejpam-6275	181	15	ring	ring	NOUN
ejpam-6275	181	16	zp[ω][x	zp[ω][x	NOUN
ejpam-6275	181	17	]	]	X
ejpam-6275	181	18	m	m	NOUN
ejpam-6275	181	19	,	,	PUNCT
ejpam-6275	181	20	then	then	ADV
ejpam-6275	181	21	c	c	PROPN
ejpam-6275	181	22	is	be	AUX
ejpam-6275	181	23	called	call	VERB
ejpam-6275	181	24	a	a	DET
ejpam-6275	181	25	bch	bch	PROPN
ejpam-6275	181	26	code	code	NOUN
ejpam-6275	181	27	of	of	ADP
ejpam-6275	181	28	length	length	NOUN
ejpam-6275	181	29	n	n	CCONJ
ejpam-6275	181	30	over	over	ADP
ejpam-6275	181	31	the	the	DET
ejpam-6275	181	32	extension	extension	NOUN
ejpam-6275	181	33	field	field	NOUN
ejpam-6275	181	34	zp[ω	zp[ω	PROPN
ejpam-6275	181	35	]	]	PUNCT
ejpam-6275	181	36	with	with	ADP
ejpam-6275	181	37	designed	design	VERB
ejpam-6275	181	38	distance	distance	NOUN
ejpam-6275	181	39	d.	d.	PROPN
ejpam-6275	181	40	now	now	ADV
ejpam-6275	181	41	,	,	PUNCT
ejpam-6275	181	42	if	if	SCONJ
ejpam-6275	181	43	the	the	DET
ejpam-6275	181	44	code	code	NOUN
ejpam-6275	181	45	length	length	NOUN
ejpam-6275	181	46	is	be	AUX
ejpam-6275	181	47	shorter	short	ADJ
ejpam-6275	181	48	than	than	ADP
ejpam-6275	181	49	the	the	DET
ejpam-6275	181	50	full	full	ADJ
ejpam-6275	181	51	length	length	NOUN
ejpam-6275	181	52	bch	bch	PROPN
ejpam-6275	181	53	code	code	PROPN
ejpam-6275	181	54	,	,	PUNCT
ejpam-6275	181	55	then	then	ADV
ejpam-6275	181	56	it	it	PRON
ejpam-6275	181	57	is	be	AUX
ejpam-6275	181	58	known	know	VERB
ejpam-6275	181	59	as	as	ADP
ejpam-6275	181	60	a	a	DET
ejpam-6275	181	61	shortened	shorten	VERB
ejpam-6275	181	62	bch	bch	PROPN
ejpam-6275	181	63	code	code	NOUN
ejpam-6275	181	64	.	.	PUNCT
ejpam-6275	182	1	if	if	SCONJ
ejpam-6275	182	2	the	the	DET
ejpam-6275	182	3	code	code	NOUN
ejpam-6275	182	4	is	be	AUX
ejpam-6275	182	5	a	a	DET
ejpam-6275	182	6	narrow	narrow	ADJ
ejpam-6275	182	7	-	-	PUNCT
ejpam-6275	182	8	sense	sense	NOUN
ejpam-6275	182	9	shortened	shorten	VERB
ejpam-6275	182	10	bch	bch	PROPN
ejpam-6275	182	11	code	code	PROPN
ejpam-6275	182	12	,	,	PUNCT
ejpam-6275	182	13	then	then	ADV
ejpam-6275	182	14	c	c	NOUN
ejpam-6275	182	15	=	=	SYM
ejpam-6275	182	16	1	1	X
ejpam-6275	182	17	.	.	PUNCT
ejpam-6275	182	18	pseudocode	pseudocode	NOUN
ejpam-6275	182	19	of	of	ADP
ejpam-6275	182	20	the	the	DET
ejpam-6275	182	21	encoding	encoding	NOUN
ejpam-6275	182	22	algorithm	algorithm	NOUN
ejpam-6275	182	23	is	be	AUX
ejpam-6275	182	24	given	give	VERB
ejpam-6275	182	25	in	in	ADP
ejpam-6275	182	26	algorithm	algorithm	NOUN
ejpam-6275	182	27	3.1	3.1	NUM
ejpam-6275	182	28	.	.	PUNCT
ejpam-6275	183	1	theorem	theorem	VERB
ejpam-6275	183	2	3.1	3.1	NUM
ejpam-6275	184	1	[	[	X
ejpam-6275	184	2	26	26	NUM
ejpam-6275	184	3	]	]	PUNCT
ejpam-6275	184	4	:	:	PUNCT
ejpam-6275	184	5	assume	assume	VERB
ejpam-6275	184	6	α	α	PRON
ejpam-6275	184	7	is	be	AUX
ejpam-6275	184	8	an	an	DET
ejpam-6275	184	9	element	element	NOUN
ejpam-6275	184	10	of	of	ADP
ejpam-6275	184	11	the	the	DET
ejpam-6275	184	12	extension	extension	NOUN
ejpam-6275	184	13	field	field	NOUN
ejpam-6275	184	14	zp[ω	zp[ω	PROPN
ejpam-6275	184	15	]	]	X
ejpam-6275	184	16	m	m	VERB
ejpam-6275	184	17	where	where	SCONJ
ejpam-6275	184	18	p	p	PRON
ejpam-6275	184	19	≡	≡	PROPN
ejpam-6275	184	20	2	2	NUM
ejpam-6275	184	21	(	(	PUNCT
ejpam-6275	184	22	mod	mod	NOUN
ejpam-6275	184	23	3	3	NUM
ejpam-6275	184	24	)	)	PUNCT
ejpam-6275	184	25	.	.	PUNCT
ejpam-6275	185	1	then	then	ADV
ejpam-6275	185	2	,	,	PUNCT
ejpam-6275	185	3	the	the	DET
ejpam-6275	185	4	elements	element	NOUN
ejpam-6275	185	5	αp0	αp0	NOUN
ejpam-6275	185	6	,	,	PUNCT
ejpam-6275	185	7	αp2	αp2	NOUN
ejpam-6275	185	8	,	,	PUNCT
ejpam-6275	185	9	αp4	αp4	PROPN
ejpam-6275	185	10	,	,	PUNCT
ejpam-6275	185	11	αp6	αp6	PROPN
ejpam-6275	185	12	,	,	PUNCT
ejpam-6275	185	13	.	.	PUNCT
ejpam-6275	185	14	.	.	PUNCT
ejpam-6275	185	15	.	.	PUNCT
ejpam-6275	186	1	have	have	VERB
ejpam-6275	186	2	minimal	minimal	ADJ
ejpam-6275	186	3	polynomials	polynomial	NOUN
ejpam-6275	186	4	over	over	ADP
ejpam-6275	186	5	the	the	DET
ejpam-6275	186	6	eisenstein	eisenstein	PROPN
ejpam-6275	186	7	field	field	NOUN
ejpam-6275	186	8	zp[ω	zp[ω	PROPN
ejpam-6275	186	9	]	]	PUNCT
ejpam-6275	186	10	.	.	PUNCT
ejpam-6275	187	1	m.	m.	PROPN
ejpam-6275	187	2	sajjad	sajjad	PROPN
ejpam-6275	187	3	et	et	PROPN
ejpam-6275	187	4	al	al	PROPN
ejpam-6275	187	5	.	.	PUNCT
ejpam-6275	187	6	/	/	SYM
ejpam-6275	187	7	eur	eur	PROPN
ejpam-6275	187	8	.	.	PUNCT
ejpam-6275	188	1	j.	j.	PROPN
ejpam-6275	188	2	pure	pure	PROPN
ejpam-6275	188	3	appl	appl	PROPN
ejpam-6275	188	4	.	.	PROPN
ejpam-6275	188	5	math	math	PROPN
ejpam-6275	188	6	,	,	PUNCT
ejpam-6275	188	7	18	18	NUM
ejpam-6275	188	8	(	(	PUNCT
ejpam-6275	188	9	3	3	NUM
ejpam-6275	188	10	)	)	PUNCT
ejpam-6275	188	11	(	(	PUNCT
ejpam-6275	188	12	2025	2025	NUM
ejpam-6275	188	13	)	)	PUNCT
ejpam-6275	188	14	,	,	PUNCT
ejpam-6275	188	15	6275	6275	NUM
ejpam-6275	188	16	10	10	NUM
ejpam-6275	188	17	of	of	ADP
ejpam-6275	188	18	36	36	NUM
ejpam-6275	188	19	illustration	illustration	NOUN
ejpam-6275	188	20	3.1	3.1	NUM
ejpam-6275	188	21	:	:	PUNCT
ejpam-6275	188	22	construct	construct	VERB
ejpam-6275	188	23	a	a	DET
ejpam-6275	188	24	shortened	shorten	VERB
ejpam-6275	188	25	bch	bch	PROPN
ejpam-6275	188	26	code	code	NOUN
ejpam-6275	188	27	for	for	ADP
ejpam-6275	188	28	a	a	DET
ejpam-6275	188	29	degree	degree	NOUN
ejpam-6275	188	30	-	-	PUNCT
ejpam-6275	188	31	two	two	NUM
ejpam-6275	188	32	polynomial	polynomial	NOUN
ejpam-6275	188	33	by	by	ADP
ejpam-6275	188	34	taking	take	VERB
ejpam-6275	188	35	n	n	NOUN
ejpam-6275	188	36	=	=	SYM
ejpam-6275	188	37	3	3	NUM
ejpam-6275	188	38	over	over	ADP
ejpam-6275	188	39	the	the	DET
ejpam-6275	188	40	eisenstein	eisenstein	PROPN
ejpam-6275	188	41	field	field	PROPN
ejpam-6275	188	42	z2[ω	z2[ω	NOUN
ejpam-6275	188	43	]	]	X
ejpam-6275	189	1	2	2	X
ejpam-6275	189	2	.	.	PUNCT
ejpam-6275	189	3	let	let	VERB
ejpam-6275	189	4	us	we	PRON
ejpam-6275	189	5	take	take	VERB
ejpam-6275	189	6	a	a	DET
ejpam-6275	189	7	degree	degree	NOUN
ejpam-6275	189	8	-	-	PUNCT
ejpam-6275	189	9	two	two	NUM
ejpam-6275	189	10	polynomial	polynomial	ADJ
ejpam-6275	189	11	f(x	f(x	PROPN
ejpam-6275	189	12	)	)	PUNCT
ejpam-6275	190	1	=	=	SYM
ejpam-6275	191	1	x2	x2	PROPN
ejpam-6275	192	1	+	+	CCONJ
ejpam-6275	192	2	ωx+	ωx+	PROPN
ejpam-6275	192	3	ω	ω	NOUN
ejpam-6275	192	4	,	,	PUNCT
ejpam-6275	192	5	which	which	PRON
ejpam-6275	192	6	is	be	AUX
ejpam-6275	192	7	primitive	primitive	ADJ
ejpam-6275	192	8	and	and	CCONJ
ejpam-6275	192	9	irreducible	irreducible	ADJ
ejpam-6275	192	10	over	over	ADP
ejpam-6275	192	11	z2[ω	z2[ω	NOUN
ejpam-6275	192	12	]	]	PUNCT
ejpam-6275	192	13	.	.	PUNCT
ejpam-6275	193	1	as	as	SCONJ
ejpam-6275	193	2	proved	prove	VERB
ejpam-6275	193	3	in	in	ADP
ejpam-6275	193	4	section	section	NOUN
ejpam-6275	193	5	2	2	NUM
ejpam-6275	193	6	,	,	PUNCT
ejpam-6275	193	7	the	the	DET
ejpam-6275	193	8	cardinality	cardinality	NOUN
ejpam-6275	193	9	of	of	ADP
ejpam-6275	193	10	the	the	DET
ejpam-6275	193	11	cyclic	cyclic	ADJ
ejpam-6275	193	12	group	group	NOUN
ejpam-6275	193	13	generated	generate	VERB
ejpam-6275	193	14	by	by	ADP
ejpam-6275	193	15	f(x	f(x	PROPN
ejpam-6275	193	16	)	)	PUNCT
ejpam-6275	193	17	is	be	AUX
ejpam-6275	193	18	15	15	NUM
ejpam-6275	193	19	,	,	PUNCT
ejpam-6275	193	20	and	and	CCONJ
ejpam-6275	193	21	since	since	SCONJ
ejpam-6275	193	22	15/3	15/3	NUM
ejpam-6275	193	23	=	=	SYM
ejpam-6275	193	24	5	5	NUM
ejpam-6275	193	25	,	,	PUNCT
ejpam-6275	193	26	our	our	PRON
ejpam-6275	193	27	required	require	VERB
ejpam-6275	193	28	cyclic	cyclic	ADJ
ejpam-6275	193	29	subgroup	subgroup	NOUN
ejpam-6275	193	30	is	be	AUX
ejpam-6275	193	31	generated	generate	VERB
ejpam-6275	193	32	by	by	ADP
ejpam-6275	193	33	⟨β	⟨β	X
ejpam-6275	193	34	=	=	SYM
ejpam-6275	193	35	α5⟩	α5⟩	PROPN
ejpam-6275	193	36	,	,	PUNCT
ejpam-6275	193	37	where	where	SCONJ
ejpam-6275	193	38	β	β	PROPN
ejpam-6275	193	39	is	be	AUX
ejpam-6275	193	40	the	the	DET
ejpam-6275	193	41	root	root	NOUN
ejpam-6275	193	42	of	of	ADP
ejpam-6275	193	43	f(x	f(x	PROPN
ejpam-6275	193	44	)	)	PUNCT
ejpam-6275	193	45	.	.	PUNCT
ejpam-6275	194	1	the	the	DET
ejpam-6275	194	2	required	require	VERB
ejpam-6275	194	3	cyclic	cyclic	ADJ
ejpam-6275	194	4	subgroup	subgroup	NOUN
ejpam-6275	194	5	is	be	AUX
ejpam-6275	194	6	g∗	g∗	NOUN
ejpam-6275	194	7	=	=	SYM
ejpam-6275	194	8	{	{	PUNCT
ejpam-6275	194	9	α5	α5	NOUN
ejpam-6275	194	10	,	,	PUNCT
ejpam-6275	194	11	α10	α10	NOUN
ejpam-6275	194	12	,	,	PUNCT
ejpam-6275	194	13	α15	α15	NOUN
ejpam-6275	194	14	=	=	SYM
ejpam-6275	194	15	1	1	NUM
ejpam-6275	194	16	}	}	PUNCT
ejpam-6275	194	17	=	=	SYM
ejpam-6275	194	18	{	{	PUNCT
ejpam-6275	194	19	ω	ω	NOUN
ejpam-6275	194	20	,	,	PUNCT
ejpam-6275	194	21	1	1	NUM
ejpam-6275	194	22	+	+	NUM
ejpam-6275	194	23	ω	ω	NUM
ejpam-6275	194	24	,	,	PUNCT
ejpam-6275	194	25	1	1	NUM
ejpam-6275	194	26	}	}	PUNCT
ejpam-6275	194	27	.	.	PUNCT
ejpam-6275	195	1	thus	thus	ADV
ejpam-6275	195	2	,	,	PUNCT
ejpam-6275	195	3	there	there	PRON
ejpam-6275	195	4	is	be	VERB
ejpam-6275	195	5	only	only	ADV
ejpam-6275	195	6	one	one	NUM
ejpam-6275	195	7	case	case	NOUN
ejpam-6275	195	8	for	for	ADP
ejpam-6275	195	9	the	the	DET
ejpam-6275	195	10	construction	construction	NOUN
ejpam-6275	195	11	of	of	ADP
ejpam-6275	195	12	shortened	shorten	VERB
ejpam-6275	195	13	bch	bch	PROPN
ejpam-6275	195	14	codes	code	NOUN
ejpam-6275	195	15	over	over	ADP
ejpam-6275	195	16	z2[ω	z2[ω	NOUN
ejpam-6275	195	17	]	]	X
ejpam-6275	195	18	2	2	NUM
ejpam-6275	195	19	with	with	ADP
ejpam-6275	195	20	parameters	parameter	NOUN
ejpam-6275	195	21	n	n	X
ejpam-6275	195	22	=	=	SYM
ejpam-6275	195	23	3	3	NUM
ejpam-6275	195	24	and	and	CCONJ
ejpam-6275	195	25	d	d	NOUN
ejpam-6275	195	26	=	=	SYM
ejpam-6275	195	27	3	3	NUM
ejpam-6275	195	28	.	.	NOUN
ejpam-6275	195	29	•	•	NOUN
ejpam-6275	195	30	for	for	ADP
ejpam-6275	195	31	i	i	PRON
ejpam-6275	195	32	=	=	NOUN
ejpam-6275	195	33	1	1	NUM
ejpam-6275	195	34	,	,	PUNCT
ejpam-6275	195	35	let	let	VERB
ejpam-6275	195	36	β	β	PRON
ejpam-6275	195	37	∈	∈	PROPN
ejpam-6275	195	38	z2[ω	z2[ω	NOUN
ejpam-6275	195	39	]	]	X
ejpam-6275	196	1	2	2	X
ejpam-6275	196	2	.	.	PUNCT
ejpam-6275	196	3	then	then	ADV
ejpam-6275	196	4	,	,	PUNCT
ejpam-6275	196	5	by	by	ADP
ejpam-6275	196	6	theorem	theorem	NOUN
ejpam-6275	196	7	3.1	3.1	NUM
ejpam-6275	196	8	,	,	PUNCT
ejpam-6275	196	9	β	β	PROPN
ejpam-6275	196	10	has	have	VERB
ejpam-6275	196	11	a	a	DET
ejpam-6275	196	12	minimal	minimal	ADJ
ejpam-6275	196	13	polynomial	polynomial	ADJ
ejpam-6275	196	14	φ1(x	φ1(x	NOUN
ejpam-6275	196	15	)	)	PUNCT
ejpam-6275	196	16	=	=	PUNCT
ejpam-6275	196	17	(	(	PUNCT
ejpam-6275	196	18	x−	x−	PROPN
ejpam-6275	196	19	β	β	X
ejpam-6275	196	20	)	)	PUNCT
ejpam-6275	197	1	=	=	SYM
ejpam-6275	197	2	x+	x+	PROPN
ejpam-6275	198	1	ω	ω	X
ejpam-6275	198	2	.	.	PROPN
ejpam-6275	198	3	•	•	NUM
ejpam-6275	198	4	for	for	ADP
ejpam-6275	198	5	i	i	PRON
ejpam-6275	198	6	=	=	SYM
ejpam-6275	198	7	2	2	NUM
ejpam-6275	198	8	,	,	PUNCT
ejpam-6275	198	9	let	let	VERB
ejpam-6275	198	10	β2	β2	PROPN
ejpam-6275	198	11	∈	∈	PROPN
ejpam-6275	198	12	z2[ω	z2[ω	NOUN
ejpam-6275	198	13	]	]	X
ejpam-6275	199	1	2	2	X
ejpam-6275	199	2	.	.	PUNCT
ejpam-6275	199	3	then	then	ADV
ejpam-6275	199	4	,	,	PUNCT
ejpam-6275	199	5	by	by	ADP
ejpam-6275	199	6	theorem	theorem	NOUN
ejpam-6275	199	7	3.1	3.1	NUM
ejpam-6275	199	8	,	,	PUNCT
ejpam-6275	199	9	β2	β2	PROPN
ejpam-6275	199	10	has	have	VERB
ejpam-6275	199	11	a	a	DET
ejpam-6275	199	12	minimal	minimal	ADJ
ejpam-6275	199	13	polynomial	polynomial	NOUN
ejpam-6275	199	14	by	by	ADP
ejpam-6275	199	15	itself	itself	PRON
ejpam-6275	199	16	:	:	PUNCT
ejpam-6275	199	17	φ2(x	φ2(x	X
ejpam-6275	199	18	)	)	PUNCT
ejpam-6275	199	19	=	=	SYM
ejpam-6275	199	20	(	(	PUNCT
ejpam-6275	199	21	x−	x−	PROPN
ejpam-6275	199	22	β2	β2	PROPN
ejpam-6275	199	23	)	)	PUNCT
ejpam-6275	200	1	=	=	PUNCT
ejpam-6275	200	2	x+	x+	PUNCT
ejpam-6275	200	3	(	(	PUNCT
ejpam-6275	200	4	1	1	NUM
ejpam-6275	200	5	+	+	NUM
ejpam-6275	200	6	ω	ω	NUM
ejpam-6275	200	7	)	)	PUNCT
ejpam-6275	200	8	.	.	PUNCT
ejpam-6275	201	1	now	now	ADV
ejpam-6275	201	2	,	,	PUNCT
ejpam-6275	201	3	the	the	DET
ejpam-6275	201	4	generator	generator	NOUN
ejpam-6275	201	5	polynomial	polynomial	NOUN
ejpam-6275	201	6	is	be	AUX
ejpam-6275	201	7	g(x	g(x	NOUN
ejpam-6275	201	8	)	)	PUNCT
ejpam-6275	202	1	=	=	SYM
ejpam-6275	202	2	φ1(x	φ1(x	NOUN
ejpam-6275	202	3	)	)	PUNCT
ejpam-6275	202	4	·	·	PUNCT
ejpam-6275	203	1	φ2(x	φ2(x	X
ejpam-6275	203	2	)	)	PUNCT
ejpam-6275	203	3	=	=	SYM
ejpam-6275	203	4	(	(	PUNCT
ejpam-6275	203	5	x+	x+	ADJ
ejpam-6275	203	6	ω)(x+	ω)(x+	X
ejpam-6275	203	7	(	(	PUNCT
ejpam-6275	203	8	1	1	NUM
ejpam-6275	203	9	+	+	NUM
ejpam-6275	203	10	ω	ω	NUM
ejpam-6275	203	11	)	)	PUNCT
ejpam-6275	203	12	)	)	PUNCT
ejpam-6275	204	1	=	=	SYM
ejpam-6275	205	1	x2	x2	PROPN
ejpam-6275	206	1	+	+	CCONJ
ejpam-6275	206	2	x+	x+	ADJ
ejpam-6275	206	3	1	1	X
ejpam-6275	206	4	.	.	PUNCT
ejpam-6275	206	5	since	since	SCONJ
ejpam-6275	206	6	k	k	PROPN
ejpam-6275	206	7	=	=	SYM
ejpam-6275	206	8	3−2	3−2	NUM
ejpam-6275	206	9	=	=	SYM
ejpam-6275	206	10	1	1	NUM
ejpam-6275	206	11	,	,	PUNCT
ejpam-6275	206	12	the	the	DET
ejpam-6275	206	13	parameters	parameter	NOUN
ejpam-6275	206	14	of	of	ADP
ejpam-6275	206	15	the	the	DET
ejpam-6275	206	16	shortened	shorten	VERB
ejpam-6275	206	17	bch	bch	PROPN
ejpam-6275	206	18	code	code	NOUN
ejpam-6275	206	19	are	be	AUX
ejpam-6275	206	20	(	(	PUNCT
ejpam-6275	206	21	n	n	X
ejpam-6275	206	22	,	,	PUNCT
ejpam-6275	206	23	k	k	NOUN
ejpam-6275	206	24	,	,	PUNCT
ejpam-6275	206	25	d	d	NOUN
ejpam-6275	206	26	)	)	PUNCT
ejpam-6275	206	27	=	=	SYM
ejpam-6275	206	28	(	(	PUNCT
ejpam-6275	206	29	3	3	NUM
ejpam-6275	206	30	,	,	PUNCT
ejpam-6275	206	31	1	1	NUM
ejpam-6275	206	32	,	,	PUNCT
ejpam-6275	206	33	3	3	NUM
ejpam-6275	206	34	)	)	PUNCT
ejpam-6275	206	35	.	.	PUNCT
ejpam-6275	207	1	illustration	illustration	NOUN
ejpam-6275	207	2	3.2	3.2	NUM
ejpam-6275	207	3	:	:	PUNCT
ejpam-6275	207	4	construct	construct	VERB
ejpam-6275	207	5	a	a	DET
ejpam-6275	207	6	shortened	shorten	VERB
ejpam-6275	207	7	bch	bch	PROPN
ejpam-6275	207	8	code	code	NOUN
ejpam-6275	207	9	for	for	ADP
ejpam-6275	207	10	a	a	DET
ejpam-6275	207	11	degree	degree	NOUN
ejpam-6275	207	12	-	-	PUNCT
ejpam-6275	207	13	two	two	NUM
ejpam-6275	207	14	polynomial	polynomial	NOUN
ejpam-6275	207	15	by	by	ADP
ejpam-6275	207	16	taking	take	VERB
ejpam-6275	207	17	n	n	NOUN
ejpam-6275	207	18	=	=	SYM
ejpam-6275	207	19	5	5	NUM
ejpam-6275	207	20	over	over	ADP
ejpam-6275	207	21	the	the	DET
ejpam-6275	207	22	eisenstein	eisenstein	PROPN
ejpam-6275	207	23	field	field	PROPN
ejpam-6275	207	24	z2[ω	z2[ω	NOUN
ejpam-6275	207	25	]	]	X
ejpam-6275	208	1	2	2	X
ejpam-6275	208	2	.	.	PUNCT
ejpam-6275	208	3	let	let	VERB
ejpam-6275	208	4	us	we	PRON
ejpam-6275	208	5	take	take	VERB
ejpam-6275	208	6	a	a	DET
ejpam-6275	208	7	degree	degree	NOUN
ejpam-6275	208	8	-	-	PUNCT
ejpam-6275	208	9	two	two	NUM
ejpam-6275	208	10	polynomial	polynomial	ADJ
ejpam-6275	208	11	f(x	f(x	PROPN
ejpam-6275	208	12	)	)	PUNCT
ejpam-6275	209	1	=	=	SYM
ejpam-6275	210	1	x2	x2	PROPN
ejpam-6275	211	1	+	+	CCONJ
ejpam-6275	211	2	ωx+	ωx+	PROPN
ejpam-6275	211	3	ω	ω	NOUN
ejpam-6275	211	4	,	,	PUNCT
ejpam-6275	211	5	which	which	PRON
ejpam-6275	211	6	is	be	AUX
ejpam-6275	211	7	primitive	primitive	ADJ
ejpam-6275	211	8	and	and	CCONJ
ejpam-6275	211	9	irreducible	irreducible	ADJ
ejpam-6275	211	10	over	over	ADP
ejpam-6275	211	11	z2[ω	z2[ω	NOUN
ejpam-6275	211	12	]	]	PUNCT
ejpam-6275	211	13	.	.	PUNCT
ejpam-6275	212	1	as	as	SCONJ
ejpam-6275	212	2	proved	prove	VERB
ejpam-6275	212	3	in	in	ADP
ejpam-6275	212	4	section	section	NOUN
ejpam-6275	212	5	2	2	NUM
ejpam-6275	212	6	,	,	PUNCT
ejpam-6275	212	7	the	the	DET
ejpam-6275	212	8	cardinality	cardinality	NOUN
ejpam-6275	212	9	of	of	ADP
ejpam-6275	212	10	the	the	DET
ejpam-6275	212	11	cyclic	cyclic	ADJ
ejpam-6275	212	12	group	group	NOUN
ejpam-6275	212	13	generated	generate	VERB
ejpam-6275	212	14	by	by	ADP
ejpam-6275	212	15	f(x	f(x	PROPN
ejpam-6275	212	16	)	)	PUNCT
ejpam-6275	212	17	is	be	AUX
ejpam-6275	212	18	15	15	NUM
ejpam-6275	212	19	and	and	CCONJ
ejpam-6275	212	20	15/5	15/5	NUM
ejpam-6275	212	21	=	=	SYM
ejpam-6275	212	22	3	3	NUM
ejpam-6275	212	23	.	.	PUNCT
ejpam-6275	213	1	so	so	ADV
ejpam-6275	213	2	,	,	PUNCT
ejpam-6275	213	3	our	our	PRON
ejpam-6275	213	4	required	require	VERB
ejpam-6275	213	5	cyclic	cyclic	ADJ
ejpam-6275	213	6	subgroup	subgroup	NOUN
ejpam-6275	213	7	is	be	AUX
ejpam-6275	213	8	generated	generate	VERB
ejpam-6275	213	9	by	by	ADP
ejpam-6275	213	10	⟨β	⟨β	X
ejpam-6275	213	11	=	=	SYM
ejpam-6275	213	12	α3⟩	α3⟩	PROPN
ejpam-6275	213	13	,	,	PUNCT
ejpam-6275	213	14	where	where	SCONJ
ejpam-6275	213	15	β	β	PROPN
ejpam-6275	213	16	is	be	AUX
ejpam-6275	213	17	the	the	DET
ejpam-6275	213	18	root	root	NOUN
ejpam-6275	213	19	of	of	ADP
ejpam-6275	213	20	f(x	f(x	PROPN
ejpam-6275	213	21	)	)	PUNCT
ejpam-6275	213	22	.	.	PUNCT
ejpam-6275	214	1	the	the	DET
ejpam-6275	214	2	required	require	VERB
ejpam-6275	214	3	cyclic	cyclic	ADJ
ejpam-6275	214	4	subgroup	subgroup	NOUN
ejpam-6275	214	5	is	be	AUX
ejpam-6275	214	6	g∗	g∗	NOUN
ejpam-6275	214	7	=	=	SYM
ejpam-6275	214	8	{	{	PUNCT
ejpam-6275	214	9	α3	α3	NOUN
ejpam-6275	214	10	,	,	PUNCT
ejpam-6275	214	11	α6	α6	NOUN
ejpam-6275	214	12	,	,	PUNCT
ejpam-6275	214	13	α9	α9	PROPN
ejpam-6275	214	14	,	,	PUNCT
ejpam-6275	214	15	α12	α12	PROPN
ejpam-6275	214	16	,	,	PUNCT
ejpam-6275	214	17	α15	α15	NOUN
ejpam-6275	214	18	=	=	SYM
ejpam-6275	214	19	1	1	NUM
ejpam-6275	214	20	}	}	PUNCT
ejpam-6275	214	21	=	=	PRON
ejpam-6275	214	22	{	{	PUNCT
ejpam-6275	214	23	1	1	NUM
ejpam-6275	214	24	+	+	NUM
ejpam-6275	214	25	ω	ω	NUM
ejpam-6275	214	26	+	+	CCONJ
ejpam-6275	214	27	α	α	NOUN
ejpam-6275	214	28	,	,	PUNCT
ejpam-6275	214	29	αω	αω	ADP
ejpam-6275	214	30	,	,	PUNCT
ejpam-6275	214	31	1	1	NUM
ejpam-6275	214	32	+	+	SYM
ejpam-6275	214	33	ω	ω	NUM
ejpam-6275	214	34	+	+	CCONJ
ejpam-6275	214	35	αω	αω	NUM
ejpam-6275	214	36	,	,	PUNCT
ejpam-6275	214	37	α+	α+	X
ejpam-6275	214	38	1	1	NUM
ejpam-6275	214	39	,	,	PUNCT
ejpam-6275	214	40	1	1	NUM
ejpam-6275	214	41	}	}	PUNCT
ejpam-6275	214	42	.	.	PUNCT
ejpam-6275	215	1	thus	thus	ADV
ejpam-6275	215	2	,	,	PUNCT
ejpam-6275	215	3	there	there	PRON
ejpam-6275	215	4	are	be	VERB
ejpam-6275	215	5	two	two	NUM
ejpam-6275	215	6	cases	case	NOUN
ejpam-6275	215	7	for	for	ADP
ejpam-6275	215	8	the	the	DET
ejpam-6275	215	9	construction	construction	NOUN
ejpam-6275	215	10	of	of	ADP
ejpam-6275	215	11	shortened	shorten	VERB
ejpam-6275	215	12	bch	bch	PROPN
ejpam-6275	215	13	codes	code	NOUN
ejpam-6275	215	14	over	over	ADP
ejpam-6275	215	15	z2[ω	z2[ω	NOUN
ejpam-6275	215	16	]	]	X
ejpam-6275	215	17	2	2	X
ejpam-6275	215	18	.	.	X
ejpam-6275	215	19	case	case	NOUN
ejpam-6275	215	20	1	1	NUM
ejpam-6275	215	21	:	:	PUNCT
ejpam-6275	215	22	for	for	ADP
ejpam-6275	215	23	n	n	NOUN
ejpam-6275	215	24	=	=	SYM
ejpam-6275	215	25	5	5	NUM
ejpam-6275	215	26	and	and	CCONJ
ejpam-6275	215	27	d	d	NOUN
ejpam-6275	215	28	=	=	SYM
ejpam-6275	215	29	3	3	NUM
ejpam-6275	215	30	,	,	PUNCT
ejpam-6275	215	31	i	i	PRON
ejpam-6275	215	32	=	=	NOUN
ejpam-6275	215	33	1	1	NUM
ejpam-6275	215	34	,	,	PUNCT
ejpam-6275	215	35	2	2	NUM
ejpam-6275	215	36	.	.	PUNCT
ejpam-6275	216	1	m.	m.	PROPN
ejpam-6275	216	2	sajjad	sajjad	PROPN
ejpam-6275	216	3	et	et	PROPN
ejpam-6275	216	4	al	al	PROPN
ejpam-6275	216	5	.	.	PUNCT
ejpam-6275	216	6	/	/	SYM
ejpam-6275	216	7	eur	eur	PROPN
ejpam-6275	216	8	.	.	PUNCT
ejpam-6275	217	1	j.	j.	PROPN
ejpam-6275	217	2	pure	pure	PROPN
ejpam-6275	217	3	appl	appl	PROPN
ejpam-6275	217	4	.	.	PROPN
ejpam-6275	217	5	math	math	PROPN
ejpam-6275	217	6	,	,	PUNCT
ejpam-6275	217	7	18	18	NUM
ejpam-6275	217	8	(	(	PUNCT
ejpam-6275	217	9	3	3	NUM
ejpam-6275	217	10	)	)	PUNCT
ejpam-6275	217	11	(	(	PUNCT
ejpam-6275	217	12	2025	2025	NUM
ejpam-6275	217	13	)	)	PUNCT
ejpam-6275	217	14	,	,	PUNCT
ejpam-6275	217	15	6275	6275	NUM
ejpam-6275	217	16	11	11	NUM
ejpam-6275	217	17	of	of	ADP
ejpam-6275	217	18	36	36	NUM
ejpam-6275	217	19	•	•	NOUN
ejpam-6275	217	20	for	for	ADP
ejpam-6275	217	21	i	i	PRON
ejpam-6275	217	22	=	=	NOUN
ejpam-6275	217	23	1	1	NUM
ejpam-6275	217	24	,	,	PUNCT
ejpam-6275	217	25	let	let	VERB
ejpam-6275	217	26	β	β	PRON
ejpam-6275	217	27	∈	∈	PROPN
ejpam-6275	217	28	z2[ω	z2[ω	NOUN
ejpam-6275	217	29	]	]	X
ejpam-6275	217	30	2	2	NUM
ejpam-6275	217	31	.	.	PUNCT
ejpam-6275	217	32	by	by	ADP
ejpam-6275	217	33	theorem	theorem	NOUN
ejpam-6275	217	34	3.1	3.1	NUM
ejpam-6275	217	35	,	,	PUNCT
ejpam-6275	217	36	β	β	NOUN
ejpam-6275	217	37	and	and	CCONJ
ejpam-6275	217	38	β4	β4	PROPN
ejpam-6275	217	39	have	have	VERB
ejpam-6275	217	40	the	the	DET
ejpam-6275	217	41	same	same	ADJ
ejpam-6275	217	42	minimal	minimal	ADJ
ejpam-6275	217	43	polynomial	polynomial	ADJ
ejpam-6275	217	44	:	:	PUNCT
ejpam-6275	217	45	φ1(x	φ1(x	NOUN
ejpam-6275	217	46	)	)	PUNCT
ejpam-6275	217	47	=	=	SYM
ejpam-6275	217	48	(	(	PUNCT
ejpam-6275	217	49	x−	x−	PROPN
ejpam-6275	217	50	β)(x−	β)(x−	PROPN
ejpam-6275	217	51	β4	β4	PROPN
ejpam-6275	217	52	)	)	PUNCT
ejpam-6275	217	53	=	=	SYM
ejpam-6275	218	1	x2	x2	NUM
ejpam-6275	218	2	−	−	PROPN
ejpam-6275	219	1	(	(	PUNCT
ejpam-6275	219	2	β	β	X
ejpam-6275	219	3	+	+	NUM
ejpam-6275	219	4	β4)x+	β4)x+	NOUN
ejpam-6275	219	5	β5	β5	NOUN
ejpam-6275	219	6	=	=	SYM
ejpam-6275	219	7	x2	x2	PROPN
ejpam-6275	220	1	+	+	CCONJ
ejpam-6275	220	2	ωx+	ωx+	NOUN
ejpam-6275	220	3	1	1	NUM
ejpam-6275	220	4	.	.	NOUN
ejpam-6275	220	5	•	•	NOUN
ejpam-6275	220	6	for	for	ADP
ejpam-6275	220	7	i	i	PRON
ejpam-6275	220	8	=	=	SYM
ejpam-6275	220	9	2	2	NUM
ejpam-6275	220	10	,	,	PUNCT
ejpam-6275	220	11	let	let	VERB
ejpam-6275	220	12	β2	β2	PROPN
ejpam-6275	220	13	∈	∈	PROPN
ejpam-6275	220	14	z2[ω	z2[ω	NOUN
ejpam-6275	220	15	]	]	X
ejpam-6275	220	16	2	2	NUM
ejpam-6275	220	17	.	.	PUNCT
ejpam-6275	220	18	by	by	ADP
ejpam-6275	220	19	theorem	theorem	ADJ
ejpam-6275	220	20	3.1	3.1	NUM
ejpam-6275	220	21	,	,	PUNCT
ejpam-6275	220	22	β2	β2	NOUN
ejpam-6275	220	23	and	and	CCONJ
ejpam-6275	220	24	β3	β3	ADJ
ejpam-6275	220	25	have	have	VERB
ejpam-6275	220	26	the	the	DET
ejpam-6275	220	27	same	same	ADJ
ejpam-6275	220	28	minimal	minimal	ADJ
ejpam-6275	220	29	polynomial	polynomial	ADJ
ejpam-6275	220	30	:	:	PUNCT
ejpam-6275	220	31	φ2(x	φ2(x	NUM
ejpam-6275	220	32	)	)	PUNCT
ejpam-6275	220	33	=	=	SYM
ejpam-6275	220	34	(	(	PUNCT
ejpam-6275	220	35	x−	x−	PROPN
ejpam-6275	220	36	β2)(x−	β2)(x−	NOUN
ejpam-6275	220	37	β3	β3	PROPN
ejpam-6275	220	38	)	)	PUNCT
ejpam-6275	220	39	=	=	SYM
ejpam-6275	221	1	x2	x2	NUM
ejpam-6275	221	2	−	−	PROPN
ejpam-6275	222	1	(	(	PUNCT
ejpam-6275	222	2	β2	β2	NOUN
ejpam-6275	222	3	+	+	PROPN
ejpam-6275	222	4	β3)x+	β3)x+	PROPN
ejpam-6275	222	5	β5	β5	NOUN
ejpam-6275	222	6	=	=	SYM
ejpam-6275	222	7	x2	x2	PROPN
ejpam-6275	223	1	+	+	CCONJ
ejpam-6275	223	2	(	(	PUNCT
ejpam-6275	223	3	1	1	NUM
ejpam-6275	223	4	+	+	SYM
ejpam-6275	223	5	ω)x+	ω)x+	NUM
ejpam-6275	223	6	1	1	NUM
ejpam-6275	223	7	.	.	PUNCT
ejpam-6275	224	1	the	the	DET
ejpam-6275	224	2	generator	generator	NOUN
ejpam-6275	224	3	polynomial	polynomial	NOUN
ejpam-6275	224	4	is	be	AUX
ejpam-6275	224	5	g(x	g(x	NOUN
ejpam-6275	224	6	)	)	PUNCT
ejpam-6275	225	1	=	=	SYM
ejpam-6275	225	2	φ1(x	φ1(x	NOUN
ejpam-6275	225	3	)	)	PUNCT
ejpam-6275	225	4	·	·	PUNCT
ejpam-6275	226	1	φ2(x	φ2(x	X
ejpam-6275	226	2	)	)	PUNCT
ejpam-6275	226	3	=	=	SYM
ejpam-6275	227	1	(	(	PUNCT
ejpam-6275	227	2	x2	x2	NOUN
ejpam-6275	227	3	+	+	CCONJ
ejpam-6275	227	4	ωx+	ωx+	NOUN
ejpam-6275	227	5	1)(x2	1)(x2	NOUN
ejpam-6275	228	1	+	+	CCONJ
ejpam-6275	229	1	(	(	PUNCT
ejpam-6275	229	2	1	1	NUM
ejpam-6275	229	3	+	+	SYM
ejpam-6275	229	4	ω)x+	ω)x+	NUM
ejpam-6275	229	5	1	1	NUM
ejpam-6275	229	6	)	)	PUNCT
ejpam-6275	229	7	=	=	SYM
ejpam-6275	229	8	x4	x4	PROPN
ejpam-6275	230	1	+	+	CCONJ
ejpam-6275	230	2	x3	x3	ADJ
ejpam-6275	230	3	+	+	CCONJ
ejpam-6275	230	4	x2	x2	PROPN
ejpam-6275	231	1	+	+	CCONJ
ejpam-6275	231	2	x+	x+	ADJ
ejpam-6275	231	3	1	1	X
ejpam-6275	231	4	.	.	PUNCT
ejpam-6275	232	1	since	since	SCONJ
ejpam-6275	232	2	k	k	PROPN
ejpam-6275	232	3	=	=	SYM
ejpam-6275	232	4	5−4	5−4	NUM
ejpam-6275	232	5	=	=	SYM
ejpam-6275	232	6	1	1	NUM
ejpam-6275	232	7	,	,	PUNCT
ejpam-6275	232	8	the	the	DET
ejpam-6275	232	9	parameters	parameter	NOUN
ejpam-6275	232	10	of	of	ADP
ejpam-6275	232	11	the	the	DET
ejpam-6275	232	12	shortened	shorten	VERB
ejpam-6275	232	13	bch	bch	PROPN
ejpam-6275	232	14	code	code	NOUN
ejpam-6275	232	15	are	be	AUX
ejpam-6275	232	16	(	(	PUNCT
ejpam-6275	232	17	n	n	X
ejpam-6275	232	18	,	,	PUNCT
ejpam-6275	232	19	k	k	NOUN
ejpam-6275	232	20	,	,	PUNCT
ejpam-6275	232	21	d	d	NOUN
ejpam-6275	232	22	)	)	PUNCT
ejpam-6275	232	23	=	=	SYM
ejpam-6275	232	24	(	(	PUNCT
ejpam-6275	232	25	5	5	NUM
ejpam-6275	232	26	,	,	PUNCT
ejpam-6275	232	27	1	1	NUM
ejpam-6275	232	28	,	,	PUNCT
ejpam-6275	232	29	3	3	NUM
ejpam-6275	232	30	)	)	PUNCT
ejpam-6275	232	31	.	.	PUNCT
ejpam-6275	233	1	case	case	NOUN
ejpam-6275	233	2	2	2	NUM
ejpam-6275	233	3	:	:	PUNCT
ejpam-6275	233	4	for	for	ADP
ejpam-6275	233	5	n	n	NOUN
ejpam-6275	233	6	=	=	SYM
ejpam-6275	233	7	5	5	NUM
ejpam-6275	233	8	and	and	CCONJ
ejpam-6275	233	9	d	d	NOUN
ejpam-6275	233	10	=	=	SYM
ejpam-6275	233	11	5	5	NUM
ejpam-6275	233	12	,	,	PUNCT
ejpam-6275	233	13	i	i	PRON
ejpam-6275	233	14	=	=	NOUN
ejpam-6275	233	15	1	1	NUM
ejpam-6275	233	16	,	,	PUNCT
ejpam-6275	233	17	2	2	NUM
ejpam-6275	233	18	,	,	PUNCT
ejpam-6275	233	19	3	3	NUM
ejpam-6275	233	20	,	,	PUNCT
ejpam-6275	233	21	4	4	NUM
ejpam-6275	233	22	.	.	NOUN
ejpam-6275	233	23	•	•	NOUN
ejpam-6275	233	24	for	for	ADP
ejpam-6275	233	25	i	i	PRON
ejpam-6275	233	26	=	=	NOUN
ejpam-6275	233	27	1	1	NUM
ejpam-6275	233	28	,	,	PUNCT
ejpam-6275	233	29	as	as	ADP
ejpam-6275	233	30	before	before	ADV
ejpam-6275	233	31	,	,	PUNCT
ejpam-6275	233	32	φ1(x	φ1(x	NOUN
ejpam-6275	233	33	)	)	PUNCT
ejpam-6275	233	34	=	=	SYM
ejpam-6275	234	1	x2	x2	NOUN
ejpam-6275	235	1	+	+	CCONJ
ejpam-6275	235	2	ωx+	ωx+	NOUN
ejpam-6275	235	3	1	1	NUM
ejpam-6275	235	4	.	.	NOUN
ejpam-6275	235	5	•	•	NOUN
ejpam-6275	235	6	for	for	ADP
ejpam-6275	235	7	i	i	PRON
ejpam-6275	235	8	=	=	SYM
ejpam-6275	235	9	2	2	NUM
ejpam-6275	235	10	,	,	PUNCT
ejpam-6275	235	11	φ2(x	φ2(x	NOUN
ejpam-6275	235	12	)	)	PUNCT
ejpam-6275	235	13	=	=	SYM
ejpam-6275	236	1	x2	x2	PROPN
ejpam-6275	237	1	+	+	CCONJ
ejpam-6275	237	2	(	(	PUNCT
ejpam-6275	237	3	1	1	NUM
ejpam-6275	237	4	+	+	SYM
ejpam-6275	237	5	ω)x+	ω)x+	NUM
ejpam-6275	237	6	1	1	NUM
ejpam-6275	237	7	.	.	NOUN
ejpam-6275	237	8	•	•	NOUN
ejpam-6275	237	9	for	for	ADP
ejpam-6275	237	10	i	i	PRON
ejpam-6275	237	11	=	=	SYM
ejpam-6275	237	12	3	3	NUM
ejpam-6275	237	13	,	,	PUNCT
ejpam-6275	237	14	by	by	ADP
ejpam-6275	237	15	theorem	theorem	ADJ
ejpam-6275	237	16	3.1	3.1	NUM
ejpam-6275	237	17	,	,	PUNCT
ejpam-6275	237	18	β3	β3	VERB
ejpam-6275	237	19	and	and	CCONJ
ejpam-6275	237	20	β2	β2	NOUN
ejpam-6275	237	21	have	have	VERB
ejpam-6275	237	22	the	the	DET
ejpam-6275	237	23	same	same	ADJ
ejpam-6275	237	24	minimal	minimal	ADJ
ejpam-6275	237	25	polynomial	polynomial	ADJ
ejpam-6275	237	26	,	,	PUNCT
ejpam-6275	237	27	so	so	ADV
ejpam-6275	237	28	φ3(x	φ3(x	NOUN
ejpam-6275	237	29	)	)	PUNCT
ejpam-6275	237	30	=	=	SYM
ejpam-6275	237	31	φ2(x	φ2(x	NUM
ejpam-6275	237	32	)	)	PUNCT
ejpam-6275	237	33	=	=	SYM
ejpam-6275	238	1	x2	x2	PROPN
ejpam-6275	239	1	+	+	CCONJ
ejpam-6275	239	2	(	(	PUNCT
ejpam-6275	239	3	1	1	NUM
ejpam-6275	239	4	+	+	SYM
ejpam-6275	239	5	ω)x+	ω)x+	NUM
ejpam-6275	239	6	1	1	NUM
ejpam-6275	239	7	.	.	NOUN
ejpam-6275	239	8	•	•	NOUN
ejpam-6275	239	9	for	for	ADP
ejpam-6275	239	10	i	i	PRON
ejpam-6275	239	11	=	=	NOUN
ejpam-6275	239	12	4	4	NUM
ejpam-6275	239	13	,	,	PUNCT
ejpam-6275	239	14	φ4(x	φ4(x	NOUN
ejpam-6275	239	15	)	)	PUNCT
ejpam-6275	239	16	=	=	SYM
ejpam-6275	239	17	φ1(x	φ1(x	NOUN
ejpam-6275	239	18	)	)	PUNCT
ejpam-6275	239	19	=	=	SYM
ejpam-6275	240	1	x2	x2	NOUN
ejpam-6275	241	1	+	+	CCONJ
ejpam-6275	241	2	ωx+	ωx+	NOUN
ejpam-6275	241	3	1	1	NUM
ejpam-6275	241	4	.	.	PUNCT
ejpam-6275	242	1	the	the	DET
ejpam-6275	242	2	generator	generator	NOUN
ejpam-6275	242	3	polynomial	polynomial	NOUN
ejpam-6275	242	4	is	be	AUX
ejpam-6275	242	5	again	again	ADV
ejpam-6275	242	6	g(x	g(x	NOUN
ejpam-6275	242	7	)	)	PUNCT
ejpam-6275	243	1	=	=	PUNCT
ejpam-6275	243	2	φ1(x	φ1(x	NOUN
ejpam-6275	243	3	)	)	PUNCT
ejpam-6275	243	4	·	·	PUNCT
ejpam-6275	244	1	φ2(x	φ2(x	X
ejpam-6275	244	2	)	)	PUNCT
ejpam-6275	244	3	=	=	SYM
ejpam-6275	245	1	(	(	PUNCT
ejpam-6275	245	2	x2	x2	NOUN
ejpam-6275	245	3	+	+	CCONJ
ejpam-6275	245	4	ωx+	ωx+	NOUN
ejpam-6275	245	5	1)(x2	1)(x2	NOUN
ejpam-6275	246	1	+	+	CCONJ
ejpam-6275	247	1	(	(	PUNCT
ejpam-6275	247	2	1	1	NUM
ejpam-6275	247	3	+	+	SYM
ejpam-6275	247	4	ω)x+	ω)x+	NUM
ejpam-6275	247	5	1	1	NUM
ejpam-6275	247	6	)	)	PUNCT
ejpam-6275	247	7	=	=	SYM
ejpam-6275	247	8	x4	x4	PROPN
ejpam-6275	248	1	+	+	CCONJ
ejpam-6275	248	2	x3	x3	ADJ
ejpam-6275	248	3	+	+	CCONJ
ejpam-6275	248	4	x2	x2	PROPN
ejpam-6275	249	1	+	+	CCONJ
ejpam-6275	249	2	x+	x+	ADJ
ejpam-6275	249	3	1	1	X
ejpam-6275	249	4	.	.	PUNCT
ejpam-6275	250	1	since	since	SCONJ
ejpam-6275	250	2	k	k	PROPN
ejpam-6275	250	3	=	=	SYM
ejpam-6275	250	4	5−4	5−4	NUM
ejpam-6275	250	5	=	=	SYM
ejpam-6275	250	6	1	1	NUM
ejpam-6275	250	7	,	,	PUNCT
ejpam-6275	250	8	the	the	DET
ejpam-6275	250	9	parameters	parameter	NOUN
ejpam-6275	250	10	of	of	ADP
ejpam-6275	250	11	this	this	DET
ejpam-6275	250	12	shortened	shorten	VERB
ejpam-6275	250	13	bch	bch	PROPN
ejpam-6275	250	14	code	code	NOUN
ejpam-6275	250	15	are	be	AUX
ejpam-6275	250	16	(	(	PUNCT
ejpam-6275	250	17	n	n	X
ejpam-6275	250	18	,	,	PUNCT
ejpam-6275	250	19	k	k	NOUN
ejpam-6275	250	20	,	,	PUNCT
ejpam-6275	250	21	d	d	NOUN
ejpam-6275	250	22	)	)	PUNCT
ejpam-6275	250	23	=	=	SYM
ejpam-6275	250	24	(	(	PUNCT
ejpam-6275	250	25	5	5	NUM
ejpam-6275	250	26	,	,	PUNCT
ejpam-6275	250	27	1	1	NUM
ejpam-6275	250	28	,	,	PUNCT
ejpam-6275	250	29	5	5	NUM
ejpam-6275	250	30	)	)	PUNCT
ejpam-6275	250	31	.	.	PUNCT
ejpam-6275	251	1	illustration	illustration	NOUN
ejpam-6275	251	2	3.3	3.3	NUM
ejpam-6275	251	3	:	:	PUNCT
ejpam-6275	251	4	construct	construct	VERB
ejpam-6275	251	5	a	a	DET
ejpam-6275	251	6	shortened	shorten	VERB
ejpam-6275	251	7	bch	bch	PROPN
ejpam-6275	251	8	code	code	NOUN
ejpam-6275	251	9	for	for	ADP
ejpam-6275	251	10	a	a	DET
ejpam-6275	251	11	degree	degree	NOUN
ejpam-6275	251	12	-	-	PUNCT
ejpam-6275	251	13	three	three	NUM
ejpam-6275	251	14	polynomial	polynomial	NOUN
ejpam-6275	251	15	by	by	ADP
ejpam-6275	251	16	taking	take	VERB
ejpam-6275	251	17	n	n	NOUN
ejpam-6275	251	18	=	=	SYM
ejpam-6275	251	19	3	3	NUM
ejpam-6275	251	20	over	over	ADP
ejpam-6275	251	21	the	the	DET
ejpam-6275	251	22	eisenstein	eisenstein	PROPN
ejpam-6275	251	23	field	field	PROPN
ejpam-6275	251	24	z2[ω	z2[ω	NOUN
ejpam-6275	251	25	]	]	X
ejpam-6275	251	26	3	3	X
ejpam-6275	251	27	.	.	PUNCT
ejpam-6275	252	1	let	let	VERB
ejpam-6275	252	2	us	we	PRON
ejpam-6275	252	3	take	take	VERB
ejpam-6275	252	4	a	a	DET
ejpam-6275	252	5	degree	degree	NOUN
ejpam-6275	252	6	-	-	PUNCT
ejpam-6275	252	7	three	three	NUM
ejpam-6275	252	8	polynomial	polynomial	ADJ
ejpam-6275	252	9	f(x	f(x	PROPN
ejpam-6275	252	10	)	)	PUNCT
ejpam-6275	252	11	=	=	PUNCT
ejpam-6275	253	1	x3	x3	ADJ
ejpam-6275	254	1	+	+	CCONJ
ejpam-6275	254	2	x2	x2	PROPN
ejpam-6275	255	1	+	+	CCONJ
ejpam-6275	255	2	x+	x+	SYM
ejpam-6275	255	3	1	1	NUM
ejpam-6275	255	4	+	+	NUM
ejpam-6275	255	5	ω	ω	NUM
ejpam-6275	255	6	,	,	PUNCT
ejpam-6275	255	7	which	which	PRON
ejpam-6275	255	8	is	be	AUX
ejpam-6275	255	9	primitive	primitive	ADJ
ejpam-6275	255	10	and	and	CCONJ
ejpam-6275	255	11	irreducible	irreducible	ADJ
ejpam-6275	255	12	over	over	ADP
ejpam-6275	255	13	z2[ω	z2[ω	NOUN
ejpam-6275	255	14	]	]	PUNCT
ejpam-6275	255	15	.	.	PUNCT
ejpam-6275	256	1	as	as	SCONJ
ejpam-6275	256	2	proved	prove	VERB
ejpam-6275	256	3	in	in	ADP
ejpam-6275	256	4	section	section	NOUN
ejpam-6275	256	5	2	2	NUM
ejpam-6275	256	6	,	,	PUNCT
ejpam-6275	256	7	the	the	DET
ejpam-6275	256	8	cardinality	cardinality	NOUN
ejpam-6275	256	9	of	of	ADP
ejpam-6275	256	10	the	the	DET
ejpam-6275	256	11	cyclic	cyclic	ADJ
ejpam-6275	256	12	group	group	NOUN
ejpam-6275	256	13	generated	generate	VERB
ejpam-6275	256	14	by	by	ADP
ejpam-6275	256	15	f(x	f(x	PROPN
ejpam-6275	256	16	)	)	PUNCT
ejpam-6275	256	17	is	be	AUX
ejpam-6275	256	18	63	63	NUM
ejpam-6275	256	19	,	,	PUNCT
ejpam-6275	256	20	and	and	CCONJ
ejpam-6275	256	21	63	63	NUM
ejpam-6275	256	22	3	3	NUM
ejpam-6275	256	23	=	=	SYM
ejpam-6275	256	24	21	21	NUM
ejpam-6275	256	25	.	.	PUNCT
ejpam-6275	257	1	m.	m.	PROPN
ejpam-6275	257	2	sajjad	sajjad	PROPN
ejpam-6275	257	3	et	et	PROPN
ejpam-6275	257	4	al	al	PROPN
ejpam-6275	257	5	.	.	PUNCT
ejpam-6275	257	6	/	/	SYM
ejpam-6275	257	7	eur	eur	PROPN
ejpam-6275	257	8	.	.	PUNCT
ejpam-6275	258	1	j.	j.	PROPN
ejpam-6275	258	2	pure	pure	PROPN
ejpam-6275	258	3	appl	appl	PROPN
ejpam-6275	258	4	.	.	PROPN
ejpam-6275	258	5	math	math	PROPN
ejpam-6275	258	6	,	,	PUNCT
ejpam-6275	258	7	18	18	NUM
ejpam-6275	258	8	(	(	PUNCT
ejpam-6275	258	9	3	3	NUM
ejpam-6275	258	10	)	)	PUNCT
ejpam-6275	258	11	(	(	PUNCT
ejpam-6275	258	12	2025	2025	NUM
ejpam-6275	258	13	)	)	PUNCT
ejpam-6275	258	14	,	,	PUNCT
ejpam-6275	258	15	6275	6275	NUM
ejpam-6275	258	16	12	12	NUM
ejpam-6275	258	17	of	of	ADP
ejpam-6275	258	18	36	36	NUM
ejpam-6275	258	19	so	so	ADV
ejpam-6275	258	20	,	,	PUNCT
ejpam-6275	258	21	our	our	PRON
ejpam-6275	258	22	required	require	VERB
ejpam-6275	258	23	cyclic	cyclic	ADJ
ejpam-6275	258	24	subgroup	subgroup	NOUN
ejpam-6275	258	25	is	be	AUX
ejpam-6275	258	26	generated	generate	VERB
ejpam-6275	258	27	by	by	ADP
ejpam-6275	258	28	⟨β	⟨β	X
ejpam-6275	258	29	=	=	SYM
ejpam-6275	258	30	α21⟩	α21⟩	PROPN
ejpam-6275	258	31	,	,	PUNCT
ejpam-6275	258	32	where	where	SCONJ
ejpam-6275	258	33	β	β	PROPN
ejpam-6275	258	34	is	be	AUX
ejpam-6275	258	35	the	the	DET
ejpam-6275	258	36	root	root	NOUN
ejpam-6275	258	37	of	of	ADP
ejpam-6275	258	38	f(x	f(x	PROPN
ejpam-6275	258	39	)	)	PUNCT
ejpam-6275	258	40	.	.	PUNCT
ejpam-6275	259	1	the	the	DET
ejpam-6275	259	2	required	require	VERB
ejpam-6275	259	3	cyclic	cyclic	ADJ
ejpam-6275	259	4	subgroup	subgroup	NOUN
ejpam-6275	259	5	is	be	AUX
ejpam-6275	259	6	g∗	g∗	NOUN
ejpam-6275	259	7	=	=	SYM
ejpam-6275	259	8	{	{	PUNCT
ejpam-6275	259	9	α21	α21	PROPN
ejpam-6275	259	10	,	,	PUNCT
ejpam-6275	259	11	α42	α42	NUM
ejpam-6275	259	12	,	,	PUNCT
ejpam-6275	259	13	α63	α63	NOUN
ejpam-6275	259	14	=	=	SYM
ejpam-6275	259	15	1	1	NUM
ejpam-6275	259	16	}	}	PUNCT
ejpam-6275	259	17	=	=	PRON
ejpam-6275	259	18	{	{	PUNCT
ejpam-6275	259	19	1	1	NUM
ejpam-6275	259	20	+	+	NUM
ejpam-6275	259	21	ω	ω	NUM
ejpam-6275	259	22	,	,	PUNCT
ejpam-6275	259	23	ω	ω	PROPN
ejpam-6275	259	24	,	,	PUNCT
ejpam-6275	259	25	1	1	NUM
ejpam-6275	259	26	}	}	PUNCT
ejpam-6275	259	27	.	.	PUNCT
ejpam-6275	260	1	hence	hence	ADV
ejpam-6275	260	2	,	,	PUNCT
ejpam-6275	260	3	there	there	PRON
ejpam-6275	260	4	is	be	VERB
ejpam-6275	260	5	only	only	ADV
ejpam-6275	260	6	one	one	NUM
ejpam-6275	260	7	case	case	NOUN
ejpam-6275	260	8	for	for	ADP
ejpam-6275	260	9	the	the	DET
ejpam-6275	260	10	construction	construction	NOUN
ejpam-6275	260	11	of	of	ADP
ejpam-6275	260	12	shortened	shorten	VERB
ejpam-6275	260	13	bch	bch	PROPN
ejpam-6275	260	14	codes	code	NOUN
ejpam-6275	260	15	over	over	ADP
ejpam-6275	260	16	the	the	DET
ejpam-6275	260	17	eisenstein	eisenstein	PROPN
ejpam-6275	260	18	field	field	PROPN
ejpam-6275	260	19	z2[ω	z2[ω	NOUN
ejpam-6275	260	20	]	]	X
ejpam-6275	260	21	3	3	X
ejpam-6275	260	22	.	.	X
ejpam-6275	260	23	for	for	ADP
ejpam-6275	260	24	n	n	NOUN
ejpam-6275	260	25	=	=	SYM
ejpam-6275	260	26	3	3	NUM
ejpam-6275	260	27	and	and	CCONJ
ejpam-6275	260	28	d	d	NOUN
ejpam-6275	260	29	=	=	SYM
ejpam-6275	260	30	3	3	NUM
ejpam-6275	260	31	,	,	PUNCT
ejpam-6275	260	32	i	i	PRON
ejpam-6275	260	33	=	=	NOUN
ejpam-6275	260	34	1	1	NUM
ejpam-6275	260	35	,	,	PUNCT
ejpam-6275	260	36	2	2	NUM
ejpam-6275	260	37	:	:	SYM
ejpam-6275	260	38	•	•	NOUN
ejpam-6275	260	39	for	for	ADP
ejpam-6275	260	40	i	i	PRON
ejpam-6275	260	41	=	=	NOUN
ejpam-6275	260	42	1	1	NUM
ejpam-6275	260	43	,	,	PUNCT
ejpam-6275	260	44	let	let	VERB
ejpam-6275	260	45	β	β	PRON
ejpam-6275	260	46	∈	∈	PROPN
ejpam-6275	260	47	z2[ω	z2[ω	NOUN
ejpam-6275	260	48	]	]	X
ejpam-6275	261	1	3	3	X
ejpam-6275	261	2	.	.	PUNCT
ejpam-6275	261	3	by	by	ADP
ejpam-6275	261	4	theorem	theorem	NOUN
ejpam-6275	261	5	3.1	3.1	NUM
ejpam-6275	261	6	,	,	PUNCT
ejpam-6275	261	7	β	β	X
ejpam-6275	261	8	has	have	VERB
ejpam-6275	261	9	minimal	minimal	ADJ
ejpam-6275	261	10	polynomial	polynomial	ADJ
ejpam-6275	261	11	itself	itself	PRON
ejpam-6275	261	12	:	:	PUNCT
ejpam-6275	261	13	φ1(x	φ1(x	NOUN
ejpam-6275	261	14	)	)	PUNCT
ejpam-6275	261	15	=	=	SYM
ejpam-6275	261	16	(	(	PUNCT
ejpam-6275	261	17	x−	x−	PROPN
ejpam-6275	261	18	β	β	X
ejpam-6275	261	19	)	)	PUNCT
ejpam-6275	262	1	=	=	PUNCT
ejpam-6275	262	2	x+	x+	PUNCT
ejpam-6275	262	3	(	(	PUNCT
ejpam-6275	262	4	1	1	NUM
ejpam-6275	262	5	+	+	NUM
ejpam-6275	262	6	ω	ω	NUM
ejpam-6275	262	7	)	)	PUNCT
ejpam-6275	262	8	.	.	PUNCT
ejpam-6275	263	1	•	•	NOUN
ejpam-6275	263	2	for	for	ADP
ejpam-6275	263	3	i	i	PRON
ejpam-6275	263	4	=	=	SYM
ejpam-6275	263	5	2	2	NUM
ejpam-6275	263	6	,	,	PUNCT
ejpam-6275	263	7	let	let	VERB
ejpam-6275	263	8	β2	β2	PROPN
ejpam-6275	263	9	∈	∈	PROPN
ejpam-6275	263	10	z2[ω	z2[ω	NOUN
ejpam-6275	263	11	]	]	X
ejpam-6275	264	1	3	3	X
ejpam-6275	264	2	.	.	PUNCT
ejpam-6275	264	3	by	by	ADP
ejpam-6275	264	4	theorem	theorem	ADJ
ejpam-6275	264	5	3.1	3.1	NUM
ejpam-6275	264	6	,	,	PUNCT
ejpam-6275	264	7	β2	β2	PROPN
ejpam-6275	264	8	has	have	VERB
ejpam-6275	264	9	minimal	minimal	ADJ
ejpam-6275	264	10	polynomial	polynomial	ADJ
ejpam-6275	264	11	itself	itself	PRON
ejpam-6275	264	12	:	:	PUNCT
ejpam-6275	264	13	φ2(x	φ2(x	X
ejpam-6275	264	14	)	)	PUNCT
ejpam-6275	264	15	=	=	SYM
ejpam-6275	264	16	(	(	PUNCT
ejpam-6275	264	17	x−	x−	PROPN
ejpam-6275	264	18	β2	β2	PROPN
ejpam-6275	264	19	)	)	PUNCT
ejpam-6275	264	20	=	=	PUNCT
ejpam-6275	265	1	x+	x+	PROPN
ejpam-6275	265	2	ω	ω	X
ejpam-6275	265	3	.	.	PUNCT
ejpam-6275	266	1	the	the	DET
ejpam-6275	266	2	generator	generator	NOUN
ejpam-6275	266	3	polynomial	polynomial	NOUN
ejpam-6275	266	4	is	be	AUX
ejpam-6275	266	5	g(x	g(x	NOUN
ejpam-6275	266	6	)	)	PUNCT
ejpam-6275	267	1	=	=	SYM
ejpam-6275	267	2	φ1(x	φ1(x	NOUN
ejpam-6275	267	3	)	)	PUNCT
ejpam-6275	267	4	·	·	PUNCT
ejpam-6275	268	1	φ2(x	φ2(x	X
ejpam-6275	268	2	)	)	PUNCT
ejpam-6275	268	3	=	=	SYM
ejpam-6275	268	4	(	(	PUNCT
ejpam-6275	268	5	x+	x+	ADJ
ejpam-6275	268	6	ω)(x+	ω)(x+	X
ejpam-6275	268	7	(	(	PUNCT
ejpam-6275	268	8	1	1	NUM
ejpam-6275	268	9	+	+	NUM
ejpam-6275	268	10	ω	ω	NUM
ejpam-6275	268	11	)	)	PUNCT
ejpam-6275	268	12	)	)	PUNCT
ejpam-6275	269	1	=	=	SYM
ejpam-6275	270	1	x2	x2	PROPN
ejpam-6275	271	1	+	+	CCONJ
ejpam-6275	271	2	x+	x+	ADJ
ejpam-6275	271	3	1	1	X
ejpam-6275	271	4	.	.	PUNCT
ejpam-6275	271	5	since	since	SCONJ
ejpam-6275	271	6	k	k	PROPN
ejpam-6275	271	7	=	=	SYM
ejpam-6275	271	8	3−2	3−2	NUM
ejpam-6275	271	9	=	=	SYM
ejpam-6275	271	10	1	1	NUM
ejpam-6275	271	11	,	,	PUNCT
ejpam-6275	271	12	the	the	DET
ejpam-6275	271	13	parameters	parameter	NOUN
ejpam-6275	271	14	of	of	ADP
ejpam-6275	271	15	this	this	DET
ejpam-6275	271	16	shortened	shorten	VERB
ejpam-6275	271	17	bch	bch	PROPN
ejpam-6275	271	18	code	code	NOUN
ejpam-6275	271	19	are	be	AUX
ejpam-6275	271	20	(	(	PUNCT
ejpam-6275	271	21	n	n	X
ejpam-6275	271	22	,	,	PUNCT
ejpam-6275	271	23	k	k	NOUN
ejpam-6275	271	24	,	,	PUNCT
ejpam-6275	271	25	d	d	NOUN
ejpam-6275	271	26	)	)	PUNCT
ejpam-6275	271	27	=	=	SYM
ejpam-6275	271	28	(	(	PUNCT
ejpam-6275	271	29	3	3	NUM
ejpam-6275	271	30	,	,	PUNCT
ejpam-6275	271	31	1	1	NUM
ejpam-6275	271	32	,	,	PUNCT
ejpam-6275	271	33	3	3	NUM
ejpam-6275	271	34	)	)	PUNCT
ejpam-6275	271	35	.	.	PUNCT
ejpam-6275	272	1	illustration	illustration	NOUN
ejpam-6275	272	2	3.4	3.4	NUM
ejpam-6275	272	3	:	:	PUNCT
ejpam-6275	272	4	construct	construct	VERB
ejpam-6275	272	5	a	a	DET
ejpam-6275	272	6	shortened	shorten	VERB
ejpam-6275	272	7	bch	bch	PROPN
ejpam-6275	272	8	code	code	NOUN
ejpam-6275	272	9	for	for	ADP
ejpam-6275	272	10	three	three	NUM
ejpam-6275	272	11	-	-	PUNCT
ejpam-6275	272	12	degree	degree	NOUN
ejpam-6275	272	13	polynomial	polynomial	NOUN
ejpam-6275	272	14	by	by	ADP
ejpam-6275	272	15	taking	take	VERB
ejpam-6275	272	16	n	n	NOUN
ejpam-6275	272	17	=	=	SYM
ejpam-6275	272	18	9	9	NUM
ejpam-6275	272	19	over	over	ADP
ejpam-6275	272	20	the	the	DET
ejpam-6275	272	21	ef	ef	PROPN
ejpam-6275	272	22	z2[ω	z2[ω	NOUN
ejpam-6275	272	23	]	]	X
ejpam-6275	273	1	3	3	X
ejpam-6275	273	2	.	.	PUNCT
ejpam-6275	273	3	let	let	VERB
ejpam-6275	273	4	us	we	PRON
ejpam-6275	273	5	take	take	VERB
ejpam-6275	273	6	f(x	f(x	PROPN
ejpam-6275	273	7	)	)	PUNCT
ejpam-6275	273	8	=	=	PUNCT
ejpam-6275	274	1	x3	x3	ADJ
ejpam-6275	275	1	+	+	CCONJ
ejpam-6275	275	2	x2	x2	PROPN
ejpam-6275	276	1	+	+	CCONJ
ejpam-6275	276	2	x+	x+	SYM
ejpam-6275	276	3	1	1	NUM
ejpam-6275	276	4	+	+	NUM
ejpam-6275	276	5	ω	ω	NUM
ejpam-6275	276	6	,	,	PUNCT
ejpam-6275	276	7	which	which	PRON
ejpam-6275	276	8	is	be	AUX
ejpam-6275	276	9	primitive	primitive	ADJ
ejpam-6275	276	10	and	and	CCONJ
ejpam-6275	276	11	irreducible	irreducible	ADJ
ejpam-6275	276	12	over	over	ADP
ejpam-6275	276	13	z2[ω	z2[ω	NOUN
ejpam-6275	276	14	]	]	PUNCT
ejpam-6275	276	15	.	.	PUNCT
ejpam-6275	277	1	as	as	SCONJ
ejpam-6275	277	2	we	we	PRON
ejpam-6275	277	3	proved	prove	VERB
ejpam-6275	277	4	in	in	ADP
ejpam-6275	277	5	section	section	NOUN
ejpam-6275	277	6	2	2	NUM
ejpam-6275	277	7	,	,	PUNCT
ejpam-6275	277	8	the	the	DET
ejpam-6275	277	9	cardinality	cardinality	NOUN
ejpam-6275	277	10	of	of	ADP
ejpam-6275	277	11	the	the	DET
ejpam-6275	277	12	cyclic	cyclic	ADJ
ejpam-6275	277	13	group	group	NOUN
ejpam-6275	277	14	of	of	ADP
ejpam-6275	277	15	f(x	f(x	PROPN
ejpam-6275	277	16	)	)	PUNCT
ejpam-6275	277	17	is	be	AUX
ejpam-6275	277	18	63	63	NUM
ejpam-6275	277	19	and	and	CCONJ
ejpam-6275	277	20	63	63	NUM
ejpam-6275	277	21	9	9	NUM
ejpam-6275	277	22	=	=	SYM
ejpam-6275	277	23	7	7	NUM
ejpam-6275	277	24	.	.	PUNCT
ejpam-6275	278	1	so	so	ADV
ejpam-6275	278	2	,	,	PUNCT
ejpam-6275	278	3	our	our	PRON
ejpam-6275	278	4	required	require	VERB
ejpam-6275	278	5	cyclic	cyclic	ADJ
ejpam-6275	278	6	subgroup	subgroup	NOUN
ejpam-6275	278	7	is	be	AUX
ejpam-6275	278	8	generated	generate	VERB
ejpam-6275	278	9	by	by	ADP
ejpam-6275	278	10	⟨β	⟨β	X
ejpam-6275	279	1	=	=	PUNCT
ejpam-6275	280	1	α7⟩	α7⟩	INTJ
ejpam-6275	280	2	by	by	ADP
ejpam-6275	280	3	taking	take	VERB
ejpam-6275	280	4	β	β	PRON
ejpam-6275	280	5	to	to	PART
ejpam-6275	280	6	be	be	AUX
ejpam-6275	280	7	the	the	DET
ejpam-6275	280	8	root	root	NOUN
ejpam-6275	280	9	of	of	ADP
ejpam-6275	280	10	f(x	f(x	PROPN
ejpam-6275	280	11	)	)	PUNCT
ejpam-6275	280	12	.	.	PUNCT
ejpam-6275	281	1	then	then	ADV
ejpam-6275	281	2	cyclic	cyclic	PROPN
ejpam-6275	281	3	subgroup	subgroup	NOUN
ejpam-6275	281	4	is	be	AUX
ejpam-6275	281	5	g8	g8	PROPN
ejpam-6275	281	6	=	=	SYM
ejpam-6275	281	7	{	{	PUNCT
ejpam-6275	281	8	β	β	X
ejpam-6275	281	9	,	,	PUNCT
ejpam-6275	281	10	β2	β2	VERB
ejpam-6275	281	11	,	,	PUNCT
ejpam-6275	281	12	β3	β3	ADJ
ejpam-6275	281	13	,	,	PUNCT
ejpam-6275	281	14	β4	β4	PROPN
ejpam-6275	281	15	,	,	PUNCT
ejpam-6275	281	16	β5	β5	NOUN
ejpam-6275	281	17	,	,	PUNCT
ejpam-6275	281	18	β6	β6	PROPN
ejpam-6275	281	19	,	,	PUNCT
ejpam-6275	281	20	β7	β7	ADJ
ejpam-6275	281	21	,	,	PUNCT
ejpam-6275	281	22	β8	β8	NOUN
ejpam-6275	281	23	,	,	PUNCT
ejpam-6275	281	24	β9	β9	NOUN
ejpam-6275	281	25	=	=	NOUN
ejpam-6275	281	26	1	1	NUM
ejpam-6275	281	27	}	}	PUNCT
ejpam-6275	281	28	=	=	PRON
ejpam-6275	281	29	{	{	PUNCT
ejpam-6275	281	30	α7	α7	NOUN
ejpam-6275	281	31	,	,	PUNCT
ejpam-6275	281	32	α14	α14	NUM
ejpam-6275	281	33	,	,	PUNCT
ejpam-6275	281	34	α21	α21	NUM
ejpam-6275	281	35	,	,	PUNCT
ejpam-6275	281	36	α28	α28	NUM
ejpam-6275	281	37	,	,	PUNCT
ejpam-6275	281	38	α35	α35	ADV
ejpam-6275	281	39	,	,	PUNCT
ejpam-6275	281	40	α42	α42	NOUN
ejpam-6275	281	41	,	,	PUNCT
ejpam-6275	281	42	α49	α49	NUM
ejpam-6275	281	43	,	,	PUNCT
ejpam-6275	281	44	α56	α56	NOUN
ejpam-6275	281	45	,	,	PUNCT
ejpam-6275	281	46	α63	α63	NOUN
ejpam-6275	281	47	=	=	SYM
ejpam-6275	281	48	1	1	NUM
ejpam-6275	281	49	}	}	PUNCT
ejpam-6275	281	50	=	=	SYM
ejpam-6275	281	51	{	{	PUNCT
ejpam-6275	281	52	α2(1	α2(1	NOUN
ejpam-6275	281	53	+	+	CCONJ
ejpam-6275	281	54	ω	ω	NUM
ejpam-6275	281	55	)	)	PUNCT
ejpam-6275	282	1	+	+	CCONJ
ejpam-6275	282	2	1	1	NUM
ejpam-6275	282	3	+	+	NUM
ejpam-6275	282	4	ω	ω	NUM
ejpam-6275	282	5	,	,	PUNCT
ejpam-6275	282	6	1	1	NUM
ejpam-6275	282	7	+	+	SYM
ejpam-6275	282	8	ω	ω	NUM
ejpam-6275	282	9	+	+	CCONJ
ejpam-6275	282	10	(	(	PUNCT
ejpam-6275	282	11	1	1	NUM
ejpam-6275	282	12	+	+	CCONJ
ejpam-6275	282	13	ω)α	ω)α	NOUN
ejpam-6275	282	14	,	,	PUNCT
ejpam-6275	282	15	1	1	NUM
ejpam-6275	282	16	+	+	SYM
ejpam-6275	282	17	ω	ω	NUM
ejpam-6275	282	18	,	,	PUNCT
ejpam-6275	282	19	ωα2	ωα2	NOUN
ejpam-6275	282	20	+	+	CCONJ
ejpam-6275	283	1	ω	ω	NUM
ejpam-6275	283	2	,	,	PUNCT
ejpam-6275	283	3	ω	ω	PROPN
ejpam-6275	283	4	+	+	CCONJ
ejpam-6275	283	5	ωα	ωα	PROPN
ejpam-6275	283	6	,	,	PUNCT
ejpam-6275	283	7	ω	ω	PROPN
ejpam-6275	283	8	,	,	PUNCT
ejpam-6275	283	9	α2	α2	PROPN
ejpam-6275	283	10	+	+	CCONJ
ejpam-6275	283	11	1	1	NUM
ejpam-6275	283	12	,	,	PUNCT
ejpam-6275	283	13	α+	α+	PRON
ejpam-6275	283	14	1	1	NUM
ejpam-6275	283	15	,	,	PUNCT
ejpam-6275	283	16	1	1	NUM
ejpam-6275	283	17	}	}	PUNCT
ejpam-6275	283	18	.	.	PUNCT
ejpam-6275	284	1	so	so	ADV
ejpam-6275	284	2	,	,	PUNCT
ejpam-6275	284	3	there	there	PRON
ejpam-6275	284	4	are	be	VERB
ejpam-6275	284	5	four	four	NUM
ejpam-6275	284	6	cases	case	NOUN
ejpam-6275	284	7	for	for	ADP
ejpam-6275	284	8	the	the	DET
ejpam-6275	284	9	construction	construction	NOUN
ejpam-6275	284	10	of	of	ADP
ejpam-6275	284	11	shortened	shorten	VERB
ejpam-6275	284	12	bch	bch	PROPN
ejpam-6275	284	13	codes	code	NOUN
ejpam-6275	284	14	over	over	ADP
ejpam-6275	284	15	the	the	DET
ejpam-6275	284	16	ef	ef	PROPN
ejpam-6275	284	17	z2[ω	z2[ω	NOUN
ejpam-6275	284	18	]	]	X
ejpam-6275	284	19	3	3	X
ejpam-6275	284	20	.	.	X
ejpam-6275	284	21	case	case	NOUN
ejpam-6275	284	22	1	1	NUM
ejpam-6275	284	23	:	:	PUNCT
ejpam-6275	284	24	for	for	ADP
ejpam-6275	284	25	n	n	NOUN
ejpam-6275	284	26	=	=	SYM
ejpam-6275	284	27	9	9	NUM
ejpam-6275	284	28	and	and	CCONJ
ejpam-6275	284	29	d	d	NOUN
ejpam-6275	284	30	=	=	SYM
ejpam-6275	284	31	3	3	NUM
ejpam-6275	284	32	,	,	PUNCT
ejpam-6275	284	33	i	i	PRON
ejpam-6275	284	34	=	=	NOUN
ejpam-6275	284	35	1	1	NUM
ejpam-6275	284	36	,	,	PUNCT
ejpam-6275	284	37	2	2	NUM
ejpam-6275	284	38	.	.	NOUN
ejpam-6275	284	39	•	•	NOUN
ejpam-6275	284	40	for	for	ADP
ejpam-6275	284	41	i	i	PRON
ejpam-6275	284	42	=	=	NOUN
ejpam-6275	284	43	1	1	NUM
ejpam-6275	284	44	,	,	PUNCT
ejpam-6275	284	45	let	let	VERB
ejpam-6275	284	46	β	β	PRON
ejpam-6275	284	47	∈	∈	PROPN
ejpam-6275	284	48	z2[ω	z2[ω	NOUN
ejpam-6275	284	49	]	]	X
ejpam-6275	285	1	3	3	X
ejpam-6275	285	2	.	.	PUNCT
ejpam-6275	285	3	by	by	ADP
ejpam-6275	285	4	theorem	theorem	ADJ
ejpam-6275	285	5	3.1	3.1	NUM
ejpam-6275	285	6	,	,	PUNCT
ejpam-6275	285	7	β	β	X
ejpam-6275	285	8	,	,	PUNCT
ejpam-6275	285	9	β4	β4	PROPN
ejpam-6275	285	10	,	,	PUNCT
ejpam-6275	285	11	β7	β7	PROPN
ejpam-6275	285	12	have	have	VERB
ejpam-6275	285	13	the	the	DET
ejpam-6275	285	14	minimal	minimal	ADJ
ejpam-6275	285	15	polynomial	polynomial	ADJ
ejpam-6275	285	16	φ1(x	φ1(x	NOUN
ejpam-6275	285	17	)	)	PUNCT
ejpam-6275	285	18	=	=	PUNCT
ejpam-6275	285	19	(	(	PUNCT
ejpam-6275	285	20	x−	x−	PROPN
ejpam-6275	285	21	β)(x−	β)(x−	PROPN
ejpam-6275	285	22	β4)(x−	β4)(x−	X
ejpam-6275	286	1	β7	β7	ADJ
ejpam-6275	286	2	)	)	PUNCT
ejpam-6275	286	3	=	=	SYM
ejpam-6275	287	1	(	(	PUNCT
ejpam-6275	287	2	x2	x2	NOUN
ejpam-6275	287	3	+	+	CCONJ
ejpam-6275	287	4	(	(	PUNCT
ejpam-6275	287	5	β	β	X
ejpam-6275	287	6	+	+	NUM
ejpam-6275	287	7	β4)x+	β4)x+	NOUN
ejpam-6275	287	8	β5)(x−	β5)(x−	NOUN
ejpam-6275	287	9	β7	β7	ADJ
ejpam-6275	287	10	)	)	PUNCT
ejpam-6275	287	11	=	=	SYM
ejpam-6275	288	1	(	(	PUNCT
ejpam-6275	288	2	x2	x2	NOUN
ejpam-6275	288	3	+	+	CCONJ
ejpam-6275	288	4	(	(	PUNCT
ejpam-6275	288	5	β	β	X
ejpam-6275	288	6	+	+	NUM
ejpam-6275	288	7	β4)x+	β4)x+	NOUN
ejpam-6275	288	8	β5)(x+	β5)(x+	X
ejpam-6275	288	9	β7	β7	ADJ
ejpam-6275	288	10	)	)	PUNCT
ejpam-6275	288	11	=	=	SYM
ejpam-6275	289	1	x3	x3	VERB
ejpam-6275	289	2	+	+	CCONJ
ejpam-6275	289	3	(	(	PUNCT
ejpam-6275	289	4	β	β	X
ejpam-6275	289	5	+	+	NUM
ejpam-6275	289	6	β4	β4	PROPN
ejpam-6275	289	7	+	+	X
ejpam-6275	289	8	β7)x2	β7)x2	X
ejpam-6275	289	9	+	+	CCONJ
ejpam-6275	289	10	(	(	PUNCT
ejpam-6275	289	11	β2	β2	NOUN
ejpam-6275	289	12	+	+	PROPN
ejpam-6275	289	13	β5	β5	NOUN
ejpam-6275	289	14	+	+	CCONJ
ejpam-6275	289	15	β8)x+	β8)x+	NOUN
ejpam-6275	289	16	β3	β3	NOUN
ejpam-6275	289	17	=	=	SYM
ejpam-6275	290	1	x3	x3	ADJ
ejpam-6275	291	1	+	+	CCONJ
ejpam-6275	291	2	0	0	NUM
ejpam-6275	291	3	·	·	PUNCT
ejpam-6275	292	1	x2	x2	PROPN
ejpam-6275	293	1	+	+	CCONJ
ejpam-6275	293	2	0	0	NUM
ejpam-6275	293	3	·	·	PUNCT
ejpam-6275	293	4	x+	x+	ADJ
ejpam-6275	293	5	β3	β3	NOUN
ejpam-6275	293	6	=	=	SYM
ejpam-6275	294	1	x3	x3	ADJ
ejpam-6275	294	2	+	+	CCONJ
ejpam-6275	294	3	1	1	NUM
ejpam-6275	294	4	+	+	NUM
ejpam-6275	294	5	ω	ω	NUM
ejpam-6275	294	6	.	.	NOUN
ejpam-6275	294	7	•	•	NUM
ejpam-6275	294	8	for	for	ADP
ejpam-6275	294	9	i	i	PRON
ejpam-6275	294	10	=	=	SYM
ejpam-6275	294	11	2	2	NUM
ejpam-6275	294	12	,	,	PUNCT
ejpam-6275	294	13	let	let	VERB
ejpam-6275	294	14	β2	β2	PROPN
ejpam-6275	294	15	∈	∈	PROPN
ejpam-6275	294	16	z2[ω	z2[ω	NOUN
ejpam-6275	294	17	]	]	X
ejpam-6275	295	1	3	3	X
ejpam-6275	295	2	.	.	PUNCT
ejpam-6275	295	3	by	by	ADP
ejpam-6275	295	4	theorem	theorem	ADJ
ejpam-6275	295	5	3.1	3.1	NUM
ejpam-6275	295	6	,	,	PUNCT
ejpam-6275	295	7	β2	β2	NOUN
ejpam-6275	295	8	,	,	PUNCT
ejpam-6275	295	9	β5	β5	PROPN
ejpam-6275	295	10	,	,	PUNCT
ejpam-6275	295	11	β8	β8	NOUN
ejpam-6275	295	12	have	have	VERB
ejpam-6275	295	13	the	the	DET
ejpam-6275	295	14	minimal	minimal	ADJ
ejpam-6275	295	15	polynomial	polynomial	ADJ
ejpam-6275	295	16	φ2(x	φ2(x	NUM
ejpam-6275	295	17	)	)	PUNCT
ejpam-6275	295	18	=	=	PUNCT
ejpam-6275	295	19	(	(	PUNCT
ejpam-6275	295	20	x−	x−	PROPN
ejpam-6275	295	21	β2)(x−	β2)(x−	PUNCT
ejpam-6275	295	22	β5)(x−	β5)(x−	NOUN
ejpam-6275	295	23	β8	β8	NOUN
ejpam-6275	295	24	)	)	PUNCT
ejpam-6275	296	1	=	=	SYM
ejpam-6275	297	1	(	(	PUNCT
ejpam-6275	297	2	x2	x2	NOUN
ejpam-6275	297	3	+	+	CCONJ
ejpam-6275	297	4	(	(	PUNCT
ejpam-6275	297	5	β2	β2	VERB
ejpam-6275	297	6	+	+	NOUN
ejpam-6275	297	7	β5)x+	β5)x+	NOUN
ejpam-6275	297	8	β7)(x+	β7)(x+	NOUN
ejpam-6275	297	9	β8	β8	NOUN
ejpam-6275	297	10	)	)	PUNCT
ejpam-6275	297	11	=	=	PUNCT
ejpam-6275	298	1	x3	x3	VERB
ejpam-6275	298	2	+	+	CCONJ
ejpam-6275	298	3	(	(	PUNCT
ejpam-6275	298	4	β2	β2	NOUN
ejpam-6275	298	5	+	+	PROPN
ejpam-6275	298	6	β5	β5	PROPN
ejpam-6275	298	7	+	+	CCONJ
ejpam-6275	298	8	β8)x2	β8)x2	NOUN
ejpam-6275	299	1	+	+	CCONJ
ejpam-6275	299	2	(	(	PUNCT
ejpam-6275	299	3	β	β	X
ejpam-6275	299	4	+	+	CCONJ
ejpam-6275	299	5	β4	β4	PROPN
ejpam-6275	299	6	+	+	CCONJ
ejpam-6275	299	7	β7)x+	β7)x+	ADJ
ejpam-6275	299	8	β6	β6	NOUN
ejpam-6275	299	9	=	=	PUNCT
ejpam-6275	299	10	x3	x3	PROPN
ejpam-6275	300	1	+	+	CCONJ
ejpam-6275	300	2	0	0	NUM
ejpam-6275	300	3	·	·	PUNCT
ejpam-6275	301	1	x2	x2	PROPN
ejpam-6275	302	1	+	+	CCONJ
ejpam-6275	302	2	0	0	NUM
ejpam-6275	302	3	·	·	PUNCT
ejpam-6275	302	4	x+	x+	ADJ
ejpam-6275	302	5	β6	β6	PROPN
ejpam-6275	302	6	=	=	PUNCT
ejpam-6275	303	1	x3	x3	PROPN
ejpam-6275	303	2	+	+	CCONJ
ejpam-6275	304	1	ω	ω	X
ejpam-6275	304	2	.	.	PUNCT
ejpam-6275	304	3	m.	m.	PROPN
ejpam-6275	304	4	sajjad	sajjad	PROPN
ejpam-6275	304	5	et	et	PROPN
ejpam-6275	304	6	al	al	PROPN
ejpam-6275	304	7	.	.	PUNCT
ejpam-6275	304	8	/	/	SYM
ejpam-6275	304	9	eur	eur	PROPN
ejpam-6275	304	10	.	.	PUNCT
ejpam-6275	305	1	j.	j.	PROPN
ejpam-6275	305	2	pure	pure	PROPN
ejpam-6275	305	3	appl	appl	PROPN
ejpam-6275	305	4	.	.	PROPN
ejpam-6275	305	5	math	math	PROPN
ejpam-6275	305	6	,	,	PUNCT
ejpam-6275	305	7	18	18	NUM
ejpam-6275	305	8	(	(	PUNCT
ejpam-6275	305	9	3	3	NUM
ejpam-6275	305	10	)	)	PUNCT
ejpam-6275	305	11	(	(	PUNCT
ejpam-6275	305	12	2025	2025	NUM
ejpam-6275	305	13	)	)	PUNCT
ejpam-6275	305	14	,	,	PUNCT
ejpam-6275	305	15	6275	6275	NUM
ejpam-6275	305	16	13	13	NUM
ejpam-6275	305	17	of	of	ADP
ejpam-6275	305	18	36	36	NUM
ejpam-6275	305	19	now	now	ADV
ejpam-6275	305	20	the	the	DET
ejpam-6275	305	21	generator	generator	NOUN
ejpam-6275	305	22	polynomial	polynomial	NOUN
ejpam-6275	305	23	is	be	AUX
ejpam-6275	305	24	g(x	g(x	NOUN
ejpam-6275	305	25	)	)	PUNCT
ejpam-6275	306	1	=	=	SYM
ejpam-6275	306	2	φ1(x	φ1(x	NOUN
ejpam-6275	306	3	)	)	PUNCT
ejpam-6275	306	4	·	·	PUNCT
ejpam-6275	307	1	φ2(x	φ2(x	X
ejpam-6275	307	2	)	)	PUNCT
ejpam-6275	307	3	=	=	SYM
ejpam-6275	307	4	(	(	PUNCT
ejpam-6275	307	5	x3	x3	ADJ
ejpam-6275	307	6	+	+	CCONJ
ejpam-6275	307	7	ω)(x3	ω)(x3	PROPN
ejpam-6275	308	1	+	+	CCONJ
ejpam-6275	308	2	1	1	NUM
ejpam-6275	308	3	+	+	NUM
ejpam-6275	308	4	ω	ω	NUM
ejpam-6275	308	5	)	)	PUNCT
ejpam-6275	308	6	=	=	SYM
ejpam-6275	308	7	x6	x6	PROPN
ejpam-6275	308	8	+	+	CCONJ
ejpam-6275	308	9	ωx3	ωx3	X
ejpam-6275	308	10	+	+	CCONJ
ejpam-6275	308	11	(	(	PUNCT
ejpam-6275	308	12	1	1	NUM
ejpam-6275	308	13	+	+	X
ejpam-6275	308	14	ω)x3	ω)x3	NOUN
ejpam-6275	308	15	+	+	CCONJ
ejpam-6275	308	16	1	1	NUM
ejpam-6275	308	17	=	=	SYM
ejpam-6275	308	18	x6	x6	PROPN
ejpam-6275	308	19	+	+	CCONJ
ejpam-6275	308	20	x3	x3	ADJ
ejpam-6275	308	21	+	+	NOUN
ejpam-6275	308	22	1	1	X
ejpam-6275	308	23	.	.	X
ejpam-6275	309	1	hence	hence	ADV
ejpam-6275	309	2	,	,	PUNCT
ejpam-6275	309	3	k	k	PROPN
ejpam-6275	309	4	=	=	SYM
ejpam-6275	309	5	9−	9−	NUM
ejpam-6275	309	6	6	6	NUM
ejpam-6275	309	7	=	=	SYM
ejpam-6275	309	8	3	3	X
ejpam-6275	309	9	.	.	PUNCT
ejpam-6275	310	1	so	so	ADV
ejpam-6275	310	2	the	the	DET
ejpam-6275	310	3	shortened	shorten	VERB
ejpam-6275	310	4	bch	bch	PROPN
ejpam-6275	310	5	code	code	NOUN
ejpam-6275	310	6	parameters	parameter	NOUN
ejpam-6275	310	7	are	be	AUX
ejpam-6275	310	8	(	(	PUNCT
ejpam-6275	310	9	n	n	X
ejpam-6275	310	10	,	,	PUNCT
ejpam-6275	310	11	k	k	NOUN
ejpam-6275	310	12	,	,	PUNCT
ejpam-6275	310	13	d	d	NOUN
ejpam-6275	310	14	)	)	PUNCT
ejpam-6275	310	15	=	=	SYM
ejpam-6275	310	16	(	(	PUNCT
ejpam-6275	310	17	9	9	NUM
ejpam-6275	310	18	,	,	PUNCT
ejpam-6275	310	19	3	3	NUM
ejpam-6275	310	20	,	,	PUNCT
ejpam-6275	310	21	3	3	NUM
ejpam-6275	310	22	)	)	PUNCT
ejpam-6275	310	23	.	.	PUNCT
ejpam-6275	311	1	case	case	NOUN
ejpam-6275	311	2	2	2	NUM
ejpam-6275	311	3	:	:	PUNCT
ejpam-6275	311	4	for	for	ADP
ejpam-6275	311	5	n	n	NOUN
ejpam-6275	311	6	=	=	SYM
ejpam-6275	311	7	9	9	NUM
ejpam-6275	311	8	and	and	CCONJ
ejpam-6275	311	9	d	d	NOUN
ejpam-6275	311	10	=	=	SYM
ejpam-6275	311	11	5	5	NUM
ejpam-6275	311	12	,	,	PUNCT
ejpam-6275	311	13	i	i	PRON
ejpam-6275	311	14	=	=	NOUN
ejpam-6275	311	15	1	1	NUM
ejpam-6275	311	16	,	,	PUNCT
ejpam-6275	311	17	2	2	NUM
ejpam-6275	311	18	,	,	PUNCT
ejpam-6275	311	19	3	3	NUM
ejpam-6275	311	20	,	,	PUNCT
ejpam-6275	311	21	4	4	NUM
ejpam-6275	311	22	.	.	NOUN
ejpam-6275	311	23	•	•	NOUN
ejpam-6275	311	24	for	for	ADP
ejpam-6275	311	25	i	i	PRON
ejpam-6275	311	26	=	=	NOUN
ejpam-6275	311	27	1	1	NUM
ejpam-6275	311	28	,	,	PUNCT
ejpam-6275	311	29	φ1(x	φ1(x	NOUN
ejpam-6275	311	30	)	)	PUNCT
ejpam-6275	311	31	=	=	SYM
ejpam-6275	312	1	x3	x3	ADJ
ejpam-6275	312	2	+	+	CCONJ
ejpam-6275	312	3	1	1	NUM
ejpam-6275	312	4	+	+	NUM
ejpam-6275	312	5	ω	ω	NUM
ejpam-6275	312	6	(	(	PUNCT
ejpam-6275	312	7	as	as	ADP
ejpam-6275	312	8	in	in	ADP
ejpam-6275	312	9	case	case	NOUN
ejpam-6275	312	10	1	1	NUM
ejpam-6275	312	11	)	)	PUNCT
ejpam-6275	312	12	.	.	PUNCT
ejpam-6275	313	1	•	•	NOUN
ejpam-6275	313	2	for	for	ADP
ejpam-6275	313	3	i	i	PRON
ejpam-6275	313	4	=	=	SYM
ejpam-6275	313	5	2	2	NUM
ejpam-6275	313	6	,	,	PUNCT
ejpam-6275	313	7	φ2(x	φ2(x	NOUN
ejpam-6275	313	8	)	)	PUNCT
ejpam-6275	313	9	=	=	SYM
ejpam-6275	314	1	x3	x3	PROPN
ejpam-6275	315	1	+	+	CCONJ
ejpam-6275	315	2	ω	ω	NUM
ejpam-6275	315	3	(	(	PUNCT
ejpam-6275	315	4	as	as	ADP
ejpam-6275	315	5	in	in	ADP
ejpam-6275	315	6	case	case	NOUN
ejpam-6275	315	7	1	1	NUM
ejpam-6275	315	8	)	)	PUNCT
ejpam-6275	315	9	.	.	PUNCT
ejpam-6275	316	1	•	•	NOUN
ejpam-6275	316	2	for	for	ADP
ejpam-6275	316	3	i	i	PRON
ejpam-6275	316	4	=	=	SYM
ejpam-6275	316	5	3	3	NUM
ejpam-6275	316	6	,	,	PUNCT
ejpam-6275	316	7	let	let	VERB
ejpam-6275	316	8	β3	β3	VERB
ejpam-6275	316	9	∈	∈	PROPN
ejpam-6275	316	10	z2[ω	z2[ω	NOUN
ejpam-6275	316	11	]	]	X
ejpam-6275	316	12	3	3	NUM
ejpam-6275	316	13	then	then	ADV
ejpam-6275	316	14	by	by	ADP
ejpam-6275	316	15	theorem	theorem	ADJ
ejpam-6275	316	16	3.1	3.1	NUM
ejpam-6275	316	17	,	,	PUNCT
ejpam-6275	316	18	φ3(x	φ3(x	PROPN
ejpam-6275	316	19	)	)	PUNCT
ejpam-6275	316	20	=	=	SYM
ejpam-6275	316	21	(	(	PUNCT
ejpam-6275	316	22	x−	x−	PROPN
ejpam-6275	316	23	β3	β3	PROPN
ejpam-6275	316	24	)	)	PUNCT
ejpam-6275	316	25	=	=	SYM
ejpam-6275	317	1	(	(	PUNCT
ejpam-6275	317	2	x+	x+	ADJ
ejpam-6275	317	3	1	1	NUM
ejpam-6275	317	4	+	+	NUM
ejpam-6275	317	5	ω	ω	NUM
ejpam-6275	317	6	)	)	PUNCT
ejpam-6275	317	7	.	.	PUNCT
ejpam-6275	318	1	•	•	NOUN
ejpam-6275	318	2	for	for	ADP
ejpam-6275	318	3	i	i	PRON
ejpam-6275	318	4	=	=	NOUN
ejpam-6275	318	5	4	4	NUM
ejpam-6275	318	6	,	,	PUNCT
ejpam-6275	318	7	φ4(x	φ4(x	NOUN
ejpam-6275	318	8	)	)	PUNCT
ejpam-6275	318	9	=	=	SYM
ejpam-6275	318	10	φ1(x	φ1(x	NOUN
ejpam-6275	318	11	)	)	PUNCT
ejpam-6275	318	12	=	=	SYM
ejpam-6275	319	1	x3	x3	ADJ
ejpam-6275	319	2	+	+	CCONJ
ejpam-6275	319	3	1	1	NUM
ejpam-6275	319	4	+	+	NUM
ejpam-6275	319	5	ω	ω	X
ejpam-6275	319	6	.	.	PUNCT
ejpam-6275	320	1	the	the	DET
ejpam-6275	320	2	generator	generator	NOUN
ejpam-6275	320	3	polynomial	polynomial	NOUN
ejpam-6275	320	4	is	be	AUX
ejpam-6275	320	5	g(x	g(x	NOUN
ejpam-6275	320	6	)	)	PUNCT
ejpam-6275	321	1	=	=	SYM
ejpam-6275	321	2	lcm(φ1(x	lcm(φ1(x	NOUN
ejpam-6275	321	3	)	)	PUNCT
ejpam-6275	321	4	,	,	PUNCT
ejpam-6275	321	5	φ2(x	φ2(x	NOUN
ejpam-6275	321	6	)	)	PUNCT
ejpam-6275	321	7	,	,	PUNCT
ejpam-6275	321	8	φ3(x	φ3(x	PROPN
ejpam-6275	321	9	)	)	PUNCT
ejpam-6275	321	10	,	,	PUNCT
ejpam-6275	321	11	φ4(x	φ4(x	NOUN
ejpam-6275	321	12	)	)	PUNCT
ejpam-6275	321	13	)	)	PUNCT
ejpam-6275	322	1	=	=	PUNCT
ejpam-6275	322	2	(	(	PUNCT
ejpam-6275	322	3	x3	x3	ADJ
ejpam-6275	322	4	+	+	CCONJ
ejpam-6275	323	1	ω)(x3	ω)(x3	PROPN
ejpam-6275	323	2	+	+	CCONJ
ejpam-6275	323	3	1	1	NUM
ejpam-6275	323	4	+	+	CCONJ
ejpam-6275	323	5	ω)(x+	ω)(x+	PROPN
ejpam-6275	323	6	1	1	NUM
ejpam-6275	323	7	+	+	NUM
ejpam-6275	323	8	ω	ω	NUM
ejpam-6275	323	9	)	)	PUNCT
ejpam-6275	323	10	=	=	SYM
ejpam-6275	323	11	(	(	PUNCT
ejpam-6275	323	12	x6	x6	NOUN
ejpam-6275	323	13	+	+	CCONJ
ejpam-6275	323	14	ωx3	ωx3	X
ejpam-6275	323	15	+	+	CCONJ
ejpam-6275	323	16	(	(	PUNCT
ejpam-6275	323	17	1	1	NUM
ejpam-6275	323	18	+	+	X
ejpam-6275	323	19	ω)x3	ω)x3	NOUN
ejpam-6275	323	20	+	+	CCONJ
ejpam-6275	323	21	1)(x+	1)(x+	NUM
ejpam-6275	323	22	1	1	NUM
ejpam-6275	323	23	+	+	NUM
ejpam-6275	323	24	ω	ω	NUM
ejpam-6275	323	25	)	)	PUNCT
ejpam-6275	323	26	=	=	SYM
ejpam-6275	324	1	x7	x7	NOUN
ejpam-6275	325	1	+	+	CCONJ
ejpam-6275	325	2	x6(ω	x6(ω	PUNCT
ejpam-6275	326	1	+	+	NOUN
ejpam-6275	326	2	1	1	NUM
ejpam-6275	326	3	)	)	PUNCT
ejpam-6275	326	4	+	+	CCONJ
ejpam-6275	326	5	x4	x4	PROPN
ejpam-6275	326	6	+	+	CCONJ
ejpam-6275	327	1	x3(1	x3(1	PROPN
ejpam-6275	327	2	+	+	NUM
ejpam-6275	327	3	ω	ω	NUM
ejpam-6275	327	4	)	)	PUNCT
ejpam-6275	328	1	+	+	CCONJ
ejpam-6275	328	2	x+	x+	SYM
ejpam-6275	328	3	1	1	NUM
ejpam-6275	328	4	+	+	NUM
ejpam-6275	328	5	ω	ω	X
ejpam-6275	328	6	.	.	PUNCT
ejpam-6275	329	1	hence	hence	ADV
ejpam-6275	329	2	,	,	PUNCT
ejpam-6275	329	3	k	k	PROPN
ejpam-6275	329	4	=	=	SYM
ejpam-6275	329	5	9−	9−	NUM
ejpam-6275	329	6	7	7	NUM
ejpam-6275	329	7	=	=	SYM
ejpam-6275	329	8	2	2	NUM
ejpam-6275	329	9	.	.	PUNCT
ejpam-6275	330	1	so	so	ADV
ejpam-6275	330	2	the	the	DET
ejpam-6275	330	3	shortened	shorten	VERB
ejpam-6275	330	4	bch	bch	PROPN
ejpam-6275	330	5	code	code	NOUN
ejpam-6275	330	6	parameters	parameter	NOUN
ejpam-6275	330	7	are	be	AUX
ejpam-6275	330	8	(	(	PUNCT
ejpam-6275	330	9	9	9	NUM
ejpam-6275	330	10	,	,	PUNCT
ejpam-6275	330	11	2	2	NUM
ejpam-6275	330	12	,	,	PUNCT
ejpam-6275	330	13	5	5	NUM
ejpam-6275	330	14	)	)	PUNCT
ejpam-6275	330	15	.	.	PUNCT
ejpam-6275	331	1	case	case	NOUN
ejpam-6275	331	2	3	3	NUM
ejpam-6275	331	3	:	:	PUNCT
ejpam-6275	331	4	for	for	ADP
ejpam-6275	331	5	n	n	NOUN
ejpam-6275	331	6	=	=	SYM
ejpam-6275	331	7	9	9	NUM
ejpam-6275	331	8	and	and	CCONJ
ejpam-6275	331	9	d	d	NOUN
ejpam-6275	331	10	=	=	SYM
ejpam-6275	331	11	7	7	NUM
ejpam-6275	331	12	,	,	PUNCT
ejpam-6275	331	13	i	i	PRON
ejpam-6275	331	14	=	=	NOUN
ejpam-6275	331	15	1	1	NUM
ejpam-6275	331	16	,	,	PUNCT
ejpam-6275	331	17	2	2	NUM
ejpam-6275	331	18	,	,	PUNCT
ejpam-6275	331	19	3	3	NUM
ejpam-6275	331	20	,	,	PUNCT
ejpam-6275	331	21	4	4	NUM
ejpam-6275	331	22	,	,	PUNCT
ejpam-6275	331	23	5	5	NUM
ejpam-6275	331	24	,	,	PUNCT
ejpam-6275	331	25	6	6	NUM
ejpam-6275	331	26	.	.	NOUN
ejpam-6275	331	27	•	•	NOUN
ejpam-6275	331	28	for	for	ADP
ejpam-6275	331	29	i	i	PRON
ejpam-6275	331	30	=	=	NOUN
ejpam-6275	331	31	1	1	NUM
ejpam-6275	331	32	,	,	PUNCT
ejpam-6275	331	33	φ1(x	φ1(x	NOUN
ejpam-6275	331	34	)	)	PUNCT
ejpam-6275	331	35	=	=	SYM
ejpam-6275	332	1	x3	x3	ADJ
ejpam-6275	332	2	+	+	CCONJ
ejpam-6275	332	3	1	1	NUM
ejpam-6275	332	4	+	+	NUM
ejpam-6275	332	5	ω	ω	NUM
ejpam-6275	332	6	.	.	NOUN
ejpam-6275	332	7	•	•	NUM
ejpam-6275	332	8	for	for	ADP
ejpam-6275	332	9	i	i	PRON
ejpam-6275	332	10	=	=	SYM
ejpam-6275	332	11	2	2	NUM
ejpam-6275	332	12	,	,	PUNCT
ejpam-6275	332	13	φ2(x	φ2(x	NOUN
ejpam-6275	332	14	)	)	PUNCT
ejpam-6275	332	15	=	=	SYM
ejpam-6275	333	1	x3	x3	PROPN
ejpam-6275	334	1	+	+	CCONJ
ejpam-6275	335	1	ω	ω	X
ejpam-6275	335	2	.	.	NOUN
ejpam-6275	335	3	•	•	NUM
ejpam-6275	335	4	for	for	ADP
ejpam-6275	335	5	i	i	PRON
ejpam-6275	335	6	=	=	SYM
ejpam-6275	335	7	3	3	NUM
ejpam-6275	335	8	,	,	PUNCT
ejpam-6275	335	9	φ3(x	φ3(x	PROPN
ejpam-6275	335	10	)	)	PUNCT
ejpam-6275	335	11	=	=	SYM
ejpam-6275	336	1	x+	x+	PUNCT
ejpam-6275	336	2	1	1	NUM
ejpam-6275	336	3	+	+	NUM
ejpam-6275	336	4	ω	ω	NUM
ejpam-6275	336	5	.	.	NOUN
ejpam-6275	336	6	•	•	NUM
ejpam-6275	336	7	for	for	ADP
ejpam-6275	336	8	i	i	PRON
ejpam-6275	336	9	=	=	NOUN
ejpam-6275	336	10	4	4	NUM
ejpam-6275	336	11	,	,	PUNCT
ejpam-6275	336	12	φ4(x	φ4(x	NOUN
ejpam-6275	336	13	)	)	PUNCT
ejpam-6275	336	14	=	=	SYM
ejpam-6275	336	15	φ1(x	φ1(x	NOUN
ejpam-6275	336	16	)	)	PUNCT
ejpam-6275	336	17	.	.	PUNCT
ejpam-6275	337	1	•	•	NOUN
ejpam-6275	337	2	for	for	ADP
ejpam-6275	337	3	i	i	PRON
ejpam-6275	337	4	=	=	SYM
ejpam-6275	337	5	5	5	NUM
ejpam-6275	337	6	,	,	PUNCT
ejpam-6275	337	7	φ5(x	φ5(x	NOUN
ejpam-6275	337	8	)	)	PUNCT
ejpam-6275	337	9	=	=	SYM
ejpam-6275	337	10	φ2(x	φ2(x	NUM
ejpam-6275	337	11	)	)	PUNCT
ejpam-6275	337	12	.	.	PUNCT
ejpam-6275	338	1	•	•	NOUN
ejpam-6275	338	2	for	for	ADP
ejpam-6275	338	3	i	i	PRON
ejpam-6275	338	4	=	=	SYM
ejpam-6275	338	5	6	6	NUM
ejpam-6275	338	6	,	,	PUNCT
ejpam-6275	338	7	let	let	VERB
ejpam-6275	338	8	β6	β6	PROPN
ejpam-6275	338	9	∈	∈	PROPN
ejpam-6275	338	10	z2[ω	z2[ω	NOUN
ejpam-6275	338	11	]	]	X
ejpam-6275	338	12	3	3	NUM
ejpam-6275	338	13	,	,	PUNCT
ejpam-6275	338	14	then	then	ADV
ejpam-6275	338	15	by	by	ADP
ejpam-6275	338	16	theorem	theorem	ADJ
ejpam-6275	338	17	3.1	3.1	NUM
ejpam-6275	338	18	,	,	PUNCT
ejpam-6275	338	19	φ6(x	φ6(x	NOUN
ejpam-6275	338	20	)	)	PUNCT
ejpam-6275	338	21	=	=	SYM
ejpam-6275	338	22	(	(	PUNCT
ejpam-6275	338	23	x−	x−	PROPN
ejpam-6275	338	24	β6	β6	PROPN
ejpam-6275	338	25	)	)	PUNCT
ejpam-6275	339	1	=	=	PRON
ejpam-6275	340	1	(	(	PUNCT
ejpam-6275	340	2	x+	x+	PROPN
ejpam-6275	340	3	ω	ω	NUM
ejpam-6275	340	4	)	)	PUNCT
ejpam-6275	340	5	.	.	PUNCT
ejpam-6275	341	1	the	the	DET
ejpam-6275	341	2	generator	generator	NOUN
ejpam-6275	341	3	polynomial	polynomial	NOUN
ejpam-6275	341	4	is	be	AUX
ejpam-6275	341	5	g(x	g(x	NOUN
ejpam-6275	341	6	)	)	PUNCT
ejpam-6275	342	1	=	=	SYM
ejpam-6275	342	2	lcm(φ1(x	lcm(φ1(x	NOUN
ejpam-6275	342	3	)	)	PUNCT
ejpam-6275	342	4	,	,	PUNCT
ejpam-6275	342	5	φ2(x	φ2(x	NOUN
ejpam-6275	342	6	)	)	PUNCT
ejpam-6275	342	7	,	,	PUNCT
ejpam-6275	342	8	φ3(x	φ3(x	PROPN
ejpam-6275	342	9	)	)	PUNCT
ejpam-6275	342	10	,	,	PUNCT
ejpam-6275	342	11	φ4(x	φ4(x	PROPN
ejpam-6275	342	12	)	)	PUNCT
ejpam-6275	342	13	,	,	PUNCT
ejpam-6275	342	14	φ5(x	φ5(x	NOUN
ejpam-6275	342	15	)	)	PUNCT
ejpam-6275	342	16	,	,	PUNCT
ejpam-6275	342	17	φ6(x	φ6(x	NOUN
ejpam-6275	342	18	)	)	PUNCT
ejpam-6275	342	19	)	)	PUNCT
ejpam-6275	343	1	=	=	SYM
ejpam-6275	343	2	(	(	PUNCT
ejpam-6275	343	3	x3+ω)(x3	x3+ω)(x3	NUM
ejpam-6275	343	4	+	+	NOUN
ejpam-6275	343	5	1+ω)(x+1+ω)(x+ω	1+ω)(x+1+ω)(x+ω	NOUN
ejpam-6275	343	6	)	)	PUNCT
ejpam-6275	343	7	=	=	SYM
ejpam-6275	343	8	(	(	PUNCT
ejpam-6275	343	9	x6	x6	NOUN
ejpam-6275	343	10	+	+	CCONJ
ejpam-6275	343	11	ωx3	ωx3	X
ejpam-6275	343	12	+	+	CCONJ
ejpam-6275	343	13	(	(	PUNCT
ejpam-6275	343	14	1	1	NUM
ejpam-6275	343	15	+	+	X
ejpam-6275	343	16	ω)x3	ω)x3	NOUN
ejpam-6275	343	17	+	+	CCONJ
ejpam-6275	343	18	1)(x+	1)(x+	NUM
ejpam-6275	343	19	1	1	NUM
ejpam-6275	343	20	+	+	CCONJ
ejpam-6275	343	21	ω)(x+	ω)(x+	PROPN
ejpam-6275	343	22	ω	ω	NOUN
ejpam-6275	343	23	)	)	PUNCT
ejpam-6275	343	24	=	=	SYM
ejpam-6275	343	25	(	(	PUNCT
ejpam-6275	343	26	x6	x6	PROPN
ejpam-6275	343	27	+	+	CCONJ
ejpam-6275	343	28	x3	x3	ADJ
ejpam-6275	343	29	+	+	CCONJ
ejpam-6275	343	30	1)(x+	1)(x+	NUM
ejpam-6275	343	31	1	1	NUM
ejpam-6275	343	32	+	+	CCONJ
ejpam-6275	343	33	ω)(x+	ω)(x+	PROPN
ejpam-6275	343	34	ω	ω	NOUN
ejpam-6275	343	35	)	)	PUNCT
ejpam-6275	343	36	=	=	PUNCT
ejpam-6275	344	1	(	(	PUNCT
ejpam-6275	344	2	x7	x7	NOUN
ejpam-6275	344	3	+	+	CCONJ
ejpam-6275	345	1	x6(ω	x6(ω	PUNCT
ejpam-6275	346	1	+	+	ADJ
ejpam-6275	346	2	1	1	NUM
ejpam-6275	346	3	)	)	PUNCT
ejpam-6275	346	4	+	+	CCONJ
ejpam-6275	346	5	x4	x4	PROPN
ejpam-6275	346	6	+	+	CCONJ
ejpam-6275	347	1	x3(1	x3(1	PROPN
ejpam-6275	347	2	+	+	NUM
ejpam-6275	347	3	ω	ω	NUM
ejpam-6275	347	4	)	)	PUNCT
ejpam-6275	348	1	+	+	CCONJ
ejpam-6275	348	2	x+	x+	SYM
ejpam-6275	348	3	1	1	NUM
ejpam-6275	348	4	+	+	NUM
ejpam-6275	348	5	ω)(x+	ω)(x+	PROPN
ejpam-6275	348	6	ω	ω	NOUN
ejpam-6275	348	7	)	)	PUNCT
ejpam-6275	348	8	=	=	SYM
ejpam-6275	348	9	x8	x8	NOUN
ejpam-6275	348	10	+	+	CCONJ
ejpam-6275	348	11	x7	x7	NOUN
ejpam-6275	348	12	+	+	CCONJ
ejpam-6275	348	13	x6	x6	PROPN
ejpam-6275	348	14	+	+	CCONJ
ejpam-6275	348	15	x5	x5	PROPN
ejpam-6275	348	16	+	+	CCONJ
ejpam-6275	348	17	x4	x4	PROPN
ejpam-6275	349	1	+	+	CCONJ
ejpam-6275	349	2	x3	x3	ADJ
ejpam-6275	349	3	+	+	CCONJ
ejpam-6275	349	4	x2	x2	PROPN
ejpam-6275	350	1	+	+	CCONJ
ejpam-6275	350	2	x+	x+	ADJ
ejpam-6275	350	3	1	1	X
ejpam-6275	350	4	.	.	PUNCT
ejpam-6275	350	5	m.	m.	PROPN
ejpam-6275	350	6	sajjad	sajjad	PROPN
ejpam-6275	350	7	et	et	PROPN
ejpam-6275	350	8	al	al	PROPN
ejpam-6275	350	9	.	.	PUNCT
ejpam-6275	350	10	/	/	SYM
ejpam-6275	350	11	eur	eur	PROPN
ejpam-6275	350	12	.	.	PUNCT
ejpam-6275	351	1	j.	j.	PROPN
ejpam-6275	351	2	pure	pure	PROPN
ejpam-6275	351	3	appl	appl	PROPN
ejpam-6275	351	4	.	.	PROPN
ejpam-6275	351	5	math	math	PROPN
ejpam-6275	351	6	,	,	PUNCT
ejpam-6275	351	7	18	18	NUM
ejpam-6275	351	8	(	(	PUNCT
ejpam-6275	351	9	3	3	NUM
ejpam-6275	351	10	)	)	PUNCT
ejpam-6275	351	11	(	(	PUNCT
ejpam-6275	351	12	2025	2025	NUM
ejpam-6275	351	13	)	)	PUNCT
ejpam-6275	351	14	,	,	PUNCT
ejpam-6275	351	15	6275	6275	NUM
ejpam-6275	351	16	14	14	NUM
ejpam-6275	351	17	of	of	ADP
ejpam-6275	351	18	36	36	NUM
ejpam-6275	351	19	hence	hence	ADV
ejpam-6275	351	20	,	,	PUNCT
ejpam-6275	351	21	k	k	PROPN
ejpam-6275	351	22	=	=	SYM
ejpam-6275	351	23	9−	9−	NUM
ejpam-6275	351	24	8	8	NUM
ejpam-6275	351	25	=	=	SYM
ejpam-6275	351	26	1	1	X
ejpam-6275	351	27	.	.	PUNCT
ejpam-6275	352	1	so	so	ADV
ejpam-6275	352	2	the	the	DET
ejpam-6275	352	3	shortened	shorten	VERB
ejpam-6275	352	4	bch	bch	PROPN
ejpam-6275	352	5	code	code	NOUN
ejpam-6275	352	6	parameters	parameter	NOUN
ejpam-6275	352	7	are	be	AUX
ejpam-6275	352	8	(	(	PUNCT
ejpam-6275	352	9	9	9	NUM
ejpam-6275	352	10	,	,	PUNCT
ejpam-6275	352	11	1	1	NUM
ejpam-6275	352	12	,	,	PUNCT
ejpam-6275	352	13	7	7	NUM
ejpam-6275	352	14	)	)	PUNCT
ejpam-6275	352	15	.	.	PUNCT
ejpam-6275	353	1	illustration	illustration	NOUN
ejpam-6275	353	2	3.5	3.5	NUM
ejpam-6275	353	3	:	:	PUNCT
ejpam-6275	353	4	construct	construct	VERB
ejpam-6275	353	5	a	a	DET
ejpam-6275	353	6	shortened	shorten	VERB
ejpam-6275	353	7	bch	bch	PROPN
ejpam-6275	353	8	code	code	NOUN
ejpam-6275	353	9	for	for	ADP
ejpam-6275	353	10	a	a	DET
ejpam-6275	353	11	three	three	NUM
ejpam-6275	353	12	-	-	PUNCT
ejpam-6275	353	13	degree	degree	NOUN
ejpam-6275	353	14	polynomial	polynomial	NOUN
ejpam-6275	353	15	by	by	ADP
ejpam-6275	353	16	taking	take	VERB
ejpam-6275	353	17	n	n	X
ejpam-6275	353	18	=	=	NUM
ejpam-6275	353	19	21	21	NUM
ejpam-6275	353	20	over	over	ADP
ejpam-6275	353	21	the	the	DET
ejpam-6275	353	22	eisenstein	eisenstein	PROPN
ejpam-6275	353	23	field	field	PROPN
ejpam-6275	353	24	z2[ω	z2[ω	NOUN
ejpam-6275	353	25	]	]	X
ejpam-6275	353	26	3	3	X
ejpam-6275	353	27	.	.	PUNCT
ejpam-6275	354	1	let	let	VERB
ejpam-6275	354	2	us	we	PRON
ejpam-6275	354	3	take	take	VERB
ejpam-6275	354	4	f(x	f(x	PROPN
ejpam-6275	354	5	)	)	PUNCT
ejpam-6275	355	1	=	=	PUNCT
ejpam-6275	356	1	(	(	PUNCT
ejpam-6275	356	2	x3	x3	VERB
ejpam-6275	356	3	+	+	CCONJ
ejpam-6275	356	4	x)2	x)2	VERB
ejpam-6275	357	1	+	+	CCONJ
ejpam-6275	357	2	x+	x+	SYM
ejpam-6275	357	3	1	1	NUM
ejpam-6275	357	4	+	+	NUM
ejpam-6275	357	5	ω	ω	NOUN
ejpam-6275	357	6	which	which	PRON
ejpam-6275	357	7	is	be	AUX
ejpam-6275	357	8	primitive	primitive	ADJ
ejpam-6275	357	9	and	and	CCONJ
ejpam-6275	357	10	irreducible	irreducible	ADJ
ejpam-6275	357	11	over	over	ADP
ejpam-6275	357	12	z2[ω	z2[ω	NOUN
ejpam-6275	357	13	]	]	PUNCT
ejpam-6275	357	14	.	.	PUNCT
ejpam-6275	358	1	as	as	SCONJ
ejpam-6275	358	2	shown	show	VERB
ejpam-6275	358	3	in	in	ADP
ejpam-6275	358	4	section	section	NOUN
ejpam-6275	358	5	2	2	NUM
ejpam-6275	358	6	,	,	PUNCT
ejpam-6275	358	7	the	the	DET
ejpam-6275	358	8	cardinality	cardinality	NOUN
ejpam-6275	358	9	of	of	ADP
ejpam-6275	358	10	the	the	DET
ejpam-6275	358	11	cyclic	cyclic	ADJ
ejpam-6275	358	12	group	group	NOUN
ejpam-6275	358	13	generated	generate	VERB
ejpam-6275	358	14	by	by	ADP
ejpam-6275	358	15	f(x	f(x	PROPN
ejpam-6275	358	16	)	)	PUNCT
ejpam-6275	358	17	is	be	AUX
ejpam-6275	358	18	63	63	NUM
ejpam-6275	358	19	.	.	PUNCT
ejpam-6275	359	1	since	since	SCONJ
ejpam-6275	359	2	63	63	NUM
ejpam-6275	359	3	21	21	NUM
ejpam-6275	359	4	=	=	SYM
ejpam-6275	359	5	3	3	NUM
ejpam-6275	359	6	,	,	PUNCT
ejpam-6275	359	7	the	the	DET
ejpam-6275	359	8	required	require	VERB
ejpam-6275	359	9	cyclic	cyclic	ADJ
ejpam-6275	359	10	subgroup	subgroup	NOUN
ejpam-6275	359	11	is	be	AUX
ejpam-6275	359	12	generated	generate	VERB
ejpam-6275	359	13	by	by	ADP
ejpam-6275	359	14	⟨β	⟨β	X
ejpam-6275	359	15	=	=	SYM
ejpam-6275	359	16	α3⟩	α3⟩	PROPN
ejpam-6275	359	17	,	,	PUNCT
ejpam-6275	359	18	where	where	SCONJ
ejpam-6275	359	19	β	β	PROPN
ejpam-6275	359	20	is	be	AUX
ejpam-6275	359	21	a	a	DET
ejpam-6275	359	22	root	root	NOUN
ejpam-6275	359	23	of	of	ADP
ejpam-6275	359	24	f(x	f(x	PROPN
ejpam-6275	359	25	)	)	PUNCT
ejpam-6275	359	26	.	.	PUNCT
ejpam-6275	360	1	then	then	ADV
ejpam-6275	360	2	the	the	DET
ejpam-6275	360	3	cyclic	cyclic	ADJ
ejpam-6275	360	4	subgroup	subgroup	NOUN
ejpam-6275	360	5	is	be	AUX
ejpam-6275	360	6	g∗	g∗	NOUN
ejpam-6275	360	7	=	=	SYM
ejpam-6275	360	8	{	{	PUNCT
ejpam-6275	360	9	β	β	X
ejpam-6275	360	10	,	,	PUNCT
ejpam-6275	360	11	β2	β2	VERB
ejpam-6275	360	12	,	,	PUNCT
ejpam-6275	360	13	.	.	PUNCT
ejpam-6275	360	14	.	.	PUNCT
ejpam-6275	360	15	.	.	PUNCT
ejpam-6275	361	1	,	,	PUNCT
ejpam-6275	361	2	β21	β21	NOUN
ejpam-6275	361	3	=	=	SYM
ejpam-6275	361	4	1	1	X
ejpam-6275	361	5	}	}	PUNCT
ejpam-6275	361	6	=	=	SYM
ejpam-6275	361	7	{	{	PUNCT
ejpam-6275	361	8	α3	α3	NOUN
ejpam-6275	361	9	,	,	PUNCT
ejpam-6275	361	10	α6	α6	NOUN
ejpam-6275	361	11	,	,	PUNCT
ejpam-6275	361	12	α9	α9	PROPN
ejpam-6275	361	13	,	,	PUNCT
ejpam-6275	361	14	.	.	PUNCT
ejpam-6275	361	15	.	.	PUNCT
ejpam-6275	362	1	.	.	PUNCT
ejpam-6275	363	1	,	,	PUNCT
ejpam-6275	363	2	α63	α63	NUM
ejpam-6275	363	3	}	}	PUNCT
ejpam-6275	363	4	.	.	PUNCT
ejpam-6275	364	1	thus	thus	ADV
ejpam-6275	364	2	,	,	PUNCT
ejpam-6275	364	3	explicitly	explicitly	ADV
ejpam-6275	364	4	:	:	PUNCT
ejpam-6275	364	5	g∗	g∗	VERB
ejpam-6275	364	6	=	=	SYM
ejpam-6275	364	7	{	{	PUNCT
ejpam-6275	364	8	α3	α3	NOUN
ejpam-6275	364	9	,	,	PUNCT
ejpam-6275	364	10	α6	α6	NOUN
ejpam-6275	364	11	,	,	PUNCT
ejpam-6275	364	12	α9	α9	PROPN
ejpam-6275	364	13	,	,	PUNCT
ejpam-6275	364	14	α12	α12	PROPN
ejpam-6275	364	15	,	,	PUNCT
ejpam-6275	364	16	α15	α15	PROPN
ejpam-6275	364	17	,	,	PUNCT
ejpam-6275	364	18	α18	α18	PROPN
ejpam-6275	364	19	,	,	PUNCT
ejpam-6275	364	20	α21	α21	NUM
ejpam-6275	364	21	,	,	PUNCT
ejpam-6275	364	22	α24	α24	NUM
ejpam-6275	364	23	,	,	PUNCT
ejpam-6275	364	24	α27	α27	NOUN
ejpam-6275	364	25	,	,	PUNCT
ejpam-6275	364	26	α30	α30	X
ejpam-6275	364	27	,	,	PUNCT
ejpam-6275	364	28	α33	α33	PROPN
ejpam-6275	364	29	,	,	PUNCT
ejpam-6275	364	30	α36	α36	PROPN
ejpam-6275	364	31	,	,	PUNCT
ejpam-6275	364	32	α39	α39	NOUN
ejpam-6275	364	33	,	,	PUNCT
ejpam-6275	364	34	α42	α42	NOUN
ejpam-6275	364	35	,	,	PUNCT
ejpam-6275	364	36	α45	α45	NOUN
ejpam-6275	364	37	,	,	PUNCT
ejpam-6275	364	38	α48	α48	NOUN
ejpam-6275	364	39	,	,	PUNCT
ejpam-6275	364	40	α51	α51	NUM
ejpam-6275	364	41	,	,	PUNCT
ejpam-6275	364	42	α54	α54	NUM
ejpam-6275	364	43	,	,	PUNCT
ejpam-6275	364	44	α57	α57	NUM
ejpam-6275	364	45	,	,	PUNCT
ejpam-6275	364	46	α60	α60	NOUN
ejpam-6275	364	47	,	,	PUNCT
ejpam-6275	364	48	α63	α63	NOUN
ejpam-6275	364	49	=	=	SYM
ejpam-6275	364	50	1	1	NUM
ejpam-6275	364	51	}	}	PUNCT
ejpam-6275	364	52	.	.	PUNCT
ejpam-6275	365	1	so	so	ADV
ejpam-6275	365	2	,	,	PUNCT
ejpam-6275	365	3	there	there	PRON
ejpam-6275	365	4	are	be	VERB
ejpam-6275	365	5	ten	ten	NUM
ejpam-6275	365	6	cases	case	NOUN
ejpam-6275	365	7	for	for	ADP
ejpam-6275	365	8	the	the	DET
ejpam-6275	365	9	construction	construction	NOUN
ejpam-6275	365	10	of	of	ADP
ejpam-6275	365	11	shortened	shorten	VERB
ejpam-6275	365	12	bch	bch	PROPN
ejpam-6275	365	13	codes	code	NOUN
ejpam-6275	365	14	over	over	ADP
ejpam-6275	365	15	the	the	DET
ejpam-6275	365	16	ef	ef	PROPN
ejpam-6275	365	17	z2[ω	z2[ω	NOUN
ejpam-6275	365	18	]	]	X
ejpam-6275	365	19	3	3	X
ejpam-6275	365	20	.	.	X
ejpam-6275	365	21	case	case	NOUN
ejpam-6275	365	22	1	1	NUM
ejpam-6275	365	23	:	:	PUNCT
ejpam-6275	365	24	for	for	ADP
ejpam-6275	365	25	n	n	NOUN
ejpam-6275	365	26	=	=	SYM
ejpam-6275	365	27	21	21	NUM
ejpam-6275	365	28	,	,	PUNCT
ejpam-6275	365	29	d	d	NOUN
ejpam-6275	365	30	=	=	SYM
ejpam-6275	365	31	3	3	NUM
ejpam-6275	365	32	,	,	PUNCT
ejpam-6275	365	33	i	i	PRON
ejpam-6275	365	34	=	=	NOUN
ejpam-6275	365	35	1	1	NUM
ejpam-6275	365	36	,	,	PUNCT
ejpam-6275	365	37	2	2	NUM
ejpam-6275	365	38	.	.	X
ejpam-6275	366	1	for	for	ADP
ejpam-6275	366	2	i	i	PRON
ejpam-6275	366	3	=	=	NOUN
ejpam-6275	366	4	1	1	NUM
ejpam-6275	366	5	,	,	PUNCT
ejpam-6275	366	6	let	let	VERB
ejpam-6275	366	7	β	β	PRON
ejpam-6275	366	8	∈	∈	PROPN
ejpam-6275	366	9	z2[ω	z2[ω	NOUN
ejpam-6275	366	10	]	]	X
ejpam-6275	366	11	3	3	X
ejpam-6275	366	12	.	.	PUNCT
ejpam-6275	366	13	by	by	ADP
ejpam-6275	366	14	theorem	theorem	NOUN
ejpam-6275	366	15	3.1	3.1	NUM
ejpam-6275	366	16	,	,	PUNCT
ejpam-6275	366	17	the	the	DET
ejpam-6275	366	18	minimal	minimal	ADJ
ejpam-6275	366	19	polynomial	polynomial	NOUN
ejpam-6275	366	20	of	of	ADP
ejpam-6275	366	21	β	β	PROPN
ejpam-6275	366	22	,	,	PUNCT
ejpam-6275	366	23	β4	β4	PROPN
ejpam-6275	366	24	,	,	PUNCT
ejpam-6275	366	25	β16	β16	PROPN
ejpam-6275	366	26	is	be	AUX
ejpam-6275	366	27	φ1(x	φ1(x	NOUN
ejpam-6275	366	28	)	)	PUNCT
ejpam-6275	366	29	=	=	SYM
ejpam-6275	366	30	(	(	PUNCT
ejpam-6275	366	31	x−β)(x−β4)(x−β16	x−β)(x−β4)(x−β16	X
ejpam-6275	366	32	)	)	PUNCT
ejpam-6275	366	33	=	=	SYM
ejpam-6275	366	34	x3+(β+β4+β16)x2+(β5+β17+β20)x+β21	x3+(β+β4+β16)x2+(β5+β17+β20)x+β21	PROPN
ejpam-6275	366	35	=	=	SYM
ejpam-6275	367	1	x3+(1+ω)x2	x3+(1+ω)x2	PUNCT
ejpam-6275	368	1	+	+	PROPN
ejpam-6275	368	2	1	1	NUM
ejpam-6275	368	3	.	.	PUNCT
ejpam-6275	369	1	for	for	ADP
ejpam-6275	369	2	i	i	PRON
ejpam-6275	369	3	=	=	SYM
ejpam-6275	369	4	2	2	NUM
ejpam-6275	369	5	,	,	PUNCT
ejpam-6275	369	6	the	the	DET
ejpam-6275	369	7	minimal	minimal	ADJ
ejpam-6275	369	8	polynomial	polynomial	NOUN
ejpam-6275	369	9	of	of	ADP
ejpam-6275	369	10	β2	β2	PROPN
ejpam-6275	369	11	,	,	PUNCT
ejpam-6275	369	12	β8	β8	PROPN
ejpam-6275	369	13	,	,	PUNCT
ejpam-6275	369	14	β11	β11	PROPN
ejpam-6275	369	15	is	be	AUX
ejpam-6275	369	16	φ2(x	φ2(x	NOUN
ejpam-6275	369	17	)	)	PUNCT
ejpam-6275	370	1	=	=	SYM
ejpam-6275	370	2	(	(	PUNCT
ejpam-6275	370	3	x−β2)(x−β8)(x−β11	x−β2)(x−β8)(x−β11	X
ejpam-6275	370	4	)	)	PUNCT
ejpam-6275	370	5	=	=	PUNCT
ejpam-6275	370	6	x3+(β2+β8+β11)x2+(β10+β13+β19)x+β21	x3+(β2+β8+β11)x2+(β10+β13+β19)x+β21	PUNCT
ejpam-6275	370	7	=	=	SYM
ejpam-6275	371	1	x3+ωx2	x3+ωx2	PROPN
ejpam-6275	371	2	+	+	PROPN
ejpam-6275	371	3	1	1	NUM
ejpam-6275	371	4	.	.	PUNCT
ejpam-6275	371	5	generator	generator	NOUN
ejpam-6275	371	6	polynomial	polynomial	ADJ
ejpam-6275	371	7	:	:	PUNCT
ejpam-6275	371	8	g(x	g(x	NOUN
ejpam-6275	371	9	)	)	PUNCT
ejpam-6275	371	10	=	=	SYM
ejpam-6275	371	11	φ1(x)φ2(x	φ1(x)φ2(x	NOUN
ejpam-6275	371	12	)	)	PUNCT
ejpam-6275	371	13	=	=	SYM
ejpam-6275	372	1	(	(	PUNCT
ejpam-6275	372	2	x3	x3	VERB
ejpam-6275	372	3	+	+	CCONJ
ejpam-6275	372	4	(	(	PUNCT
ejpam-6275	372	5	1	1	NUM
ejpam-6275	372	6	+	+	X
ejpam-6275	372	7	ω)x2	ω)x2	NUM
ejpam-6275	372	8	+	+	CCONJ
ejpam-6275	372	9	1)(x3	1)(x3	NUM
ejpam-6275	373	1	+	+	CCONJ
ejpam-6275	373	2	ωx2	ωx2	PRON
ejpam-6275	373	3	+	+	CCONJ
ejpam-6275	373	4	1	1	X
ejpam-6275	373	5	)	)	PUNCT
ejpam-6275	373	6	=	=	SYM
ejpam-6275	373	7	x6	x6	PROPN
ejpam-6275	373	8	+	+	CCONJ
ejpam-6275	373	9	x5	x5	PROPN
ejpam-6275	373	10	+	+	CCONJ
ejpam-6275	373	11	x4	x4	PROPN
ejpam-6275	374	1	+	+	CCONJ
ejpam-6275	374	2	x2	x2	PROPN
ejpam-6275	375	1	+	+	CCONJ
ejpam-6275	375	2	x+	x+	ADJ
ejpam-6275	375	3	1	1	X
ejpam-6275	375	4	.	.	X
ejpam-6275	375	5	therefore	therefore	ADV
ejpam-6275	375	6	,	,	PUNCT
ejpam-6275	375	7	k	k	PROPN
ejpam-6275	375	8	=	=	PUNCT
ejpam-6275	375	9	21−	21−	NUM
ejpam-6275	375	10	6	6	NUM
ejpam-6275	375	11	=	=	SYM
ejpam-6275	375	12	15	15	NUM
ejpam-6275	375	13	,	,	PUNCT
ejpam-6275	375	14	and	and	CCONJ
ejpam-6275	375	15	the	the	DET
ejpam-6275	375	16	bch	bch	PROPN
ejpam-6275	375	17	code	code	PROPN
ejpam-6275	375	18	is	be	AUX
ejpam-6275	375	19	(	(	PUNCT
ejpam-6275	375	20	n	n	X
ejpam-6275	375	21	,	,	PUNCT
ejpam-6275	375	22	k	k	NOUN
ejpam-6275	375	23	,	,	PUNCT
ejpam-6275	375	24	d	d	NOUN
ejpam-6275	375	25	)	)	PUNCT
ejpam-6275	375	26	=	=	SYM
ejpam-6275	375	27	(	(	PUNCT
ejpam-6275	375	28	21	21	NUM
ejpam-6275	375	29	,	,	PUNCT
ejpam-6275	375	30	15	15	NUM
ejpam-6275	375	31	,	,	PUNCT
ejpam-6275	375	32	3	3	NUM
ejpam-6275	375	33	)	)	PUNCT
ejpam-6275	375	34	.	.	PUNCT
ejpam-6275	376	1	case	case	NOUN
ejpam-6275	376	2	2	2	NUM
ejpam-6275	376	3	:	:	PUNCT
ejpam-6275	376	4	for	for	ADP
ejpam-6275	376	5	n	n	NOUN
ejpam-6275	376	6	=	=	SYM
ejpam-6275	376	7	21	21	NUM
ejpam-6275	376	8	,	,	PUNCT
ejpam-6275	376	9	d	d	NOUN
ejpam-6275	376	10	=	=	SYM
ejpam-6275	376	11	5	5	NUM
ejpam-6275	376	12	,	,	PUNCT
ejpam-6275	376	13	i	i	PRON
ejpam-6275	376	14	=	=	NOUN
ejpam-6275	376	15	1	1	NUM
ejpam-6275	376	16	,	,	PUNCT
ejpam-6275	376	17	2	2	NUM
ejpam-6275	376	18	,	,	PUNCT
ejpam-6275	376	19	3	3	NUM
ejpam-6275	376	20	,	,	PUNCT
ejpam-6275	376	21	4	4	NUM
ejpam-6275	376	22	.	.	PUNCT
ejpam-6275	377	1	the	the	DET
ejpam-6275	377	2	minimal	minimal	ADJ
ejpam-6275	377	3	polynomials	polynomial	NOUN
ejpam-6275	377	4	are	be	AUX
ejpam-6275	377	5	:	:	PUNCT
ejpam-6275	377	6	φ1(x	φ1(x	NOUN
ejpam-6275	377	7	)	)	PUNCT
ejpam-6275	377	8	=	=	SYM
ejpam-6275	378	1	x3	x3	VERB
ejpam-6275	378	2	+	+	CCONJ
ejpam-6275	378	3	(	(	PUNCT
ejpam-6275	378	4	1	1	NUM
ejpam-6275	378	5	+	+	X
ejpam-6275	378	6	ω)x2	ω)x2	ADJ
ejpam-6275	378	7	+	+	CCONJ
ejpam-6275	378	8	1	1	NUM
ejpam-6275	378	9	,	,	PUNCT
ejpam-6275	378	10	φ2(x	φ2(x	NOUN
ejpam-6275	378	11	)	)	PUNCT
ejpam-6275	378	12	=	=	PUNCT
ejpam-6275	379	1	x3	x3	VERB
ejpam-6275	380	1	+	+	CCONJ
ejpam-6275	380	2	ωx2	ωx2	PRON
ejpam-6275	380	3	+	+	CCONJ
ejpam-6275	380	4	1	1	NUM
ejpam-6275	380	5	,	,	PUNCT
ejpam-6275	380	6	φ3(x	φ3(x	PROPN
ejpam-6275	380	7	)	)	PUNCT
ejpam-6275	380	8	=	=	SYM
ejpam-6275	380	9	(	(	PUNCT
ejpam-6275	381	1	x−	x−	PROPN
ejpam-6275	381	2	β3)(x−	β3)(x−	PROPN
ejpam-6275	381	3	β6)(x−	β6)(x−	X
ejpam-6275	381	4	β12	β12	PROPN
ejpam-6275	381	5	)	)	PUNCT
ejpam-6275	382	1	=	=	PUNCT
ejpam-6275	382	2	x3	x3	VERB
ejpam-6275	382	3	+	+	CCONJ
ejpam-6275	382	4	x+	x+	ADJ
ejpam-6275	382	5	1	1	NUM
ejpam-6275	382	6	,	,	PUNCT
ejpam-6275	382	7	φ4(x	φ4(x	NOUN
ejpam-6275	382	8	)	)	PUNCT
ejpam-6275	382	9	=	=	SYM
ejpam-6275	382	10	φ2(x	φ2(x	NUM
ejpam-6275	382	11	)	)	PUNCT
ejpam-6275	382	12	.	.	PUNCT
ejpam-6275	383	1	generator	generator	NOUN
ejpam-6275	383	2	polynomial	polynomial	ADJ
ejpam-6275	383	3	:	:	PUNCT
ejpam-6275	383	4	g(x	g(x	NOUN
ejpam-6275	383	5	)	)	PUNCT
ejpam-6275	383	6	=	=	PUNCT
ejpam-6275	384	1	φ1(x)φ2(x)φ3(x	φ1(x)φ2(x)φ3(x	PROPN
ejpam-6275	384	2	)	)	PUNCT
ejpam-6275	384	3	=	=	SYM
ejpam-6275	384	4	(	(	PUNCT
ejpam-6275	384	5	x6	x6	PROPN
ejpam-6275	384	6	+	+	CCONJ
ejpam-6275	384	7	x5	x5	PROPN
ejpam-6275	384	8	+	+	CCONJ
ejpam-6275	384	9	x4	x4	PROPN
ejpam-6275	385	1	+	+	CCONJ
ejpam-6275	385	2	x2	x2	PROPN
ejpam-6275	386	1	+	+	CCONJ
ejpam-6275	386	2	x+	x+	ADJ
ejpam-6275	386	3	1)(x3	1)(x3	NUM
ejpam-6275	386	4	+	+	CCONJ
ejpam-6275	386	5	x+	x+	ADJ
ejpam-6275	386	6	1	1	X
ejpam-6275	386	7	)	)	PUNCT
ejpam-6275	386	8	=	=	SYM
ejpam-6275	386	9	x9	x9	NOUN
ejpam-6275	386	10	+	+	NUM
ejpam-6275	386	11	x8	x8	PROPN
ejpam-6275	386	12	+	+	CCONJ
ejpam-6275	386	13	x5	x5	NOUN
ejpam-6275	386	14	+	+	CCONJ
ejpam-6275	386	15	1	1	NUM
ejpam-6275	386	16	.	.	PUNCT
ejpam-6275	387	1	m.	m.	PROPN
ejpam-6275	387	2	sajjad	sajjad	PROPN
ejpam-6275	387	3	et	et	PROPN
ejpam-6275	387	4	al	al	PROPN
ejpam-6275	387	5	.	.	PUNCT
ejpam-6275	387	6	/	/	SYM
ejpam-6275	387	7	eur	eur	PROPN
ejpam-6275	387	8	.	.	PUNCT
ejpam-6275	388	1	j.	j.	PROPN
ejpam-6275	388	2	pure	pure	PROPN
ejpam-6275	388	3	appl	appl	PROPN
ejpam-6275	388	4	.	.	PROPN
ejpam-6275	388	5	math	math	PROPN
ejpam-6275	388	6	,	,	PUNCT
ejpam-6275	388	7	18	18	NUM
ejpam-6275	388	8	(	(	PUNCT
ejpam-6275	388	9	3	3	NUM
ejpam-6275	388	10	)	)	PUNCT
ejpam-6275	388	11	(	(	PUNCT
ejpam-6275	388	12	2025	2025	NUM
ejpam-6275	388	13	)	)	PUNCT
ejpam-6275	388	14	,	,	PUNCT
ejpam-6275	388	15	6275	6275	NUM
ejpam-6275	388	16	15	15	NUM
ejpam-6275	388	17	of	of	ADP
ejpam-6275	388	18	36	36	NUM
ejpam-6275	388	19	therefore	therefore	ADV
ejpam-6275	388	20	,	,	PUNCT
ejpam-6275	388	21	k	k	PROPN
ejpam-6275	388	22	=	=	PUNCT
ejpam-6275	388	23	21−	21−	NOUN
ejpam-6275	388	24	9	9	NUM
ejpam-6275	388	25	=	=	SYM
ejpam-6275	388	26	12	12	NUM
ejpam-6275	388	27	,	,	PUNCT
ejpam-6275	388	28	and	and	CCONJ
ejpam-6275	388	29	the	the	DET
ejpam-6275	388	30	bch	bch	PROPN
ejpam-6275	388	31	code	code	PROPN
ejpam-6275	388	32	is	be	AUX
ejpam-6275	388	33	(	(	PUNCT
ejpam-6275	388	34	n	n	X
ejpam-6275	388	35	,	,	PUNCT
ejpam-6275	388	36	k	k	NOUN
ejpam-6275	388	37	,	,	PUNCT
ejpam-6275	388	38	d	d	NOUN
ejpam-6275	388	39	)	)	PUNCT
ejpam-6275	388	40	=	=	SYM
ejpam-6275	388	41	(	(	PUNCT
ejpam-6275	388	42	21	21	NUM
ejpam-6275	388	43	,	,	PUNCT
ejpam-6275	388	44	12	12	NUM
ejpam-6275	388	45	,	,	PUNCT
ejpam-6275	388	46	5	5	NUM
ejpam-6275	388	47	)	)	PUNCT
ejpam-6275	388	48	.	.	PUNCT
ejpam-6275	389	1	case	case	NOUN
ejpam-6275	389	2	3	3	NUM
ejpam-6275	389	3	:	:	PUNCT
ejpam-6275	389	4	for	for	ADP
ejpam-6275	389	5	n	n	NOUN
ejpam-6275	389	6	=	=	SYM
ejpam-6275	389	7	21	21	NUM
ejpam-6275	389	8	,	,	PUNCT
ejpam-6275	389	9	d	d	NOUN
ejpam-6275	389	10	=	=	SYM
ejpam-6275	389	11	7	7	NUM
ejpam-6275	389	12	,	,	PUNCT
ejpam-6275	389	13	i	i	PRON
ejpam-6275	389	14	=	=	NOUN
ejpam-6275	389	15	1	1	NUM
ejpam-6275	389	16	,	,	PUNCT
ejpam-6275	389	17	2	2	NUM
ejpam-6275	389	18	,	,	PUNCT
ejpam-6275	389	19	3	3	NUM
ejpam-6275	389	20	,	,	PUNCT
ejpam-6275	389	21	4	4	NUM
ejpam-6275	389	22	,	,	PUNCT
ejpam-6275	389	23	5	5	NUM
ejpam-6275	389	24	,	,	PUNCT
ejpam-6275	389	25	6	6	NUM
ejpam-6275	389	26	.	.	PUNCT
ejpam-6275	390	1	the	the	DET
ejpam-6275	390	2	minimal	minimal	ADJ
ejpam-6275	390	3	polynomials	polynomial	NOUN
ejpam-6275	390	4	are	be	AUX
ejpam-6275	390	5	:	:	PUNCT
ejpam-6275	390	6	φ1(x	φ1(x	NOUN
ejpam-6275	390	7	)	)	PUNCT
ejpam-6275	390	8	=	=	SYM
ejpam-6275	391	1	x3	x3	VERB
ejpam-6275	391	2	+	+	CCONJ
ejpam-6275	391	3	(	(	PUNCT
ejpam-6275	391	4	1	1	NUM
ejpam-6275	391	5	+	+	X
ejpam-6275	391	6	ω)x2	ω)x2	ADJ
ejpam-6275	391	7	+	+	CCONJ
ejpam-6275	391	8	1	1	NUM
ejpam-6275	391	9	,	,	PUNCT
ejpam-6275	391	10	φ2(x	φ2(x	NOUN
ejpam-6275	391	11	)	)	PUNCT
ejpam-6275	391	12	=	=	PUNCT
ejpam-6275	392	1	x3	x3	VERB
ejpam-6275	393	1	+	+	CCONJ
ejpam-6275	393	2	ωx2	ωx2	PRON
ejpam-6275	393	3	+	+	CCONJ
ejpam-6275	393	4	1	1	NUM
ejpam-6275	393	5	,	,	PUNCT
ejpam-6275	393	6	φ3(x	φ3(x	PROPN
ejpam-6275	393	7	)	)	PUNCT
ejpam-6275	393	8	=	=	SYM
ejpam-6275	394	1	x3	x3	VERB
ejpam-6275	394	2	+	+	CCONJ
ejpam-6275	394	3	x+	x+	ADJ
ejpam-6275	394	4	1	1	NUM
ejpam-6275	394	5	,	,	PUNCT
ejpam-6275	394	6	φ4(x	φ4(x	NOUN
ejpam-6275	394	7	)	)	PUNCT
ejpam-6275	394	8	=	=	SYM
ejpam-6275	394	9	φ2(x	φ2(x	NUM
ejpam-6275	394	10	)	)	PUNCT
ejpam-6275	394	11	,	,	PUNCT
ejpam-6275	394	12	φ5(x	φ5(x	NOUN
ejpam-6275	394	13	)	)	PUNCT
ejpam-6275	394	14	=	=	SYM
ejpam-6275	394	15	(	(	PUNCT
ejpam-6275	394	16	x−	x−	PROPN
ejpam-6275	394	17	β5)(x−	β5)(x−	PROPN
ejpam-6275	394	18	β17)(x−	β17)(x−	PROPN
ejpam-6275	394	19	β20	β20	PROPN
ejpam-6275	394	20	)	)	PUNCT
ejpam-6275	394	21	=	=	SYM
ejpam-6275	395	1	x3	x3	VERB
ejpam-6275	395	2	+	+	CCONJ
ejpam-6275	395	3	(	(	PUNCT
ejpam-6275	395	4	1	1	NUM
ejpam-6275	395	5	+	+	SYM
ejpam-6275	395	6	ω)x+	ω)x+	NUM
ejpam-6275	395	7	1	1	NUM
ejpam-6275	395	8	,	,	PUNCT
ejpam-6275	395	9	φ6(x	φ6(x	NOUN
ejpam-6275	395	10	)	)	PUNCT
ejpam-6275	395	11	=	=	SYM
ejpam-6275	395	12	φ3(x	φ3(x	PROPN
ejpam-6275	395	13	)	)	PUNCT
ejpam-6275	395	14	.	.	PUNCT
ejpam-6275	396	1	generator	generator	NOUN
ejpam-6275	396	2	polynomial	polynomial	ADJ
ejpam-6275	396	3	:	:	PUNCT
ejpam-6275	396	4	g(x	g(x	NOUN
ejpam-6275	396	5	)	)	PUNCT
ejpam-6275	396	6	=	=	SYM
ejpam-6275	396	7	φ1(x)φ2(x)φ3(x)φ5(x	φ1(x)φ2(x)φ3(x)φ5(x	NOUN
ejpam-6275	396	8	)	)	PUNCT
ejpam-6275	396	9	.	.	PUNCT
ejpam-6275	397	1	[	[	X
ejpam-6275	397	2	polynomial	polynomial	ADJ
ejpam-6275	397	3	multiplication	multiplication	NOUN
ejpam-6275	397	4	can	can	AUX
ejpam-6275	397	5	be	be	AUX
ejpam-6275	397	6	performed	perform	VERB
ejpam-6275	397	7	explicitly	explicitly	ADV
ejpam-6275	397	8	if	if	SCONJ
ejpam-6275	397	9	required	require	VERB
ejpam-6275	397	10	.	.	PUNCT
ejpam-6275	397	11	]	]	PUNCT
ejpam-6275	398	1	therefore	therefore	ADV
ejpam-6275	398	2	,	,	PUNCT
ejpam-6275	398	3	k	k	PROPN
ejpam-6275	398	4	=	=	SYM
ejpam-6275	398	5	21−	21−	PROPN
ejpam-6275	398	6	(	(	PUNCT
ejpam-6275	398	7	degg(x	degg(x	NOUN
ejpam-6275	398	8	)	)	PUNCT
ejpam-6275	398	9	)	)	PUNCT
ejpam-6275	398	10	,	,	PUNCT
ejpam-6275	398	11	and	and	CCONJ
ejpam-6275	398	12	the	the	DET
ejpam-6275	398	13	bch	bch	PROPN
ejpam-6275	398	14	code	code	PROPN
ejpam-6275	398	15	is	be	AUX
ejpam-6275	398	16	(	(	PUNCT
ejpam-6275	398	17	n	n	X
ejpam-6275	398	18	,	,	PUNCT
ejpam-6275	398	19	k	k	NOUN
ejpam-6275	398	20	,	,	PUNCT
ejpam-6275	398	21	d	d	NOUN
ejpam-6275	398	22	)	)	PUNCT
ejpam-6275	398	23	=	=	SYM
ejpam-6275	398	24	(	(	PUNCT
ejpam-6275	398	25	21	21	NUM
ejpam-6275	398	26	,	,	PUNCT
ejpam-6275	398	27	9	9	NUM
ejpam-6275	398	28	,	,	PUNCT
ejpam-6275	398	29	7	7	NUM
ejpam-6275	398	30	)	)	PUNCT
ejpam-6275	398	31	.	.	PUNCT
ejpam-6275	399	1	case	case	NOUN
ejpam-6275	399	2	4	4	NUM
ejpam-6275	399	3	:	:	PUNCT
ejpam-6275	399	4	for	for	ADP
ejpam-6275	399	5	n	n	NOUN
ejpam-6275	399	6	=	=	SYM
ejpam-6275	399	7	21	21	NUM
ejpam-6275	399	8	and	and	CCONJ
ejpam-6275	399	9	d	d	NOUN
ejpam-6275	399	10	=	=	SYM
ejpam-6275	399	11	9	9	NUM
ejpam-6275	399	12	,	,	PUNCT
ejpam-6275	399	13	i	i	PRON
ejpam-6275	399	14	=	=	NOUN
ejpam-6275	399	15	1	1	NUM
ejpam-6275	399	16	,	,	PUNCT
ejpam-6275	399	17	2	2	NUM
ejpam-6275	399	18	,	,	PUNCT
ejpam-6275	399	19	3	3	NUM
ejpam-6275	399	20	,	,	PUNCT
ejpam-6275	399	21	4	4	NUM
ejpam-6275	399	22	,	,	PUNCT
ejpam-6275	399	23	5	5	NUM
ejpam-6275	399	24	,	,	PUNCT
ejpam-6275	399	25	6	6	NUM
ejpam-6275	399	26	,	,	PUNCT
ejpam-6275	399	27	7	7	NUM
ejpam-6275	399	28	,	,	PUNCT
ejpam-6275	399	29	8	8	NUM
ejpam-6275	399	30	.	.	NOUN
ejpam-6275	399	31	•	•	NOUN
ejpam-6275	399	32	for	for	ADP
ejpam-6275	399	33	i	i	PRON
ejpam-6275	399	34	=	=	NOUN
ejpam-6275	399	35	1	1	NUM
ejpam-6275	399	36	,	,	PUNCT
ejpam-6275	399	37	let	let	VERB
ejpam-6275	399	38	β	β	PRON
ejpam-6275	399	39	∈	∈	PROPN
ejpam-6275	399	40	z2[ω	z2[ω	NOUN
ejpam-6275	399	41	]	]	X
ejpam-6275	399	42	3	3	NUM
ejpam-6275	399	43	,	,	PUNCT
ejpam-6275	399	44	then	then	ADV
ejpam-6275	399	45	by	by	ADP
ejpam-6275	399	46	theorem	theorem	ADJ
ejpam-6275	399	47	3.1	3.1	NUM
ejpam-6275	399	48	,	,	PUNCT
ejpam-6275	399	49	β	β	X
ejpam-6275	399	50	,	,	PUNCT
ejpam-6275	399	51	β4	β4	PROPN
ejpam-6275	399	52	,	,	PUNCT
ejpam-6275	399	53	β16	β16	PROPN
ejpam-6275	399	54	has	have	VERB
ejpam-6275	399	55	the	the	DET
ejpam-6275	399	56	minimal	minimal	ADJ
ejpam-6275	399	57	polynomial	polynomial	ADJ
ejpam-6275	399	58	φ1(x	φ1(x	NOUN
ejpam-6275	399	59	)	)	PUNCT
ejpam-6275	399	60	=	=	SYM
ejpam-6275	399	61	(	(	PUNCT
ejpam-6275	399	62	x−β)(x−β4)(x−β16	x−β)(x−β4)(x−β16	X
ejpam-6275	399	63	)	)	PUNCT
ejpam-6275	399	64	=	=	SYM
ejpam-6275	399	65	x3+(β+β4+β16)x2+(β5+β17+β20)x+β21	x3+(β+β4+β16)x2+(β5+β17+β20)x+β21	PROPN
ejpam-6275	399	66	=	=	SYM
ejpam-6275	399	67	x3+(1+ω)x2	x3+(1+ω)x2	PUNCT
ejpam-6275	400	1	+	+	PROPN
ejpam-6275	400	2	1	1	NUM
ejpam-6275	400	3	.	.	NOUN
ejpam-6275	400	4	•	•	NOUN
ejpam-6275	400	5	for	for	ADP
ejpam-6275	400	6	i	i	PRON
ejpam-6275	400	7	=	=	SYM
ejpam-6275	400	8	2	2	NUM
ejpam-6275	400	9	,	,	PUNCT
ejpam-6275	400	10	let	let	VERB
ejpam-6275	400	11	β2	β2	PROPN
ejpam-6275	400	12	∈	∈	PROPN
ejpam-6275	400	13	z2[ω	z2[ω	NOUN
ejpam-6275	400	14	]	]	X
ejpam-6275	400	15	3	3	NUM
ejpam-6275	400	16	,	,	PUNCT
ejpam-6275	400	17	then	then	ADV
ejpam-6275	400	18	φ2(x	φ2(x	NUM
ejpam-6275	400	19	)	)	PUNCT
ejpam-6275	400	20	=	=	SYM
ejpam-6275	400	21	(	(	PUNCT
ejpam-6275	400	22	x−β2)(x−β8)(x−β11	x−β2)(x−β8)(x−β11	X
ejpam-6275	400	23	)	)	PUNCT
ejpam-6275	400	24	=	=	PUNCT
ejpam-6275	400	25	x3+(β2+β8+β11)x2+(β10+β13+β19)x+β21	x3+(β2+β8+β11)x2+(β10+β13+β19)x+β21	PUNCT
ejpam-6275	400	26	=	=	SYM
ejpam-6275	401	1	x3+ωx2	x3+ωx2	PROPN
ejpam-6275	402	1	+	+	PROPN
ejpam-6275	402	2	1	1	NUM
ejpam-6275	402	3	.	.	NOUN
ejpam-6275	402	4	•	•	NOUN
ejpam-6275	402	5	for	for	ADP
ejpam-6275	402	6	i	i	PRON
ejpam-6275	402	7	=	=	SYM
ejpam-6275	402	8	3	3	NUM
ejpam-6275	402	9	,	,	PUNCT
ejpam-6275	402	10	let	let	VERB
ejpam-6275	402	11	β3	β3	VERB
ejpam-6275	402	12	∈	∈	PROPN
ejpam-6275	402	13	z2[ω	z2[ω	NOUN
ejpam-6275	402	14	]	]	X
ejpam-6275	402	15	3	3	NUM
ejpam-6275	402	16	,	,	PUNCT
ejpam-6275	402	17	then	then	ADV
ejpam-6275	402	18	φ3(x	φ3(x	NUM
ejpam-6275	402	19	)	)	PUNCT
ejpam-6275	403	1	=	=	SYM
ejpam-6275	403	2	(	(	PUNCT
ejpam-6275	403	3	x−β3)(x−β6)(x−β12	x−β3)(x−β6)(x−β12	X
ejpam-6275	403	4	)	)	PUNCT
ejpam-6275	403	5	=	=	SYM
ejpam-6275	403	6	x3+(β3+β6+β12)x2+(β9+β18+β15)x+β21	x3+(β3+β6+β12)x2+(β9+β18+β15)x+β21	X
ejpam-6275	403	7	=	=	PUNCT
ejpam-6275	404	1	x3+x+1	x3+x+1	NOUN
ejpam-6275	404	2	.	.	NOUN
ejpam-6275	405	1	•	•	NOUN
ejpam-6275	405	2	for	for	ADP
ejpam-6275	405	3	i	i	PRON
ejpam-6275	405	4	=	=	NOUN
ejpam-6275	405	5	4	4	NUM
ejpam-6275	405	6	,	,	PUNCT
ejpam-6275	405	7	let	let	VERB
ejpam-6275	405	8	β4	β4	PROPN
ejpam-6275	405	9	∈	∈	PROPN
ejpam-6275	405	10	z2[ω	z2[ω	NOUN
ejpam-6275	405	11	]	]	X
ejpam-6275	405	12	3	3	NUM
ejpam-6275	405	13	,	,	PUNCT
ejpam-6275	405	14	then	then	ADV
ejpam-6275	405	15	φ4(x	φ4(x	NOUN
ejpam-6275	405	16	)	)	PUNCT
ejpam-6275	405	17	=	=	SYM
ejpam-6275	405	18	(	(	PUNCT
ejpam-6275	405	19	x−	x−	PROPN
ejpam-6275	405	20	β2)(x−	β2)(x−	PROPN
ejpam-6275	405	21	β8)(x−	β8)(x−	X
ejpam-6275	405	22	β11	β11	PROPN
ejpam-6275	405	23	)	)	PUNCT
ejpam-6275	405	24	=	=	PUNCT
ejpam-6275	406	1	x3	x3	VERB
ejpam-6275	407	1	+	+	CCONJ
ejpam-6275	407	2	ωx2	ωx2	PRON
ejpam-6275	407	3	+	+	CCONJ
ejpam-6275	407	4	1	1	NUM
ejpam-6275	407	5	=	=	SYM
ejpam-6275	407	6	φ2(x	φ2(x	NUM
ejpam-6275	407	7	)	)	PUNCT
ejpam-6275	407	8	.	.	PUNCT
ejpam-6275	408	1	•	•	NOUN
ejpam-6275	408	2	for	for	ADP
ejpam-6275	408	3	i	i	PRON
ejpam-6275	408	4	=	=	SYM
ejpam-6275	408	5	5	5	NUM
ejpam-6275	408	6	,	,	PUNCT
ejpam-6275	408	7	let	let	VERB
ejpam-6275	408	8	β5	β5	NOUN
ejpam-6275	408	9	∈	∈	PROPN
ejpam-6275	408	10	z2[ω	z2[ω	NOUN
ejpam-6275	408	11	]	]	X
ejpam-6275	408	12	3	3	NUM
ejpam-6275	408	13	,	,	PUNCT
ejpam-6275	408	14	then	then	ADV
ejpam-6275	408	15	φ5(x	φ5(x	NOUN
ejpam-6275	408	16	)	)	PUNCT
ejpam-6275	408	17	=	=	SYM
ejpam-6275	408	18	(	(	PUNCT
ejpam-6275	408	19	x−β5)(x−β17)(x−β20	x−β5)(x−β17)(x−β20	PROPN
ejpam-6275	408	20	)	)	PUNCT
ejpam-6275	408	21	=	=	SYM
ejpam-6275	409	1	x3+(β5+β17+β20)x2+(β+β4+β16)x+β21	x3+(β5+β17+β20)x2+(β+β4+β16)x+β21	PROPN
ejpam-6275	409	2	=	=	PUNCT
ejpam-6275	409	3	x3+(1+ω)x+1	x3+(1+ω)x+1	PROPN
ejpam-6275	409	4	.	.	PUNCT
ejpam-6275	410	1	m.	m.	PROPN
ejpam-6275	410	2	sajjad	sajjad	PROPN
ejpam-6275	410	3	et	et	PROPN
ejpam-6275	410	4	al	al	PROPN
ejpam-6275	410	5	.	.	PUNCT
ejpam-6275	410	6	/	/	SYM
ejpam-6275	410	7	eur	eur	PROPN
ejpam-6275	410	8	.	.	PUNCT
ejpam-6275	411	1	j.	j.	PROPN
ejpam-6275	411	2	pure	pure	PROPN
ejpam-6275	411	3	appl	appl	PROPN
ejpam-6275	411	4	.	.	PROPN
ejpam-6275	411	5	math	math	PROPN
ejpam-6275	411	6	,	,	PUNCT
ejpam-6275	411	7	18	18	NUM
ejpam-6275	411	8	(	(	PUNCT
ejpam-6275	411	9	3	3	NUM
ejpam-6275	411	10	)	)	PUNCT
ejpam-6275	411	11	(	(	PUNCT
ejpam-6275	411	12	2025	2025	NUM
ejpam-6275	411	13	)	)	PUNCT
ejpam-6275	411	14	,	,	PUNCT
ejpam-6275	411	15	6275	6275	NUM
ejpam-6275	411	16	16	16	NUM
ejpam-6275	411	17	of	of	ADP
ejpam-6275	411	18	36	36	NUM
ejpam-6275	411	19	•	•	NOUN
ejpam-6275	411	20	for	for	ADP
ejpam-6275	411	21	i	i	PRON
ejpam-6275	411	22	=	=	SYM
ejpam-6275	411	23	6	6	NUM
ejpam-6275	411	24	,	,	PUNCT
ejpam-6275	411	25	let	let	VERB
ejpam-6275	411	26	β6	β6	PROPN
ejpam-6275	411	27	∈	∈	PROPN
ejpam-6275	411	28	z2[ω	z2[ω	NOUN
ejpam-6275	411	29	]	]	X
ejpam-6275	411	30	3	3	NUM
ejpam-6275	411	31	,	,	PUNCT
ejpam-6275	411	32	then	then	ADV
ejpam-6275	411	33	φ6(x	φ6(x	PROPN
ejpam-6275	411	34	)	)	PUNCT
ejpam-6275	411	35	=	=	SYM
ejpam-6275	411	36	(	(	PUNCT
ejpam-6275	411	37	x−	x−	PROPN
ejpam-6275	411	38	β3)(x−	β3)(x−	PROPN
ejpam-6275	411	39	β6)(x−	β6)(x−	X
ejpam-6275	411	40	β12	β12	PROPN
ejpam-6275	411	41	)	)	PUNCT
ejpam-6275	411	42	=	=	PUNCT
ejpam-6275	412	1	x3	x3	VERB
ejpam-6275	413	1	+	+	CCONJ
ejpam-6275	413	2	x+	x+	SYM
ejpam-6275	413	3	1	1	X
ejpam-6275	413	4	=	=	SYM
ejpam-6275	413	5	φ3(x	φ3(x	PROPN
ejpam-6275	413	6	)	)	PUNCT
ejpam-6275	413	7	.	.	PUNCT
ejpam-6275	414	1	•	•	NOUN
ejpam-6275	414	2	for	for	ADP
ejpam-6275	414	3	i	i	PRON
ejpam-6275	414	4	=	=	SYM
ejpam-6275	414	5	7	7	NUM
ejpam-6275	414	6	,	,	PUNCT
ejpam-6275	414	7	let	let	VERB
ejpam-6275	414	8	β7	β7	ADJ
ejpam-6275	414	9	∈	∈	PROPN
ejpam-6275	414	10	z2[ω	z2[ω	NOUN
ejpam-6275	414	11	]	]	X
ejpam-6275	414	12	3	3	NUM
ejpam-6275	414	13	,	,	PUNCT
ejpam-6275	414	14	then	then	ADV
ejpam-6275	414	15	φ7(x	φ7(x	PROPN
ejpam-6275	414	16	)	)	PUNCT
ejpam-6275	414	17	=	=	SYM
ejpam-6275	414	18	(	(	PUNCT
ejpam-6275	414	19	x−	x−	PROPN
ejpam-6275	414	20	β7	β7	PROPN
ejpam-6275	414	21	)	)	PUNCT
ejpam-6275	415	1	=	=	PUNCT
ejpam-6275	415	2	x+	x+	PUNCT
ejpam-6275	416	1	1	1	NUM
ejpam-6275	416	2	+	+	NUM
ejpam-6275	416	3	ω	ω	NUM
ejpam-6275	416	4	.	.	NOUN
ejpam-6275	416	5	•	•	NUM
ejpam-6275	416	6	for	for	ADP
ejpam-6275	416	7	i	i	PRON
ejpam-6275	416	8	=	=	NOUN
ejpam-6275	416	9	8	8	NUM
ejpam-6275	416	10	,	,	PUNCT
ejpam-6275	416	11	let	let	VERB
ejpam-6275	416	12	β8	β8	PROPN
ejpam-6275	416	13	∈	∈	PROPN
ejpam-6275	416	14	z2[ω	z2[ω	NOUN
ejpam-6275	416	15	]	]	X
ejpam-6275	416	16	3	3	NUM
ejpam-6275	416	17	,	,	PUNCT
ejpam-6275	416	18	then	then	ADV
ejpam-6275	416	19	φ8(x	φ8(x	NOUN
ejpam-6275	416	20	)	)	PUNCT
ejpam-6275	416	21	=	=	SYM
ejpam-6275	416	22	(	(	PUNCT
ejpam-6275	416	23	x−	x−	PROPN
ejpam-6275	416	24	β2)(x−	β2)(x−	PROPN
ejpam-6275	416	25	β8)(x−	β8)(x−	X
ejpam-6275	416	26	β11	β11	PROPN
ejpam-6275	416	27	)	)	PUNCT
ejpam-6275	416	28	=	=	PUNCT
ejpam-6275	417	1	x3	x3	VERB
ejpam-6275	418	1	+	+	CCONJ
ejpam-6275	418	2	ωx2	ωx2	PRON
ejpam-6275	418	3	+	+	CCONJ
ejpam-6275	418	4	1	1	NUM
ejpam-6275	418	5	=	=	SYM
ejpam-6275	418	6	φ2(x	φ2(x	NUM
ejpam-6275	418	7	)	)	PUNCT
ejpam-6275	418	8	.	.	PUNCT
ejpam-6275	419	1	now	now	ADV
ejpam-6275	419	2	generator	generator	NOUN
ejpam-6275	419	3	polynomial	polynomial	NOUN
ejpam-6275	419	4	is	be	AUX
ejpam-6275	419	5	g(x	g(x	NOUN
ejpam-6275	419	6	)	)	PUNCT
ejpam-6275	420	1	=	=	SYM
ejpam-6275	420	2	φ1(x)φ2(x)φ3(x)φ5(x)φ7(x	φ1(x)φ2(x)φ3(x)φ5(x)φ7(x	NOUN
ejpam-6275	420	3	)	)	PUNCT
ejpam-6275	420	4	=	=	SYM
ejpam-6275	420	5	(	(	PUNCT
ejpam-6275	420	6	x12	x12	NUM
ejpam-6275	420	7	+	+	CCONJ
ejpam-6275	420	8	x11	x11	NOUN
ejpam-6275	421	1	+	+	CCONJ
ejpam-6275	421	2	(	(	PUNCT
ejpam-6275	421	3	1	1	NUM
ejpam-6275	421	4	+	+	CCONJ
ejpam-6275	421	5	ω)x10	ω)x10	NUM
ejpam-6275	421	6	+	+	CCONJ
ejpam-6275	421	7	ωx9	ωx9	NOUN
ejpam-6275	421	8	+	+	CCONJ
ejpam-6275	421	9	(	(	PUNCT
ejpam-6275	421	10	1	1	NUM
ejpam-6275	421	11	+	+	NUM
ejpam-6275	421	12	ω)x6	ω)x6	PROPN
ejpam-6275	421	13	+	+	CCONJ
ejpam-6275	421	14	x5	x5	PROPN
ejpam-6275	421	15	+	+	CCONJ
ejpam-6275	421	16	x3	x3	ADJ
ejpam-6275	421	17	+	+	CCONJ
ejpam-6275	421	18	(	(	PUNCT
ejpam-6275	421	19	1	1	NUM
ejpam-6275	421	20	+	+	NUM
ejpam-6275	421	21	ω)x+	ω)x+	NUM
ejpam-6275	421	22	1)(x+	1)(x+	NUM
ejpam-6275	421	23	1	1	NUM
ejpam-6275	421	24	+	+	NUM
ejpam-6275	421	25	ω	ω	NUM
ejpam-6275	421	26	)	)	PUNCT
ejpam-6275	421	27	=	=	PUNCT
ejpam-6275	422	1	x13+ωx12+x9+(1+ω)x7+(1+ω)x6+(1+ω)x5+x4+(1+ω)x3+(1+ω)x2+(1+ω)x+1+ω	x13+ωx12+x9+(1+ω)x7+(1+ω)x6+(1+ω)x5+x4+(1+ω)x3+(1+ω)x2+(1+ω)x+1+ω	PROPN
ejpam-6275	422	2	.	.	PUNCT
ejpam-6275	423	1	now	now	ADV
ejpam-6275	423	2	k	k	X
ejpam-6275	423	3	=	=	PUNCT
ejpam-6275	423	4	21−	21−	NUM
ejpam-6275	423	5	13	13	NUM
ejpam-6275	423	6	=	=	SYM
ejpam-6275	423	7	8	8	NUM
ejpam-6275	423	8	.	.	PUNCT
ejpam-6275	424	1	so	so	ADV
ejpam-6275	424	2	(	(	PUNCT
ejpam-6275	424	3	n	n	CCONJ
ejpam-6275	424	4	,	,	PUNCT
ejpam-6275	424	5	k	k	NOUN
ejpam-6275	424	6	,	,	PUNCT
ejpam-6275	424	7	d	d	NOUN
ejpam-6275	424	8	)	)	PUNCT
ejpam-6275	424	9	=	=	SYM
ejpam-6275	424	10	(	(	PUNCT
ejpam-6275	424	11	21	21	NUM
ejpam-6275	424	12	,	,	PUNCT
ejpam-6275	424	13	8	8	NUM
ejpam-6275	424	14	,	,	PUNCT
ejpam-6275	424	15	9	9	NUM
ejpam-6275	424	16	)	)	PUNCT
ejpam-6275	424	17	is	be	AUX
ejpam-6275	424	18	a	a	DET
ejpam-6275	424	19	shortened	shorten	VERB
ejpam-6275	424	20	bch	bch	PROPN
ejpam-6275	424	21	code	code	NOUN
ejpam-6275	424	22	.	.	PUNCT
ejpam-6275	425	1	case	case	NOUN
ejpam-6275	425	2	5	5	NUM
ejpam-6275	425	3	:	:	PUNCT
ejpam-6275	425	4	for	for	ADP
ejpam-6275	425	5	n	n	NOUN
ejpam-6275	425	6	=	=	SYM
ejpam-6275	425	7	21	21	NUM
ejpam-6275	425	8	and	and	CCONJ
ejpam-6275	425	9	d	d	NOUN
ejpam-6275	425	10	=	=	SYM
ejpam-6275	425	11	11	11	NUM
ejpam-6275	425	12	,	,	PUNCT
ejpam-6275	425	13	i	i	PRON
ejpam-6275	425	14	=	=	NOUN
ejpam-6275	425	15	1	1	NUM
ejpam-6275	425	16	,	,	PUNCT
ejpam-6275	425	17	2	2	NUM
ejpam-6275	425	18	,	,	PUNCT
ejpam-6275	425	19	3	3	NUM
ejpam-6275	425	20	,	,	PUNCT
ejpam-6275	425	21	.	.	PUNCT
ejpam-6275	425	22	.	.	PUNCT
ejpam-6275	426	1	.	.	PUNCT
ejpam-6275	427	1	,	,	PUNCT
ejpam-6275	428	1	10	10	NUM
ejpam-6275	428	2	.	.	NOUN
ejpam-6275	428	3	•	•	NUM
ejpam-6275	428	4	for	for	ADP
ejpam-6275	428	5	i	i	PRON
ejpam-6275	428	6	=	=	NOUN
ejpam-6275	428	7	1	1	NUM
ejpam-6275	428	8	,	,	PUNCT
ejpam-6275	428	9	let	let	VERB
ejpam-6275	428	10	β	β	PRON
ejpam-6275	428	11	∈	∈	PROPN
ejpam-6275	428	12	z2[ω	z2[ω	NOUN
ejpam-6275	428	13	]	]	X
ejpam-6275	428	14	3	3	NUM
ejpam-6275	428	15	,	,	PUNCT
ejpam-6275	428	16	then	then	ADV
ejpam-6275	428	17	by	by	ADP
ejpam-6275	428	18	theorem	theorem	ADJ
ejpam-6275	428	19	3.1	3.1	NUM
ejpam-6275	428	20	,	,	PUNCT
ejpam-6275	428	21	β	β	X
ejpam-6275	428	22	,	,	PUNCT
ejpam-6275	428	23	β4	β4	PROPN
ejpam-6275	428	24	and	and	CCONJ
ejpam-6275	428	25	β16	β16	PROPN
ejpam-6275	428	26	have	have	VERB
ejpam-6275	428	27	the	the	DET
ejpam-6275	428	28	minimal	minimal	ADJ
ejpam-6275	428	29	polynomial	polynomial	ADJ
ejpam-6275	428	30	ϕ1(x	ϕ1(x	NUM
ejpam-6275	428	31	)	)	PUNCT
ejpam-6275	428	32	=	=	SYM
ejpam-6275	428	33	(	(	PUNCT
ejpam-6275	428	34	x−β)(x−β4)(x−β16	x−β)(x−β4)(x−β16	X
ejpam-6275	428	35	)	)	PUNCT
ejpam-6275	429	1	=	=	SYM
ejpam-6275	429	2	x3+(β+β4+β16)x2+(β5+β17+β20)x+β21	x3+(β+β4+β16)x2+(β5+β17+β20)x+β21	PROPN
ejpam-6275	429	3	=	=	SYM
ejpam-6275	430	1	x3+(1+ω)x2	x3+(1+ω)x2	PUNCT
ejpam-6275	430	2	+	+	PROPN
ejpam-6275	430	3	1	1	NUM
ejpam-6275	430	4	.	.	NOUN
ejpam-6275	430	5	•	•	NOUN
ejpam-6275	430	6	for	for	ADP
ejpam-6275	430	7	i	i	PRON
ejpam-6275	430	8	=	=	SYM
ejpam-6275	430	9	2	2	NUM
ejpam-6275	430	10	,	,	PUNCT
ejpam-6275	430	11	let	let	VERB
ejpam-6275	430	12	β2	β2	PROPN
ejpam-6275	430	13	∈	∈	PROPN
ejpam-6275	430	14	z2[ω	z2[ω	NOUN
ejpam-6275	430	15	]	]	X
ejpam-6275	430	16	3	3	NUM
ejpam-6275	430	17	,	,	PUNCT
ejpam-6275	430	18	then	then	ADV
ejpam-6275	430	19	by	by	ADP
ejpam-6275	430	20	theorem	theorem	ADJ
ejpam-6275	430	21	3.1	3.1	NUM
ejpam-6275	430	22	,	,	PUNCT
ejpam-6275	430	23	β2	β2	NOUN
ejpam-6275	430	24	,	,	PUNCT
ejpam-6275	430	25	β8	β8	NOUN
ejpam-6275	430	26	,	,	PUNCT
ejpam-6275	430	27	and	and	CCONJ
ejpam-6275	430	28	β11	β11	NOUN
ejpam-6275	430	29	have	have	VERB
ejpam-6275	430	30	the	the	DET
ejpam-6275	430	31	minimal	minimal	ADJ
ejpam-6275	430	32	polynomial	polynomial	ADJ
ejpam-6275	430	33	ϕ2(x	ϕ2(x	PROPN
ejpam-6275	430	34	)	)	PUNCT
ejpam-6275	431	1	=	=	SYM
ejpam-6275	431	2	(	(	PUNCT
ejpam-6275	431	3	x−β2)(x−β8)(x−β11	x−β2)(x−β8)(x−β11	X
ejpam-6275	431	4	)	)	PUNCT
ejpam-6275	431	5	=	=	PUNCT
ejpam-6275	431	6	x3+(β2+β8+β11)x2+(β10+β13+β19)x+β21	x3+(β2+β8+β11)x2+(β10+β13+β19)x+β21	PUNCT
ejpam-6275	431	7	=	=	SYM
ejpam-6275	432	1	x3+ωx2	x3+ωx2	PROPN
ejpam-6275	433	1	+	+	PROPN
ejpam-6275	433	2	1	1	NUM
ejpam-6275	433	3	.	.	NOUN
ejpam-6275	433	4	•	•	NOUN
ejpam-6275	433	5	for	for	ADP
ejpam-6275	433	6	i	i	PRON
ejpam-6275	433	7	=	=	SYM
ejpam-6275	433	8	3	3	NUM
ejpam-6275	433	9	,	,	PUNCT
ejpam-6275	433	10	let	let	VERB
ejpam-6275	433	11	β3	β3	VERB
ejpam-6275	433	12	∈	∈	PROPN
ejpam-6275	433	13	z2[ω	z2[ω	NOUN
ejpam-6275	433	14	]	]	X
ejpam-6275	433	15	3	3	NUM
ejpam-6275	433	16	,	,	PUNCT
ejpam-6275	433	17	then	then	ADV
ejpam-6275	433	18	by	by	ADP
ejpam-6275	433	19	theorem	theorem	ADJ
ejpam-6275	433	20	3.1	3.1	NUM
ejpam-6275	433	21	,	,	PUNCT
ejpam-6275	433	22	β3	β3	ADJ
ejpam-6275	433	23	,	,	PUNCT
ejpam-6275	433	24	β6	β6	PROPN
ejpam-6275	433	25	,	,	PUNCT
ejpam-6275	433	26	and	and	CCONJ
ejpam-6275	433	27	β12	β12	CCONJ
ejpam-6275	433	28	have	have	VERB
ejpam-6275	433	29	the	the	DET
ejpam-6275	433	30	minimal	minimal	ADJ
ejpam-6275	433	31	polynomial	polynomial	ADJ
ejpam-6275	433	32	ϕ3(x	ϕ3(x	PROPN
ejpam-6275	433	33	)	)	PUNCT
ejpam-6275	434	1	=	=	PUNCT
ejpam-6275	434	2	(	(	PUNCT
ejpam-6275	434	3	x−β3)(x−β6)(x−β12	x−β3)(x−β6)(x−β12	X
ejpam-6275	434	4	)	)	PUNCT
ejpam-6275	434	5	=	=	SYM
ejpam-6275	434	6	x3+(β3+β6+β12)x2+(β9+β18+β15)x+β21	x3+(β3+β6+β12)x2+(β9+β18+β15)x+β21	X
ejpam-6275	435	1	=	=	PUNCT
ejpam-6275	435	2	x3+x+1	x3+x+1	NOUN
ejpam-6275	435	3	.	.	NOUN
ejpam-6275	436	1	•	•	NOUN
ejpam-6275	436	2	for	for	ADP
ejpam-6275	436	3	i	i	PRON
ejpam-6275	436	4	=	=	NOUN
ejpam-6275	436	5	4	4	NUM
ejpam-6275	436	6	,	,	PUNCT
ejpam-6275	436	7	let	let	VERB
ejpam-6275	436	8	β4	β4	PROPN
ejpam-6275	436	9	∈	∈	PROPN
ejpam-6275	436	10	z2[ω	z2[ω	NOUN
ejpam-6275	436	11	]	]	X
ejpam-6275	436	12	3	3	NUM
ejpam-6275	436	13	,	,	PUNCT
ejpam-6275	436	14	then	then	ADV
ejpam-6275	436	15	by	by	ADP
ejpam-6275	436	16	theorem	theorem	ADJ
ejpam-6275	436	17	3.1	3.1	NUM
ejpam-6275	436	18	,	,	PUNCT
ejpam-6275	436	19	β2	β2	NOUN
ejpam-6275	436	20	,	,	PUNCT
ejpam-6275	436	21	β8	β8	NOUN
ejpam-6275	436	22	,	,	PUNCT
ejpam-6275	436	23	and	and	CCONJ
ejpam-6275	436	24	β11	β11	NOUN
ejpam-6275	436	25	have	have	VERB
ejpam-6275	436	26	the	the	DET
ejpam-6275	436	27	minimal	minimal	ADJ
ejpam-6275	436	28	polynomial	polynomial	ADJ
ejpam-6275	436	29	ϕ4(x	ϕ4(x	NOUN
ejpam-6275	436	30	)	)	PUNCT
ejpam-6275	436	31	=	=	SYM
ejpam-6275	436	32	(	(	PUNCT
ejpam-6275	436	33	x−	x−	PROPN
ejpam-6275	436	34	β2)(x−	β2)(x−	PROPN
ejpam-6275	436	35	β8)(x−	β8)(x−	X
ejpam-6275	436	36	β11	β11	PROPN
ejpam-6275	436	37	)	)	PUNCT
ejpam-6275	436	38	=	=	PUNCT
ejpam-6275	437	1	x3	x3	VERB
ejpam-6275	438	1	+	+	CCONJ
ejpam-6275	438	2	ωx2	ωx2	PRON
ejpam-6275	438	3	+	+	CCONJ
ejpam-6275	438	4	1	1	NUM
ejpam-6275	438	5	=	=	SYM
ejpam-6275	438	6	ϕ2(x	ϕ2(x	PROPN
ejpam-6275	438	7	)	)	PUNCT
ejpam-6275	438	8	.	.	PUNCT
ejpam-6275	439	1	m.	m.	PROPN
ejpam-6275	439	2	sajjad	sajjad	PROPN
ejpam-6275	439	3	et	et	PROPN
ejpam-6275	439	4	al	al	PROPN
ejpam-6275	439	5	.	.	PUNCT
ejpam-6275	439	6	/	/	SYM
ejpam-6275	439	7	eur	eur	PROPN
ejpam-6275	439	8	.	.	PUNCT
ejpam-6275	440	1	j.	j.	PROPN
ejpam-6275	440	2	pure	pure	PROPN
ejpam-6275	440	3	appl	appl	PROPN
ejpam-6275	440	4	.	.	PROPN
ejpam-6275	440	5	math	math	PROPN
ejpam-6275	440	6	,	,	PUNCT
ejpam-6275	440	7	18	18	NUM
ejpam-6275	440	8	(	(	PUNCT
ejpam-6275	440	9	3	3	NUM
ejpam-6275	440	10	)	)	PUNCT
ejpam-6275	440	11	(	(	PUNCT
ejpam-6275	440	12	2025	2025	NUM
ejpam-6275	440	13	)	)	PUNCT
ejpam-6275	440	14	,	,	PUNCT
ejpam-6275	440	15	6275	6275	NUM
ejpam-6275	440	16	17	17	NUM
ejpam-6275	440	17	of	of	ADP
ejpam-6275	440	18	36	36	NUM
ejpam-6275	440	19	•	•	NOUN
ejpam-6275	440	20	for	for	ADP
ejpam-6275	440	21	i	i	PRON
ejpam-6275	440	22	=	=	SYM
ejpam-6275	440	23	5	5	NUM
ejpam-6275	440	24	,	,	PUNCT
ejpam-6275	440	25	let	let	VERB
ejpam-6275	440	26	β5	β5	NOUN
ejpam-6275	440	27	∈	∈	PROPN
ejpam-6275	440	28	z2[ω	z2[ω	NOUN
ejpam-6275	440	29	]	]	X
ejpam-6275	440	30	3	3	NUM
ejpam-6275	440	31	,	,	PUNCT
ejpam-6275	440	32	then	then	ADV
ejpam-6275	440	33	by	by	ADP
ejpam-6275	440	34	theorem	theorem	ADJ
ejpam-6275	440	35	3.1	3.1	NUM
ejpam-6275	440	36	β5	β5	NOUN
ejpam-6275	440	37	,	,	PUNCT
ejpam-6275	440	38	β17	β17	NOUN
ejpam-6275	440	39	,	,	PUNCT
ejpam-6275	440	40	and	and	CCONJ
ejpam-6275	440	41	β20	β20	PROPN
ejpam-6275	440	42	have	have	VERB
ejpam-6275	440	43	the	the	DET
ejpam-6275	440	44	minimal	minimal	ADJ
ejpam-6275	440	45	polynomial	polynomial	ADJ
ejpam-6275	440	46	ϕ5(x	ϕ5(x	NOUN
ejpam-6275	440	47	)	)	PUNCT
ejpam-6275	440	48	=	=	SYM
ejpam-6275	440	49	(	(	PUNCT
ejpam-6275	440	50	x−β5)(x−β17)(x−β20	x−β5)(x−β17)(x−β20	PROPN
ejpam-6275	440	51	)	)	PUNCT
ejpam-6275	441	1	=	=	SYM
ejpam-6275	441	2	x3+(β5+β17+β20)x2+(β+β4+β16)x+β21	x3+(β5+β17+β20)x2+(β+β4+β16)x+β21	PROPN
ejpam-6275	442	1	=	=	PUNCT
ejpam-6275	443	1	x3+(1+ω)x+1	x3+(1+ω)x+1	PROPN
ejpam-6275	443	2	.	.	NOUN
ejpam-6275	444	1	•	•	NOUN
ejpam-6275	444	2	for	for	ADP
ejpam-6275	444	3	i	i	PRON
ejpam-6275	444	4	=	=	SYM
ejpam-6275	444	5	6	6	NUM
ejpam-6275	444	6	,	,	PUNCT
ejpam-6275	444	7	let	let	VERB
ejpam-6275	444	8	β6	β6	PROPN
ejpam-6275	444	9	∈	∈	PROPN
ejpam-6275	444	10	z2[ω	z2[ω	NOUN
ejpam-6275	444	11	]	]	X
ejpam-6275	444	12	3	3	NUM
ejpam-6275	444	13	,	,	PUNCT
ejpam-6275	444	14	then	then	ADV
ejpam-6275	444	15	by	by	ADP
ejpam-6275	444	16	theorem	theorem	ADJ
ejpam-6275	444	17	3.1	3.1	NUM
ejpam-6275	444	18	,	,	PUNCT
ejpam-6275	444	19	β3	β3	ADJ
ejpam-6275	444	20	,	,	PUNCT
ejpam-6275	444	21	β6	β6	PROPN
ejpam-6275	444	22	,	,	PUNCT
ejpam-6275	444	23	and	and	CCONJ
ejpam-6275	444	24	β12	β12	CCONJ
ejpam-6275	444	25	have	have	VERB
ejpam-6275	444	26	the	the	DET
ejpam-6275	444	27	minimal	minimal	ADJ
ejpam-6275	444	28	polynomial	polynomial	ADJ
ejpam-6275	444	29	ϕ6(x	ϕ6(x	PROPN
ejpam-6275	444	30	)	)	PUNCT
ejpam-6275	444	31	=	=	SYM
ejpam-6275	444	32	(	(	PUNCT
ejpam-6275	444	33	x−	x−	PROPN
ejpam-6275	444	34	β3)(x−	β3)(x−	PROPN
ejpam-6275	444	35	β6)(x−	β6)(x−	X
ejpam-6275	444	36	β12	β12	PROPN
ejpam-6275	444	37	)	)	PUNCT
ejpam-6275	444	38	=	=	PUNCT
ejpam-6275	445	1	x3	x3	VERB
ejpam-6275	446	1	+	+	CCONJ
ejpam-6275	446	2	x+	x+	SYM
ejpam-6275	446	3	1	1	X
ejpam-6275	446	4	=	=	SYM
ejpam-6275	446	5	ϕ3(x	ϕ3(x	PROPN
ejpam-6275	446	6	)	)	PUNCT
ejpam-6275	446	7	.	.	PUNCT
ejpam-6275	447	1	•	•	NOUN
ejpam-6275	447	2	for	for	ADP
ejpam-6275	447	3	i	i	PRON
ejpam-6275	447	4	=	=	SYM
ejpam-6275	447	5	7	7	NUM
ejpam-6275	447	6	,	,	PUNCT
ejpam-6275	447	7	let	let	VERB
ejpam-6275	447	8	β7	β7	ADJ
ejpam-6275	447	9	∈	∈	PROPN
ejpam-6275	447	10	z2[ω	z2[ω	NOUN
ejpam-6275	447	11	]	]	X
ejpam-6275	447	12	3	3	NUM
ejpam-6275	447	13	,	,	PUNCT
ejpam-6275	447	14	then	then	ADV
ejpam-6275	447	15	by	by	ADP
ejpam-6275	447	16	theorem	theorem	ADJ
ejpam-6275	447	17	3.1	3.1	NUM
ejpam-6275	447	18	,	,	PUNCT
ejpam-6275	447	19	β7	β7	PROPN
ejpam-6275	447	20	has	have	VERB
ejpam-6275	447	21	the	the	DET
ejpam-6275	447	22	minimal	minimal	ADJ
ejpam-6275	447	23	polynomial	polynomial	ADJ
ejpam-6275	447	24	ϕ7(x	ϕ7(x	NOUN
ejpam-6275	447	25	)	)	PUNCT
ejpam-6275	447	26	=	=	PUNCT
ejpam-6275	447	27	(	(	PUNCT
ejpam-6275	447	28	x−	x−	PROPN
ejpam-6275	447	29	β7	β7	PROPN
ejpam-6275	447	30	)	)	PUNCT
ejpam-6275	448	1	=	=	PUNCT
ejpam-6275	448	2	x+	x+	PUNCT
ejpam-6275	449	1	1	1	NUM
ejpam-6275	449	2	+	+	NUM
ejpam-6275	449	3	ω	ω	NUM
ejpam-6275	449	4	.	.	NOUN
ejpam-6275	449	5	•	•	NUM
ejpam-6275	449	6	for	for	ADP
ejpam-6275	449	7	i	i	PRON
ejpam-6275	449	8	=	=	NOUN
ejpam-6275	449	9	8	8	NUM
ejpam-6275	449	10	,	,	PUNCT
ejpam-6275	449	11	let	let	VERB
ejpam-6275	449	12	β8	β8	PROPN
ejpam-6275	449	13	∈	∈	PROPN
ejpam-6275	449	14	z2[ω	z2[ω	NOUN
ejpam-6275	449	15	]	]	X
ejpam-6275	449	16	3	3	NUM
ejpam-6275	449	17	,	,	PUNCT
ejpam-6275	449	18	then	then	ADV
ejpam-6275	449	19	by	by	ADP
ejpam-6275	449	20	theorem	theorem	ADJ
ejpam-6275	449	21	3.1	3.1	NUM
ejpam-6275	449	22	,	,	PUNCT
ejpam-6275	449	23	β2	β2	NOUN
ejpam-6275	449	24	,	,	PUNCT
ejpam-6275	449	25	β8	β8	NOUN
ejpam-6275	449	26	,	,	PUNCT
ejpam-6275	449	27	and	and	CCONJ
ejpam-6275	449	28	β11	β11	NOUN
ejpam-6275	449	29	have	have	VERB
ejpam-6275	449	30	the	the	DET
ejpam-6275	449	31	minimal	minimal	ADJ
ejpam-6275	449	32	polynomial	polynomial	ADJ
ejpam-6275	449	33	ϕ8(x	ϕ8(x	PROPN
ejpam-6275	449	34	)	)	PUNCT
ejpam-6275	449	35	=	=	PUNCT
ejpam-6275	449	36	(	(	PUNCT
ejpam-6275	449	37	x−	x−	PROPN
ejpam-6275	449	38	β2)(x−	β2)(x−	PROPN
ejpam-6275	449	39	β8)(x−	β8)(x−	X
ejpam-6275	449	40	β11	β11	PROPN
ejpam-6275	449	41	)	)	PUNCT
ejpam-6275	449	42	=	=	PUNCT
ejpam-6275	450	1	x3	x3	VERB
ejpam-6275	451	1	+	+	CCONJ
ejpam-6275	451	2	ωx2	ωx2	PRON
ejpam-6275	451	3	+	+	CCONJ
ejpam-6275	451	4	1	1	NUM
ejpam-6275	451	5	=	=	SYM
ejpam-6275	451	6	ϕ2(x	ϕ2(x	PROPN
ejpam-6275	451	7	)	)	PUNCT
ejpam-6275	451	8	.	.	PUNCT
ejpam-6275	452	1	•	•	NOUN
ejpam-6275	452	2	for	for	ADP
ejpam-6275	452	3	i	i	PRON
ejpam-6275	452	4	=	=	NOUN
ejpam-6275	452	5	9	9	NUM
ejpam-6275	452	6	,	,	PUNCT
ejpam-6275	452	7	let	let	VERB
ejpam-6275	452	8	β9	β9	PROPN
ejpam-6275	452	9	∈	∈	PROPN
ejpam-6275	452	10	z2[ω	z2[ω	NOUN
ejpam-6275	452	11	]	]	X
ejpam-6275	452	12	3	3	NUM
ejpam-6275	452	13	,	,	PUNCT
ejpam-6275	452	14	then	then	ADV
ejpam-6275	452	15	by	by	ADP
ejpam-6275	452	16	theorem	theorem	ADJ
ejpam-6275	452	17	3.1	3.1	NUM
ejpam-6275	452	18	,	,	PUNCT
ejpam-6275	452	19	β9	β9	PROPN
ejpam-6275	452	20	,	,	PUNCT
ejpam-6275	452	21	β18	β18	ADJ
ejpam-6275	452	22	,	,	PUNCT
ejpam-6275	452	23	and	and	CCONJ
ejpam-6275	452	24	β15	β15	NOUN
ejpam-6275	452	25	have	have	VERB
ejpam-6275	452	26	the	the	DET
ejpam-6275	452	27	minimal	minimal	ADJ
ejpam-6275	452	28	polynomial	polynomial	ADJ
ejpam-6275	452	29	ϕ9(x	ϕ9(x	NOUN
ejpam-6275	452	30	)	)	PUNCT
ejpam-6275	453	1	=	=	SYM
ejpam-6275	453	2	(	(	PUNCT
ejpam-6275	453	3	x−β9)(x−β18)(x−β15	x−β9)(x−β18)(x−β15	PROPN
ejpam-6275	453	4	)	)	PUNCT
ejpam-6275	453	5	=	=	SYM
ejpam-6275	453	6	x3+(β9+β15+β18)x2+(β3+β6+β12)x+β21	x3+(β9+β15+β18)x2+(β3+β6+β12)x+β21	X
ejpam-6275	453	7	=	=	PUNCT
ejpam-6275	454	1	x3+x2	x3+x2	NOUN
ejpam-6275	454	2	+	+	PROPN
ejpam-6275	454	3	1	1	NUM
ejpam-6275	454	4	.	.	NOUN
ejpam-6275	454	5	•	•	NOUN
ejpam-6275	454	6	for	for	ADP
ejpam-6275	454	7	i	i	PRON
ejpam-6275	454	8	=	=	NOUN
ejpam-6275	454	9	10	10	NUM
ejpam-6275	454	10	,	,	PUNCT
ejpam-6275	454	11	let	let	VERB
ejpam-6275	454	12	β10	β10	VERB
ejpam-6275	454	13	∈	∈	PROPN
ejpam-6275	454	14	z2[ω	z2[ω	NOUN
ejpam-6275	454	15	]	]	X
ejpam-6275	454	16	3	3	NUM
ejpam-6275	454	17	,	,	PUNCT
ejpam-6275	454	18	then	then	ADV
ejpam-6275	454	19	by	by	ADP
ejpam-6275	454	20	theorem	theorem	ADJ
ejpam-6275	454	21	3.1	3.1	NUM
ejpam-6275	454	22	,	,	PUNCT
ejpam-6275	454	23	β10	β10	NOUN
ejpam-6275	454	24	,	,	PUNCT
ejpam-6275	454	25	β13	β13	NOUN
ejpam-6275	454	26	,	,	PUNCT
ejpam-6275	454	27	and	and	CCONJ
ejpam-6275	454	28	β19	β19	NOUN
ejpam-6275	454	29	have	have	VERB
ejpam-6275	454	30	the	the	DET
ejpam-6275	454	31	minimal	minimal	ADJ
ejpam-6275	454	32	polynomial	polynomial	ADJ
ejpam-6275	454	33	ϕ10(x	ϕ10(x	NOUN
ejpam-6275	454	34	)	)	PUNCT
ejpam-6275	455	1	=	=	SYM
ejpam-6275	455	2	(	(	PUNCT
ejpam-6275	455	3	x−β10)(x−β13)(x−β19	x−β10)(x−β13)(x−β19	PROPN
ejpam-6275	455	4	)	)	PUNCT
ejpam-6275	455	5	=	=	SYM
ejpam-6275	455	6	x3+(β10+β13+β19)x2+(β2+β8+β11)x+β21	x3+(β10+β13+β19)x2+(β2+β8+β11)x+β21	NOUN
ejpam-6275	455	7	=	=	PUNCT
ejpam-6275	456	1	x3+ωx+1	x3+ωx+1	PROPN
ejpam-6275	456	2	.	.	PUNCT
ejpam-6275	457	1	now	now	ADV
ejpam-6275	457	2	the	the	DET
ejpam-6275	457	3	generator	generator	NOUN
ejpam-6275	457	4	polynomial	polynomial	NOUN
ejpam-6275	457	5	is	be	AUX
ejpam-6275	457	6	g(x	g(x	NOUN
ejpam-6275	457	7	)	)	PUNCT
ejpam-6275	457	8	=	=	SYM
ejpam-6275	457	9	ϕ1(x)ϕ2(x)ϕ3(x)ϕ5(x)ϕ7(x)ϕ9(x)ϕ10(x	ϕ1(x)ϕ2(x)ϕ3(x)ϕ5(x)ϕ7(x)ϕ9(x)ϕ10(x	NOUN
ejpam-6275	457	10	)	)	PUNCT
ejpam-6275	457	11	=	=	SYM
ejpam-6275	458	1	x19+(1+ω)x18+x16+(1+ω)x15+(1+ω)x14+(1+ω)x13+x8+ωx7+(1+ω)x6+x5+(1+ω)x4+x2+ωx+1+ω	x19+(1+ω)x18+x16+(1+ω)x15+(1+ω)x14+(1+ω)x13+x8+ωx7+(1+ω)x6+x5+(1+ω)x4+x2+ωx+1+ω	PROPN
ejpam-6275	458	2	.	.	PUNCT
ejpam-6275	459	1	now	now	ADV
ejpam-6275	459	2	k	k	X
ejpam-6275	459	3	=	=	SYM
ejpam-6275	459	4	21−	21−	NUM
ejpam-6275	459	5	19	19	NUM
ejpam-6275	459	6	=	=	SYM
ejpam-6275	459	7	2	2	NUM
ejpam-6275	459	8	.	.	PUNCT
ejpam-6275	460	1	so	so	ADV
ejpam-6275	460	2	the	the	DET
ejpam-6275	460	3	parameters	parameter	NOUN
ejpam-6275	460	4	are	be	AUX
ejpam-6275	460	5	:	:	PUNCT
ejpam-6275	460	6	(	(	PUNCT
ejpam-6275	460	7	n	n	X
ejpam-6275	460	8	,	,	PUNCT
ejpam-6275	460	9	k	k	NOUN
ejpam-6275	460	10	,	,	PUNCT
ejpam-6275	460	11	d	d	NOUN
ejpam-6275	460	12	)	)	PUNCT
ejpam-6275	460	13	=	=	SYM
ejpam-6275	460	14	(	(	PUNCT
ejpam-6275	460	15	21	21	NUM
ejpam-6275	460	16	,	,	PUNCT
ejpam-6275	460	17	2	2	NUM
ejpam-6275	460	18	,	,	PUNCT
ejpam-6275	460	19	11	11	NUM
ejpam-6275	460	20	)	)	PUNCT
ejpam-6275	460	21	shortened	shorten	VERB
ejpam-6275	460	22	bch	bch	PROPN
ejpam-6275	460	23	code	code	PROPN
ejpam-6275	460	24	.	.	PUNCT
ejpam-6275	461	1	case	case	NOUN
ejpam-6275	461	2	6	6	NUM
ejpam-6275	461	3	:	:	PUNCT
ejpam-6275	461	4	for	for	ADP
ejpam-6275	461	5	n	n	NOUN
ejpam-6275	461	6	=	=	SYM
ejpam-6275	461	7	21	21	NUM
ejpam-6275	461	8	and	and	CCONJ
ejpam-6275	461	9	d	d	NOUN
ejpam-6275	461	10	=	=	SYM
ejpam-6275	461	11	13	13	NUM
ejpam-6275	461	12	,	,	PUNCT
ejpam-6275	461	13	i	i	PRON
ejpam-6275	461	14	=	=	NOUN
ejpam-6275	461	15	1	1	NUM
ejpam-6275	461	16	,	,	PUNCT
ejpam-6275	461	17	2	2	NUM
ejpam-6275	461	18	,	,	PUNCT
ejpam-6275	461	19	.	.	PUNCT
ejpam-6275	461	20	.	.	PUNCT
ejpam-6275	462	1	.	.	PUNCT
ejpam-6275	463	1	,	,	PUNCT
ejpam-6275	463	2	12	12	NUM
ejpam-6275	463	3	.	.	NOUN
ejpam-6275	463	4	•	•	NOUN
ejpam-6275	463	5	for	for	ADP
ejpam-6275	463	6	i	i	PRON
ejpam-6275	463	7	=	=	NOUN
ejpam-6275	463	8	1	1	NUM
ejpam-6275	463	9	,	,	PUNCT
ejpam-6275	463	10	let	let	VERB
ejpam-6275	463	11	β	β	PRON
ejpam-6275	463	12	∈	∈	PROPN
ejpam-6275	463	13	z2[ω	z2[ω	NOUN
ejpam-6275	463	14	]	]	X
ejpam-6275	463	15	3	3	NUM
ejpam-6275	463	16	,	,	PUNCT
ejpam-6275	463	17	then	then	ADV
ejpam-6275	463	18	by	by	ADP
ejpam-6275	463	19	theorem	theorem	ADJ
ejpam-6275	463	20	3.1	3.1	NUM
ejpam-6275	463	21	,	,	PUNCT
ejpam-6275	463	22	β	β	X
ejpam-6275	463	23	,	,	PUNCT
ejpam-6275	463	24	β4	β4	PROPN
ejpam-6275	463	25	,	,	PUNCT
ejpam-6275	463	26	and	and	CCONJ
ejpam-6275	463	27	β16	β16	PROPN
ejpam-6275	463	28	have	have	VERB
ejpam-6275	463	29	the	the	DET
ejpam-6275	463	30	minimal	minimal	ADJ
ejpam-6275	463	31	polynomial	polynomial	ADJ
ejpam-6275	463	32	φ1(x	φ1(x	NOUN
ejpam-6275	463	33	)	)	PUNCT
ejpam-6275	463	34	=	=	SYM
ejpam-6275	463	35	(	(	PUNCT
ejpam-6275	463	36	x−β)(x−β4)(x−β16	x−β)(x−β4)(x−β16	X
ejpam-6275	463	37	)	)	PUNCT
ejpam-6275	464	1	=	=	SYM
ejpam-6275	464	2	x3+(β+β4+β16)x2+(β5+β17+β20)x+β21	x3+(β+β4+β16)x2+(β5+β17+β20)x+β21	PROPN
ejpam-6275	464	3	=	=	SYM
ejpam-6275	465	1	x3+(1+ω)x2	x3+(1+ω)x2	PUNCT
ejpam-6275	465	2	+	+	PROPN
ejpam-6275	465	3	1	1	NUM
ejpam-6275	465	4	.	.	PUNCT
ejpam-6275	465	5	m.	m.	PROPN
ejpam-6275	465	6	sajjad	sajjad	PROPN
ejpam-6275	465	7	et	et	PROPN
ejpam-6275	465	8	al	al	PROPN
ejpam-6275	465	9	.	.	PUNCT
ejpam-6275	465	10	/	/	SYM
ejpam-6275	465	11	eur	eur	PROPN
ejpam-6275	465	12	.	.	PUNCT
ejpam-6275	466	1	j.	j.	PROPN
ejpam-6275	466	2	pure	pure	PROPN
ejpam-6275	466	3	appl	appl	PROPN
ejpam-6275	466	4	.	.	PROPN
ejpam-6275	466	5	math	math	PROPN
ejpam-6275	466	6	,	,	PUNCT
ejpam-6275	466	7	18	18	NUM
ejpam-6275	466	8	(	(	PUNCT
ejpam-6275	466	9	3	3	NUM
ejpam-6275	466	10	)	)	PUNCT
ejpam-6275	466	11	(	(	PUNCT
ejpam-6275	466	12	2025	2025	NUM
ejpam-6275	466	13	)	)	PUNCT
ejpam-6275	466	14	,	,	PUNCT
ejpam-6275	466	15	6275	6275	NUM
ejpam-6275	466	16	18	18	NUM
ejpam-6275	466	17	of	of	ADP
ejpam-6275	466	18	36	36	NUM
ejpam-6275	466	19	•	•	NOUN
ejpam-6275	466	20	for	for	ADP
ejpam-6275	466	21	i	i	PRON
ejpam-6275	466	22	=	=	SYM
ejpam-6275	466	23	2	2	NUM
ejpam-6275	466	24	,	,	PUNCT
ejpam-6275	466	25	let	let	VERB
ejpam-6275	466	26	β2	β2	PROPN
ejpam-6275	466	27	∈	∈	PROPN
ejpam-6275	466	28	z2[ω	z2[ω	NOUN
ejpam-6275	466	29	]	]	X
ejpam-6275	466	30	3	3	NUM
ejpam-6275	466	31	,	,	PUNCT
ejpam-6275	466	32	then	then	ADV
ejpam-6275	466	33	by	by	ADP
ejpam-6275	466	34	theorem	theorem	ADJ
ejpam-6275	466	35	3.1	3.1	NUM
ejpam-6275	466	36	,	,	PUNCT
ejpam-6275	466	37	β2	β2	NOUN
ejpam-6275	466	38	,	,	PUNCT
ejpam-6275	466	39	β8	β8	NOUN
ejpam-6275	466	40	,	,	PUNCT
ejpam-6275	466	41	and	and	CCONJ
ejpam-6275	466	42	β11	β11	NOUN
ejpam-6275	466	43	have	have	VERB
ejpam-6275	466	44	the	the	DET
ejpam-6275	466	45	minimal	minimal	ADJ
ejpam-6275	466	46	polynomial	polynomial	ADJ
ejpam-6275	466	47	φ2(x	φ2(x	NUM
ejpam-6275	466	48	)	)	PUNCT
ejpam-6275	466	49	=	=	PUNCT
ejpam-6275	466	50	(	(	PUNCT
ejpam-6275	466	51	x−β2)(x−β8)(x−β11	x−β2)(x−β8)(x−β11	X
ejpam-6275	466	52	)	)	PUNCT
ejpam-6275	466	53	=	=	PUNCT
ejpam-6275	466	54	x3+(β2+β8+β11)x2+(β10+β13+β19)x+β21	x3+(β2+β8+β11)x2+(β10+β13+β19)x+β21	PUNCT
ejpam-6275	466	55	=	=	SYM
ejpam-6275	467	1	x3+ωx2	x3+ωx2	PROPN
ejpam-6275	468	1	+	+	PROPN
ejpam-6275	468	2	1	1	NUM
ejpam-6275	468	3	.	.	NOUN
ejpam-6275	468	4	•	•	NOUN
ejpam-6275	468	5	for	for	ADP
ejpam-6275	468	6	i	i	PRON
ejpam-6275	468	7	=	=	SYM
ejpam-6275	468	8	3	3	NUM
ejpam-6275	468	9	,	,	PUNCT
ejpam-6275	468	10	let	let	VERB
ejpam-6275	468	11	β3	β3	VERB
ejpam-6275	468	12	∈	∈	PROPN
ejpam-6275	468	13	z2[ω	z2[ω	NOUN
ejpam-6275	468	14	]	]	X
ejpam-6275	468	15	3	3	NUM
ejpam-6275	468	16	,	,	PUNCT
ejpam-6275	468	17	then	then	ADV
ejpam-6275	468	18	by	by	ADP
ejpam-6275	468	19	theorem	theorem	ADJ
ejpam-6275	468	20	3.1	3.1	NUM
ejpam-6275	468	21	,	,	PUNCT
ejpam-6275	468	22	β3	β3	ADJ
ejpam-6275	468	23	,	,	PUNCT
ejpam-6275	468	24	β6	β6	PROPN
ejpam-6275	468	25	,	,	PUNCT
ejpam-6275	468	26	and	and	CCONJ
ejpam-6275	468	27	β12	β12	CCONJ
ejpam-6275	468	28	have	have	VERB
ejpam-6275	468	29	the	the	DET
ejpam-6275	468	30	minimal	minimal	ADJ
ejpam-6275	468	31	polynomial	polynomial	ADJ
ejpam-6275	468	32	φ3(x	φ3(x	PROPN
ejpam-6275	468	33	)	)	PUNCT
ejpam-6275	468	34	=	=	SYM
ejpam-6275	468	35	(	(	PUNCT
ejpam-6275	468	36	x−β3)(x−β6)(x−β12	x−β3)(x−β6)(x−β12	X
ejpam-6275	468	37	)	)	PUNCT
ejpam-6275	468	38	=	=	SYM
ejpam-6275	468	39	x3+(β3+β6+β12)x2+(β9+β18+β15)x+β21	x3+(β3+β6+β12)x2+(β9+β18+β15)x+β21	X
ejpam-6275	469	1	=	=	PUNCT
ejpam-6275	469	2	x3+x+1	x3+x+1	NOUN
ejpam-6275	469	3	.	.	NOUN
ejpam-6275	470	1	•	•	NOUN
ejpam-6275	470	2	for	for	ADP
ejpam-6275	470	3	i	i	PRON
ejpam-6275	470	4	=	=	NOUN
ejpam-6275	470	5	4	4	NUM
ejpam-6275	470	6	,	,	PUNCT
ejpam-6275	470	7	let	let	VERB
ejpam-6275	470	8	β4	β4	PROPN
ejpam-6275	470	9	∈	∈	PROPN
ejpam-6275	470	10	z2[ω	z2[ω	NOUN
ejpam-6275	470	11	]	]	X
ejpam-6275	470	12	3	3	NUM
ejpam-6275	470	13	,	,	PUNCT
ejpam-6275	470	14	then	then	ADV
ejpam-6275	470	15	by	by	ADP
ejpam-6275	470	16	theorem	theorem	ADJ
ejpam-6275	470	17	3.1	3.1	NUM
ejpam-6275	470	18	,	,	PUNCT
ejpam-6275	470	19	β2	β2	NOUN
ejpam-6275	470	20	,	,	PUNCT
ejpam-6275	470	21	β8	β8	NOUN
ejpam-6275	470	22	,	,	PUNCT
ejpam-6275	470	23	and	and	CCONJ
ejpam-6275	470	24	β11	β11	NOUN
ejpam-6275	470	25	have	have	VERB
ejpam-6275	470	26	the	the	DET
ejpam-6275	470	27	minimal	minimal	ADJ
ejpam-6275	470	28	polynomial	polynomial	ADJ
ejpam-6275	470	29	φ4(x	φ4(x	NOUN
ejpam-6275	470	30	)	)	PUNCT
ejpam-6275	470	31	=	=	SYM
ejpam-6275	470	32	(	(	PUNCT
ejpam-6275	470	33	x−	x−	PROPN
ejpam-6275	470	34	β2)(x−	β2)(x−	PROPN
ejpam-6275	470	35	β8)(x−	β8)(x−	X
ejpam-6275	470	36	β11	β11	PROPN
ejpam-6275	470	37	)	)	PUNCT
ejpam-6275	470	38	=	=	PUNCT
ejpam-6275	471	1	x3	x3	VERB
ejpam-6275	472	1	+	+	CCONJ
ejpam-6275	472	2	ωx2	ωx2	PRON
ejpam-6275	472	3	+	+	CCONJ
ejpam-6275	472	4	1	1	NUM
ejpam-6275	472	5	=	=	SYM
ejpam-6275	472	6	φ2(x	φ2(x	NUM
ejpam-6275	472	7	)	)	PUNCT
ejpam-6275	472	8	.	.	PUNCT
ejpam-6275	473	1	•	•	NOUN
ejpam-6275	473	2	for	for	ADP
ejpam-6275	473	3	i	i	PRON
ejpam-6275	473	4	=	=	SYM
ejpam-6275	473	5	5	5	NUM
ejpam-6275	473	6	,	,	PUNCT
ejpam-6275	473	7	let	let	VERB
ejpam-6275	473	8	β5	β5	NOUN
ejpam-6275	473	9	∈	∈	PROPN
ejpam-6275	473	10	z2[ω	z2[ω	NOUN
ejpam-6275	473	11	]	]	X
ejpam-6275	473	12	3	3	NUM
ejpam-6275	473	13	,	,	PUNCT
ejpam-6275	473	14	then	then	ADV
ejpam-6275	473	15	by	by	ADP
ejpam-6275	473	16	theorem	theorem	ADJ
ejpam-6275	473	17	3.1	3.1	NUM
ejpam-6275	473	18	,	,	PUNCT
ejpam-6275	473	19	β5	β5	NOUN
ejpam-6275	473	20	,	,	PUNCT
ejpam-6275	473	21	β17	β17	NUM
ejpam-6275	473	22	,	,	PUNCT
ejpam-6275	473	23	and	and	CCONJ
ejpam-6275	473	24	β20	β20	PROPN
ejpam-6275	473	25	have	have	VERB
ejpam-6275	473	26	the	the	DET
ejpam-6275	473	27	minimal	minimal	ADJ
ejpam-6275	473	28	polynomial	polynomial	ADJ
ejpam-6275	473	29	φ5(x	φ5(x	NOUN
ejpam-6275	473	30	)	)	PUNCT
ejpam-6275	474	1	=	=	SYM
ejpam-6275	474	2	(	(	PUNCT
ejpam-6275	474	3	x−β5)(x−β17)(x−β20	x−β5)(x−β17)(x−β20	PROPN
ejpam-6275	474	4	)	)	PUNCT
ejpam-6275	475	1	=	=	SYM
ejpam-6275	475	2	x3+(β5+β17+β20)x2+(β+β4+β16)x+β21	x3+(β5+β17+β20)x2+(β+β4+β16)x+β21	PROPN
ejpam-6275	476	1	=	=	PUNCT
ejpam-6275	477	1	x3+(1+ω)x+1	x3+(1+ω)x+1	PROPN
ejpam-6275	477	2	.	.	NOUN
ejpam-6275	478	1	•	•	NOUN
ejpam-6275	478	2	for	for	ADP
ejpam-6275	478	3	i	i	PRON
ejpam-6275	478	4	=	=	SYM
ejpam-6275	478	5	6	6	NUM
ejpam-6275	478	6	,	,	PUNCT
ejpam-6275	478	7	let	let	VERB
ejpam-6275	478	8	β6	β6	PROPN
ejpam-6275	478	9	∈	∈	PROPN
ejpam-6275	478	10	z2[ω	z2[ω	NOUN
ejpam-6275	478	11	]	]	X
ejpam-6275	478	12	3	3	NUM
ejpam-6275	478	13	,	,	PUNCT
ejpam-6275	478	14	then	then	ADV
ejpam-6275	478	15	by	by	ADP
ejpam-6275	478	16	theorem	theorem	ADJ
ejpam-6275	478	17	3.1	3.1	NUM
ejpam-6275	478	18	,	,	PUNCT
ejpam-6275	478	19	β3	β3	ADJ
ejpam-6275	478	20	,	,	PUNCT
ejpam-6275	478	21	β6	β6	PROPN
ejpam-6275	478	22	,	,	PUNCT
ejpam-6275	478	23	and	and	CCONJ
ejpam-6275	478	24	β12	β12	CCONJ
ejpam-6275	478	25	have	have	VERB
ejpam-6275	478	26	the	the	DET
ejpam-6275	478	27	minimal	minimal	ADJ
ejpam-6275	478	28	polynomial	polynomial	ADJ
ejpam-6275	478	29	φ6(x	φ6(x	NOUN
ejpam-6275	478	30	)	)	PUNCT
ejpam-6275	478	31	=	=	SYM
ejpam-6275	478	32	(	(	PUNCT
ejpam-6275	479	1	x−	x−	PROPN
ejpam-6275	479	2	β3)(x−	β3)(x−	PROPN
ejpam-6275	479	3	β6)(x−	β6)(x−	X
ejpam-6275	479	4	β12	β12	PROPN
ejpam-6275	479	5	)	)	PUNCT
ejpam-6275	480	1	=	=	PUNCT
ejpam-6275	480	2	x3	x3	VERB
ejpam-6275	480	3	+	+	CCONJ
ejpam-6275	480	4	x+	x+	SYM
ejpam-6275	480	5	1	1	X
ejpam-6275	480	6	=	=	SYM
ejpam-6275	480	7	φ3(x	φ3(x	PROPN
ejpam-6275	480	8	)	)	PUNCT
ejpam-6275	480	9	.	.	PUNCT
ejpam-6275	481	1	•	•	NOUN
ejpam-6275	481	2	for	for	ADP
ejpam-6275	481	3	i	i	PRON
ejpam-6275	481	4	=	=	SYM
ejpam-6275	481	5	7	7	NUM
ejpam-6275	481	6	,	,	PUNCT
ejpam-6275	481	7	let	let	VERB
ejpam-6275	481	8	β7	β7	ADJ
ejpam-6275	481	9	∈	∈	PROPN
ejpam-6275	481	10	z2[ω	z2[ω	NOUN
ejpam-6275	481	11	]	]	X
ejpam-6275	481	12	3	3	NUM
ejpam-6275	481	13	,	,	PUNCT
ejpam-6275	481	14	then	then	ADV
ejpam-6275	481	15	by	by	ADP
ejpam-6275	481	16	theorem	theorem	ADJ
ejpam-6275	481	17	3.1	3.1	NUM
ejpam-6275	481	18	,	,	PUNCT
ejpam-6275	481	19	β7	β7	PROPN
ejpam-6275	481	20	has	have	VERB
ejpam-6275	481	21	the	the	DET
ejpam-6275	481	22	minimal	minimal	ADJ
ejpam-6275	481	23	polynomial	polynomial	ADJ
ejpam-6275	481	24	φ7(x	φ7(x	NOUN
ejpam-6275	481	25	)	)	PUNCT
ejpam-6275	481	26	=	=	PUNCT
ejpam-6275	481	27	(	(	PUNCT
ejpam-6275	481	28	x−	x−	PROPN
ejpam-6275	481	29	β7	β7	PROPN
ejpam-6275	481	30	)	)	PUNCT
ejpam-6275	482	1	=	=	PUNCT
ejpam-6275	482	2	x+	x+	PUNCT
ejpam-6275	483	1	1	1	NUM
ejpam-6275	483	2	+	+	NUM
ejpam-6275	483	3	ω	ω	NUM
ejpam-6275	483	4	.	.	NOUN
ejpam-6275	483	5	•	•	NUM
ejpam-6275	483	6	for	for	ADP
ejpam-6275	483	7	i	i	PRON
ejpam-6275	483	8	=	=	NOUN
ejpam-6275	483	9	8	8	NUM
ejpam-6275	483	10	,	,	PUNCT
ejpam-6275	483	11	let	let	VERB
ejpam-6275	483	12	β8	β8	PROPN
ejpam-6275	483	13	∈	∈	PROPN
ejpam-6275	483	14	z2[ω	z2[ω	NOUN
ejpam-6275	483	15	]	]	X
ejpam-6275	483	16	3	3	NUM
ejpam-6275	483	17	,	,	PUNCT
ejpam-6275	483	18	then	then	ADV
ejpam-6275	483	19	by	by	ADP
ejpam-6275	483	20	theorem	theorem	ADJ
ejpam-6275	483	21	3.1	3.1	NUM
ejpam-6275	483	22	,	,	PUNCT
ejpam-6275	483	23	β2	β2	NOUN
ejpam-6275	483	24	,	,	PUNCT
ejpam-6275	483	25	β8	β8	NOUN
ejpam-6275	483	26	,	,	PUNCT
ejpam-6275	483	27	and	and	CCONJ
ejpam-6275	483	28	β11	β11	NOUN
ejpam-6275	483	29	have	have	VERB
ejpam-6275	483	30	the	the	DET
ejpam-6275	483	31	minimal	minimal	ADJ
ejpam-6275	483	32	polynomial	polynomial	ADJ
ejpam-6275	483	33	φ8(x	φ8(x	NOUN
ejpam-6275	483	34	)	)	PUNCT
ejpam-6275	483	35	=	=	SYM
ejpam-6275	484	1	x3	x3	VERB
ejpam-6275	485	1	+	+	CCONJ
ejpam-6275	485	2	ωx2	ωx2	PRON
ejpam-6275	485	3	+	+	CCONJ
ejpam-6275	485	4	1	1	NUM
ejpam-6275	485	5	=	=	SYM
ejpam-6275	485	6	φ2(x	φ2(x	NUM
ejpam-6275	485	7	)	)	PUNCT
ejpam-6275	485	8	.	.	PUNCT
ejpam-6275	486	1	•	•	NOUN
ejpam-6275	486	2	for	for	ADP
ejpam-6275	486	3	i	i	PRON
ejpam-6275	486	4	=	=	NOUN
ejpam-6275	486	5	9	9	NUM
ejpam-6275	486	6	,	,	PUNCT
ejpam-6275	486	7	let	let	VERB
ejpam-6275	486	8	β9	β9	PROPN
ejpam-6275	486	9	∈	∈	PROPN
ejpam-6275	486	10	z2[ω	z2[ω	NOUN
ejpam-6275	486	11	]	]	X
ejpam-6275	486	12	3	3	NUM
ejpam-6275	486	13	,	,	PUNCT
ejpam-6275	486	14	then	then	ADV
ejpam-6275	486	15	by	by	ADP
ejpam-6275	486	16	theorem	theorem	ADJ
ejpam-6275	486	17	3.1	3.1	NUM
ejpam-6275	486	18	,	,	PUNCT
ejpam-6275	486	19	β9	β9	PROPN
ejpam-6275	486	20	,	,	PUNCT
ejpam-6275	486	21	β18	β18	ADJ
ejpam-6275	486	22	,	,	PUNCT
ejpam-6275	486	23	and	and	CCONJ
ejpam-6275	486	24	β15	β15	NOUN
ejpam-6275	486	25	have	have	VERB
ejpam-6275	486	26	the	the	DET
ejpam-6275	486	27	minimal	minimal	ADJ
ejpam-6275	486	28	polynomial	polynomial	ADJ
ejpam-6275	486	29	φ9(x	φ9(x	NUM
ejpam-6275	486	30	)	)	PUNCT
ejpam-6275	486	31	=	=	SYM
ejpam-6275	487	1	x3	x3	PROPN
ejpam-6275	488	1	+	+	CCONJ
ejpam-6275	488	2	x2	x2	PROPN
ejpam-6275	489	1	+	+	CCONJ
ejpam-6275	490	1	1	1	NUM
ejpam-6275	490	2	.	.	NUM
ejpam-6275	490	3	•	•	NOUN
ejpam-6275	490	4	for	for	ADP
ejpam-6275	490	5	i	i	PRON
ejpam-6275	490	6	=	=	NOUN
ejpam-6275	490	7	10	10	NUM
ejpam-6275	490	8	,	,	PUNCT
ejpam-6275	490	9	let	let	VERB
ejpam-6275	490	10	β10	β10	VERB
ejpam-6275	490	11	∈	∈	PROPN
ejpam-6275	490	12	z2[ω	z2[ω	NOUN
ejpam-6275	490	13	]	]	X
ejpam-6275	490	14	3	3	NUM
ejpam-6275	490	15	,	,	PUNCT
ejpam-6275	490	16	then	then	ADV
ejpam-6275	490	17	by	by	ADP
ejpam-6275	490	18	theorem	theorem	ADJ
ejpam-6275	490	19	3.1	3.1	NUM
ejpam-6275	490	20	,	,	PUNCT
ejpam-6275	490	21	β10	β10	NOUN
ejpam-6275	490	22	,	,	PUNCT
ejpam-6275	490	23	β13	β13	NOUN
ejpam-6275	490	24	,	,	PUNCT
ejpam-6275	490	25	and	and	CCONJ
ejpam-6275	490	26	β19	β19	NOUN
ejpam-6275	490	27	have	have	VERB
ejpam-6275	490	28	the	the	DET
ejpam-6275	490	29	minimal	minimal	ADJ
ejpam-6275	490	30	polynomial	polynomial	ADJ
ejpam-6275	490	31	φ10(x	φ10(x	NOUN
ejpam-6275	490	32	)	)	PUNCT
ejpam-6275	490	33	=	=	SYM
ejpam-6275	491	1	x3	x3	ADJ
ejpam-6275	491	2	+	+	CCONJ
ejpam-6275	491	3	ωx+	ωx+	NOUN
ejpam-6275	491	4	1	1	NUM
ejpam-6275	491	5	.	.	PUNCT
ejpam-6275	492	1	m.	m.	PROPN
ejpam-6275	492	2	sajjad	sajjad	PROPN
ejpam-6275	492	3	et	et	PROPN
ejpam-6275	492	4	al	al	PROPN
ejpam-6275	492	5	.	.	PUNCT
ejpam-6275	492	6	/	/	SYM
ejpam-6275	492	7	eur	eur	PROPN
ejpam-6275	492	8	.	.	PUNCT
ejpam-6275	493	1	j.	j.	PROPN
ejpam-6275	493	2	pure	pure	PROPN
ejpam-6275	493	3	appl	appl	PROPN
ejpam-6275	493	4	.	.	PROPN
ejpam-6275	493	5	math	math	PROPN
ejpam-6275	493	6	,	,	PUNCT
ejpam-6275	493	7	18	18	NUM
ejpam-6275	493	8	(	(	PUNCT
ejpam-6275	493	9	3	3	NUM
ejpam-6275	493	10	)	)	PUNCT
ejpam-6275	493	11	(	(	PUNCT
ejpam-6275	493	12	2025	2025	NUM
ejpam-6275	493	13	)	)	PUNCT
ejpam-6275	493	14	,	,	PUNCT
ejpam-6275	493	15	6275	6275	NUM
ejpam-6275	493	16	19	19	NUM
ejpam-6275	493	17	of	of	ADP
ejpam-6275	493	18	36	36	NUM
ejpam-6275	493	19	•	•	NOUN
ejpam-6275	493	20	for	for	ADP
ejpam-6275	493	21	i	i	PRON
ejpam-6275	493	22	=	=	NOUN
ejpam-6275	493	23	11	11	NUM
ejpam-6275	493	24	,	,	PUNCT
ejpam-6275	493	25	let	let	VERB
ejpam-6275	493	26	β11	β11	PROPN
ejpam-6275	493	27	∈	∈	PROPN
ejpam-6275	493	28	z2[ω	z2[ω	NOUN
ejpam-6275	493	29	]	]	X
ejpam-6275	493	30	3	3	NUM
ejpam-6275	493	31	,	,	PUNCT
ejpam-6275	493	32	then	then	ADV
ejpam-6275	493	33	by	by	ADP
ejpam-6275	493	34	theorem	theorem	ADJ
ejpam-6275	493	35	3.1	3.1	NUM
ejpam-6275	493	36	,	,	PUNCT
ejpam-6275	493	37	β2	β2	NOUN
ejpam-6275	493	38	,	,	PUNCT
ejpam-6275	493	39	β8	β8	NOUN
ejpam-6275	493	40	,	,	PUNCT
ejpam-6275	493	41	and	and	CCONJ
ejpam-6275	493	42	β11	β11	NOUN
ejpam-6275	493	43	have	have	VERB
ejpam-6275	493	44	the	the	DET
ejpam-6275	493	45	minimal	minimal	ADJ
ejpam-6275	493	46	polynomial	polynomial	ADJ
ejpam-6275	493	47	φ11(x	φ11(x	NOUN
ejpam-6275	493	48	)	)	PUNCT
ejpam-6275	493	49	=	=	SYM
ejpam-6275	494	1	x3	x3	VERB
ejpam-6275	495	1	+	+	CCONJ
ejpam-6275	495	2	ωx2	ωx2	PRON
ejpam-6275	495	3	+	+	CCONJ
ejpam-6275	495	4	1	1	NUM
ejpam-6275	495	5	=	=	SYM
ejpam-6275	495	6	φ2(x	φ2(x	NUM
ejpam-6275	495	7	)	)	PUNCT
ejpam-6275	495	8	.	.	PUNCT
ejpam-6275	496	1	•	•	NOUN
ejpam-6275	496	2	for	for	ADP
ejpam-6275	496	3	i	i	PRON
ejpam-6275	496	4	=	=	NOUN
ejpam-6275	496	5	12	12	NUM
ejpam-6275	496	6	,	,	PUNCT
ejpam-6275	496	7	let	let	VERB
ejpam-6275	496	8	β12	β12	PRON
ejpam-6275	496	9	∈	∈	PROPN
ejpam-6275	496	10	z2[ω	z2[ω	NOUN
ejpam-6275	496	11	]	]	X
ejpam-6275	496	12	3	3	NUM
ejpam-6275	496	13	,	,	PUNCT
ejpam-6275	496	14	then	then	ADV
ejpam-6275	496	15	by	by	ADP
ejpam-6275	496	16	theorem	theorem	ADJ
ejpam-6275	496	17	3.1	3.1	NUM
ejpam-6275	496	18	,	,	PUNCT
ejpam-6275	496	19	β3	β3	ADJ
ejpam-6275	496	20	,	,	PUNCT
ejpam-6275	496	21	β6	β6	PROPN
ejpam-6275	496	22	,	,	PUNCT
ejpam-6275	496	23	and	and	CCONJ
ejpam-6275	496	24	β12	β12	CCONJ
ejpam-6275	496	25	have	have	VERB
ejpam-6275	496	26	the	the	DET
ejpam-6275	496	27	minimal	minimal	ADJ
ejpam-6275	496	28	polynomial	polynomial	ADJ
ejpam-6275	496	29	φ12(x	φ12(x	NOUN
ejpam-6275	496	30	)	)	PUNCT
ejpam-6275	496	31	=	=	PUNCT
ejpam-6275	497	1	x3	x3	VERB
ejpam-6275	498	1	+	+	CCONJ
ejpam-6275	498	2	x+	x+	SYM
ejpam-6275	498	3	1	1	X
ejpam-6275	498	4	=	=	SYM
ejpam-6275	498	5	φ3(x	φ3(x	PROPN
ejpam-6275	498	6	)	)	PUNCT
ejpam-6275	498	7	.	.	PUNCT
ejpam-6275	499	1	now	now	ADV
ejpam-6275	499	2	,	,	PUNCT
ejpam-6275	499	3	the	the	DET
ejpam-6275	499	4	generator	generator	NOUN
ejpam-6275	499	5	polynomial	polynomial	NOUN
ejpam-6275	499	6	is	be	AUX
ejpam-6275	499	7	g(x	g(x	NOUN
ejpam-6275	499	8	)	)	PUNCT
ejpam-6275	500	1	=	=	SYM
ejpam-6275	500	2	φ1(x)φ2(x)φ2(x)φ5(x)φ7(x)φ9(x)φ10(x	φ1(x)φ2(x)φ2(x)φ5(x)φ7(x)φ9(x)φ10(x	X
ejpam-6275	500	3	)	)	PUNCT
ejpam-6275	500	4	=	=	SYM
ejpam-6275	501	1	x19+(1+ω)x18+x16+(1+ω)x15+(1+ω)x14+(1+ω)x13+x8+ωx7+(1+ω)x6+x5+(1+ω)x4+x2+ωx+1+ω	x19+(1+ω)x18+x16+(1+ω)x15+(1+ω)x14+(1+ω)x13+x8+ωx7+(1+ω)x6+x5+(1+ω)x4+x2+ωx+1+ω	NOUN
ejpam-6275	501	2	.	.	PUNCT
ejpam-6275	502	1	hence	hence	ADV
ejpam-6275	502	2	,	,	PUNCT
ejpam-6275	502	3	k	k	PROPN
ejpam-6275	502	4	=	=	SYM
ejpam-6275	502	5	21−	21−	NUM
ejpam-6275	502	6	19	19	NUM
ejpam-6275	502	7	=	=	SYM
ejpam-6275	502	8	2	2	NUM
ejpam-6275	502	9	.	.	PUNCT
ejpam-6275	503	1	so	so	ADV
ejpam-6275	503	2	the	the	DET
ejpam-6275	503	3	parameters	parameter	NOUN
ejpam-6275	503	4	of	of	ADP
ejpam-6275	503	5	the	the	DET
ejpam-6275	503	6	shortened	shorten	VERB
ejpam-6275	503	7	bch	bch	PROPN
ejpam-6275	503	8	code	code	NOUN
ejpam-6275	503	9	are	be	AUX
ejpam-6275	503	10	:	:	PUNCT
ejpam-6275	503	11	(	(	PUNCT
ejpam-6275	503	12	n	n	X
ejpam-6275	503	13	,	,	PUNCT
ejpam-6275	503	14	k	k	NOUN
ejpam-6275	503	15	,	,	PUNCT
ejpam-6275	503	16	d	d	NOUN
ejpam-6275	503	17	)	)	PUNCT
ejpam-6275	503	18	=	=	SYM
ejpam-6275	503	19	(	(	PUNCT
ejpam-6275	503	20	21	21	NUM
ejpam-6275	503	21	,	,	PUNCT
ejpam-6275	503	22	2	2	NUM
ejpam-6275	503	23	,	,	PUNCT
ejpam-6275	503	24	13	13	NUM
ejpam-6275	503	25	)	)	PUNCT
ejpam-6275	503	26	.	.	PUNCT
ejpam-6275	504	1	case	case	NOUN
ejpam-6275	504	2	7	7	NUM
ejpam-6275	504	3	:	:	PUNCT
ejpam-6275	504	4	for	for	ADP
ejpam-6275	504	5	n	n	NOUN
ejpam-6275	504	6	=	=	SYM
ejpam-6275	504	7	21	21	NUM
ejpam-6275	504	8	and	and	CCONJ
ejpam-6275	504	9	d	d	NOUN
ejpam-6275	504	10	=	=	SYM
ejpam-6275	504	11	15	15	NUM
ejpam-6275	504	12	,	,	PUNCT
ejpam-6275	504	13	i	i	PRON
ejpam-6275	504	14	=	=	NOUN
ejpam-6275	504	15	1	1	NUM
ejpam-6275	504	16	,	,	PUNCT
ejpam-6275	504	17	2	2	NUM
ejpam-6275	504	18	,	,	PUNCT
ejpam-6275	504	19	3	3	NUM
ejpam-6275	504	20	,	,	PUNCT
ejpam-6275	504	21	.	.	PUNCT
ejpam-6275	504	22	.	.	PUNCT
ejpam-6275	505	1	.	.	PUNCT
ejpam-6275	506	1	,	,	PUNCT
ejpam-6275	507	1	14	14	NUM
ejpam-6275	507	2	.	.	NOUN
ejpam-6275	507	3	•	•	NUM
ejpam-6275	507	4	for	for	ADP
ejpam-6275	507	5	i	i	PRON
ejpam-6275	507	6	=	=	NOUN
ejpam-6275	507	7	1	1	NUM
ejpam-6275	507	8	,	,	PUNCT
ejpam-6275	507	9	let	let	VERB
ejpam-6275	507	10	β	β	PRON
ejpam-6275	507	11	∈	∈	PROPN
ejpam-6275	507	12	z2[ω	z2[ω	NOUN
ejpam-6275	507	13	]	]	X
ejpam-6275	508	1	3	3	X
ejpam-6275	508	2	.	.	PUNCT
ejpam-6275	508	3	then	then	ADV
ejpam-6275	508	4	by	by	ADP
ejpam-6275	508	5	theorem	theorem	NOUN
ejpam-6275	508	6	3.1	3.1	NUM
ejpam-6275	508	7	,	,	PUNCT
ejpam-6275	508	8	β	β	X
ejpam-6275	508	9	,	,	PUNCT
ejpam-6275	508	10	β4	β4	PROPN
ejpam-6275	508	11	and	and	CCONJ
ejpam-6275	508	12	β16	β16	PROPN
ejpam-6275	508	13	have	have	VERB
ejpam-6275	508	14	the	the	DET
ejpam-6275	508	15	minimal	minimal	ADJ
ejpam-6275	508	16	polynomial	polynomial	ADJ
ejpam-6275	508	17	φ1(x	φ1(x	NOUN
ejpam-6275	508	18	)	)	PUNCT
ejpam-6275	508	19	=	=	SYM
ejpam-6275	508	20	(	(	PUNCT
ejpam-6275	508	21	x−β)(x−β4)(x−β16	x−β)(x−β4)(x−β16	X
ejpam-6275	508	22	)	)	PUNCT
ejpam-6275	508	23	=	=	SYM
ejpam-6275	508	24	x3+(β+β4+β16)x2+(β5+β17+β20)x+β21	x3+(β+β4+β16)x2+(β5+β17+β20)x+β21	PROPN
ejpam-6275	508	25	=	=	SYM
ejpam-6275	509	1	x3+(1+ω)x2	x3+(1+ω)x2	PUNCT
ejpam-6275	509	2	+	+	PROPN
ejpam-6275	509	3	1	1	NUM
ejpam-6275	509	4	.	.	NOUN
ejpam-6275	509	5	•	•	NOUN
ejpam-6275	509	6	for	for	ADP
ejpam-6275	509	7	i	i	PRON
ejpam-6275	509	8	=	=	SYM
ejpam-6275	509	9	2	2	NUM
ejpam-6275	509	10	,	,	PUNCT
ejpam-6275	509	11	let	let	VERB
ejpam-6275	509	12	β2	β2	PROPN
ejpam-6275	509	13	∈	∈	PROPN
ejpam-6275	509	14	z2[ω	z2[ω	NOUN
ejpam-6275	509	15	]	]	X
ejpam-6275	509	16	3	3	X
ejpam-6275	509	17	.	.	PUNCT
ejpam-6275	509	18	then	then	ADV
ejpam-6275	509	19	by	by	ADP
ejpam-6275	509	20	theorem	theorem	ADJ
ejpam-6275	509	21	3.1	3.1	NUM
ejpam-6275	509	22	,	,	PUNCT
ejpam-6275	509	23	β2	β2	ADJ
ejpam-6275	509	24	,	,	PUNCT
ejpam-6275	509	25	β8	β8	NOUN
ejpam-6275	509	26	and	and	CCONJ
ejpam-6275	509	27	β11	β11	NOUN
ejpam-6275	509	28	have	have	VERB
ejpam-6275	509	29	the	the	DET
ejpam-6275	509	30	minimal	minimal	ADJ
ejpam-6275	509	31	polynomial	polynomial	ADJ
ejpam-6275	509	32	φ2(x	φ2(x	NUM
ejpam-6275	509	33	)	)	PUNCT
ejpam-6275	509	34	=	=	PUNCT
ejpam-6275	509	35	(	(	PUNCT
ejpam-6275	509	36	x−	x−	PROPN
ejpam-6275	509	37	β2)(x−	β2)(x−	PROPN
ejpam-6275	509	38	β8)(x−	β8)(x−	X
ejpam-6275	509	39	β11	β11	PROPN
ejpam-6275	509	40	)	)	PUNCT
ejpam-6275	509	41	=	=	PUNCT
ejpam-6275	510	1	x3	x3	VERB
ejpam-6275	511	1	+	+	CCONJ
ejpam-6275	511	2	ωx2	ωx2	PRON
ejpam-6275	511	3	+	+	X
ejpam-6275	511	4	1	1	NUM
ejpam-6275	511	5	.	.	NUM
ejpam-6275	511	6	•	•	NOUN
ejpam-6275	511	7	for	for	ADP
ejpam-6275	511	8	i	i	PRON
ejpam-6275	511	9	=	=	SYM
ejpam-6275	511	10	3	3	NUM
ejpam-6275	511	11	,	,	PUNCT
ejpam-6275	511	12	let	let	VERB
ejpam-6275	511	13	β3	β3	VERB
ejpam-6275	511	14	∈	∈	PROPN
ejpam-6275	511	15	z2[ω	z2[ω	NOUN
ejpam-6275	511	16	]	]	X
ejpam-6275	511	17	3	3	X
ejpam-6275	511	18	.	.	PUNCT
ejpam-6275	511	19	then	then	ADV
ejpam-6275	511	20	by	by	ADP
ejpam-6275	511	21	theorem	theorem	ADJ
ejpam-6275	511	22	3.1	3.1	NUM
ejpam-6275	511	23	,	,	PUNCT
ejpam-6275	511	24	β3	β3	ADJ
ejpam-6275	511	25	,	,	PUNCT
ejpam-6275	511	26	β6	β6	PROPN
ejpam-6275	511	27	and	and	CCONJ
ejpam-6275	511	28	β12	β12	NOUN
ejpam-6275	511	29	have	have	VERB
ejpam-6275	511	30	the	the	DET
ejpam-6275	511	31	minimal	minimal	ADJ
ejpam-6275	511	32	polynomial	polynomial	ADJ
ejpam-6275	511	33	φ3(x	φ3(x	PROPN
ejpam-6275	511	34	)	)	PUNCT
ejpam-6275	511	35	=	=	SYM
ejpam-6275	511	36	(	(	PUNCT
ejpam-6275	511	37	x−	x−	PROPN
ejpam-6275	511	38	β3)(x−	β3)(x−	PROPN
ejpam-6275	511	39	β6)(x−	β6)(x−	X
ejpam-6275	511	40	β12	β12	PROPN
ejpam-6275	511	41	)	)	PUNCT
ejpam-6275	511	42	=	=	PUNCT
ejpam-6275	512	1	x3	x3	VERB
ejpam-6275	512	2	+	+	CCONJ
ejpam-6275	512	3	x+	x+	ADJ
ejpam-6275	512	4	1	1	NUM
ejpam-6275	512	5	.	.	NUM
ejpam-6275	512	6	•	•	NOUN
ejpam-6275	512	7	for	for	ADP
ejpam-6275	512	8	i	i	PRON
ejpam-6275	512	9	=	=	NOUN
ejpam-6275	512	10	4	4	NUM
ejpam-6275	512	11	,	,	PUNCT
ejpam-6275	512	12	let	let	VERB
ejpam-6275	512	13	β4	β4	PROPN
ejpam-6275	512	14	∈	∈	PROPN
ejpam-6275	512	15	z2[ω	z2[ω	NOUN
ejpam-6275	512	16	]	]	X
ejpam-6275	513	1	3	3	X
ejpam-6275	513	2	.	.	PUNCT
ejpam-6275	513	3	then	then	ADV
ejpam-6275	513	4	by	by	ADP
ejpam-6275	513	5	theorem	theorem	ADJ
ejpam-6275	513	6	3.1	3.1	NUM
ejpam-6275	513	7	,	,	PUNCT
ejpam-6275	513	8	β2	β2	ADJ
ejpam-6275	513	9	,	,	PUNCT
ejpam-6275	513	10	β8	β8	NOUN
ejpam-6275	513	11	and	and	CCONJ
ejpam-6275	513	12	β11	β11	NOUN
ejpam-6275	513	13	have	have	VERB
ejpam-6275	513	14	the	the	DET
ejpam-6275	513	15	minimal	minimal	ADJ
ejpam-6275	513	16	polynomial	polynomial	ADJ
ejpam-6275	513	17	φ4(x	φ4(x	NOUN
ejpam-6275	513	18	)	)	PUNCT
ejpam-6275	513	19	=	=	SYM
ejpam-6275	513	20	(	(	PUNCT
ejpam-6275	513	21	x−	x−	PROPN
ejpam-6275	513	22	β2)(x−	β2)(x−	PROPN
ejpam-6275	513	23	β8)(x−	β8)(x−	X
ejpam-6275	513	24	β11	β11	PROPN
ejpam-6275	513	25	)	)	PUNCT
ejpam-6275	513	26	=	=	PUNCT
ejpam-6275	514	1	x3	x3	VERB
ejpam-6275	515	1	+	+	CCONJ
ejpam-6275	515	2	ωx2	ωx2	PRON
ejpam-6275	515	3	+	+	CCONJ
ejpam-6275	515	4	1	1	NUM
ejpam-6275	515	5	=	=	SYM
ejpam-6275	515	6	φ2(x	φ2(x	NUM
ejpam-6275	515	7	)	)	PUNCT
ejpam-6275	515	8	.	.	PUNCT
ejpam-6275	516	1	m.	m.	PROPN
ejpam-6275	516	2	sajjad	sajjad	PROPN
ejpam-6275	516	3	et	et	PROPN
ejpam-6275	516	4	al	al	PROPN
ejpam-6275	516	5	.	.	PUNCT
ejpam-6275	516	6	/	/	SYM
ejpam-6275	516	7	eur	eur	PROPN
ejpam-6275	516	8	.	.	PUNCT
ejpam-6275	517	1	j.	j.	PROPN
ejpam-6275	517	2	pure	pure	PROPN
ejpam-6275	517	3	appl	appl	PROPN
ejpam-6275	517	4	.	.	PROPN
ejpam-6275	517	5	math	math	PROPN
ejpam-6275	517	6	,	,	PUNCT
ejpam-6275	517	7	18	18	NUM
ejpam-6275	517	8	(	(	PUNCT
ejpam-6275	517	9	3	3	NUM
ejpam-6275	517	10	)	)	PUNCT
ejpam-6275	517	11	(	(	PUNCT
ejpam-6275	517	12	2025	2025	NUM
ejpam-6275	517	13	)	)	PUNCT
ejpam-6275	517	14	,	,	PUNCT
ejpam-6275	517	15	6275	6275	NUM
ejpam-6275	517	16	20	20	NUM
ejpam-6275	517	17	of	of	ADP
ejpam-6275	517	18	36	36	NUM
ejpam-6275	517	19	•	•	NOUN
ejpam-6275	517	20	for	for	ADP
ejpam-6275	517	21	i	i	PRON
ejpam-6275	517	22	=	=	SYM
ejpam-6275	517	23	5	5	NUM
ejpam-6275	517	24	,	,	PUNCT
ejpam-6275	517	25	let	let	VERB
ejpam-6275	517	26	β5	β5	NOUN
ejpam-6275	517	27	∈	∈	PROPN
ejpam-6275	517	28	z2[ω	z2[ω	NOUN
ejpam-6275	517	29	]	]	X
ejpam-6275	518	1	3	3	X
ejpam-6275	518	2	.	.	PUNCT
ejpam-6275	518	3	then	then	ADV
ejpam-6275	518	4	by	by	ADP
ejpam-6275	518	5	theorem	theorem	ADJ
ejpam-6275	518	6	3.1	3.1	NUM
ejpam-6275	518	7	,	,	PUNCT
ejpam-6275	518	8	β5	β5	NOUN
ejpam-6275	518	9	,	,	PUNCT
ejpam-6275	518	10	β17	β17	NUM
ejpam-6275	518	11	and	and	CCONJ
ejpam-6275	518	12	β20	β20	PROPN
ejpam-6275	518	13	have	have	VERB
ejpam-6275	518	14	the	the	DET
ejpam-6275	518	15	minimal	minimal	ADJ
ejpam-6275	518	16	polynomial	polynomial	ADJ
ejpam-6275	518	17	φ5(x	φ5(x	NOUN
ejpam-6275	518	18	)	)	PUNCT
ejpam-6275	518	19	=	=	SYM
ejpam-6275	518	20	(	(	PUNCT
ejpam-6275	518	21	x−	x−	PROPN
ejpam-6275	518	22	β5)(x−	β5)(x−	PROPN
ejpam-6275	518	23	β17)(x−	β17)(x−	PROPN
ejpam-6275	518	24	β20	β20	PROPN
ejpam-6275	518	25	)	)	PUNCT
ejpam-6275	518	26	=	=	SYM
ejpam-6275	519	1	x3	x3	VERB
ejpam-6275	519	2	+	+	CCONJ
ejpam-6275	519	3	(	(	PUNCT
ejpam-6275	519	4	1	1	NUM
ejpam-6275	519	5	+	+	SYM
ejpam-6275	519	6	ω)x+	ω)x+	NUM
ejpam-6275	519	7	1	1	NUM
ejpam-6275	519	8	.	.	NOUN
ejpam-6275	519	9	•	•	NOUN
ejpam-6275	519	10	for	for	ADP
ejpam-6275	519	11	i	i	PRON
ejpam-6275	519	12	=	=	SYM
ejpam-6275	519	13	6	6	NUM
ejpam-6275	519	14	,	,	PUNCT
ejpam-6275	519	15	let	let	VERB
ejpam-6275	519	16	β6	β6	PROPN
ejpam-6275	519	17	∈	∈	PROPN
ejpam-6275	519	18	z2[ω	z2[ω	NOUN
ejpam-6275	519	19	]	]	X
ejpam-6275	520	1	3	3	X
ejpam-6275	520	2	.	.	PUNCT
ejpam-6275	520	3	then	then	ADV
ejpam-6275	520	4	by	by	ADP
ejpam-6275	520	5	theorem	theorem	ADJ
ejpam-6275	520	6	3.1	3.1	NUM
ejpam-6275	520	7	,	,	PUNCT
ejpam-6275	520	8	β3	β3	ADJ
ejpam-6275	520	9	,	,	PUNCT
ejpam-6275	520	10	β6	β6	PROPN
ejpam-6275	520	11	and	and	CCONJ
ejpam-6275	520	12	β12	β12	NOUN
ejpam-6275	520	13	have	have	VERB
ejpam-6275	520	14	the	the	DET
ejpam-6275	520	15	minimal	minimal	ADJ
ejpam-6275	520	16	polynomial	polynomial	ADJ
ejpam-6275	520	17	φ6(x	φ6(x	NOUN
ejpam-6275	520	18	)	)	PUNCT
ejpam-6275	520	19	=	=	SYM
ejpam-6275	520	20	(	(	PUNCT
ejpam-6275	521	1	x−	x−	PROPN
ejpam-6275	521	2	β3)(x−	β3)(x−	PROPN
ejpam-6275	521	3	β6)(x−	β6)(x−	X
ejpam-6275	521	4	β12	β12	PROPN
ejpam-6275	521	5	)	)	PUNCT
ejpam-6275	522	1	=	=	PUNCT
ejpam-6275	522	2	x3	x3	VERB
ejpam-6275	522	3	+	+	CCONJ
ejpam-6275	522	4	x+	x+	SYM
ejpam-6275	522	5	1	1	X
ejpam-6275	522	6	=	=	SYM
ejpam-6275	522	7	φ3(x	φ3(x	PROPN
ejpam-6275	522	8	)	)	PUNCT
ejpam-6275	522	9	.	.	PUNCT
ejpam-6275	523	1	•	•	NOUN
ejpam-6275	523	2	for	for	ADP
ejpam-6275	523	3	i	i	PRON
ejpam-6275	523	4	=	=	SYM
ejpam-6275	523	5	7	7	NUM
ejpam-6275	523	6	,	,	PUNCT
ejpam-6275	523	7	let	let	VERB
ejpam-6275	523	8	β7	β7	ADJ
ejpam-6275	523	9	∈	∈	PROPN
ejpam-6275	523	10	z2[ω	z2[ω	NOUN
ejpam-6275	523	11	]	]	X
ejpam-6275	523	12	3	3	X
ejpam-6275	523	13	.	.	PUNCT
ejpam-6275	523	14	then	then	ADV
ejpam-6275	523	15	by	by	ADP
ejpam-6275	523	16	theorem	theorem	ADJ
ejpam-6275	523	17	3.1	3.1	NUM
ejpam-6275	523	18	,	,	PUNCT
ejpam-6275	523	19	β7	β7	PROPN
ejpam-6275	523	20	has	have	VERB
ejpam-6275	523	21	the	the	DET
ejpam-6275	523	22	minimal	minimal	ADJ
ejpam-6275	523	23	polynomial	polynomial	ADJ
ejpam-6275	523	24	φ7(x	φ7(x	NOUN
ejpam-6275	523	25	)	)	PUNCT
ejpam-6275	523	26	=	=	PUNCT
ejpam-6275	523	27	(	(	PUNCT
ejpam-6275	523	28	x−	x−	PROPN
ejpam-6275	523	29	β7	β7	PROPN
ejpam-6275	523	30	)	)	PUNCT
ejpam-6275	524	1	=	=	PUNCT
ejpam-6275	524	2	x+	x+	PUNCT
ejpam-6275	525	1	1	1	NUM
ejpam-6275	525	2	+	+	NUM
ejpam-6275	525	3	ω	ω	NUM
ejpam-6275	525	4	.	.	NOUN
ejpam-6275	525	5	•	•	NUM
ejpam-6275	525	6	for	for	ADP
ejpam-6275	525	7	i	i	PRON
ejpam-6275	525	8	=	=	NOUN
ejpam-6275	525	9	8	8	NUM
ejpam-6275	525	10	,	,	PUNCT
ejpam-6275	525	11	let	let	VERB
ejpam-6275	525	12	β8	β8	PROPN
ejpam-6275	525	13	∈	∈	PROPN
ejpam-6275	525	14	z2[ω	z2[ω	NOUN
ejpam-6275	525	15	]	]	X
ejpam-6275	525	16	3	3	X
ejpam-6275	525	17	.	.	PUNCT
ejpam-6275	525	18	then	then	ADV
ejpam-6275	525	19	by	by	ADP
ejpam-6275	525	20	theorem	theorem	ADJ
ejpam-6275	525	21	3.1	3.1	NUM
ejpam-6275	525	22	,	,	PUNCT
ejpam-6275	525	23	β2	β2	ADJ
ejpam-6275	525	24	,	,	PUNCT
ejpam-6275	525	25	β8	β8	NOUN
ejpam-6275	525	26	and	and	CCONJ
ejpam-6275	525	27	β11	β11	NOUN
ejpam-6275	525	28	have	have	VERB
ejpam-6275	525	29	the	the	DET
ejpam-6275	525	30	minimal	minimal	ADJ
ejpam-6275	525	31	polynomial	polynomial	ADJ
ejpam-6275	525	32	φ8(x	φ8(x	NOUN
ejpam-6275	525	33	)	)	PUNCT
ejpam-6275	525	34	=	=	SYM
ejpam-6275	525	35	(	(	PUNCT
ejpam-6275	525	36	x−	x−	PROPN
ejpam-6275	525	37	β2)(x−	β2)(x−	PROPN
ejpam-6275	525	38	β8)(x−	β8)(x−	X
ejpam-6275	525	39	β11	β11	PROPN
ejpam-6275	525	40	)	)	PUNCT
ejpam-6275	525	41	=	=	PUNCT
ejpam-6275	526	1	x3	x3	VERB
ejpam-6275	527	1	+	+	CCONJ
ejpam-6275	527	2	ωx2	ωx2	PRON
ejpam-6275	527	3	+	+	CCONJ
ejpam-6275	527	4	1	1	NUM
ejpam-6275	527	5	=	=	SYM
ejpam-6275	527	6	φ2(x	φ2(x	NUM
ejpam-6275	527	7	)	)	PUNCT
ejpam-6275	527	8	.	.	PUNCT
ejpam-6275	528	1	•	•	NOUN
ejpam-6275	528	2	for	for	ADP
ejpam-6275	528	3	i	i	PRON
ejpam-6275	528	4	=	=	NOUN
ejpam-6275	528	5	9	9	NUM
ejpam-6275	528	6	,	,	PUNCT
ejpam-6275	528	7	let	let	VERB
ejpam-6275	528	8	β9	β9	PROPN
ejpam-6275	528	9	∈	∈	PROPN
ejpam-6275	528	10	z2[ω	z2[ω	NOUN
ejpam-6275	528	11	]	]	X
ejpam-6275	528	12	3	3	X
ejpam-6275	528	13	.	.	PUNCT
ejpam-6275	528	14	then	then	ADV
ejpam-6275	528	15	by	by	ADP
ejpam-6275	528	16	theorem	theorem	ADJ
ejpam-6275	528	17	3.1	3.1	NUM
ejpam-6275	528	18	,	,	PUNCT
ejpam-6275	528	19	β9	β9	PROPN
ejpam-6275	528	20	,	,	PUNCT
ejpam-6275	528	21	β18	β18	NUM
ejpam-6275	528	22	and	and	CCONJ
ejpam-6275	528	23	β15	β15	NOUN
ejpam-6275	528	24	have	have	VERB
ejpam-6275	528	25	the	the	DET
ejpam-6275	528	26	minimal	minimal	ADJ
ejpam-6275	528	27	polynomial	polynomial	ADJ
ejpam-6275	528	28	φ9(x	φ9(x	NUM
ejpam-6275	528	29	)	)	PUNCT
ejpam-6275	528	30	=	=	SYM
ejpam-6275	528	31	(	(	PUNCT
ejpam-6275	528	32	x−	x−	PROPN
ejpam-6275	528	33	β9)(x−	β9)(x−	PUNCT
ejpam-6275	528	34	β18)(x−	β18)(x−	NOUN
ejpam-6275	528	35	β15	β15	NOUN
ejpam-6275	528	36	)	)	PUNCT
ejpam-6275	528	37	=	=	SYM
ejpam-6275	529	1	x3	x3	VERB
ejpam-6275	530	1	+	+	CCONJ
ejpam-6275	530	2	x2	x2	PROPN
ejpam-6275	531	1	+	+	CCONJ
ejpam-6275	532	1	1	1	NUM
ejpam-6275	532	2	.	.	NUM
ejpam-6275	532	3	•	•	NOUN
ejpam-6275	532	4	for	for	ADP
ejpam-6275	532	5	i	i	PRON
ejpam-6275	532	6	=	=	NOUN
ejpam-6275	532	7	10	10	NUM
ejpam-6275	532	8	,	,	PUNCT
ejpam-6275	532	9	let	let	VERB
ejpam-6275	532	10	β10	β10	VERB
ejpam-6275	532	11	∈	∈	PROPN
ejpam-6275	532	12	z2[ω	z2[ω	NOUN
ejpam-6275	532	13	]	]	X
ejpam-6275	533	1	3	3	X
ejpam-6275	533	2	.	.	PUNCT
ejpam-6275	533	3	then	then	ADV
ejpam-6275	533	4	by	by	ADP
ejpam-6275	533	5	theorem	theorem	ADJ
ejpam-6275	533	6	3.1	3.1	NUM
ejpam-6275	533	7	,	,	PUNCT
ejpam-6275	533	8	β10	β10	NOUN
ejpam-6275	533	9	,	,	PUNCT
ejpam-6275	533	10	β13	β13	NOUN
ejpam-6275	533	11	and	and	CCONJ
ejpam-6275	533	12	β19	β19	NOUN
ejpam-6275	533	13	have	have	VERB
ejpam-6275	533	14	the	the	DET
ejpam-6275	533	15	minimal	minimal	ADJ
ejpam-6275	533	16	polynomial	polynomial	ADJ
ejpam-6275	533	17	φ10(x	φ10(x	NOUN
ejpam-6275	533	18	)	)	PUNCT
ejpam-6275	534	1	=	=	PUNCT
ejpam-6275	534	2	(	(	PUNCT
ejpam-6275	534	3	x−	x−	PROPN
ejpam-6275	534	4	β10)(x−	β10)(x−	PROPN
ejpam-6275	534	5	β13)(x−	β13)(x−	PROPN
ejpam-6275	534	6	β19	β19	ADV
ejpam-6275	534	7	)	)	PUNCT
ejpam-6275	534	8	=	=	SYM
ejpam-6275	535	1	x3	x3	ADJ
ejpam-6275	535	2	+	+	CCONJ
ejpam-6275	535	3	ωx+	ωx+	NOUN
ejpam-6275	535	4	1	1	NUM
ejpam-6275	535	5	.	.	NOUN
ejpam-6275	535	6	•	•	NOUN
ejpam-6275	535	7	for	for	ADP
ejpam-6275	535	8	i	i	PRON
ejpam-6275	535	9	=	=	NOUN
ejpam-6275	535	10	11	11	NUM
ejpam-6275	535	11	,	,	PUNCT
ejpam-6275	535	12	let	let	VERB
ejpam-6275	535	13	β11	β11	PROPN
ejpam-6275	535	14	∈	∈	PROPN
ejpam-6275	535	15	z2[ω	z2[ω	NOUN
ejpam-6275	535	16	]	]	X
ejpam-6275	536	1	3	3	X
ejpam-6275	536	2	.	.	PUNCT
ejpam-6275	536	3	then	then	ADV
ejpam-6275	536	4	by	by	ADP
ejpam-6275	536	5	theorem	theorem	ADJ
ejpam-6275	536	6	3.1	3.1	NUM
ejpam-6275	536	7	,	,	PUNCT
ejpam-6275	536	8	β2	β2	ADJ
ejpam-6275	536	9	,	,	PUNCT
ejpam-6275	536	10	β8	β8	NOUN
ejpam-6275	536	11	and	and	CCONJ
ejpam-6275	536	12	β11	β11	NOUN
ejpam-6275	536	13	have	have	VERB
ejpam-6275	536	14	the	the	DET
ejpam-6275	536	15	minimal	minimal	ADJ
ejpam-6275	536	16	polynomial	polynomial	ADJ
ejpam-6275	536	17	φ11(x	φ11(x	NOUN
ejpam-6275	536	18	)	)	PUNCT
ejpam-6275	536	19	=	=	PUNCT
ejpam-6275	536	20	(	(	PUNCT
ejpam-6275	536	21	x−	x−	PROPN
ejpam-6275	536	22	β2)(x−	β2)(x−	PROPN
ejpam-6275	536	23	β8)(x−	β8)(x−	X
ejpam-6275	536	24	β11	β11	PROPN
ejpam-6275	536	25	)	)	PUNCT
ejpam-6275	536	26	=	=	SYM
ejpam-6275	536	27	φ2(x	φ2(x	NUM
ejpam-6275	536	28	)	)	PUNCT
ejpam-6275	536	29	.	.	PUNCT
ejpam-6275	537	1	•	•	NOUN
ejpam-6275	537	2	for	for	ADP
ejpam-6275	537	3	i	i	PRON
ejpam-6275	537	4	=	=	NOUN
ejpam-6275	537	5	12	12	NUM
ejpam-6275	537	6	,	,	PUNCT
ejpam-6275	537	7	let	let	VERB
ejpam-6275	537	8	β12	β12	PRON
ejpam-6275	537	9	∈	∈	PROPN
ejpam-6275	537	10	z2[ω	z2[ω	NOUN
ejpam-6275	537	11	]	]	X
ejpam-6275	538	1	3	3	X
ejpam-6275	538	2	.	.	PUNCT
ejpam-6275	538	3	then	then	ADV
ejpam-6275	538	4	by	by	ADP
ejpam-6275	538	5	theorem	theorem	ADJ
ejpam-6275	538	6	3.1	3.1	NUM
ejpam-6275	538	7	,	,	PUNCT
ejpam-6275	538	8	β3	β3	ADJ
ejpam-6275	538	9	,	,	PUNCT
ejpam-6275	538	10	β6	β6	PROPN
ejpam-6275	538	11	and	and	CCONJ
ejpam-6275	538	12	β12	β12	NOUN
ejpam-6275	538	13	have	have	VERB
ejpam-6275	538	14	the	the	DET
ejpam-6275	538	15	minimal	minimal	ADJ
ejpam-6275	538	16	polynomial	polynomial	ADJ
ejpam-6275	538	17	φ12(x	φ12(x	NOUN
ejpam-6275	538	18	)	)	PUNCT
ejpam-6275	538	19	=	=	PUNCT
ejpam-6275	538	20	(	(	PUNCT
ejpam-6275	538	21	x−	x−	PROPN
ejpam-6275	538	22	β3)(x−	β3)(x−	PROPN
ejpam-6275	538	23	β6)(x−	β6)(x−	X
ejpam-6275	538	24	β12	β12	PROPN
ejpam-6275	538	25	)	)	PUNCT
ejpam-6275	538	26	=	=	PUNCT
ejpam-6275	539	1	x3	x3	VERB
ejpam-6275	540	1	+	+	CCONJ
ejpam-6275	540	2	x+	x+	SYM
ejpam-6275	540	3	1	1	X
ejpam-6275	540	4	=	=	SYM
ejpam-6275	540	5	φ3(x	φ3(x	PROPN
ejpam-6275	540	6	)	)	PUNCT
ejpam-6275	540	7	.	.	PUNCT
ejpam-6275	541	1	•	•	NOUN
ejpam-6275	541	2	for	for	ADP
ejpam-6275	541	3	i	i	PRON
ejpam-6275	541	4	=	=	NOUN
ejpam-6275	541	5	13	13	NUM
ejpam-6275	541	6	,	,	PUNCT
ejpam-6275	541	7	let	let	VERB
ejpam-6275	541	8	β13	β13	PRON
ejpam-6275	541	9	∈	∈	PROPN
ejpam-6275	541	10	z2[ω	z2[ω	NOUN
ejpam-6275	541	11	]	]	X
ejpam-6275	542	1	3	3	X
ejpam-6275	542	2	.	.	PUNCT
ejpam-6275	542	3	then	then	ADV
ejpam-6275	542	4	by	by	ADP
ejpam-6275	542	5	theorem	theorem	ADJ
ejpam-6275	542	6	3.1	3.1	NUM
ejpam-6275	542	7	,	,	PUNCT
ejpam-6275	542	8	β10	β10	NOUN
ejpam-6275	542	9	,	,	PUNCT
ejpam-6275	542	10	β13	β13	NOUN
ejpam-6275	542	11	and	and	CCONJ
ejpam-6275	542	12	β19	β19	NOUN
ejpam-6275	542	13	have	have	VERB
ejpam-6275	542	14	the	the	DET
ejpam-6275	542	15	minimal	minimal	ADJ
ejpam-6275	542	16	polynomial	polynomial	ADJ
ejpam-6275	542	17	φ13(x	φ13(x	NOUN
ejpam-6275	542	18	)	)	PUNCT
ejpam-6275	543	1	=	=	PUNCT
ejpam-6275	543	2	(	(	PUNCT
ejpam-6275	543	3	x−	x−	PROPN
ejpam-6275	543	4	β10)(x−	β10)(x−	PROPN
ejpam-6275	543	5	β13)(x−	β13)(x−	PROPN
ejpam-6275	543	6	β19	β19	ADV
ejpam-6275	543	7	)	)	PUNCT
ejpam-6275	543	8	=	=	SYM
ejpam-6275	544	1	x3	x3	ADJ
ejpam-6275	544	2	+	+	CCONJ
ejpam-6275	544	3	ωx+	ωx+	NOUN
ejpam-6275	544	4	1	1	NUM
ejpam-6275	544	5	=	=	SYM
ejpam-6275	544	6	φ10(x	φ10(x	NOUN
ejpam-6275	544	7	)	)	PUNCT
ejpam-6275	544	8	.	.	PUNCT
ejpam-6275	545	1	m.	m.	PROPN
ejpam-6275	545	2	sajjad	sajjad	PROPN
ejpam-6275	545	3	et	et	PROPN
ejpam-6275	545	4	al	al	PROPN
ejpam-6275	545	5	.	.	PUNCT
ejpam-6275	545	6	/	/	SYM
ejpam-6275	545	7	eur	eur	PROPN
ejpam-6275	545	8	.	.	PUNCT
ejpam-6275	546	1	j.	j.	PROPN
ejpam-6275	546	2	pure	pure	PROPN
ejpam-6275	546	3	appl	appl	PROPN
ejpam-6275	546	4	.	.	PROPN
ejpam-6275	546	5	math	math	PROPN
ejpam-6275	546	6	,	,	PUNCT
ejpam-6275	546	7	18	18	NUM
ejpam-6275	546	8	(	(	PUNCT
ejpam-6275	546	9	3	3	NUM
ejpam-6275	546	10	)	)	PUNCT
ejpam-6275	546	11	(	(	PUNCT
ejpam-6275	546	12	2025	2025	NUM
ejpam-6275	546	13	)	)	PUNCT
ejpam-6275	546	14	,	,	PUNCT
ejpam-6275	546	15	6275	6275	NUM
ejpam-6275	546	16	21	21	NUM
ejpam-6275	546	17	of	of	ADP
ejpam-6275	546	18	36	36	NUM
ejpam-6275	546	19	•	•	NOUN
ejpam-6275	546	20	for	for	ADP
ejpam-6275	546	21	i	i	PRON
ejpam-6275	546	22	=	=	NOUN
ejpam-6275	546	23	14	14	NUM
ejpam-6275	546	24	,	,	PUNCT
ejpam-6275	546	25	let	let	VERB
ejpam-6275	546	26	β14	β14	PRON
ejpam-6275	546	27	∈	∈	PROPN
ejpam-6275	546	28	z2[ω	z2[ω	NOUN
ejpam-6275	546	29	]	]	X
ejpam-6275	547	1	3	3	X
ejpam-6275	547	2	.	.	PUNCT
ejpam-6275	547	3	then	then	ADV
ejpam-6275	547	4	by	by	ADP
ejpam-6275	547	5	theorem	theorem	NOUN
ejpam-6275	547	6	3.1	3.1	NUM
ejpam-6275	547	7	,	,	PUNCT
ejpam-6275	547	8	β14	β14	X
ejpam-6275	547	9	has	have	VERB
ejpam-6275	547	10	the	the	DET
ejpam-6275	547	11	minimal	minimal	ADJ
ejpam-6275	547	12	polynomial	polynomial	ADJ
ejpam-6275	547	13	φ14(x	φ14(x	NOUN
ejpam-6275	547	14	)	)	PUNCT
ejpam-6275	547	15	=	=	PUNCT
ejpam-6275	547	16	(	(	PUNCT
ejpam-6275	547	17	x−	x−	PROPN
ejpam-6275	547	18	β14	β14	ADJ
ejpam-6275	547	19	)	)	PUNCT
ejpam-6275	548	1	=	=	SYM
ejpam-6275	548	2	x+	x+	PROPN
ejpam-6275	549	1	ω	ω	X
ejpam-6275	549	2	.	.	PUNCT
ejpam-6275	550	1	now	now	ADV
ejpam-6275	550	2	the	the	DET
ejpam-6275	550	3	generator	generator	NOUN
ejpam-6275	550	4	polynomial	polynomial	NOUN
ejpam-6275	550	5	is	be	AUX
ejpam-6275	550	6	g(x	g(x	NOUN
ejpam-6275	550	7	)	)	PUNCT
ejpam-6275	551	1	=	=	SYM
ejpam-6275	551	2	φ1(x	φ1(x	NOUN
ejpam-6275	551	3	)	)	PUNCT
ejpam-6275	551	4	·	·	PUNCT
ejpam-6275	552	1	φ2(x	φ2(x	X
ejpam-6275	552	2	)	)	PUNCT
ejpam-6275	552	3	·	·	PUNCT
ejpam-6275	553	1	φ3(x	φ3(x	X
ejpam-6275	553	2	)	)	PUNCT
ejpam-6275	553	3	·	·	PUNCT
ejpam-6275	553	4	φ5(x	φ5(x	NOUN
ejpam-6275	553	5	)	)	PUNCT
ejpam-6275	553	6	·	·	PUNCT
ejpam-6275	553	7	φ7(x	φ7(x	X
ejpam-6275	553	8	)	)	PUNCT
ejpam-6275	553	9	·	·	PUNCT
ejpam-6275	553	10	φ9(x	φ9(x	NUM
ejpam-6275	553	11	)	)	PUNCT
ejpam-6275	553	12	·	·	PUNCT
ejpam-6275	553	13	φ10(x	φ10(x	NOUN
ejpam-6275	553	14	)	)	PUNCT
ejpam-6275	553	15	·	·	PUNCT
ejpam-6275	553	16	φ14(x	φ14(x	NOUN
ejpam-6275	553	17	)	)	PUNCT
ejpam-6275	553	18	.	.	PUNCT
ejpam-6275	554	1	expanding	expand	VERB
ejpam-6275	554	2	,	,	PUNCT
ejpam-6275	554	3	g(x	g(x	NOUN
ejpam-6275	554	4	)	)	PUNCT
ejpam-6275	554	5	=	=	SYM
ejpam-6275	555	1	x20	x20	NOUN
ejpam-6275	555	2	+	+	CCONJ
ejpam-6275	555	3	x18	x18	NOUN
ejpam-6275	556	1	+	+	CCONJ
ejpam-6275	556	2	x17	x17	NOUN
ejpam-6275	556	3	+	+	CCONJ
ejpam-6275	556	4	x16	x16	NOUN
ejpam-6275	556	5	+	+	CCONJ
ejpam-6275	556	6	ωx15	ωx15	PROPN
ejpam-6275	556	7	+	+	CCONJ
ejpam-6275	556	8	ωx14	ωx14	PROPN
ejpam-6275	556	9	+	+	NUM
ejpam-6275	556	10	x13	x13	NOUN
ejpam-6275	556	11	+	+	NUM
ejpam-6275	556	12	x9	x9	NOUN
ejpam-6275	556	13	+	+	CCONJ
ejpam-6275	556	14	x5	x5	NOUN
ejpam-6275	556	15	+	+	CCONJ
ejpam-6275	556	16	x4	x4	PROPN
ejpam-6275	557	1	+	+	CCONJ
ejpam-6275	557	2	x3	x3	ADJ
ejpam-6275	557	3	+	+	NOUN
ejpam-6275	557	4	1	1	X
ejpam-6275	557	5	.	.	PUNCT
ejpam-6275	557	6	since	since	SCONJ
ejpam-6275	557	7	k	k	PROPN
ejpam-6275	557	8	=	=	SYM
ejpam-6275	557	9	21	21	NUM
ejpam-6275	557	10	−	−	NUM
ejpam-6275	557	11	20	20	NUM
ejpam-6275	557	12	=	=	SYM
ejpam-6275	557	13	1	1	NUM
ejpam-6275	557	14	,	,	PUNCT
ejpam-6275	557	15	the	the	DET
ejpam-6275	557	16	code	code	NOUN
ejpam-6275	557	17	parameters	parameter	NOUN
ejpam-6275	557	18	are	be	AUX
ejpam-6275	557	19	(	(	PUNCT
ejpam-6275	557	20	n	n	X
ejpam-6275	557	21	,	,	PUNCT
ejpam-6275	557	22	k	k	NOUN
ejpam-6275	557	23	,	,	PUNCT
ejpam-6275	557	24	d	d	NOUN
ejpam-6275	557	25	)	)	PUNCT
ejpam-6275	558	1	=	=	SYM
ejpam-6275	558	2	(	(	PUNCT
ejpam-6275	558	3	21	21	NUM
ejpam-6275	558	4	,	,	PUNCT
ejpam-6275	558	5	1	1	NUM
ejpam-6275	558	6	,	,	PUNCT
ejpam-6275	558	7	15	15	NUM
ejpam-6275	558	8	)	)	PUNCT
ejpam-6275	558	9	,	,	PUNCT
ejpam-6275	558	10	which	which	PRON
ejpam-6275	558	11	is	be	AUX
ejpam-6275	558	12	a	a	DET
ejpam-6275	558	13	shortened	shorten	VERB
ejpam-6275	558	14	bch	bch	PROPN
ejpam-6275	558	15	code	code	NOUN
ejpam-6275	558	16	.	.	PUNCT
ejpam-6275	559	1	case	case	NOUN
ejpam-6275	559	2	8	8	NUM
ejpam-6275	559	3	:	:	PUNCT
ejpam-6275	559	4	for	for	ADP
ejpam-6275	559	5	n	n	NOUN
ejpam-6275	559	6	=	=	SYM
ejpam-6275	559	7	21	21	NUM
ejpam-6275	559	8	and	and	CCONJ
ejpam-6275	559	9	d	d	NOUN
ejpam-6275	559	10	=	=	SYM
ejpam-6275	559	11	17	17	NUM
ejpam-6275	559	12	,	,	PUNCT
ejpam-6275	559	13	i	i	PRON
ejpam-6275	559	14	=	=	NOUN
ejpam-6275	559	15	1	1	NUM
ejpam-6275	559	16	,	,	PUNCT
ejpam-6275	559	17	2	2	NUM
ejpam-6275	559	18	,	,	PUNCT
ejpam-6275	559	19	3	3	NUM
ejpam-6275	559	20	,	,	PUNCT
ejpam-6275	559	21	.	.	PUNCT
ejpam-6275	559	22	.	.	PUNCT
ejpam-6275	560	1	.	.	PUNCT
ejpam-6275	561	1	,	,	PUNCT
ejpam-6275	561	2	16	16	NUM
ejpam-6275	561	3	.	.	NOUN
ejpam-6275	561	4	•	•	NOUN
ejpam-6275	561	5	for	for	ADP
ejpam-6275	561	6	i	i	PRON
ejpam-6275	561	7	=	=	NOUN
ejpam-6275	561	8	1	1	NUM
ejpam-6275	561	9	,	,	PUNCT
ejpam-6275	561	10	let	let	VERB
ejpam-6275	561	11	β	β	PRON
ejpam-6275	561	12	∈	∈	PROPN
ejpam-6275	561	13	z2[ω	z2[ω	NOUN
ejpam-6275	561	14	]	]	X
ejpam-6275	562	1	3	3	X
ejpam-6275	562	2	.	.	PUNCT
ejpam-6275	562	3	then	then	ADV
ejpam-6275	562	4	by	by	ADP
ejpam-6275	562	5	theorem	theorem	NOUN
ejpam-6275	562	6	3.1	3.1	NUM
ejpam-6275	562	7	,	,	PUNCT
ejpam-6275	562	8	β	β	X
ejpam-6275	562	9	,	,	PUNCT
ejpam-6275	562	10	β4	β4	PROPN
ejpam-6275	562	11	and	and	CCONJ
ejpam-6275	562	12	β16	β16	PROPN
ejpam-6275	562	13	have	have	VERB
ejpam-6275	562	14	the	the	DET
ejpam-6275	562	15	minimal	minimal	ADJ
ejpam-6275	562	16	polynomial	polynomial	ADJ
ejpam-6275	562	17	φ1(x	φ1(x	NOUN
ejpam-6275	562	18	)	)	PUNCT
ejpam-6275	562	19	=	=	PUNCT
ejpam-6275	562	20	(	(	PUNCT
ejpam-6275	562	21	x−	x−	PROPN
ejpam-6275	562	22	β)(x−	β)(x−	PROPN
ejpam-6275	562	23	β4)(x−	β4)(x−	CCONJ
ejpam-6275	562	24	β16	β16	PROPN
ejpam-6275	562	25	)	)	PUNCT
ejpam-6275	562	26	=	=	PUNCT
ejpam-6275	563	1	x3	x3	VERB
ejpam-6275	563	2	+	+	CCONJ
ejpam-6275	564	1	(	(	PUNCT
ejpam-6275	564	2	1	1	NUM
ejpam-6275	564	3	+	+	X
ejpam-6275	564	4	ω)x2	ω)x2	NUM
ejpam-6275	564	5	+	+	CCONJ
ejpam-6275	564	6	1	1	NUM
ejpam-6275	564	7	.	.	NOUN
ejpam-6275	564	8	•	•	NOUN
ejpam-6275	564	9	for	for	ADP
ejpam-6275	564	10	i	i	PRON
ejpam-6275	564	11	=	=	SYM
ejpam-6275	564	12	2	2	NUM
ejpam-6275	564	13	,	,	PUNCT
ejpam-6275	564	14	let	let	VERB
ejpam-6275	564	15	β2	β2	PROPN
ejpam-6275	564	16	∈	∈	PROPN
ejpam-6275	564	17	z2[ω	z2[ω	NOUN
ejpam-6275	564	18	]	]	X
ejpam-6275	565	1	3	3	X
ejpam-6275	565	2	.	.	PUNCT
ejpam-6275	565	3	then	then	ADV
ejpam-6275	565	4	by	by	ADP
ejpam-6275	565	5	theorem	theorem	ADJ
ejpam-6275	565	6	3.1	3.1	NUM
ejpam-6275	565	7	,	,	PUNCT
ejpam-6275	565	8	β2	β2	ADJ
ejpam-6275	565	9	,	,	PUNCT
ejpam-6275	565	10	β8	β8	NOUN
ejpam-6275	565	11	and	and	CCONJ
ejpam-6275	565	12	β11	β11	NOUN
ejpam-6275	565	13	have	have	VERB
ejpam-6275	565	14	the	the	DET
ejpam-6275	565	15	minimal	minimal	ADJ
ejpam-6275	565	16	polynomial	polynomial	ADJ
ejpam-6275	565	17	φ2(x	φ2(x	NUM
ejpam-6275	565	18	)	)	PUNCT
ejpam-6275	565	19	=	=	PUNCT
ejpam-6275	565	20	(	(	PUNCT
ejpam-6275	565	21	x−	x−	PROPN
ejpam-6275	565	22	β2)(x−	β2)(x−	PROPN
ejpam-6275	565	23	β8)(x−	β8)(x−	X
ejpam-6275	565	24	β11	β11	PROPN
ejpam-6275	565	25	)	)	PUNCT
ejpam-6275	565	26	=	=	PUNCT
ejpam-6275	566	1	x3	x3	VERB
ejpam-6275	567	1	+	+	CCONJ
ejpam-6275	567	2	ωx2	ωx2	PRON
ejpam-6275	567	3	+	+	X
ejpam-6275	567	4	1	1	NUM
ejpam-6275	567	5	.	.	NUM
ejpam-6275	567	6	•	•	NOUN
ejpam-6275	567	7	for	for	ADP
ejpam-6275	567	8	i	i	PRON
ejpam-6275	567	9	=	=	SYM
ejpam-6275	567	10	3	3	NUM
ejpam-6275	567	11	,	,	PUNCT
ejpam-6275	567	12	let	let	VERB
ejpam-6275	567	13	β3	β3	VERB
ejpam-6275	567	14	∈	∈	PROPN
ejpam-6275	567	15	z2[ω	z2[ω	NOUN
ejpam-6275	567	16	]	]	X
ejpam-6275	567	17	3	3	X
ejpam-6275	567	18	.	.	PUNCT
ejpam-6275	567	19	then	then	ADV
ejpam-6275	567	20	by	by	ADP
ejpam-6275	567	21	theorem	theorem	ADJ
ejpam-6275	567	22	3.1	3.1	NUM
ejpam-6275	567	23	,	,	PUNCT
ejpam-6275	567	24	β3	β3	ADJ
ejpam-6275	567	25	,	,	PUNCT
ejpam-6275	567	26	β6	β6	PROPN
ejpam-6275	567	27	and	and	CCONJ
ejpam-6275	567	28	β12	β12	NOUN
ejpam-6275	567	29	have	have	VERB
ejpam-6275	567	30	the	the	DET
ejpam-6275	567	31	minimal	minimal	ADJ
ejpam-6275	567	32	polynomial	polynomial	ADJ
ejpam-6275	567	33	φ3(x	φ3(x	PROPN
ejpam-6275	567	34	)	)	PUNCT
ejpam-6275	567	35	=	=	SYM
ejpam-6275	567	36	(	(	PUNCT
ejpam-6275	567	37	x−	x−	PROPN
ejpam-6275	567	38	β3)(x−	β3)(x−	PROPN
ejpam-6275	567	39	β6)(x−	β6)(x−	X
ejpam-6275	567	40	β12	β12	PROPN
ejpam-6275	567	41	)	)	PUNCT
ejpam-6275	567	42	=	=	PUNCT
ejpam-6275	568	1	x3	x3	VERB
ejpam-6275	568	2	+	+	CCONJ
ejpam-6275	568	3	x+	x+	ADJ
ejpam-6275	568	4	1	1	NUM
ejpam-6275	568	5	.	.	NUM
ejpam-6275	568	6	•	•	NOUN
ejpam-6275	568	7	for	for	ADP
ejpam-6275	568	8	i	i	PRON
ejpam-6275	568	9	=	=	NOUN
ejpam-6275	568	10	4	4	NUM
ejpam-6275	568	11	,	,	PUNCT
ejpam-6275	568	12	let	let	VERB
ejpam-6275	568	13	β4	β4	PROPN
ejpam-6275	568	14	∈	∈	PROPN
ejpam-6275	568	15	z2[ω	z2[ω	NOUN
ejpam-6275	568	16	]	]	X
ejpam-6275	569	1	3	3	X
ejpam-6275	569	2	.	.	PUNCT
ejpam-6275	569	3	then	then	ADV
ejpam-6275	569	4	by	by	ADP
ejpam-6275	569	5	theorem	theorem	ADJ
ejpam-6275	569	6	3.1	3.1	NUM
ejpam-6275	569	7	,	,	PUNCT
ejpam-6275	569	8	β2	β2	ADJ
ejpam-6275	569	9	,	,	PUNCT
ejpam-6275	569	10	β8	β8	NOUN
ejpam-6275	569	11	and	and	CCONJ
ejpam-6275	569	12	β11	β11	NOUN
ejpam-6275	569	13	have	have	VERB
ejpam-6275	569	14	the	the	DET
ejpam-6275	569	15	minimal	minimal	ADJ
ejpam-6275	569	16	polynomial	polynomial	ADJ
ejpam-6275	569	17	φ4(x	φ4(x	NOUN
ejpam-6275	569	18	)	)	PUNCT
ejpam-6275	569	19	=	=	SYM
ejpam-6275	569	20	(	(	PUNCT
ejpam-6275	569	21	x−	x−	PROPN
ejpam-6275	569	22	β2)(x−	β2)(x−	PROPN
ejpam-6275	569	23	β8)(x−	β8)(x−	X
ejpam-6275	569	24	β11	β11	PROPN
ejpam-6275	569	25	)	)	PUNCT
ejpam-6275	569	26	=	=	SYM
ejpam-6275	569	27	φ2(x	φ2(x	NUM
ejpam-6275	569	28	)	)	PUNCT
ejpam-6275	569	29	.	.	PUNCT
ejpam-6275	570	1	•	•	NOUN
ejpam-6275	570	2	for	for	ADP
ejpam-6275	570	3	i	i	PRON
ejpam-6275	570	4	=	=	SYM
ejpam-6275	570	5	5	5	NUM
ejpam-6275	570	6	,	,	PUNCT
ejpam-6275	570	7	let	let	VERB
ejpam-6275	570	8	β5	β5	NOUN
ejpam-6275	570	9	∈	∈	PROPN
ejpam-6275	570	10	z2[ω	z2[ω	NOUN
ejpam-6275	570	11	]	]	X
ejpam-6275	571	1	3	3	X
ejpam-6275	571	2	.	.	PUNCT
ejpam-6275	571	3	then	then	ADV
ejpam-6275	571	4	by	by	ADP
ejpam-6275	571	5	theorem	theorem	ADJ
ejpam-6275	571	6	3.1	3.1	NUM
ejpam-6275	571	7	,	,	PUNCT
ejpam-6275	571	8	β5	β5	NOUN
ejpam-6275	571	9	,	,	PUNCT
ejpam-6275	571	10	β17	β17	NUM
ejpam-6275	571	11	and	and	CCONJ
ejpam-6275	571	12	β20	β20	PROPN
ejpam-6275	571	13	have	have	VERB
ejpam-6275	571	14	the	the	DET
ejpam-6275	571	15	minimal	minimal	ADJ
ejpam-6275	571	16	polynomial	polynomial	ADJ
ejpam-6275	571	17	φ5(x	φ5(x	NOUN
ejpam-6275	571	18	)	)	PUNCT
ejpam-6275	571	19	=	=	SYM
ejpam-6275	571	20	(	(	PUNCT
ejpam-6275	571	21	x−	x−	PROPN
ejpam-6275	571	22	β5)(x−	β5)(x−	PROPN
ejpam-6275	571	23	β17)(x−	β17)(x−	PROPN
ejpam-6275	571	24	β20	β20	PROPN
ejpam-6275	571	25	)	)	PUNCT
ejpam-6275	571	26	=	=	SYM
ejpam-6275	572	1	x3	x3	VERB
ejpam-6275	572	2	+	+	CCONJ
ejpam-6275	572	3	(	(	PUNCT
ejpam-6275	572	4	1	1	NUM
ejpam-6275	572	5	+	+	SYM
ejpam-6275	572	6	ω)x+	ω)x+	NUM
ejpam-6275	572	7	1	1	NUM
ejpam-6275	572	8	.	.	NOUN
ejpam-6275	572	9	•	•	NOUN
ejpam-6275	572	10	for	for	ADP
ejpam-6275	572	11	i	i	PRON
ejpam-6275	572	12	=	=	SYM
ejpam-6275	572	13	6	6	NUM
ejpam-6275	572	14	,	,	PUNCT
ejpam-6275	572	15	let	let	VERB
ejpam-6275	572	16	β6	β6	PROPN
ejpam-6275	572	17	∈	∈	PROPN
ejpam-6275	572	18	z2[ω	z2[ω	NOUN
ejpam-6275	572	19	]	]	X
ejpam-6275	573	1	3	3	X
ejpam-6275	573	2	.	.	PUNCT
ejpam-6275	573	3	then	then	ADV
ejpam-6275	573	4	by	by	ADP
ejpam-6275	573	5	theorem	theorem	ADJ
ejpam-6275	573	6	3.1	3.1	NUM
ejpam-6275	573	7	,	,	PUNCT
ejpam-6275	573	8	β3	β3	ADJ
ejpam-6275	573	9	,	,	PUNCT
ejpam-6275	573	10	β6	β6	PROPN
ejpam-6275	573	11	and	and	CCONJ
ejpam-6275	573	12	β12	β12	NOUN
ejpam-6275	573	13	have	have	VERB
ejpam-6275	573	14	the	the	DET
ejpam-6275	573	15	minimal	minimal	ADJ
ejpam-6275	573	16	polynomial	polynomial	ADJ
ejpam-6275	573	17	φ6(x	φ6(x	NOUN
ejpam-6275	573	18	)	)	PUNCT
ejpam-6275	573	19	=	=	SYM
ejpam-6275	573	20	(	(	PUNCT
ejpam-6275	574	1	x−	x−	PROPN
ejpam-6275	574	2	β3)(x−	β3)(x−	PROPN
ejpam-6275	574	3	β6)(x−	β6)(x−	X
ejpam-6275	574	4	β12	β12	PROPN
ejpam-6275	574	5	)	)	PUNCT
ejpam-6275	575	1	=	=	SYM
ejpam-6275	575	2	φ3(x	φ3(x	PROPN
ejpam-6275	575	3	)	)	PUNCT
ejpam-6275	575	4	.	.	PUNCT
ejpam-6275	576	1	m.	m.	PROPN
ejpam-6275	576	2	sajjad	sajjad	PROPN
ejpam-6275	576	3	et	et	PROPN
ejpam-6275	576	4	al	al	PROPN
ejpam-6275	576	5	.	.	PUNCT
ejpam-6275	576	6	/	/	SYM
ejpam-6275	576	7	eur	eur	PROPN
ejpam-6275	576	8	.	.	PUNCT
ejpam-6275	577	1	j.	j.	PROPN
ejpam-6275	577	2	pure	pure	PROPN
ejpam-6275	577	3	appl	appl	PROPN
ejpam-6275	577	4	.	.	PROPN
ejpam-6275	577	5	math	math	PROPN
ejpam-6275	577	6	,	,	PUNCT
ejpam-6275	577	7	18	18	NUM
ejpam-6275	577	8	(	(	PUNCT
ejpam-6275	577	9	3	3	NUM
ejpam-6275	577	10	)	)	PUNCT
ejpam-6275	577	11	(	(	PUNCT
ejpam-6275	577	12	2025	2025	NUM
ejpam-6275	577	13	)	)	PUNCT
ejpam-6275	577	14	,	,	PUNCT
ejpam-6275	577	15	6275	6275	NUM
ejpam-6275	577	16	22	22	NUM
ejpam-6275	577	17	of	of	ADP
ejpam-6275	577	18	36	36	NUM
ejpam-6275	577	19	•	•	NOUN
ejpam-6275	577	20	for	for	ADP
ejpam-6275	577	21	i	i	PRON
ejpam-6275	577	22	=	=	SYM
ejpam-6275	577	23	7	7	NUM
ejpam-6275	577	24	,	,	PUNCT
ejpam-6275	577	25	let	let	VERB
ejpam-6275	577	26	β7	β7	ADJ
ejpam-6275	577	27	∈	∈	PROPN
ejpam-6275	577	28	z2[ω	z2[ω	NOUN
ejpam-6275	577	29	]	]	X
ejpam-6275	577	30	3	3	X
ejpam-6275	577	31	.	.	PUNCT
ejpam-6275	577	32	then	then	ADV
ejpam-6275	577	33	by	by	ADP
ejpam-6275	577	34	theorem	theorem	ADJ
ejpam-6275	577	35	3.1	3.1	NUM
ejpam-6275	577	36	,	,	PUNCT
ejpam-6275	577	37	β7	β7	PROPN
ejpam-6275	577	38	has	have	VERB
ejpam-6275	577	39	the	the	DET
ejpam-6275	577	40	minimal	minimal	ADJ
ejpam-6275	577	41	polynomial	polynomial	ADJ
ejpam-6275	577	42	φ7(x	φ7(x	NOUN
ejpam-6275	577	43	)	)	PUNCT
ejpam-6275	577	44	=	=	PUNCT
ejpam-6275	577	45	(	(	PUNCT
ejpam-6275	577	46	x−	x−	PROPN
ejpam-6275	577	47	β7	β7	PROPN
ejpam-6275	577	48	)	)	PUNCT
ejpam-6275	578	1	=	=	PUNCT
ejpam-6275	578	2	x+	x+	PUNCT
ejpam-6275	579	1	1	1	NUM
ejpam-6275	579	2	+	+	NUM
ejpam-6275	579	3	ω	ω	NUM
ejpam-6275	579	4	.	.	NOUN
ejpam-6275	579	5	•	•	NUM
ejpam-6275	579	6	for	for	ADP
ejpam-6275	579	7	i	i	PRON
ejpam-6275	579	8	=	=	NOUN
ejpam-6275	579	9	8	8	NUM
ejpam-6275	579	10	,	,	PUNCT
ejpam-6275	579	11	let	let	VERB
ejpam-6275	579	12	β8	β8	PROPN
ejpam-6275	579	13	∈	∈	PROPN
ejpam-6275	579	14	z2[ω	z2[ω	NOUN
ejpam-6275	579	15	]	]	X
ejpam-6275	579	16	3	3	X
ejpam-6275	579	17	.	.	PUNCT
ejpam-6275	579	18	then	then	ADV
ejpam-6275	579	19	by	by	ADP
ejpam-6275	579	20	theorem	theorem	ADJ
ejpam-6275	579	21	3.1	3.1	NUM
ejpam-6275	579	22	,	,	PUNCT
ejpam-6275	579	23	β2	β2	ADJ
ejpam-6275	579	24	,	,	PUNCT
ejpam-6275	579	25	β8	β8	NOUN
ejpam-6275	579	26	and	and	CCONJ
ejpam-6275	579	27	β11	β11	NOUN
ejpam-6275	579	28	have	have	VERB
ejpam-6275	579	29	the	the	DET
ejpam-6275	579	30	minimal	minimal	ADJ
ejpam-6275	579	31	polynomial	polynomial	ADJ
ejpam-6275	579	32	φ8(x	φ8(x	NOUN
ejpam-6275	579	33	)	)	PUNCT
ejpam-6275	579	34	=	=	SYM
ejpam-6275	579	35	(	(	PUNCT
ejpam-6275	579	36	x−	x−	PROPN
ejpam-6275	579	37	β2)(x−	β2)(x−	PROPN
ejpam-6275	579	38	β8)(x−	β8)(x−	X
ejpam-6275	579	39	β11	β11	PROPN
ejpam-6275	579	40	)	)	PUNCT
ejpam-6275	579	41	=	=	SYM
ejpam-6275	580	1	φ2(x	φ2(x	NUM
ejpam-6275	580	2	)	)	PUNCT
ejpam-6275	580	3	.	.	PUNCT
ejpam-6275	581	1	•	•	NOUN
ejpam-6275	581	2	for	for	ADP
ejpam-6275	581	3	i	i	PRON
ejpam-6275	581	4	=	=	NOUN
ejpam-6275	581	5	9	9	NUM
ejpam-6275	581	6	,	,	PUNCT
ejpam-6275	581	7	let	let	VERB
ejpam-6275	581	8	β9	β9	PROPN
ejpam-6275	581	9	∈	∈	PROPN
ejpam-6275	581	10	z2[ω	z2[ω	NOUN
ejpam-6275	581	11	]	]	X
ejpam-6275	581	12	3	3	X
ejpam-6275	581	13	.	.	PUNCT
ejpam-6275	581	14	then	then	ADV
ejpam-6275	581	15	by	by	ADP
ejpam-6275	581	16	theorem	theorem	ADJ
ejpam-6275	581	17	3.1	3.1	NUM
ejpam-6275	581	18	,	,	PUNCT
ejpam-6275	581	19	β9	β9	PROPN
ejpam-6275	581	20	,	,	PUNCT
ejpam-6275	581	21	β18	β18	NUM
ejpam-6275	581	22	and	and	CCONJ
ejpam-6275	581	23	β15	β15	NOUN
ejpam-6275	581	24	have	have	VERB
ejpam-6275	581	25	the	the	DET
ejpam-6275	581	26	minimal	minimal	ADJ
ejpam-6275	581	27	polynomial	polynomial	ADJ
ejpam-6275	581	28	φ9(x	φ9(x	NUM
ejpam-6275	581	29	)	)	PUNCT
ejpam-6275	581	30	=	=	SYM
ejpam-6275	581	31	(	(	PUNCT
ejpam-6275	581	32	x−	x−	PROPN
ejpam-6275	581	33	β9)(x−	β9)(x−	PUNCT
ejpam-6275	581	34	β18)(x−	β18)(x−	NOUN
ejpam-6275	581	35	β15	β15	NOUN
ejpam-6275	581	36	)	)	PUNCT
ejpam-6275	581	37	=	=	SYM
ejpam-6275	582	1	x3	x3	VERB
ejpam-6275	583	1	+	+	CCONJ
ejpam-6275	583	2	x2	x2	PROPN
ejpam-6275	584	1	+	+	CCONJ
ejpam-6275	585	1	1	1	NUM
ejpam-6275	585	2	.	.	NUM
ejpam-6275	585	3	•	•	NOUN
ejpam-6275	585	4	for	for	ADP
ejpam-6275	585	5	i	i	PRON
ejpam-6275	585	6	=	=	NOUN
ejpam-6275	585	7	10	10	NUM
ejpam-6275	585	8	,	,	PUNCT
ejpam-6275	585	9	let	let	VERB
ejpam-6275	585	10	β10	β10	VERB
ejpam-6275	585	11	∈	∈	PROPN
ejpam-6275	585	12	z2[ω	z2[ω	NOUN
ejpam-6275	585	13	]	]	X
ejpam-6275	586	1	3	3	X
ejpam-6275	586	2	.	.	PUNCT
ejpam-6275	586	3	then	then	ADV
ejpam-6275	586	4	by	by	ADP
ejpam-6275	586	5	theorem	theorem	ADJ
ejpam-6275	586	6	3.1	3.1	NUM
ejpam-6275	586	7	,	,	PUNCT
ejpam-6275	586	8	β10	β10	NOUN
ejpam-6275	586	9	,	,	PUNCT
ejpam-6275	586	10	β13	β13	NOUN
ejpam-6275	586	11	and	and	CCONJ
ejpam-6275	586	12	β19	β19	NOUN
ejpam-6275	586	13	have	have	VERB
ejpam-6275	586	14	the	the	DET
ejpam-6275	586	15	minimal	minimal	ADJ
ejpam-6275	586	16	polynomial	polynomial	ADJ
ejpam-6275	586	17	φ10(x	φ10(x	NOUN
ejpam-6275	586	18	)	)	PUNCT
ejpam-6275	587	1	=	=	PUNCT
ejpam-6275	587	2	(	(	PUNCT
ejpam-6275	587	3	x−	x−	PROPN
ejpam-6275	587	4	β10)(x−	β10)(x−	PROPN
ejpam-6275	587	5	β13)(x−	β13)(x−	PROPN
ejpam-6275	587	6	β19	β19	ADV
ejpam-6275	587	7	)	)	PUNCT
ejpam-6275	587	8	=	=	SYM
ejpam-6275	588	1	x3	x3	ADJ
ejpam-6275	588	2	+	+	CCONJ
ejpam-6275	588	3	ωx+	ωx+	NOUN
ejpam-6275	588	4	1	1	NUM
ejpam-6275	588	5	.	.	NOUN
ejpam-6275	588	6	•	•	NOUN
ejpam-6275	588	7	for	for	ADP
ejpam-6275	588	8	i	i	PRON
ejpam-6275	588	9	=	=	NOUN
ejpam-6275	588	10	11	11	NUM
ejpam-6275	588	11	,	,	PUNCT
ejpam-6275	588	12	let	let	VERB
ejpam-6275	588	13	β11	β11	PROPN
ejpam-6275	588	14	∈	∈	PROPN
ejpam-6275	588	15	z2[ω	z2[ω	NOUN
ejpam-6275	588	16	]	]	X
ejpam-6275	589	1	3	3	X
ejpam-6275	589	2	.	.	PUNCT
ejpam-6275	589	3	then	then	ADV
ejpam-6275	589	4	by	by	ADP
ejpam-6275	589	5	theorem	theorem	ADJ
ejpam-6275	589	6	3.1	3.1	NUM
ejpam-6275	589	7	,	,	PUNCT
ejpam-6275	589	8	β2	β2	ADJ
ejpam-6275	589	9	,	,	PUNCT
ejpam-6275	589	10	β8	β8	NOUN
ejpam-6275	589	11	and	and	CCONJ
ejpam-6275	589	12	β11	β11	NOUN
ejpam-6275	589	13	have	have	VERB
ejpam-6275	589	14	the	the	DET
ejpam-6275	589	15	minimal	minimal	ADJ
ejpam-6275	589	16	polynomial	polynomial	ADJ
ejpam-6275	589	17	φ11(x	φ11(x	NOUN
ejpam-6275	589	18	)	)	PUNCT
ejpam-6275	589	19	=	=	PUNCT
ejpam-6275	589	20	(	(	PUNCT
ejpam-6275	589	21	x−	x−	PROPN
ejpam-6275	589	22	β2)(x−	β2)(x−	PROPN
ejpam-6275	589	23	β8)(x−	β8)(x−	X
ejpam-6275	589	24	β11	β11	PROPN
ejpam-6275	589	25	)	)	PUNCT
ejpam-6275	589	26	=	=	SYM
ejpam-6275	589	27	φ2(x	φ2(x	NUM
ejpam-6275	589	28	)	)	PUNCT
ejpam-6275	589	29	.	.	PUNCT
ejpam-6275	590	1	•	•	NOUN
ejpam-6275	590	2	for	for	ADP
ejpam-6275	590	3	i	i	PRON
ejpam-6275	590	4	=	=	NOUN
ejpam-6275	590	5	12	12	NUM
ejpam-6275	590	6	,	,	PUNCT
ejpam-6275	590	7	let	let	VERB
ejpam-6275	590	8	β12	β12	PRON
ejpam-6275	590	9	∈	∈	PROPN
ejpam-6275	590	10	z2[ω	z2[ω	NOUN
ejpam-6275	590	11	]	]	X
ejpam-6275	591	1	3	3	X
ejpam-6275	591	2	.	.	PUNCT
ejpam-6275	591	3	then	then	ADV
ejpam-6275	591	4	by	by	ADP
ejpam-6275	591	5	theorem	theorem	ADJ
ejpam-6275	591	6	3.1	3.1	NUM
ejpam-6275	591	7	,	,	PUNCT
ejpam-6275	591	8	β3	β3	ADJ
ejpam-6275	591	9	,	,	PUNCT
ejpam-6275	591	10	β6	β6	PROPN
ejpam-6275	591	11	and	and	CCONJ
ejpam-6275	591	12	β12	β12	NOUN
ejpam-6275	591	13	have	have	VERB
ejpam-6275	591	14	the	the	DET
ejpam-6275	591	15	minimal	minimal	ADJ
ejpam-6275	591	16	polynomial	polynomial	ADJ
ejpam-6275	591	17	φ12(x	φ12(x	NOUN
ejpam-6275	591	18	)	)	PUNCT
ejpam-6275	591	19	=	=	PUNCT
ejpam-6275	591	20	(	(	PUNCT
ejpam-6275	591	21	x−	x−	PROPN
ejpam-6275	591	22	β3)(x−	β3)(x−	PROPN
ejpam-6275	591	23	β6)(x−	β6)(x−	X
ejpam-6275	591	24	β12	β12	PROPN
ejpam-6275	591	25	)	)	PUNCT
ejpam-6275	591	26	=	=	PUNCT
ejpam-6275	592	1	x3	x3	VERB
ejpam-6275	593	1	+	+	CCONJ
ejpam-6275	593	2	x+	x+	SYM
ejpam-6275	593	3	1	1	X
ejpam-6275	593	4	=	=	SYM
ejpam-6275	593	5	φ3(x	φ3(x	PROPN
ejpam-6275	593	6	)	)	PUNCT
ejpam-6275	593	7	.	.	PUNCT
ejpam-6275	594	1	•	•	NOUN
ejpam-6275	594	2	for	for	ADP
ejpam-6275	594	3	i	i	PRON
ejpam-6275	594	4	=	=	NOUN
ejpam-6275	594	5	13	13	NUM
ejpam-6275	594	6	,	,	PUNCT
ejpam-6275	594	7	let	let	VERB
ejpam-6275	594	8	β13	β13	PRON
ejpam-6275	594	9	∈	∈	PROPN
ejpam-6275	594	10	z2[ω	z2[ω	NOUN
ejpam-6275	594	11	]	]	X
ejpam-6275	595	1	3	3	X
ejpam-6275	595	2	.	.	PUNCT
ejpam-6275	595	3	then	then	ADV
ejpam-6275	595	4	by	by	ADP
ejpam-6275	595	5	theorem	theorem	ADJ
ejpam-6275	595	6	3.1	3.1	NUM
ejpam-6275	595	7	,	,	PUNCT
ejpam-6275	595	8	β10	β10	NOUN
ejpam-6275	595	9	,	,	PUNCT
ejpam-6275	595	10	β13	β13	NOUN
ejpam-6275	595	11	and	and	CCONJ
ejpam-6275	595	12	β19	β19	NOUN
ejpam-6275	595	13	have	have	VERB
ejpam-6275	595	14	the	the	DET
ejpam-6275	595	15	minimal	minimal	ADJ
ejpam-6275	595	16	polynomial	polynomial	ADJ
ejpam-6275	595	17	φ13(x	φ13(x	NOUN
ejpam-6275	595	18	)	)	PUNCT
ejpam-6275	596	1	=	=	PUNCT
ejpam-6275	596	2	(	(	PUNCT
ejpam-6275	596	3	x−	x−	PROPN
ejpam-6275	596	4	β10)(x−	β10)(x−	PROPN
ejpam-6275	596	5	β13)(x−	β13)(x−	PROPN
ejpam-6275	596	6	β19	β19	ADV
ejpam-6275	596	7	)	)	PUNCT
ejpam-6275	597	1	=	=	SYM
ejpam-6275	597	2	=	=	SYM
ejpam-6275	597	3	x3	x3	VERB
ejpam-6275	597	4	+	+	CCONJ
ejpam-6275	597	5	ωx+	ωx+	NOUN
ejpam-6275	597	6	1	1	NUM
ejpam-6275	597	7	=	=	SYM
ejpam-6275	597	8	φ10(x	φ10(x	NOUN
ejpam-6275	597	9	)	)	PUNCT
ejpam-6275	597	10	.	.	PUNCT
ejpam-6275	598	1	•	•	NOUN
ejpam-6275	598	2	for	for	ADP
ejpam-6275	598	3	i	i	PRON
ejpam-6275	598	4	=	=	NOUN
ejpam-6275	598	5	14	14	NUM
ejpam-6275	598	6	,	,	PUNCT
ejpam-6275	598	7	let	let	VERB
ejpam-6275	598	8	β14	β14	PRON
ejpam-6275	598	9	∈	∈	PROPN
ejpam-6275	598	10	z2[ω	z2[ω	NOUN
ejpam-6275	598	11	]	]	X
ejpam-6275	599	1	3	3	X
ejpam-6275	599	2	.	.	PUNCT
ejpam-6275	599	3	then	then	ADV
ejpam-6275	599	4	by	by	ADP
ejpam-6275	599	5	theorem	theorem	NOUN
ejpam-6275	599	6	3.1	3.1	NUM
ejpam-6275	599	7	,	,	PUNCT
ejpam-6275	599	8	β14	β14	X
ejpam-6275	599	9	has	have	VERB
ejpam-6275	599	10	the	the	DET
ejpam-6275	599	11	minimal	minimal	ADJ
ejpam-6275	599	12	polynomial	polynomial	ADJ
ejpam-6275	599	13	φ14(x	φ14(x	NOUN
ejpam-6275	599	14	)	)	PUNCT
ejpam-6275	599	15	=	=	PUNCT
ejpam-6275	599	16	(	(	PUNCT
ejpam-6275	599	17	x−	x−	PROPN
ejpam-6275	599	18	β14	β14	ADJ
ejpam-6275	599	19	)	)	PUNCT
ejpam-6275	600	1	=	=	SYM
ejpam-6275	600	2	x+	x+	PROPN
ejpam-6275	601	1	ω	ω	X
ejpam-6275	601	2	.	.	PROPN
ejpam-6275	601	3	•	•	NUM
ejpam-6275	601	4	for	for	ADP
ejpam-6275	601	5	i	i	PRON
ejpam-6275	601	6	=	=	NOUN
ejpam-6275	601	7	15	15	NUM
ejpam-6275	601	8	,	,	PUNCT
ejpam-6275	601	9	let	let	VERB
ejpam-6275	601	10	β15	β15	NOUN
ejpam-6275	601	11	∈	∈	PROPN
ejpam-6275	601	12	z2[ω	z2[ω	NOUN
ejpam-6275	601	13	]	]	X
ejpam-6275	601	14	3	3	X
ejpam-6275	601	15	.	.	PUNCT
ejpam-6275	601	16	then	then	ADV
ejpam-6275	601	17	by	by	ADP
ejpam-6275	601	18	theorem	theorem	ADJ
ejpam-6275	601	19	3.1	3.1	NUM
ejpam-6275	601	20	,	,	PUNCT
ejpam-6275	601	21	β9	β9	PROPN
ejpam-6275	601	22	,	,	PUNCT
ejpam-6275	601	23	β18	β18	NUM
ejpam-6275	601	24	and	and	CCONJ
ejpam-6275	601	25	β15	β15	NOUN
ejpam-6275	601	26	have	have	VERB
ejpam-6275	601	27	the	the	DET
ejpam-6275	601	28	minimal	minimal	ADJ
ejpam-6275	601	29	polynomial	polynomial	ADJ
ejpam-6275	601	30	φ15(x	φ15(x	NOUN
ejpam-6275	601	31	)	)	PUNCT
ejpam-6275	601	32	=	=	PUNCT
ejpam-6275	601	33	(	(	PUNCT
ejpam-6275	601	34	x−	x−	PROPN
ejpam-6275	601	35	β9)(x−	β9)(x−	PUNCT
ejpam-6275	601	36	β18)(x−	β18)(x−	NOUN
ejpam-6275	601	37	β15	β15	NOUN
ejpam-6275	601	38	)	)	PUNCT
ejpam-6275	602	1	=	=	SYM
ejpam-6275	602	2	φ9(x	φ9(x	NOUN
ejpam-6275	602	3	)	)	PUNCT
ejpam-6275	602	4	.	.	PUNCT
ejpam-6275	603	1	m.	m.	PROPN
ejpam-6275	603	2	sajjad	sajjad	PROPN
ejpam-6275	603	3	et	et	PROPN
ejpam-6275	603	4	al	al	PROPN
ejpam-6275	603	5	.	.	PUNCT
ejpam-6275	603	6	/	/	SYM
ejpam-6275	603	7	eur	eur	PROPN
ejpam-6275	603	8	.	.	PUNCT
ejpam-6275	604	1	j.	j.	PROPN
ejpam-6275	604	2	pure	pure	PROPN
ejpam-6275	604	3	appl	appl	PROPN
ejpam-6275	604	4	.	.	PROPN
ejpam-6275	604	5	math	math	PROPN
ejpam-6275	604	6	,	,	PUNCT
ejpam-6275	604	7	18	18	NUM
ejpam-6275	604	8	(	(	PUNCT
ejpam-6275	604	9	3	3	NUM
ejpam-6275	604	10	)	)	PUNCT
ejpam-6275	604	11	(	(	PUNCT
ejpam-6275	604	12	2025	2025	NUM
ejpam-6275	604	13	)	)	PUNCT
ejpam-6275	604	14	,	,	PUNCT
ejpam-6275	604	15	6275	6275	NUM
ejpam-6275	604	16	23	23	NUM
ejpam-6275	604	17	of	of	ADP
ejpam-6275	604	18	36	36	NUM
ejpam-6275	604	19	•	•	NOUN
ejpam-6275	604	20	for	for	ADP
ejpam-6275	604	21	i	i	PRON
ejpam-6275	604	22	=	=	NOUN
ejpam-6275	604	23	16	16	NUM
ejpam-6275	604	24	,	,	PUNCT
ejpam-6275	604	25	let	let	VERB
ejpam-6275	604	26	β16	β16	PROPN
ejpam-6275	604	27	∈	∈	PROPN
ejpam-6275	604	28	z2[ω	z2[ω	NOUN
ejpam-6275	604	29	]	]	X
ejpam-6275	604	30	3	3	X
ejpam-6275	604	31	.	.	PUNCT
ejpam-6275	604	32	then	then	ADV
ejpam-6275	604	33	by	by	ADP
ejpam-6275	604	34	theorem	theorem	NOUN
ejpam-6275	604	35	3.1	3.1	NUM
ejpam-6275	604	36	,	,	PUNCT
ejpam-6275	604	37	β	β	X
ejpam-6275	604	38	,	,	PUNCT
ejpam-6275	604	39	β4	β4	PROPN
ejpam-6275	604	40	and	and	CCONJ
ejpam-6275	604	41	β16	β16	PROPN
ejpam-6275	604	42	have	have	VERB
ejpam-6275	604	43	the	the	DET
ejpam-6275	604	44	minimal	minimal	ADJ
ejpam-6275	604	45	polynomial	polynomial	ADJ
ejpam-6275	604	46	φ16(x	φ16(x	NOUN
ejpam-6275	604	47	)	)	PUNCT
ejpam-6275	604	48	=	=	PUNCT
ejpam-6275	604	49	(	(	PUNCT
ejpam-6275	604	50	x−	x−	PROPN
ejpam-6275	604	51	β)(x−	β)(x−	PROPN
ejpam-6275	604	52	β4)(x−	β4)(x−	CCONJ
ejpam-6275	604	53	β16	β16	PROPN
ejpam-6275	604	54	)	)	PUNCT
ejpam-6275	604	55	=	=	PUNCT
ejpam-6275	605	1	x3	x3	VERB
ejpam-6275	605	2	+	+	CCONJ
ejpam-6275	605	3	(	(	PUNCT
ejpam-6275	605	4	1	1	NUM
ejpam-6275	605	5	+	+	X
ejpam-6275	605	6	ω)x2	ω)x2	NUM
ejpam-6275	605	7	+	+	CCONJ
ejpam-6275	605	8	1	1	NUM
ejpam-6275	605	9	=	=	SYM
ejpam-6275	605	10	φ1(x	φ1(x	NOUN
ejpam-6275	605	11	)	)	PUNCT
ejpam-6275	605	12	.	.	PUNCT
ejpam-6275	606	1	now	now	ADV
ejpam-6275	606	2	the	the	DET
ejpam-6275	606	3	generator	generator	NOUN
ejpam-6275	606	4	polynomial	polynomial	NOUN
ejpam-6275	606	5	is	be	AUX
ejpam-6275	606	6	g(x	g(x	NOUN
ejpam-6275	606	7	)	)	PUNCT
ejpam-6275	607	1	=	=	SYM
ejpam-6275	607	2	φ1(x	φ1(x	NOUN
ejpam-6275	607	3	)	)	PUNCT
ejpam-6275	607	4	·	·	PUNCT
ejpam-6275	608	1	φ2(x	φ2(x	X
ejpam-6275	608	2	)	)	PUNCT
ejpam-6275	608	3	·	·	PUNCT
ejpam-6275	609	1	φ3(x	φ3(x	X
ejpam-6275	609	2	)	)	PUNCT
ejpam-6275	609	3	·	·	PUNCT
ejpam-6275	609	4	φ5(x	φ5(x	NOUN
ejpam-6275	609	5	)	)	PUNCT
ejpam-6275	609	6	·	·	PUNCT
ejpam-6275	609	7	φ7(x	φ7(x	X
ejpam-6275	609	8	)	)	PUNCT
ejpam-6275	609	9	·	·	PUNCT
ejpam-6275	609	10	φ9(x	φ9(x	NUM
ejpam-6275	609	11	)	)	PUNCT
ejpam-6275	609	12	·	·	PUNCT
ejpam-6275	609	13	φ10(x	φ10(x	NOUN
ejpam-6275	609	14	)	)	PUNCT
ejpam-6275	609	15	·	·	PUNCT
ejpam-6275	609	16	φ14(x	φ14(x	NOUN
ejpam-6275	609	17	)	)	PUNCT
ejpam-6275	609	18	=	=	SYM
ejpam-6275	609	19	(	(	PUNCT
ejpam-6275	609	20	x19	x19	NOUN
ejpam-6275	609	21	+	+	CCONJ
ejpam-6275	609	22	(	(	PUNCT
ejpam-6275	609	23	1	1	NUM
ejpam-6275	609	24	+	+	NOUN
ejpam-6275	609	25	ω)x18	ω)x18	PRON
ejpam-6275	609	26	+	+	CCONJ
ejpam-6275	609	27	x16	x16	NOUN
ejpam-6275	609	28	+	+	CCONJ
ejpam-6275	609	29	(	(	PUNCT
ejpam-6275	609	30	1	1	NUM
ejpam-6275	609	31	+	+	CCONJ
ejpam-6275	609	32	ω)x15	ω)x15	NUM
ejpam-6275	609	33	+	+	CCONJ
ejpam-6275	609	34	(	(	PUNCT
ejpam-6275	609	35	1	1	NUM
ejpam-6275	609	36	+	+	NUM
ejpam-6275	609	37	ω)x14	ω)x14	PRON
ejpam-6275	609	38	+	+	CCONJ
ejpam-6275	609	39	(	(	PUNCT
ejpam-6275	609	40	1	1	NUM
ejpam-6275	609	41	+	+	CCONJ
ejpam-6275	609	42	ω)x13	ω)x13	NUM
ejpam-6275	609	43	+	+	NUM
ejpam-6275	609	44	x8	x8	NOUN
ejpam-6275	609	45	+	+	CCONJ
ejpam-6275	609	46	ωx7	ωx7	X
ejpam-6275	609	47	+	+	CCONJ
ejpam-6275	609	48	(	(	PUNCT
ejpam-6275	609	49	1	1	NUM
ejpam-6275	609	50	+	+	NUM
ejpam-6275	609	51	ω)x6	ω)x6	PROPN
ejpam-6275	609	52	+	+	CCONJ
ejpam-6275	609	53	x5	x5	NOUN
ejpam-6275	609	54	+	+	CCONJ
ejpam-6275	609	55	(	(	PUNCT
ejpam-6275	609	56	1	1	NUM
ejpam-6275	609	57	+	+	NUM
ejpam-6275	609	58	ω)x4	ω)x4	PROPN
ejpam-6275	609	59	+	+	CCONJ
ejpam-6275	609	60	x2	x2	PROPN
ejpam-6275	610	1	+	+	CCONJ
ejpam-6275	610	2	ωx+	ωx+	NOUN
ejpam-6275	610	3	1	1	NUM
ejpam-6275	610	4	+	+	NUM
ejpam-6275	610	5	ω	ω	NUM
ejpam-6275	610	6	)	)	PUNCT
ejpam-6275	610	7	(	(	PUNCT
ejpam-6275	610	8	ω	ω	NOUN
ejpam-6275	610	9	+	+	CCONJ
ejpam-6275	610	10	x	x	X
ejpam-6275	610	11	)	)	PUNCT
ejpam-6275	610	12	=	=	SYM
ejpam-6275	610	13	x20	x20	NOUN
ejpam-6275	611	1	+	+	CCONJ
ejpam-6275	611	2	x18	x18	NOUN
ejpam-6275	611	3	+	+	CCONJ
ejpam-6275	611	4	x17	x17	NOUN
ejpam-6275	612	1	+	+	CCONJ
ejpam-6275	612	2	x16	x16	NOUN
ejpam-6275	612	3	+	+	CCONJ
ejpam-6275	613	1	ωx15	ωx15	PROPN
ejpam-6275	613	2	+	+	CCONJ
ejpam-6275	613	3	ωx14	ωx14	PROPN
ejpam-6275	613	4	+	+	NUM
ejpam-6275	613	5	x13	x13	NOUN
ejpam-6275	613	6	+	+	NUM
ejpam-6275	613	7	x9	x9	NOUN
ejpam-6275	613	8	+	+	CCONJ
ejpam-6275	613	9	x5	x5	NOUN
ejpam-6275	613	10	+	+	CCONJ
ejpam-6275	613	11	x4	x4	PROPN
ejpam-6275	614	1	+	+	CCONJ
ejpam-6275	614	2	x3	x3	ADJ
ejpam-6275	614	3	+	+	NOUN
ejpam-6275	614	4	1	1	X
ejpam-6275	614	5	.	.	PUNCT
ejpam-6275	615	1	now	now	ADV
ejpam-6275	615	2	,	,	PUNCT
ejpam-6275	615	3	k	k	PROPN
ejpam-6275	615	4	=	=	SYM
ejpam-6275	615	5	21	21	NUM
ejpam-6275	615	6	−	−	NUM
ejpam-6275	615	7	20	20	NUM
ejpam-6275	615	8	=	=	SYM
ejpam-6275	615	9	1	1	NUM
ejpam-6275	615	10	.	.	PUNCT
ejpam-6275	616	1	so	so	ADV
ejpam-6275	616	2	the	the	DET
ejpam-6275	616	3	parameters	parameter	NOUN
ejpam-6275	616	4	of	of	ADP
ejpam-6275	616	5	the	the	DET
ejpam-6275	616	6	shortened	shorten	VERB
ejpam-6275	616	7	bch	bch	PROPN
ejpam-6275	616	8	code	code	NOUN
ejpam-6275	616	9	are	be	AUX
ejpam-6275	616	10	(	(	PUNCT
ejpam-6275	616	11	n	n	X
ejpam-6275	616	12	,	,	PUNCT
ejpam-6275	616	13	k	k	NOUN
ejpam-6275	616	14	,	,	PUNCT
ejpam-6275	616	15	d	d	NOUN
ejpam-6275	616	16	)	)	PUNCT
ejpam-6275	616	17	=	=	SYM
ejpam-6275	616	18	(	(	PUNCT
ejpam-6275	616	19	21	21	NUM
ejpam-6275	616	20	,	,	PUNCT
ejpam-6275	616	21	1	1	NUM
ejpam-6275	616	22	,	,	PUNCT
ejpam-6275	616	23	17	17	NUM
ejpam-6275	616	24	)	)	PUNCT
ejpam-6275	616	25	.	.	PUNCT
ejpam-6275	617	1	case	case	NOUN
ejpam-6275	617	2	9	9	NUM
ejpam-6275	617	3	:	:	PUNCT
ejpam-6275	617	4	for	for	ADP
ejpam-6275	617	5	n	n	NOUN
ejpam-6275	617	6	=	=	SYM
ejpam-6275	617	7	21	21	NUM
ejpam-6275	617	8	and	and	CCONJ
ejpam-6275	617	9	d	d	NOUN
ejpam-6275	617	10	=	=	SYM
ejpam-6275	617	11	19	19	NUM
ejpam-6275	617	12	,	,	PUNCT
ejpam-6275	617	13	i	i	PRON
ejpam-6275	617	14	=	=	NOUN
ejpam-6275	617	15	1	1	NUM
ejpam-6275	617	16	,	,	PUNCT
ejpam-6275	617	17	2	2	NUM
ejpam-6275	617	18	,	,	PUNCT
ejpam-6275	617	19	3	3	NUM
ejpam-6275	617	20	,	,	PUNCT
ejpam-6275	617	21	.	.	PUNCT
ejpam-6275	617	22	.	.	PUNCT
ejpam-6275	618	1	.	.	PUNCT
ejpam-6275	619	1	,	,	PUNCT
ejpam-6275	619	2	18	18	NUM
ejpam-6275	619	3	.	.	NOUN
ejpam-6275	619	4	•	•	NOUN
ejpam-6275	619	5	for	for	ADP
ejpam-6275	619	6	i	i	PRON
ejpam-6275	619	7	=	=	NOUN
ejpam-6275	619	8	1	1	NUM
ejpam-6275	619	9	,	,	PUNCT
ejpam-6275	619	10	let	let	VERB
ejpam-6275	619	11	β	β	PRON
ejpam-6275	619	12	∈	∈	PROPN
ejpam-6275	619	13	z2[ω	z2[ω	NOUN
ejpam-6275	619	14	]	]	X
ejpam-6275	619	15	3	3	NUM
ejpam-6275	619	16	,	,	PUNCT
ejpam-6275	619	17	then	then	ADV
ejpam-6275	619	18	by	by	ADP
ejpam-6275	619	19	theorem	theorem	ADJ
ejpam-6275	619	20	3.1	3.1	NUM
ejpam-6275	619	21	,	,	PUNCT
ejpam-6275	619	22	β	β	X
ejpam-6275	619	23	,	,	PUNCT
ejpam-6275	619	24	β4	β4	PROPN
ejpam-6275	619	25	and	and	CCONJ
ejpam-6275	619	26	β16	β16	PROPN
ejpam-6275	619	27	have	have	VERB
ejpam-6275	619	28	the	the	DET
ejpam-6275	619	29	minimal	minimal	ADJ
ejpam-6275	619	30	polynomial	polynomial	ADJ
ejpam-6275	619	31	ϕ1(x	ϕ1(x	NUM
ejpam-6275	619	32	)	)	PUNCT
ejpam-6275	619	33	=	=	PUNCT
ejpam-6275	619	34	(	(	PUNCT
ejpam-6275	619	35	x−	x−	PROPN
ejpam-6275	619	36	β)(x−	β)(x−	PROPN
ejpam-6275	619	37	β4)(x−	β4)(x−	CCONJ
ejpam-6275	619	38	β16	β16	PROPN
ejpam-6275	619	39	)	)	PUNCT
ejpam-6275	620	1	=	=	PUNCT
ejpam-6275	621	1	x3	x3	VERB
ejpam-6275	621	2	+	+	CCONJ
ejpam-6275	621	3	(	(	PUNCT
ejpam-6275	621	4	β	β	X
ejpam-6275	621	5	+	+	CCONJ
ejpam-6275	621	6	β4	β4	PROPN
ejpam-6275	621	7	+	+	CCONJ
ejpam-6275	621	8	β16)x2	β16)x2	NOUN
ejpam-6275	621	9	+	+	CCONJ
ejpam-6275	621	10	(	(	PUNCT
ejpam-6275	621	11	β5	β5	NOUN
ejpam-6275	621	12	+	+	CCONJ
ejpam-6275	621	13	β17	β17	NUM
ejpam-6275	621	14	+	+	NUM
ejpam-6275	621	15	β20)x+	β20)x+	NOUN
ejpam-6275	621	16	β21	β21	NOUN
ejpam-6275	622	1	=	=	SYM
ejpam-6275	622	2	x3	x3	VERB
ejpam-6275	622	3	+	+	CCONJ
ejpam-6275	623	1	(	(	PUNCT
ejpam-6275	623	2	1	1	NUM
ejpam-6275	623	3	+	+	X
ejpam-6275	623	4	ω)x2	ω)x2	NUM
ejpam-6275	623	5	+	+	CCONJ
ejpam-6275	623	6	1	1	NUM
ejpam-6275	623	7	.	.	NOUN
ejpam-6275	623	8	•	•	NOUN
ejpam-6275	623	9	for	for	ADP
ejpam-6275	623	10	i	i	PRON
ejpam-6275	623	11	=	=	SYM
ejpam-6275	623	12	2	2	NUM
ejpam-6275	623	13	,	,	PUNCT
ejpam-6275	623	14	let	let	VERB
ejpam-6275	623	15	β2	β2	PROPN
ejpam-6275	623	16	∈	∈	PROPN
ejpam-6275	623	17	z2[ω	z2[ω	NOUN
ejpam-6275	623	18	]	]	X
ejpam-6275	623	19	3	3	NUM
ejpam-6275	623	20	,	,	PUNCT
ejpam-6275	623	21	then	then	ADV
ejpam-6275	623	22	β2	β2	PROPN
ejpam-6275	623	23	,	,	PUNCT
ejpam-6275	623	24	β8	β8	PROPN
ejpam-6275	623	25	,	,	PUNCT
ejpam-6275	623	26	β11	β11	NOUN
ejpam-6275	623	27	have	have	VERB
ejpam-6275	623	28	ϕ2(x	ϕ2(x	PROPN
ejpam-6275	623	29	)	)	PUNCT
ejpam-6275	623	30	=	=	SYM
ejpam-6275	623	31	(	(	PUNCT
ejpam-6275	623	32	x−	x−	PROPN
ejpam-6275	623	33	β2)(x−	β2)(x−	PROPN
ejpam-6275	623	34	β8)(x−	β8)(x−	X
ejpam-6275	623	35	β11	β11	PROPN
ejpam-6275	623	36	)	)	PUNCT
ejpam-6275	623	37	=	=	PUNCT
ejpam-6275	624	1	x3	x3	VERB
ejpam-6275	624	2	+	+	CCONJ
ejpam-6275	624	3	(	(	PUNCT
ejpam-6275	624	4	β2	β2	VERB
ejpam-6275	624	5	+	+	NOUN
ejpam-6275	624	6	β8	β8	PROPN
ejpam-6275	624	7	+	+	CCONJ
ejpam-6275	624	8	β11)x2	β11)x2	X
ejpam-6275	625	1	+	+	CCONJ
ejpam-6275	625	2	(	(	PUNCT
ejpam-6275	625	3	β10	β10	NOUN
ejpam-6275	625	4	+	+	CCONJ
ejpam-6275	625	5	β13	β13	NOUN
ejpam-6275	625	6	+	+	CCONJ
ejpam-6275	625	7	β19)x+	β19)x+	VERB
ejpam-6275	625	8	β21	β21	NOUN
ejpam-6275	626	1	=	=	SYM
ejpam-6275	626	2	x3	x3	PROPN
ejpam-6275	627	1	+	+	CCONJ
ejpam-6275	627	2	ωx2	ωx2	PRON
ejpam-6275	627	3	+	+	X
ejpam-6275	627	4	1	1	NUM
ejpam-6275	627	5	.	.	NUM
ejpam-6275	627	6	•	•	NOUN
ejpam-6275	627	7	for	for	ADP
ejpam-6275	627	8	i	i	PRON
ejpam-6275	627	9	=	=	SYM
ejpam-6275	627	10	3	3	NUM
ejpam-6275	627	11	,	,	PUNCT
ejpam-6275	627	12	let	let	VERB
ejpam-6275	627	13	β3	β3	VERB
ejpam-6275	627	14	∈	∈	PROPN
ejpam-6275	627	15	z2[ω	z2[ω	NOUN
ejpam-6275	627	16	]	]	X
ejpam-6275	627	17	3	3	NUM
ejpam-6275	627	18	,	,	PUNCT
ejpam-6275	627	19	then	then	ADV
ejpam-6275	627	20	β3	β3	VERB
ejpam-6275	627	21	,	,	PUNCT
ejpam-6275	627	22	β6	β6	PROPN
ejpam-6275	627	23	,	,	PUNCT
ejpam-6275	627	24	β12	β12	PRON
ejpam-6275	627	25	have	have	VERB
ejpam-6275	627	26	ϕ3(x	ϕ3(x	PROPN
ejpam-6275	627	27	)	)	PUNCT
ejpam-6275	627	28	=	=	SYM
ejpam-6275	628	1	(	(	PUNCT
ejpam-6275	628	2	x−	x−	PROPN
ejpam-6275	628	3	β3)(x−	β3)(x−	PROPN
ejpam-6275	628	4	β6)(x−	β6)(x−	X
ejpam-6275	628	5	β12	β12	PROPN
ejpam-6275	628	6	)	)	PUNCT
ejpam-6275	629	1	=	=	SYM
ejpam-6275	630	1	x3	x3	VERB
ejpam-6275	630	2	+	+	CCONJ
ejpam-6275	630	3	(	(	PUNCT
ejpam-6275	630	4	β3	β3	VERB
ejpam-6275	630	5	+	+	NUM
ejpam-6275	630	6	β6	β6	PROPN
ejpam-6275	630	7	+	+	CCONJ
ejpam-6275	630	8	β12)x2	β12)x2	SYM
ejpam-6275	631	1	+	+	CCONJ
ejpam-6275	631	2	(	(	PUNCT
ejpam-6275	631	3	β9	β9	ADJ
ejpam-6275	631	4	+	+	CCONJ
ejpam-6275	631	5	β18	β18	ADJ
ejpam-6275	631	6	+	+	NUM
ejpam-6275	631	7	β15)x+	β15)x+	NOUN
ejpam-6275	631	8	β21	β21	NOUN
ejpam-6275	631	9	=	=	SYM
ejpam-6275	631	10	x3	x3	VERB
ejpam-6275	631	11	+	+	CCONJ
ejpam-6275	631	12	x+	x+	ADJ
ejpam-6275	631	13	1	1	NUM
ejpam-6275	631	14	.	.	NUM
ejpam-6275	631	15	•	•	NOUN
ejpam-6275	631	16	for	for	ADP
ejpam-6275	631	17	i	i	PRON
ejpam-6275	631	18	=	=	NOUN
ejpam-6275	631	19	4	4	NUM
ejpam-6275	631	20	,	,	PUNCT
ejpam-6275	631	21	ϕ4(x	ϕ4(x	PRON
ejpam-6275	631	22	)	)	PUNCT
ejpam-6275	631	23	=	=	SYM
ejpam-6275	631	24	ϕ2(x	ϕ2(x	PROPN
ejpam-6275	631	25	)	)	PUNCT
ejpam-6275	631	26	=	=	PUNCT
ejpam-6275	632	1	x3	x3	VERB
ejpam-6275	633	1	+	+	CCONJ
ejpam-6275	633	2	ωx2	ωx2	PRON
ejpam-6275	633	3	+	+	X
ejpam-6275	633	4	1	1	NUM
ejpam-6275	633	5	.	.	NUM
ejpam-6275	633	6	•	•	NOUN
ejpam-6275	633	7	for	for	ADP
ejpam-6275	633	8	i	i	PRON
ejpam-6275	633	9	=	=	SYM
ejpam-6275	633	10	5	5	NUM
ejpam-6275	633	11	,	,	PUNCT
ejpam-6275	633	12	ϕ5(x	ϕ5(x	NOUN
ejpam-6275	633	13	)	)	PUNCT
ejpam-6275	633	14	=	=	SYM
ejpam-6275	633	15	(	(	PUNCT
ejpam-6275	633	16	x−	x−	PROPN
ejpam-6275	633	17	β5)(x−	β5)(x−	PROPN
ejpam-6275	633	18	β17)(x−	β17)(x−	PROPN
ejpam-6275	633	19	β20	β20	PROPN
ejpam-6275	633	20	)	)	PUNCT
ejpam-6275	633	21	=	=	SYM
ejpam-6275	634	1	x3	x3	VERB
ejpam-6275	634	2	+	+	CCONJ
ejpam-6275	634	3	(	(	PUNCT
ejpam-6275	634	4	β5	β5	NOUN
ejpam-6275	634	5	+	+	CCONJ
ejpam-6275	634	6	β17	β17	NUM
ejpam-6275	634	7	+	+	NOUN
ejpam-6275	634	8	β20)x2	β20)x2	PUNCT
ejpam-6275	634	9	+	+	PUNCT
ejpam-6275	634	10	(	(	PUNCT
ejpam-6275	634	11	β	β	X
ejpam-6275	634	12	+	+	CCONJ
ejpam-6275	634	13	β4	β4	PROPN
ejpam-6275	634	14	+	+	CCONJ
ejpam-6275	634	15	β16)x+	β16)x+	NOUN
ejpam-6275	634	16	β21	β21	NOUN
ejpam-6275	635	1	=	=	SYM
ejpam-6275	635	2	x3	x3	VERB
ejpam-6275	636	1	+	+	CCONJ
ejpam-6275	636	2	(	(	PUNCT
ejpam-6275	636	3	1	1	NUM
ejpam-6275	636	4	+	+	SYM
ejpam-6275	636	5	ω)x+	ω)x+	NUM
ejpam-6275	636	6	1	1	NUM
ejpam-6275	636	7	.	.	PUNCT
ejpam-6275	636	8	m.	m.	PROPN
ejpam-6275	636	9	sajjad	sajjad	PROPN
ejpam-6275	636	10	et	et	PROPN
ejpam-6275	636	11	al	al	PROPN
ejpam-6275	636	12	.	.	PUNCT
ejpam-6275	636	13	/	/	SYM
ejpam-6275	636	14	eur	eur	PROPN
ejpam-6275	636	15	.	.	PUNCT
ejpam-6275	637	1	j.	j.	PROPN
ejpam-6275	637	2	pure	pure	PROPN
ejpam-6275	637	3	appl	appl	PROPN
ejpam-6275	637	4	.	.	PROPN
ejpam-6275	637	5	math	math	PROPN
ejpam-6275	637	6	,	,	PUNCT
ejpam-6275	637	7	18	18	NUM
ejpam-6275	637	8	(	(	PUNCT
ejpam-6275	637	9	3	3	NUM
ejpam-6275	637	10	)	)	PUNCT
ejpam-6275	637	11	(	(	PUNCT
ejpam-6275	637	12	2025	2025	NUM
ejpam-6275	637	13	)	)	PUNCT
ejpam-6275	637	14	,	,	PUNCT
ejpam-6275	637	15	6275	6275	NUM
ejpam-6275	637	16	24	24	NUM
ejpam-6275	637	17	of	of	ADP
ejpam-6275	637	18	36	36	NUM
ejpam-6275	637	19	•	•	NOUN
ejpam-6275	637	20	for	for	ADP
ejpam-6275	637	21	i	i	PRON
ejpam-6275	637	22	=	=	NOUN
ejpam-6275	637	23	6	6	NUM
ejpam-6275	637	24	,	,	PUNCT
ejpam-6275	637	25	ϕ6(x	ϕ6(x	PROPN
ejpam-6275	637	26	)	)	PUNCT
ejpam-6275	637	27	=	=	SYM
ejpam-6275	637	28	ϕ3(x	ϕ3(x	PROPN
ejpam-6275	637	29	)	)	PUNCT
ejpam-6275	637	30	=	=	SYM
ejpam-6275	638	1	x3	x3	VERB
ejpam-6275	638	2	+	+	CCONJ
ejpam-6275	638	3	x+	x+	ADJ
ejpam-6275	638	4	1	1	NUM
ejpam-6275	638	5	.	.	NUM
ejpam-6275	638	6	•	•	NOUN
ejpam-6275	638	7	for	for	ADP
ejpam-6275	638	8	i	i	PRON
ejpam-6275	638	9	=	=	SYM
ejpam-6275	638	10	7	7	NUM
ejpam-6275	638	11	,	,	PUNCT
ejpam-6275	638	12	ϕ7(x	ϕ7(x	NOUN
ejpam-6275	638	13	)	)	PUNCT
ejpam-6275	638	14	=	=	SYM
ejpam-6275	638	15	(	(	PUNCT
ejpam-6275	638	16	x−	x−	PROPN
ejpam-6275	638	17	β7	β7	PROPN
ejpam-6275	638	18	)	)	PUNCT
ejpam-6275	639	1	=	=	PUNCT
ejpam-6275	639	2	x+	x+	PUNCT
ejpam-6275	640	1	1	1	NUM
ejpam-6275	640	2	+	+	NUM
ejpam-6275	640	3	ω	ω	NUM
ejpam-6275	640	4	.	.	NOUN
ejpam-6275	640	5	•	•	NUM
ejpam-6275	640	6	for	for	ADP
ejpam-6275	640	7	i	i	PRON
ejpam-6275	640	8	=	=	NOUN
ejpam-6275	640	9	8	8	NUM
ejpam-6275	640	10	,	,	PUNCT
ejpam-6275	640	11	ϕ8(x	ϕ8(x	NOUN
ejpam-6275	641	1	)	)	PUNCT
ejpam-6275	641	2	=	=	SYM
ejpam-6275	641	3	ϕ2(x	ϕ2(x	PROPN
ejpam-6275	641	4	)	)	PUNCT
ejpam-6275	641	5	=	=	PUNCT
ejpam-6275	642	1	x3	x3	VERB
ejpam-6275	643	1	+	+	CCONJ
ejpam-6275	643	2	ωx2	ωx2	PRON
ejpam-6275	643	3	+	+	X
ejpam-6275	643	4	1	1	NUM
ejpam-6275	643	5	.	.	NUM
ejpam-6275	643	6	•	•	NOUN
ejpam-6275	643	7	for	for	ADP
ejpam-6275	643	8	i	i	PRON
ejpam-6275	643	9	=	=	NOUN
ejpam-6275	643	10	9	9	NUM
ejpam-6275	643	11	,	,	PUNCT
ejpam-6275	643	12	ϕ9(x	ϕ9(x	PRON
ejpam-6275	643	13	)	)	PUNCT
ejpam-6275	643	14	=	=	SYM
ejpam-6275	643	15	(	(	PUNCT
ejpam-6275	643	16	x−	x−	PROPN
ejpam-6275	643	17	β9)(x−	β9)(x−	PUNCT
ejpam-6275	643	18	β18)(x−	β18)(x−	NOUN
ejpam-6275	643	19	β15	β15	NOUN
ejpam-6275	643	20	)	)	PUNCT
ejpam-6275	643	21	=	=	SYM
ejpam-6275	644	1	x3	x3	VERB
ejpam-6275	644	2	+	+	CCONJ
ejpam-6275	644	3	(	(	PUNCT
ejpam-6275	644	4	β9	β9	NOUN
ejpam-6275	644	5	+	+	CCONJ
ejpam-6275	644	6	β15	β15	NOUN
ejpam-6275	644	7	+	+	X
ejpam-6275	644	8	β18)x2	β18)x2	NOUN
ejpam-6275	644	9	+	+	CCONJ
ejpam-6275	644	10	(	(	PUNCT
ejpam-6275	644	11	β3	β3	VERB
ejpam-6275	644	12	+	+	NUM
ejpam-6275	644	13	β6	β6	NOUN
ejpam-6275	644	14	+	+	CCONJ
ejpam-6275	644	15	β12)x+	β12)x+	ADJ
ejpam-6275	644	16	β21	β21	NOUN
ejpam-6275	645	1	=	=	SYM
ejpam-6275	645	2	x3	x3	PROPN
ejpam-6275	646	1	+	+	CCONJ
ejpam-6275	646	2	x2	x2	PROPN
ejpam-6275	647	1	+	+	CCONJ
ejpam-6275	648	1	1	1	NUM
ejpam-6275	648	2	.	.	NUM
ejpam-6275	648	3	•	•	NOUN
ejpam-6275	648	4	for	for	ADP
ejpam-6275	648	5	i	i	PRON
ejpam-6275	648	6	=	=	NOUN
ejpam-6275	648	7	10	10	NUM
ejpam-6275	648	8	,	,	PUNCT
ejpam-6275	648	9	ϕ10(x	ϕ10(x	NUM
ejpam-6275	648	10	)	)	PUNCT
ejpam-6275	648	11	=	=	SYM
ejpam-6275	648	12	(	(	PUNCT
ejpam-6275	648	13	x−	x−	PROPN
ejpam-6275	648	14	β10)(x−	β10)(x−	PROPN
ejpam-6275	648	15	β13)(x−	β13)(x−	PROPN
ejpam-6275	648	16	β19	β19	ADV
ejpam-6275	648	17	)	)	PUNCT
ejpam-6275	648	18	=	=	SYM
ejpam-6275	649	1	x3	x3	VERB
ejpam-6275	649	2	+	+	CCONJ
ejpam-6275	649	3	(	(	PUNCT
ejpam-6275	649	4	β10	β10	NOUN
ejpam-6275	650	1	+	+	CCONJ
ejpam-6275	650	2	β13	β13	NOUN
ejpam-6275	651	1	+	+	CCONJ
ejpam-6275	651	2	β19)x2	β19)x2	X
ejpam-6275	651	3	+	+	CCONJ
ejpam-6275	651	4	(	(	PUNCT
ejpam-6275	651	5	β2	β2	VERB
ejpam-6275	651	6	+	+	NOUN
ejpam-6275	651	7	β8	β8	NOUN
ejpam-6275	651	8	+	+	CCONJ
ejpam-6275	651	9	β11)x+	β11)x+	NOUN
ejpam-6275	651	10	β21	β21	NOUN
ejpam-6275	652	1	=	=	SYM
ejpam-6275	652	2	x3	x3	PROPN
ejpam-6275	653	1	+	+	CCONJ
ejpam-6275	653	2	ωx+	ωx+	NOUN
ejpam-6275	653	3	1	1	NUM
ejpam-6275	653	4	.	.	NOUN
ejpam-6275	653	5	•	•	NOUN
ejpam-6275	653	6	for	for	ADP
ejpam-6275	653	7	i	i	PRON
ejpam-6275	653	8	=	=	NOUN
ejpam-6275	653	9	11	11	NUM
ejpam-6275	653	10	,	,	PUNCT
ejpam-6275	653	11	ϕ11(x	ϕ11(x	NOUN
ejpam-6275	653	12	)	)	PUNCT
ejpam-6275	654	1	=	=	PUNCT
ejpam-6275	654	2	ϕ2(x	ϕ2(x	PROPN
ejpam-6275	654	3	)	)	PUNCT
ejpam-6275	654	4	=	=	PUNCT
ejpam-6275	655	1	x3	x3	VERB
ejpam-6275	656	1	+	+	CCONJ
ejpam-6275	656	2	ωx2	ωx2	PRON
ejpam-6275	656	3	+	+	X
ejpam-6275	656	4	1	1	NUM
ejpam-6275	656	5	.	.	NUM
ejpam-6275	656	6	•	•	NOUN
ejpam-6275	656	7	for	for	ADP
ejpam-6275	656	8	i	i	PRON
ejpam-6275	656	9	=	=	NOUN
ejpam-6275	656	10	12	12	NUM
ejpam-6275	656	11	,	,	PUNCT
ejpam-6275	656	12	ϕ12(x	ϕ12(x	PROPN
ejpam-6275	656	13	)	)	PUNCT
ejpam-6275	656	14	=	=	SYM
ejpam-6275	657	1	ϕ3(x	ϕ3(x	PROPN
ejpam-6275	657	2	)	)	PUNCT
ejpam-6275	657	3	=	=	SYM
ejpam-6275	658	1	x3	x3	VERB
ejpam-6275	658	2	+	+	CCONJ
ejpam-6275	658	3	x+	x+	ADJ
ejpam-6275	658	4	1	1	NUM
ejpam-6275	658	5	.	.	NUM
ejpam-6275	658	6	•	•	NOUN
ejpam-6275	658	7	for	for	ADP
ejpam-6275	658	8	i	i	PRON
ejpam-6275	658	9	=	=	NOUN
ejpam-6275	658	10	13	13	NUM
ejpam-6275	658	11	,	,	PUNCT
ejpam-6275	658	12	ϕ13(x	ϕ13(x	PUNCT
ejpam-6275	658	13	)	)	PUNCT
ejpam-6275	658	14	=	=	SYM
ejpam-6275	658	15	ϕ10(x	ϕ10(x	X
ejpam-6275	658	16	)	)	PUNCT
ejpam-6275	658	17	=	=	SYM
ejpam-6275	659	1	x3	x3	ADJ
ejpam-6275	659	2	+	+	CCONJ
ejpam-6275	659	3	ωx+	ωx+	NOUN
ejpam-6275	659	4	1	1	NUM
ejpam-6275	659	5	.	.	NOUN
ejpam-6275	659	6	•	•	NOUN
ejpam-6275	659	7	for	for	ADP
ejpam-6275	659	8	i	i	PRON
ejpam-6275	659	9	=	=	NOUN
ejpam-6275	659	10	14	14	NUM
ejpam-6275	659	11	,	,	PUNCT
ejpam-6275	659	12	ϕ14(x	ϕ14(x	NOUN
ejpam-6275	659	13	)	)	PUNCT
ejpam-6275	659	14	=	=	PUNCT
ejpam-6275	660	1	(	(	PUNCT
ejpam-6275	660	2	x−	x−	PROPN
ejpam-6275	660	3	β14	β14	ADJ
ejpam-6275	660	4	)	)	PUNCT
ejpam-6275	661	1	=	=	SYM
ejpam-6275	661	2	x+	x+	PROPN
ejpam-6275	662	1	ω	ω	X
ejpam-6275	662	2	.	.	PROPN
ejpam-6275	662	3	•	•	NUM
ejpam-6275	662	4	for	for	ADP
ejpam-6275	662	5	i	i	PRON
ejpam-6275	662	6	=	=	NOUN
ejpam-6275	662	7	15	15	NUM
ejpam-6275	662	8	,	,	PUNCT
ejpam-6275	662	9	ϕ15(x	ϕ15(x	NUM
ejpam-6275	662	10	)	)	PUNCT
ejpam-6275	663	1	=	=	SYM
ejpam-6275	663	2	ϕ9(x	ϕ9(x	X
ejpam-6275	663	3	)	)	PUNCT
ejpam-6275	663	4	=	=	PUNCT
ejpam-6275	664	1	x3	x3	ADJ
ejpam-6275	665	1	+	+	CCONJ
ejpam-6275	665	2	x2	x2	PROPN
ejpam-6275	666	1	+	+	CCONJ
ejpam-6275	667	1	1	1	NUM
ejpam-6275	667	2	.	.	NUM
ejpam-6275	667	3	•	•	NOUN
ejpam-6275	667	4	for	for	ADP
ejpam-6275	667	5	i	i	PRON
ejpam-6275	667	6	=	=	NOUN
ejpam-6275	667	7	16	16	NUM
ejpam-6275	667	8	,	,	PUNCT
ejpam-6275	667	9	ϕ16(x	ϕ16(x	PROPN
ejpam-6275	667	10	)	)	PUNCT
ejpam-6275	667	11	=	=	SYM
ejpam-6275	667	12	ϕ1(x	ϕ1(x	X
ejpam-6275	667	13	)	)	PUNCT
ejpam-6275	667	14	=	=	SYM
ejpam-6275	668	1	x3	x3	VERB
ejpam-6275	668	2	+	+	CCONJ
ejpam-6275	669	1	(	(	PUNCT
ejpam-6275	669	2	1	1	NUM
ejpam-6275	669	3	+	+	X
ejpam-6275	669	4	ω)x2	ω)x2	NUM
ejpam-6275	669	5	+	+	CCONJ
ejpam-6275	669	6	1	1	NUM
ejpam-6275	669	7	.	.	NOUN
ejpam-6275	669	8	•	•	NOUN
ejpam-6275	669	9	for	for	ADP
ejpam-6275	669	10	i	i	PRON
ejpam-6275	669	11	=	=	NOUN
ejpam-6275	669	12	17	17	NUM
ejpam-6275	669	13	,	,	PUNCT
ejpam-6275	669	14	ϕ17(x	ϕ17(x	NUM
ejpam-6275	669	15	)	)	PUNCT
ejpam-6275	670	1	=	=	PUNCT
ejpam-6275	670	2	ϕ5(x	ϕ5(x	X
ejpam-6275	670	3	)	)	PUNCT
ejpam-6275	670	4	=	=	SYM
ejpam-6275	671	1	x3	x3	VERB
ejpam-6275	671	2	+	+	CCONJ
ejpam-6275	671	3	(	(	PUNCT
ejpam-6275	671	4	1	1	NUM
ejpam-6275	671	5	+	+	SYM
ejpam-6275	671	6	ω)x+	ω)x+	NUM
ejpam-6275	671	7	1	1	NUM
ejpam-6275	671	8	.	.	NOUN
ejpam-6275	671	9	•	•	NOUN
ejpam-6275	671	10	for	for	ADP
ejpam-6275	671	11	i	i	PRON
ejpam-6275	671	12	=	=	NOUN
ejpam-6275	671	13	18	18	NUM
ejpam-6275	671	14	,	,	PUNCT
ejpam-6275	671	15	ϕ18(x	ϕ18(x	NOUN
ejpam-6275	671	16	)	)	PUNCT
ejpam-6275	672	1	=	=	PUNCT
ejpam-6275	672	2	ϕ9(x	ϕ9(x	X
ejpam-6275	672	3	)	)	PUNCT
ejpam-6275	672	4	=	=	PUNCT
ejpam-6275	673	1	x3	x3	ADJ
ejpam-6275	674	1	+	+	CCONJ
ejpam-6275	674	2	x2	x2	PROPN
ejpam-6275	675	1	+	+	NOUN
ejpam-6275	675	2	1	1	X
ejpam-6275	675	3	.	.	PUNCT
ejpam-6275	676	1	now	now	ADV
ejpam-6275	676	2	the	the	DET
ejpam-6275	676	3	generator	generator	NOUN
ejpam-6275	676	4	polynomial	polynomial	NOUN
ejpam-6275	676	5	is	be	AUX
ejpam-6275	676	6	g(x	g(x	NOUN
ejpam-6275	676	7	)	)	PUNCT
ejpam-6275	677	1	=	=	SYM
ejpam-6275	677	2	ϕ1(x	ϕ1(x	X
ejpam-6275	677	3	)	)	PUNCT
ejpam-6275	677	4	·	·	PUNCT
ejpam-6275	678	1	ϕ2(x	ϕ2(x	X
ejpam-6275	678	2	)	)	PUNCT
ejpam-6275	678	3	·	·	PUNCT
ejpam-6275	679	1	ϕ3(x	ϕ3(x	X
ejpam-6275	679	2	)	)	PUNCT
ejpam-6275	679	3	·	·	PUNCT
ejpam-6275	680	1	ϕ5(x	ϕ5(x	X
ejpam-6275	680	2	)	)	PUNCT
ejpam-6275	680	3	·	·	PUNCT
ejpam-6275	680	4	ϕ7(x	ϕ7(x	X
ejpam-6275	680	5	)	)	PUNCT
ejpam-6275	680	6	·	·	PUNCT
ejpam-6275	681	1	ϕ9(x	ϕ9(x	X
ejpam-6275	681	2	)	)	PUNCT
ejpam-6275	681	3	·	·	PUNCT
ejpam-6275	681	4	ϕ10(x	ϕ10(x	X
ejpam-6275	681	5	)	)	PUNCT
ejpam-6275	681	6	·	·	PUNCT
ejpam-6275	681	7	ϕ14(x	ϕ14(x	X
ejpam-6275	681	8	)	)	PUNCT
ejpam-6275	681	9	=	=	SYM
ejpam-6275	682	1	(	(	PUNCT
ejpam-6275	682	2	x19	x19	NOUN
ejpam-6275	682	3	+	+	CCONJ
ejpam-6275	682	4	(	(	PUNCT
ejpam-6275	682	5	1	1	NUM
ejpam-6275	682	6	+	+	NOUN
ejpam-6275	682	7	ω)x18	ω)x18	PRON
ejpam-6275	682	8	+	+	CCONJ
ejpam-6275	682	9	x16	x16	NOUN
ejpam-6275	683	1	+	+	CCONJ
ejpam-6275	683	2	(	(	PUNCT
ejpam-6275	683	3	1	1	NUM
ejpam-6275	683	4	+	+	CCONJ
ejpam-6275	683	5	ω)x15	ω)x15	NUM
ejpam-6275	683	6	+	+	CCONJ
ejpam-6275	683	7	(	(	PUNCT
ejpam-6275	683	8	1	1	NUM
ejpam-6275	683	9	+	+	NUM
ejpam-6275	683	10	ω)x14	ω)x14	PRON
ejpam-6275	684	1	+	+	CCONJ
ejpam-6275	684	2	(	(	PUNCT
ejpam-6275	684	3	1	1	NUM
ejpam-6275	684	4	+	+	CCONJ
ejpam-6275	684	5	ω)x13	ω)x13	NUM
ejpam-6275	684	6	+	+	NUM
ejpam-6275	684	7	x8	x8	NOUN
ejpam-6275	684	8	+	+	CCONJ
ejpam-6275	684	9	ωx7	ωx7	X
ejpam-6275	684	10	+	+	CCONJ
ejpam-6275	684	11	(	(	PUNCT
ejpam-6275	684	12	1	1	NUM
ejpam-6275	684	13	+	+	NUM
ejpam-6275	684	14	ω)x6	ω)x6	PROPN
ejpam-6275	684	15	+	+	CCONJ
ejpam-6275	684	16	x5	x5	NOUN
ejpam-6275	684	17	+	+	CCONJ
ejpam-6275	684	18	(	(	PUNCT
ejpam-6275	684	19	1	1	NUM
ejpam-6275	684	20	+	+	NUM
ejpam-6275	684	21	ω)x4	ω)x4	PROPN
ejpam-6275	684	22	+	+	CCONJ
ejpam-6275	684	23	x2	x2	PROPN
ejpam-6275	685	1	+	+	CCONJ
ejpam-6275	685	2	ωx+	ωx+	NOUN
ejpam-6275	685	3	1	1	NUM
ejpam-6275	685	4	+	+	CCONJ
ejpam-6275	685	5	ω)(ω	ω)(ω	VERB
ejpam-6275	685	6	+	+	CCONJ
ejpam-6275	685	7	x	x	X
ejpam-6275	685	8	)	)	PUNCT
ejpam-6275	685	9	=	=	SYM
ejpam-6275	685	10	x20	x20	NOUN
ejpam-6275	685	11	+	+	CCONJ
ejpam-6275	685	12	x18	x18	NOUN
ejpam-6275	686	1	+	+	CCONJ
ejpam-6275	686	2	x17	x17	NOUN
ejpam-6275	686	3	+	+	CCONJ
ejpam-6275	686	4	x16	x16	NOUN
ejpam-6275	686	5	+	+	CCONJ
ejpam-6275	686	6	ωx15	ωx15	PROPN
ejpam-6275	686	7	+	+	CCONJ
ejpam-6275	686	8	ωx14	ωx14	PROPN
ejpam-6275	686	9	+	+	NUM
ejpam-6275	686	10	x13	x13	NOUN
ejpam-6275	686	11	+	+	NUM
ejpam-6275	686	12	x9	x9	NOUN
ejpam-6275	686	13	+	+	CCONJ
ejpam-6275	686	14	x5	x5	NOUN
ejpam-6275	686	15	+	+	CCONJ
ejpam-6275	686	16	x4	x4	PROPN
ejpam-6275	687	1	+	+	CCONJ
ejpam-6275	687	2	x3	x3	ADJ
ejpam-6275	687	3	+	+	NOUN
ejpam-6275	687	4	1	1	X
ejpam-6275	687	5	.	.	PUNCT
ejpam-6275	687	6	thus	thus	ADV
ejpam-6275	687	7	,	,	PUNCT
ejpam-6275	687	8	the	the	DET
ejpam-6275	687	9	dimension	dimension	NOUN
ejpam-6275	687	10	of	of	ADP
ejpam-6275	687	11	the	the	DET
ejpam-6275	687	12	code	code	NOUN
ejpam-6275	687	13	is	be	AUX
ejpam-6275	687	14	k	k	NOUN
ejpam-6275	687	15	=	=	PUNCT
ejpam-6275	687	16	21−	21−	NUM
ejpam-6275	687	17	20	20	NUM
ejpam-6275	687	18	=	=	SYM
ejpam-6275	687	19	1	1	NUM
ejpam-6275	687	20	.	.	PUNCT
ejpam-6275	687	21	case	case	NOUN
ejpam-6275	687	22	10	10	NUM
ejpam-6275	687	23	:	:	PUNCT
ejpam-6275	687	24	for	for	ADP
ejpam-6275	687	25	n	n	NOUN
ejpam-6275	687	26	=	=	SYM
ejpam-6275	687	27	21	21	NUM
ejpam-6275	687	28	and	and	CCONJ
ejpam-6275	687	29	d	d	NOUN
ejpam-6275	687	30	=	=	SYM
ejpam-6275	687	31	21	21	NUM
ejpam-6275	687	32	,	,	PUNCT
ejpam-6275	687	33	i	i	PRON
ejpam-6275	687	34	=	=	NOUN
ejpam-6275	687	35	1	1	NUM
ejpam-6275	687	36	,	,	PUNCT
ejpam-6275	687	37	2	2	NUM
ejpam-6275	687	38	,	,	PUNCT
ejpam-6275	687	39	3	3	NUM
ejpam-6275	687	40	,	,	PUNCT
ejpam-6275	687	41	.	.	PUNCT
ejpam-6275	687	42	.	.	PUNCT
ejpam-6275	688	1	.	.	PUNCT
ejpam-6275	689	1	,	,	PUNCT
ejpam-6275	689	2	20	20	X
ejpam-6275	689	3	.	.	PUNCT
ejpam-6275	690	1	m.	m.	PROPN
ejpam-6275	690	2	sajjad	sajjad	PROPN
ejpam-6275	690	3	et	et	PROPN
ejpam-6275	690	4	al	al	PROPN
ejpam-6275	690	5	.	.	PUNCT
ejpam-6275	690	6	/	/	SYM
ejpam-6275	690	7	eur	eur	PROPN
ejpam-6275	690	8	.	.	PUNCT
ejpam-6275	691	1	j.	j.	PROPN
ejpam-6275	691	2	pure	pure	PROPN
ejpam-6275	691	3	appl	appl	PROPN
ejpam-6275	691	4	.	.	PROPN
ejpam-6275	691	5	math	math	PROPN
ejpam-6275	691	6	,	,	PUNCT
ejpam-6275	691	7	18	18	NUM
ejpam-6275	691	8	(	(	PUNCT
ejpam-6275	691	9	3	3	NUM
ejpam-6275	691	10	)	)	PUNCT
ejpam-6275	691	11	(	(	PUNCT
ejpam-6275	691	12	2025	2025	NUM
ejpam-6275	691	13	)	)	PUNCT
ejpam-6275	691	14	,	,	PUNCT
ejpam-6275	691	15	6275	6275	NUM
ejpam-6275	691	16	25	25	NUM
ejpam-6275	691	17	of	of	ADP
ejpam-6275	691	18	36	36	NUM
ejpam-6275	691	19	•	•	NOUN
ejpam-6275	691	20	for	for	ADP
ejpam-6275	691	21	i	i	PRON
ejpam-6275	691	22	=	=	NOUN
ejpam-6275	691	23	1	1	NUM
ejpam-6275	691	24	,	,	PUNCT
ejpam-6275	691	25	let	let	VERB
ejpam-6275	691	26	β	β	PRON
ejpam-6275	691	27	∈	∈	PROPN
ejpam-6275	691	28	z2[ω	z2[ω	NOUN
ejpam-6275	691	29	]	]	X
ejpam-6275	691	30	3	3	NUM
ejpam-6275	691	31	,	,	PUNCT
ejpam-6275	691	32	then	then	ADV
ejpam-6275	691	33	by	by	ADP
ejpam-6275	691	34	theorem	theorem	ADJ
ejpam-6275	691	35	3.1	3.1	NUM
ejpam-6275	691	36	,	,	PUNCT
ejpam-6275	691	37	β	β	X
ejpam-6275	691	38	,	,	PUNCT
ejpam-6275	691	39	β4	β4	PROPN
ejpam-6275	691	40	,	,	PUNCT
ejpam-6275	691	41	and	and	CCONJ
ejpam-6275	691	42	β16	β16	PROPN
ejpam-6275	691	43	have	have	VERB
ejpam-6275	691	44	the	the	DET
ejpam-6275	691	45	minimal	minimal	ADJ
ejpam-6275	691	46	polynomial	polynomial	ADJ
ejpam-6275	691	47	φ1(x	φ1(x	NOUN
ejpam-6275	691	48	)	)	PUNCT
ejpam-6275	691	49	=	=	SYM
ejpam-6275	691	50	(	(	PUNCT
ejpam-6275	691	51	x−β)(x−β4)(x−β16	x−β)(x−β4)(x−β16	X
ejpam-6275	691	52	)	)	PUNCT
ejpam-6275	691	53	=	=	SYM
ejpam-6275	691	54	x3+(β+β4+β16)x2+(β5+β17+β20)x+β21	x3+(β+β4+β16)x2+(β5+β17+β20)x+β21	PROPN
ejpam-6275	691	55	=	=	SYM
ejpam-6275	691	56	x3+(1+ω)x2	x3+(1+ω)x2	PUNCT
ejpam-6275	692	1	+	+	PROPN
ejpam-6275	692	2	1	1	NUM
ejpam-6275	692	3	.	.	NOUN
ejpam-6275	692	4	•	•	NOUN
ejpam-6275	692	5	for	for	ADP
ejpam-6275	692	6	i	i	PRON
ejpam-6275	692	7	=	=	SYM
ejpam-6275	692	8	2	2	NUM
ejpam-6275	692	9	,	,	PUNCT
ejpam-6275	692	10	let	let	VERB
ejpam-6275	692	11	β2	β2	PROPN
ejpam-6275	692	12	∈	∈	PROPN
ejpam-6275	692	13	z2[ω	z2[ω	NOUN
ejpam-6275	692	14	]	]	X
ejpam-6275	692	15	3	3	NUM
ejpam-6275	692	16	,	,	PUNCT
ejpam-6275	692	17	then	then	ADV
ejpam-6275	692	18	by	by	ADP
ejpam-6275	692	19	theorem	theorem	ADJ
ejpam-6275	692	20	3.1	3.1	NUM
ejpam-6275	692	21	,	,	PUNCT
ejpam-6275	692	22	β2	β2	NOUN
ejpam-6275	692	23	,	,	PUNCT
ejpam-6275	692	24	β8	β8	NOUN
ejpam-6275	692	25	,	,	PUNCT
ejpam-6275	692	26	and	and	CCONJ
ejpam-6275	692	27	β11	β11	NOUN
ejpam-6275	692	28	have	have	VERB
ejpam-6275	692	29	the	the	DET
ejpam-6275	692	30	minimal	minimal	ADJ
ejpam-6275	692	31	polynomial	polynomial	ADJ
ejpam-6275	692	32	φ2(x	φ2(x	NUM
ejpam-6275	692	33	)	)	PUNCT
ejpam-6275	692	34	=	=	PUNCT
ejpam-6275	692	35	(	(	PUNCT
ejpam-6275	692	36	x−	x−	PROPN
ejpam-6275	692	37	β2)(x−	β2)(x−	PROPN
ejpam-6275	692	38	β8)(x−	β8)(x−	X
ejpam-6275	692	39	β11	β11	PROPN
ejpam-6275	692	40	)	)	PUNCT
ejpam-6275	692	41	=	=	PUNCT
ejpam-6275	693	1	x3	x3	VERB
ejpam-6275	694	1	+	+	CCONJ
ejpam-6275	694	2	ωx2	ωx2	PRON
ejpam-6275	694	3	+	+	X
ejpam-6275	694	4	1	1	NUM
ejpam-6275	694	5	.	.	NUM
ejpam-6275	694	6	•	•	NOUN
ejpam-6275	694	7	for	for	ADP
ejpam-6275	694	8	i	i	PRON
ejpam-6275	694	9	=	=	SYM
ejpam-6275	694	10	3	3	NUM
ejpam-6275	694	11	,	,	PUNCT
ejpam-6275	694	12	let	let	VERB
ejpam-6275	694	13	β3	β3	VERB
ejpam-6275	694	14	∈	∈	PROPN
ejpam-6275	694	15	z2[ω	z2[ω	NOUN
ejpam-6275	694	16	]	]	X
ejpam-6275	694	17	3	3	NUM
ejpam-6275	694	18	,	,	PUNCT
ejpam-6275	694	19	then	then	ADV
ejpam-6275	694	20	by	by	ADP
ejpam-6275	694	21	theorem	theorem	ADJ
ejpam-6275	694	22	3.1	3.1	NUM
ejpam-6275	694	23	,	,	PUNCT
ejpam-6275	694	24	β3	β3	ADJ
ejpam-6275	694	25	,	,	PUNCT
ejpam-6275	694	26	β6	β6	PROPN
ejpam-6275	694	27	,	,	PUNCT
ejpam-6275	694	28	and	and	CCONJ
ejpam-6275	694	29	β12	β12	CCONJ
ejpam-6275	694	30	have	have	VERB
ejpam-6275	694	31	the	the	DET
ejpam-6275	694	32	minimal	minimal	ADJ
ejpam-6275	694	33	polynomial	polynomial	ADJ
ejpam-6275	694	34	φ3(x	φ3(x	PROPN
ejpam-6275	694	35	)	)	PUNCT
ejpam-6275	694	36	=	=	SYM
ejpam-6275	694	37	(	(	PUNCT
ejpam-6275	694	38	x−	x−	PROPN
ejpam-6275	694	39	β3)(x−	β3)(x−	PROPN
ejpam-6275	694	40	β6)(x−	β6)(x−	X
ejpam-6275	694	41	β12	β12	PROPN
ejpam-6275	694	42	)	)	PUNCT
ejpam-6275	694	43	=	=	PUNCT
ejpam-6275	695	1	x3	x3	VERB
ejpam-6275	695	2	+	+	CCONJ
ejpam-6275	695	3	x+	x+	ADJ
ejpam-6275	695	4	1	1	NUM
ejpam-6275	695	5	.	.	NUM
ejpam-6275	695	6	•	•	NOUN
ejpam-6275	695	7	for	for	ADP
ejpam-6275	695	8	i	i	PRON
ejpam-6275	695	9	=	=	NOUN
ejpam-6275	695	10	4	4	NUM
ejpam-6275	695	11	,	,	PUNCT
ejpam-6275	695	12	let	let	VERB
ejpam-6275	695	13	β4	β4	PROPN
ejpam-6275	695	14	∈	∈	PROPN
ejpam-6275	695	15	z2[ω	z2[ω	NOUN
ejpam-6275	695	16	]	]	X
ejpam-6275	695	17	3	3	NUM
ejpam-6275	695	18	,	,	PUNCT
ejpam-6275	695	19	then	then	ADV
ejpam-6275	695	20	by	by	ADP
ejpam-6275	695	21	theorem	theorem	ADJ
ejpam-6275	695	22	3.1	3.1	NUM
ejpam-6275	695	23	,	,	PUNCT
ejpam-6275	695	24	β2	β2	NOUN
ejpam-6275	695	25	,	,	PUNCT
ejpam-6275	695	26	β8	β8	NOUN
ejpam-6275	695	27	,	,	PUNCT
ejpam-6275	695	28	and	and	CCONJ
ejpam-6275	695	29	β11	β11	NOUN
ejpam-6275	695	30	have	have	VERB
ejpam-6275	695	31	the	the	DET
ejpam-6275	695	32	minimal	minimal	ADJ
ejpam-6275	695	33	polynomial	polynomial	ADJ
ejpam-6275	695	34	φ4(x	φ4(x	NOUN
ejpam-6275	695	35	)	)	PUNCT
ejpam-6275	695	36	=	=	SYM
ejpam-6275	695	37	φ2(x	φ2(x	NUM
ejpam-6275	695	38	)	)	PUNCT
ejpam-6275	695	39	=	=	PUNCT
ejpam-6275	696	1	x3	x3	VERB
ejpam-6275	697	1	+	+	CCONJ
ejpam-6275	697	2	ωx2	ωx2	PRON
ejpam-6275	697	3	+	+	X
ejpam-6275	697	4	1	1	NUM
ejpam-6275	697	5	.	.	NUM
ejpam-6275	697	6	•	•	NOUN
ejpam-6275	697	7	for	for	ADP
ejpam-6275	697	8	i	i	PRON
ejpam-6275	697	9	=	=	SYM
ejpam-6275	697	10	5	5	NUM
ejpam-6275	697	11	,	,	PUNCT
ejpam-6275	697	12	let	let	VERB
ejpam-6275	697	13	β5	β5	NOUN
ejpam-6275	697	14	∈	∈	PROPN
ejpam-6275	697	15	z2[ω	z2[ω	NOUN
ejpam-6275	697	16	]	]	X
ejpam-6275	697	17	3	3	NUM
ejpam-6275	697	18	,	,	PUNCT
ejpam-6275	697	19	then	then	ADV
ejpam-6275	697	20	by	by	ADP
ejpam-6275	697	21	theorem	theorem	ADJ
ejpam-6275	697	22	3.1	3.1	NUM
ejpam-6275	697	23	,	,	PUNCT
ejpam-6275	697	24	β5	β5	NOUN
ejpam-6275	697	25	,	,	PUNCT
ejpam-6275	697	26	β17	β17	NUM
ejpam-6275	697	27	,	,	PUNCT
ejpam-6275	697	28	and	and	CCONJ
ejpam-6275	697	29	β20	β20	PROPN
ejpam-6275	697	30	have	have	VERB
ejpam-6275	697	31	the	the	DET
ejpam-6275	697	32	minimal	minimal	ADJ
ejpam-6275	697	33	polynomial	polynomial	ADJ
ejpam-6275	697	34	φ5(x	φ5(x	NOUN
ejpam-6275	697	35	)	)	PUNCT
ejpam-6275	697	36	=	=	SYM
ejpam-6275	697	37	(	(	PUNCT
ejpam-6275	697	38	x−	x−	PROPN
ejpam-6275	697	39	β5)(x−	β5)(x−	PROPN
ejpam-6275	697	40	β17)(x−	β17)(x−	PROPN
ejpam-6275	697	41	β20	β20	PROPN
ejpam-6275	697	42	)	)	PUNCT
ejpam-6275	697	43	=	=	SYM
ejpam-6275	698	1	x3	x3	VERB
ejpam-6275	698	2	+	+	CCONJ
ejpam-6275	698	3	(	(	PUNCT
ejpam-6275	698	4	1	1	NUM
ejpam-6275	698	5	+	+	SYM
ejpam-6275	698	6	ω)x+	ω)x+	NUM
ejpam-6275	698	7	1	1	NUM
ejpam-6275	698	8	.	.	NOUN
ejpam-6275	698	9	•	•	NOUN
ejpam-6275	698	10	for	for	ADP
ejpam-6275	698	11	i	i	PRON
ejpam-6275	698	12	=	=	SYM
ejpam-6275	698	13	6	6	NUM
ejpam-6275	698	14	,	,	PUNCT
ejpam-6275	698	15	let	let	VERB
ejpam-6275	698	16	β6	β6	PROPN
ejpam-6275	698	17	∈	∈	PROPN
ejpam-6275	698	18	z2[ω	z2[ω	NOUN
ejpam-6275	698	19	]	]	X
ejpam-6275	698	20	3	3	NUM
ejpam-6275	698	21	,	,	PUNCT
ejpam-6275	698	22	then	then	ADV
ejpam-6275	698	23	φ6(x	φ6(x	PROPN
ejpam-6275	698	24	)	)	PUNCT
ejpam-6275	698	25	=	=	SYM
ejpam-6275	698	26	φ3(x	φ3(x	PROPN
ejpam-6275	698	27	)	)	PUNCT
ejpam-6275	698	28	=	=	SYM
ejpam-6275	699	1	x3	x3	VERB
ejpam-6275	699	2	+	+	CCONJ
ejpam-6275	699	3	x+	x+	ADJ
ejpam-6275	699	4	1	1	NUM
ejpam-6275	699	5	.	.	NUM
ejpam-6275	699	6	•	•	NOUN
ejpam-6275	699	7	for	for	ADP
ejpam-6275	699	8	i	i	PRON
ejpam-6275	699	9	=	=	SYM
ejpam-6275	699	10	7	7	NUM
ejpam-6275	699	11	,	,	PUNCT
ejpam-6275	699	12	let	let	VERB
ejpam-6275	699	13	β7	β7	ADJ
ejpam-6275	699	14	∈	∈	PROPN
ejpam-6275	699	15	z2[ω	z2[ω	NOUN
ejpam-6275	699	16	]	]	X
ejpam-6275	699	17	3	3	NUM
ejpam-6275	699	18	,	,	PUNCT
ejpam-6275	699	19	then	then	ADV
ejpam-6275	699	20	φ7(x	φ7(x	PROPN
ejpam-6275	699	21	)	)	PUNCT
ejpam-6275	699	22	=	=	SYM
ejpam-6275	700	1	x+	x+	PUNCT
ejpam-6275	700	2	1	1	NUM
ejpam-6275	700	3	+	+	NUM
ejpam-6275	700	4	ω	ω	NUM
ejpam-6275	700	5	.	.	NOUN
ejpam-6275	700	6	•	•	NUM
ejpam-6275	700	7	for	for	ADP
ejpam-6275	700	8	i	i	PRON
ejpam-6275	700	9	=	=	NOUN
ejpam-6275	700	10	8	8	NUM
ejpam-6275	700	11	,	,	PUNCT
ejpam-6275	700	12	let	let	VERB
ejpam-6275	700	13	β8	β8	PROPN
ejpam-6275	700	14	∈	∈	PROPN
ejpam-6275	700	15	z2[ω	z2[ω	NOUN
ejpam-6275	700	16	]	]	X
ejpam-6275	700	17	3	3	NUM
ejpam-6275	700	18	,	,	PUNCT
ejpam-6275	700	19	then	then	ADV
ejpam-6275	700	20	φ8(x	φ8(x	NOUN
ejpam-6275	700	21	)	)	PUNCT
ejpam-6275	700	22	=	=	SYM
ejpam-6275	700	23	φ2(x	φ2(x	NUM
ejpam-6275	700	24	)	)	PUNCT
ejpam-6275	700	25	=	=	PUNCT
ejpam-6275	701	1	x3	x3	VERB
ejpam-6275	702	1	+	+	CCONJ
ejpam-6275	702	2	ωx2	ωx2	PRON
ejpam-6275	702	3	+	+	X
ejpam-6275	702	4	1	1	NUM
ejpam-6275	702	5	.	.	NUM
ejpam-6275	702	6	•	•	NOUN
ejpam-6275	702	7	for	for	ADP
ejpam-6275	702	8	i	i	PRON
ejpam-6275	702	9	=	=	NOUN
ejpam-6275	702	10	9	9	NUM
ejpam-6275	702	11	,	,	PUNCT
ejpam-6275	702	12	let	let	VERB
ejpam-6275	702	13	β9	β9	PROPN
ejpam-6275	702	14	∈	∈	PROPN
ejpam-6275	702	15	z2[ω	z2[ω	NOUN
ejpam-6275	702	16	]	]	X
ejpam-6275	702	17	3	3	NUM
ejpam-6275	702	18	,	,	PUNCT
ejpam-6275	702	19	then	then	ADV
ejpam-6275	702	20	φ9(x	φ9(x	NUM
ejpam-6275	702	21	)	)	PUNCT
ejpam-6275	702	22	=	=	SYM
ejpam-6275	703	1	x3	x3	PROPN
ejpam-6275	704	1	+	+	CCONJ
ejpam-6275	704	2	x2	x2	PROPN
ejpam-6275	705	1	+	+	CCONJ
ejpam-6275	706	1	1	1	NUM
ejpam-6275	706	2	.	.	NUM
ejpam-6275	706	3	•	•	NOUN
ejpam-6275	706	4	for	for	ADP
ejpam-6275	706	5	i	i	PRON
ejpam-6275	706	6	=	=	NOUN
ejpam-6275	706	7	10	10	NUM
ejpam-6275	706	8	,	,	PUNCT
ejpam-6275	706	9	let	let	VERB
ejpam-6275	706	10	β10	β10	VERB
ejpam-6275	706	11	∈	∈	PROPN
ejpam-6275	706	12	z2[ω	z2[ω	NOUN
ejpam-6275	706	13	]	]	X
ejpam-6275	706	14	3	3	NUM
ejpam-6275	706	15	,	,	PUNCT
ejpam-6275	706	16	then	then	ADV
ejpam-6275	706	17	φ10(x	φ10(x	NOUN
ejpam-6275	706	18	)	)	PUNCT
ejpam-6275	706	19	=	=	SYM
ejpam-6275	707	1	x3	x3	ADJ
ejpam-6275	707	2	+	+	CCONJ
ejpam-6275	707	3	ωx+	ωx+	NOUN
ejpam-6275	707	4	1	1	NUM
ejpam-6275	707	5	.	.	NOUN
ejpam-6275	707	6	•	•	NOUN
ejpam-6275	707	7	for	for	ADP
ejpam-6275	707	8	i	i	PRON
ejpam-6275	707	9	=	=	NOUN
ejpam-6275	707	10	11	11	NUM
ejpam-6275	707	11	,	,	PUNCT
ejpam-6275	707	12	let	let	VERB
ejpam-6275	707	13	β11	β11	PROPN
ejpam-6275	707	14	∈	∈	PROPN
ejpam-6275	707	15	z2[ω	z2[ω	NOUN
ejpam-6275	707	16	]	]	X
ejpam-6275	707	17	3	3	NUM
ejpam-6275	707	18	,	,	PUNCT
ejpam-6275	707	19	then	then	ADV
ejpam-6275	707	20	φ11(x	φ11(x	NOUN
ejpam-6275	707	21	)	)	PUNCT
ejpam-6275	707	22	=	=	SYM
ejpam-6275	708	1	φ2(x	φ2(x	NUM
ejpam-6275	708	2	)	)	PUNCT
ejpam-6275	708	3	=	=	PUNCT
ejpam-6275	709	1	x3	x3	VERB
ejpam-6275	710	1	+	+	CCONJ
ejpam-6275	710	2	ωx2	ωx2	PRON
ejpam-6275	710	3	+	+	CCONJ
ejpam-6275	710	4	1	1	X
ejpam-6275	710	5	.	.	X
ejpam-6275	710	6	m.	m.	PROPN
ejpam-6275	710	7	sajjad	sajjad	PROPN
ejpam-6275	710	8	et	et	PROPN
ejpam-6275	710	9	al	al	PROPN
ejpam-6275	710	10	.	.	PUNCT
ejpam-6275	710	11	/	/	SYM
ejpam-6275	710	12	eur	eur	PROPN
ejpam-6275	710	13	.	.	PUNCT
ejpam-6275	711	1	j.	j.	PROPN
ejpam-6275	711	2	pure	pure	PROPN
ejpam-6275	711	3	appl	appl	PROPN
ejpam-6275	711	4	.	.	PROPN
ejpam-6275	711	5	math	math	PROPN
ejpam-6275	711	6	,	,	PUNCT
ejpam-6275	711	7	18	18	NUM
ejpam-6275	711	8	(	(	PUNCT
ejpam-6275	711	9	3	3	NUM
ejpam-6275	711	10	)	)	PUNCT
ejpam-6275	711	11	(	(	PUNCT
ejpam-6275	711	12	2025	2025	NUM
ejpam-6275	711	13	)	)	PUNCT
ejpam-6275	711	14	,	,	PUNCT
ejpam-6275	711	15	6275	6275	NUM
ejpam-6275	711	16	26	26	NUM
ejpam-6275	711	17	of	of	ADP
ejpam-6275	711	18	36	36	NUM
ejpam-6275	711	19	•	•	NOUN
ejpam-6275	711	20	for	for	ADP
ejpam-6275	711	21	i	i	PRON
ejpam-6275	711	22	=	=	NOUN
ejpam-6275	711	23	12	12	NUM
ejpam-6275	711	24	,	,	PUNCT
ejpam-6275	711	25	let	let	VERB
ejpam-6275	711	26	β12	β12	PRON
ejpam-6275	711	27	∈	∈	PROPN
ejpam-6275	711	28	z2[ω	z2[ω	NOUN
ejpam-6275	711	29	]	]	X
ejpam-6275	711	30	3	3	NUM
ejpam-6275	711	31	,	,	PUNCT
ejpam-6275	711	32	then	then	ADV
ejpam-6275	711	33	φ12(x	φ12(x	NOUN
ejpam-6275	711	34	)	)	PUNCT
ejpam-6275	711	35	=	=	SYM
ejpam-6275	712	1	φ3(x	φ3(x	PROPN
ejpam-6275	712	2	)	)	PUNCT
ejpam-6275	712	3	=	=	SYM
ejpam-6275	713	1	x3	x3	VERB
ejpam-6275	713	2	+	+	CCONJ
ejpam-6275	713	3	x+	x+	ADJ
ejpam-6275	713	4	1	1	NUM
ejpam-6275	713	5	.	.	NUM
ejpam-6275	713	6	•	•	NOUN
ejpam-6275	713	7	for	for	ADP
ejpam-6275	713	8	i	i	PRON
ejpam-6275	713	9	=	=	NOUN
ejpam-6275	713	10	13	13	NUM
ejpam-6275	713	11	,	,	PUNCT
ejpam-6275	713	12	let	let	VERB
ejpam-6275	713	13	β13	β13	PRON
ejpam-6275	713	14	∈	∈	PROPN
ejpam-6275	713	15	z2[ω	z2[ω	NOUN
ejpam-6275	713	16	]	]	X
ejpam-6275	713	17	3	3	NUM
ejpam-6275	713	18	,	,	PUNCT
ejpam-6275	713	19	then	then	ADV
ejpam-6275	713	20	φ13(x	φ13(x	NUM
ejpam-6275	713	21	)	)	PUNCT
ejpam-6275	713	22	=	=	SYM
ejpam-6275	713	23	φ10(x	φ10(x	NOUN
ejpam-6275	713	24	)	)	PUNCT
ejpam-6275	713	25	=	=	SYM
ejpam-6275	714	1	x3	x3	ADJ
ejpam-6275	714	2	+	+	CCONJ
ejpam-6275	714	3	ωx+	ωx+	NOUN
ejpam-6275	714	4	1	1	NUM
ejpam-6275	714	5	.	.	NOUN
ejpam-6275	714	6	•	•	NOUN
ejpam-6275	714	7	for	for	ADP
ejpam-6275	714	8	i	i	PRON
ejpam-6275	714	9	=	=	NOUN
ejpam-6275	714	10	14	14	NUM
ejpam-6275	714	11	,	,	PUNCT
ejpam-6275	714	12	let	let	VERB
ejpam-6275	714	13	β14	β14	PRON
ejpam-6275	714	14	∈	∈	PROPN
ejpam-6275	714	15	z2[ω	z2[ω	NOUN
ejpam-6275	714	16	]	]	X
ejpam-6275	714	17	3	3	NUM
ejpam-6275	714	18	,	,	PUNCT
ejpam-6275	714	19	then	then	ADV
ejpam-6275	714	20	φ14(x	φ14(x	NOUN
ejpam-6275	714	21	)	)	PUNCT
ejpam-6275	714	22	=	=	SYM
ejpam-6275	715	1	x+	x+	PROPN
ejpam-6275	715	2	ω	ω	X
ejpam-6275	715	3	.	.	PROPN
ejpam-6275	715	4	•	•	NUM
ejpam-6275	715	5	for	for	ADP
ejpam-6275	715	6	i	i	PRON
ejpam-6275	715	7	=	=	NOUN
ejpam-6275	715	8	15	15	NUM
ejpam-6275	715	9	,	,	PUNCT
ejpam-6275	715	10	let	let	VERB
ejpam-6275	715	11	β15	β15	NOUN
ejpam-6275	715	12	∈	∈	PROPN
ejpam-6275	715	13	z2[ω	z2[ω	NOUN
ejpam-6275	715	14	]	]	X
ejpam-6275	715	15	3	3	NUM
ejpam-6275	715	16	,	,	PUNCT
ejpam-6275	715	17	then	then	ADV
ejpam-6275	715	18	φ15(x	φ15(x	NUM
ejpam-6275	715	19	)	)	PUNCT
ejpam-6275	716	1	=	=	SYM
ejpam-6275	716	2	φ9(x	φ9(x	NOUN
ejpam-6275	716	3	)	)	PUNCT
ejpam-6275	716	4	=	=	SYM
ejpam-6275	717	1	x3	x3	PROPN
ejpam-6275	718	1	+	+	CCONJ
ejpam-6275	718	2	x2	x2	PROPN
ejpam-6275	719	1	+	+	CCONJ
ejpam-6275	720	1	1	1	NUM
ejpam-6275	720	2	.	.	NUM
ejpam-6275	720	3	•	•	NOUN
ejpam-6275	720	4	for	for	ADP
ejpam-6275	720	5	i	i	PRON
ejpam-6275	720	6	=	=	SYM
ejpam-6275	720	7	16	16	NUM
ejpam-6275	720	8	,	,	PUNCT
ejpam-6275	720	9	let	let	VERB
ejpam-6275	720	10	β16	β16	PROPN
ejpam-6275	720	11	∈	∈	PROPN
ejpam-6275	720	12	z2[ω	z2[ω	NOUN
ejpam-6275	720	13	]	]	X
ejpam-6275	720	14	3	3	NUM
ejpam-6275	720	15	,	,	PUNCT
ejpam-6275	720	16	then	then	ADV
ejpam-6275	720	17	φ16(x	φ16(x	PROPN
ejpam-6275	720	18	)	)	PUNCT
ejpam-6275	720	19	=	=	PUNCT
ejpam-6275	721	1	φ1(x	φ1(x	NOUN
ejpam-6275	721	2	)	)	PUNCT
ejpam-6275	721	3	=	=	SYM
ejpam-6275	722	1	x3	x3	VERB
ejpam-6275	722	2	+	+	CCONJ
ejpam-6275	723	1	(	(	PUNCT
ejpam-6275	723	2	1	1	NUM
ejpam-6275	723	3	+	+	X
ejpam-6275	723	4	ω)x2	ω)x2	NUM
ejpam-6275	723	5	+	+	CCONJ
ejpam-6275	723	6	1	1	NUM
ejpam-6275	723	7	.	.	NOUN
ejpam-6275	723	8	•	•	NOUN
ejpam-6275	723	9	for	for	ADP
ejpam-6275	723	10	i	i	PRON
ejpam-6275	723	11	=	=	SYM
ejpam-6275	723	12	17	17	NUM
ejpam-6275	723	13	,	,	PUNCT
ejpam-6275	723	14	let	let	VERB
ejpam-6275	723	15	β17	β17	NUM
ejpam-6275	723	16	∈	∈	PROPN
ejpam-6275	723	17	z2[ω	z2[ω	NOUN
ejpam-6275	723	18	]	]	X
ejpam-6275	723	19	3	3	NUM
ejpam-6275	723	20	,	,	PUNCT
ejpam-6275	723	21	then	then	ADV
ejpam-6275	723	22	φ17(x	φ17(x	NOUN
ejpam-6275	723	23	)	)	PUNCT
ejpam-6275	723	24	=	=	PUNCT
ejpam-6275	724	1	φ5(x	φ5(x	PROPN
ejpam-6275	724	2	)	)	PUNCT
ejpam-6275	724	3	=	=	SYM
ejpam-6275	725	1	x3	x3	VERB
ejpam-6275	725	2	+	+	CCONJ
ejpam-6275	725	3	(	(	PUNCT
ejpam-6275	725	4	1	1	NUM
ejpam-6275	725	5	+	+	SYM
ejpam-6275	725	6	ω)x+	ω)x+	NUM
ejpam-6275	725	7	1	1	NUM
ejpam-6275	725	8	.	.	NOUN
ejpam-6275	725	9	•	•	NOUN
ejpam-6275	725	10	for	for	ADP
ejpam-6275	725	11	i	i	PRON
ejpam-6275	725	12	=	=	NOUN
ejpam-6275	725	13	18	18	NUM
ejpam-6275	725	14	,	,	PUNCT
ejpam-6275	725	15	let	let	VERB
ejpam-6275	725	16	β18	β18	NUM
ejpam-6275	725	17	∈	∈	PROPN
ejpam-6275	725	18	z2[ω	z2[ω	NOUN
ejpam-6275	725	19	]	]	X
ejpam-6275	725	20	3	3	NUM
ejpam-6275	725	21	,	,	PUNCT
ejpam-6275	725	22	then	then	ADV
ejpam-6275	725	23	φ18(x	φ18(x	NUM
ejpam-6275	725	24	)	)	PUNCT
ejpam-6275	726	1	=	=	SYM
ejpam-6275	726	2	φ9(x	φ9(x	NOUN
ejpam-6275	726	3	)	)	PUNCT
ejpam-6275	726	4	=	=	SYM
ejpam-6275	727	1	x3	x3	PROPN
ejpam-6275	728	1	+	+	CCONJ
ejpam-6275	728	2	x2	x2	PROPN
ejpam-6275	729	1	+	+	CCONJ
ejpam-6275	730	1	1	1	NUM
ejpam-6275	730	2	.	.	NUM
ejpam-6275	730	3	•	•	NOUN
ejpam-6275	730	4	for	for	ADP
ejpam-6275	730	5	i	i	PRON
ejpam-6275	730	6	=	=	SYM
ejpam-6275	730	7	19	19	NUM
ejpam-6275	730	8	,	,	PUNCT
ejpam-6275	730	9	let	let	VERB
ejpam-6275	730	10	β19	β19	PRON
ejpam-6275	730	11	∈	∈	PROPN
ejpam-6275	730	12	z2[ω	z2[ω	NOUN
ejpam-6275	730	13	]	]	X
ejpam-6275	730	14	3	3	NUM
ejpam-6275	730	15	,	,	PUNCT
ejpam-6275	730	16	then	then	ADV
ejpam-6275	730	17	φ19(x	φ19(x	PRON
ejpam-6275	730	18	)	)	PUNCT
ejpam-6275	730	19	=	=	SYM
ejpam-6275	730	20	φ10(x	φ10(x	NOUN
ejpam-6275	730	21	)	)	PUNCT
ejpam-6275	730	22	=	=	SYM
ejpam-6275	731	1	x3	x3	ADJ
ejpam-6275	731	2	+	+	CCONJ
ejpam-6275	731	3	ωx+	ωx+	NOUN
ejpam-6275	731	4	1	1	NUM
ejpam-6275	731	5	.	.	NOUN
ejpam-6275	731	6	•	•	NOUN
ejpam-6275	731	7	for	for	ADP
ejpam-6275	731	8	i	i	PRON
ejpam-6275	731	9	=	=	NOUN
ejpam-6275	731	10	20	20	NUM
ejpam-6275	731	11	,	,	PUNCT
ejpam-6275	731	12	let	let	VERB
ejpam-6275	731	13	β20	β20	PRON
ejpam-6275	731	14	∈	∈	PROPN
ejpam-6275	731	15	z2[ω	z2[ω	NOUN
ejpam-6275	731	16	]	]	X
ejpam-6275	731	17	3	3	NUM
ejpam-6275	731	18	,	,	PUNCT
ejpam-6275	731	19	then	then	ADV
ejpam-6275	731	20	φ20(x	φ20(x	VERB
ejpam-6275	731	21	)	)	PUNCT
ejpam-6275	732	1	=	=	PUNCT
ejpam-6275	732	2	φ5(x	φ5(x	PROPN
ejpam-6275	732	3	)	)	PUNCT
ejpam-6275	732	4	=	=	SYM
ejpam-6275	733	1	x3	x3	VERB
ejpam-6275	733	2	+	+	CCONJ
ejpam-6275	733	3	(	(	PUNCT
ejpam-6275	733	4	1	1	NUM
ejpam-6275	733	5	+	+	SYM
ejpam-6275	733	6	ω)x+	ω)x+	NUM
ejpam-6275	733	7	1	1	NUM
ejpam-6275	733	8	.	.	PUNCT
ejpam-6275	734	1	now	now	ADV
ejpam-6275	734	2	the	the	DET
ejpam-6275	734	3	generator	generator	NOUN
ejpam-6275	734	4	polynomial	polynomial	NOUN
ejpam-6275	734	5	is	be	AUX
ejpam-6275	734	6	g(x	g(x	NOUN
ejpam-6275	734	7	)	)	PUNCT
ejpam-6275	735	1	=	=	SYM
ejpam-6275	735	2	ϕ1(x	ϕ1(x	X
ejpam-6275	735	3	)	)	PUNCT
ejpam-6275	735	4	·	·	PUNCT
ejpam-6275	736	1	ϕ2(x	ϕ2(x	X
ejpam-6275	736	2	)	)	PUNCT
ejpam-6275	736	3	·	·	PUNCT
ejpam-6275	737	1	ϕ3(x	ϕ3(x	X
ejpam-6275	737	2	)	)	PUNCT
ejpam-6275	737	3	·	·	PUNCT
ejpam-6275	738	1	ϕ5(x	ϕ5(x	X
ejpam-6275	738	2	)	)	PUNCT
ejpam-6275	738	3	·	·	PUNCT
ejpam-6275	738	4	ϕ7(x	ϕ7(x	X
ejpam-6275	738	5	)	)	PUNCT
ejpam-6275	738	6	·	·	PUNCT
ejpam-6275	739	1	ϕ9(x	ϕ9(x	X
ejpam-6275	739	2	)	)	PUNCT
ejpam-6275	739	3	·	·	PUNCT
ejpam-6275	739	4	ϕ10(x	ϕ10(x	X
ejpam-6275	739	5	)	)	PUNCT
ejpam-6275	739	6	·	·	PUNCT
ejpam-6275	739	7	ϕ14(x	ϕ14(x	X
ejpam-6275	739	8	)	)	PUNCT
ejpam-6275	739	9	=	=	SYM
ejpam-6275	740	1	x20	x20	NOUN
ejpam-6275	740	2	+	+	CCONJ
ejpam-6275	740	3	x18	x18	NOUN
ejpam-6275	741	1	+	+	CCONJ
ejpam-6275	741	2	x17	x17	NOUN
ejpam-6275	741	3	+	+	CCONJ
ejpam-6275	741	4	x16	x16	NOUN
ejpam-6275	741	5	+	+	CCONJ
ejpam-6275	741	6	ωx15	ωx15	PROPN
ejpam-6275	741	7	+	+	CCONJ
ejpam-6275	741	8	ωx14	ωx14	PROPN
ejpam-6275	741	9	+	+	NUM
ejpam-6275	741	10	x13	x13	NOUN
ejpam-6275	741	11	+	+	NUM
ejpam-6275	741	12	x9	x9	NOUN
ejpam-6275	741	13	+	+	CCONJ
ejpam-6275	741	14	x5	x5	NOUN
ejpam-6275	741	15	+	+	CCONJ
ejpam-6275	741	16	x4	x4	PROPN
ejpam-6275	742	1	+	+	CCONJ
ejpam-6275	742	2	x3	x3	ADJ
ejpam-6275	742	3	+	+	NOUN
ejpam-6275	742	4	1	1	X
ejpam-6275	742	5	.	.	PUNCT
ejpam-6275	743	1	now	now	ADV
ejpam-6275	743	2	,	,	PUNCT
ejpam-6275	743	3	k	k	PROPN
ejpam-6275	743	4	=	=	SYM
ejpam-6275	743	5	21	21	NUM
ejpam-6275	743	6	−	−	NUM
ejpam-6275	743	7	20	20	NUM
ejpam-6275	743	8	=	=	SYM
ejpam-6275	743	9	1	1	NUM
ejpam-6275	743	10	.	.	PUNCT
ejpam-6275	744	1	so	so	ADV
ejpam-6275	744	2	the	the	DET
ejpam-6275	744	3	parameters	parameter	NOUN
ejpam-6275	744	4	of	of	ADP
ejpam-6275	744	5	the	the	DET
ejpam-6275	744	6	shortened	shorten	VERB
ejpam-6275	744	7	bch	bch	PROPN
ejpam-6275	744	8	code	code	NOUN
ejpam-6275	744	9	are	be	AUX
ejpam-6275	744	10	(	(	PUNCT
ejpam-6275	744	11	n	n	X
ejpam-6275	744	12	,	,	PUNCT
ejpam-6275	744	13	k	k	NOUN
ejpam-6275	744	14	,	,	PUNCT
ejpam-6275	744	15	d	d	NOUN
ejpam-6275	744	16	)	)	PUNCT
ejpam-6275	744	17	=	=	SYM
ejpam-6275	744	18	(	(	PUNCT
ejpam-6275	744	19	21	21	NUM
ejpam-6275	744	20	,	,	PUNCT
ejpam-6275	744	21	1	1	NUM
ejpam-6275	744	22	,	,	PUNCT
ejpam-6275	744	23	21	21	NUM
ejpam-6275	744	24	)	)	PUNCT
ejpam-6275	744	25	.	.	PUNCT
ejpam-6275	745	1	4	4	NUM
ejpam-6275	745	2	.	.	X
ejpam-6275	745	3	4	4	NUM
ejpam-6275	745	4	.	.	X
ejpam-6275	745	5	decoding	decode	VERB
ejpam-6275	745	6	algorithm	algorithm	NOUN
ejpam-6275	745	7	for	for	ADP
ejpam-6275	745	8	shortened	shorten	VERB
ejpam-6275	745	9	bch	bch	PROPN
ejpam-6275	745	10	codes	code	NOUN
ejpam-6275	745	11	over	over	ADP
ejpam-6275	745	12	the	the	DET
ejpam-6275	745	13	eisenstein	eisenstein	NOUN
ejpam-6275	745	14	fields	field	VERB
ejpam-6275	745	15	this	this	DET
ejpam-6275	745	16	section	section	NOUN
ejpam-6275	745	17	is	be	AUX
ejpam-6275	745	18	based	base	VERB
ejpam-6275	745	19	on	on	ADP
ejpam-6275	745	20	the	the	DET
ejpam-6275	745	21	procedure	procedure	NOUN
ejpam-6275	745	22	of	of	ADP
ejpam-6275	745	23	decoding	decode	VERB
ejpam-6275	745	24	shortened	shorten	VERB
ejpam-6275	745	25	bch	bch	PROPN
ejpam-6275	745	26	codes	code	NOUN
ejpam-6275	745	27	over	over	ADP
ejpam-6275	745	28	eisenstein	eisenstein	NOUN
ejpam-6275	745	29	fields	field	NOUN
ejpam-6275	745	30	of	of	ADP
ejpam-6275	745	31	length	length	NOUN
ejpam-6275	745	32	n	n	CCONJ
ejpam-6275	745	33	,	,	PUNCT
ejpam-6275	745	34	using	use	VERB
ejpam-6275	745	35	a	a	DET
ejpam-6275	745	36	modified	modify	VERB
ejpam-6275	745	37	berlekamp	berlekamp	NOUN
ejpam-6275	745	38	–	–	PUNCT
ejpam-6275	745	39	massey	massey	NOUN
ejpam-6275	745	40	algorithm	algorithm	NOUN
ejpam-6275	745	41	(	(	PUNCT
ejpam-6275	745	42	bma	bma	PROPN
ejpam-6275	745	43	)	)	PUNCT
ejpam-6275	745	44	.	.	PUNCT
ejpam-6275	746	1	the	the	DET
ejpam-6275	746	2	procedure	procedure	NOUN
ejpam-6275	746	3	follows	follow	VERB
ejpam-6275	746	4	the	the	DET
ejpam-6275	746	5	same	same	ADJ
ejpam-6275	746	6	principles	principle	NOUN
ejpam-6275	746	7	as	as	ADP
ejpam-6275	746	8	decoding	decode	VERB
ejpam-6275	746	9	conventional	conventional	ADJ
ejpam-6275	746	10	bch	bch	PROPN
ejpam-6275	746	11	codes	code	NOUN
ejpam-6275	746	12	.	.	PUNCT
ejpam-6275	747	1	theorem	theorem	VERB
ejpam-6275	747	2	4.1	4.1	NUM
ejpam-6275	747	3	[	[	SYM
ejpam-6275	747	4	6	6	NUM
ejpam-6275	747	5	,	,	PUNCT
ejpam-6275	747	6	theorem	theorem	VERB
ejpam-6275	747	7	4.2	4.2	NUM
ejpam-6275	747	8	]	]	PUNCT
ejpam-6275	747	9	:	:	PUNCT
ejpam-6275	747	10	let	let	VERB
ejpam-6275	747	11	c	c	PART
ejpam-6275	747	12	be	be	AUX
ejpam-6275	747	13	a	a	DET
ejpam-6275	747	14	bch	bch	NOUN
ejpam-6275	747	15	or	or	CCONJ
ejpam-6275	747	16	shortened	shorten	VERB
ejpam-6275	747	17	bch	bch	PROPN
ejpam-6275	747	18	code	code	NOUN
ejpam-6275	747	19	of	of	ADP
ejpam-6275	747	20	length	length	NOUN
ejpam-6275	747	21	n	n	CCONJ
ejpam-6275	747	22	,	,	PUNCT
ejpam-6275	747	23	defined	define	VERB
ejpam-6275	747	24	over	over	ADP
ejpam-6275	747	25	the	the	DET
ejpam-6275	747	26	eisenstein	eisenstein	PROPN
ejpam-6275	747	27	field	field	NOUN
ejpam-6275	747	28	zp[ω	zp[ω	PROPN
ejpam-6275	747	29	]	]	X
ejpam-6275	747	30	,	,	PUNCT
ejpam-6275	747	31	with	with	ADP
ejpam-6275	747	32	designed	design	VERB
ejpam-6275	747	33	distance	distance	NOUN
ejpam-6275	747	34	d.	d.	PROPN
ejpam-6275	747	35	then	then	ADV
ejpam-6275	747	36	,	,	PUNCT
ejpam-6275	747	37	the	the	DET
ejpam-6275	747	38	code	code	NOUN
ejpam-6275	747	39	c	c	PROPN
ejpam-6275	747	40	is	be	AUX
ejpam-6275	747	41	the	the	DET
ejpam-6275	747	42	null	null	ADJ
ejpam-6275	747	43	space	space	NOUN
ejpam-6275	747	44	of	of	ADP
ejpam-6275	747	45	the	the	DET
ejpam-6275	747	46	matrix	matrix	NOUN
ejpam-6275	748	1	n	n	NOUN
ejpam-6275	748	2	,	,	PUNCT
ejpam-6275	749	1	where	where	SCONJ
ejpam-6275	749	2	m.	m.	NOUN
ejpam-6275	749	3	sajjad	sajjad	PROPN
ejpam-6275	749	4	et	et	PROPN
ejpam-6275	749	5	al	al	PROPN
ejpam-6275	749	6	.	.	PUNCT
ejpam-6275	749	7	/	/	SYM
ejpam-6275	749	8	eur	eur	PROPN
ejpam-6275	749	9	.	.	PUNCT
ejpam-6275	750	1	j.	j.	PROPN
ejpam-6275	750	2	pure	pure	PROPN
ejpam-6275	750	3	appl	appl	PROPN
ejpam-6275	750	4	.	.	PROPN
ejpam-6275	750	5	math	math	PROPN
ejpam-6275	750	6	,	,	PUNCT
ejpam-6275	750	7	18	18	NUM
ejpam-6275	750	8	(	(	PUNCT
ejpam-6275	750	9	3	3	NUM
ejpam-6275	750	10	)	)	PUNCT
ejpam-6275	750	11	(	(	PUNCT
ejpam-6275	750	12	2025	2025	NUM
ejpam-6275	750	13	)	)	PUNCT
ejpam-6275	750	14	,	,	PUNCT
ejpam-6275	750	15	6275	6275	NUM
ejpam-6275	750	16	27	27	NUM
ejpam-6275	750	17	of	of	ADP
ejpam-6275	750	18	36	36	NUM
ejpam-6275	750	19	n	n	NOUN
ejpam-6275	750	20	=	=	SYM
ejpam-6275	750	21			PROPN
ejpam-6275	750	22	1	1	NUM
ejpam-6275	750	23	βc	βc	INTJ
ejpam-6275	750	24	β2c	β2c	PUNCT
ejpam-6275	750	25	·	·	PUNCT
ejpam-6275	750	26	·	·	PUNCT
ejpam-6275	750	27	·	·	PUNCT
ejpam-6275	751	1	β(n−1)c	β(n−1)c	NOUN
ejpam-6275	751	2	1	1	NUM
ejpam-6275	751	3	βc+1	βc+1	X
ejpam-6275	751	4	β2(c+1	β2(c+1	NOUN
ejpam-6275	751	5	)	)	PUNCT
ejpam-6275	751	6	·	·	PUNCT
ejpam-6275	751	7	·	·	PUNCT
ejpam-6275	751	8	·	·	PUNCT
ejpam-6275	752	1	β(n−1)(c+1	β(n−1)(c+1	X
ejpam-6275	752	2	)	)	PUNCT
ejpam-6275	752	3	...	...	PUNCT
ejpam-6275	752	4	...	...	PUNCT
ejpam-6275	752	5	...	...	PUNCT
ejpam-6275	752	6	.	.	PUNCT
ejpam-6275	752	7	.	.	PUNCT
ejpam-6275	752	8	.	.	PUNCT
ejpam-6275	753	1	...	...	PUNCT
ejpam-6275	754	1	1	1	NUM
ejpam-6275	754	2	βc+d−2	βc+d−2	ADV
ejpam-6275	754	3	β2(c+d−2	β2(c+d−2	ADJ
ejpam-6275	754	4	)	)	PUNCT
ejpam-6275	754	5	·	·	PUNCT
ejpam-6275	754	6	·	·	PUNCT
ejpam-6275	754	7	·	·	PUNCT
ejpam-6275	754	8	β(n−1)(c+d−2	β(n−1)(c+d−2	X
ejpam-6275	754	9	)	)	PUNCT
ejpam-6275	754	10			NOUN
ejpam-6275	754	11	.	.	PUNCT
ejpam-6275	755	1	here	here	ADV
ejpam-6275	755	2	,	,	PUNCT
ejpam-6275	755	3	β	β	X
ejpam-6275	755	4	is	be	AUX
ejpam-6275	755	5	a	a	DET
ejpam-6275	755	6	primitive	primitive	ADJ
ejpam-6275	755	7	element	element	NOUN
ejpam-6275	755	8	in	in	ADP
ejpam-6275	755	9	the	the	DET
ejpam-6275	755	10	extension	extension	NOUN
ejpam-6275	755	11	field	field	NOUN
ejpam-6275	755	12	containing	contain	VERB
ejpam-6275	755	13	zp[ω	zp[ω	PROPN
ejpam-6275	755	14	]	]	PUNCT
ejpam-6275	755	15	,	,	PUNCT
ejpam-6275	755	16	and	and	CCONJ
ejpam-6275	755	17	the	the	DET
ejpam-6275	755	18	matrix	matrix	NOUN
ejpam-6275	755	19	n	n	ADP
ejpam-6275	755	20	consists	consist	VERB
ejpam-6275	755	21	of	of	ADP
ejpam-6275	755	22	d−	d−	PROPN
ejpam-6275	755	23	1	1	NUM
ejpam-6275	755	24	rows	row	NOUN
ejpam-6275	755	25	and	and	CCONJ
ejpam-6275	755	26	n	n	DET
ejpam-6275	755	27	columns	column	NOUN
ejpam-6275	755	28	.	.	PUNCT
ejpam-6275	756	1	the	the	DET
ejpam-6275	756	2	codewords	codeword	NOUN
ejpam-6275	756	3	of	of	ADP
ejpam-6275	756	4	c	c	PROPN
ejpam-6275	756	5	are	be	AUX
ejpam-6275	756	6	those	those	DET
ejpam-6275	756	7	vectors	vector	NOUN
ejpam-6275	756	8	v	v	ADP
ejpam-6275	756	9	∈	∈	PROPN
ejpam-6275	756	10	zp[ω	zp[ω	PROPN
ejpam-6275	756	11	]	]	X
ejpam-6275	756	12	n	n	CCONJ
ejpam-6275	756	13	such	such	ADJ
ejpam-6275	756	14	that	that	SCONJ
ejpam-6275	756	15	nvt	nvt	PROPN
ejpam-6275	756	16	=	=	SYM
ejpam-6275	756	17	0	0	NUM
ejpam-6275	756	18	.	.	NOUN
ejpam-6275	756	19	4.1	4.1	NUM
ejpam-6275	756	20	.	.	PUNCT
ejpam-6275	757	1	algorithm	algorithm	NOUN
ejpam-6275	757	2	for	for	ADP
ejpam-6275	757	3	decoding	decode	VERB
ejpam-6275	757	4	of	of	ADP
ejpam-6275	757	5	shortened	shorten	VERB
ejpam-6275	757	6	bch	bch	PROPN
ejpam-6275	757	7	codes	code	NOUN
ejpam-6275	757	8	over	over	ADP
ejpam-6275	757	9	the	the	DET
ejpam-6275	757	10	eisenstein	eisenstein	PROPN
ejpam-6275	757	11	fields	field	NOUN
ejpam-6275	757	12	assume	assume	VERB
ejpam-6275	757	13	that	that	SCONJ
ejpam-6275	757	14	c	c	PROPN
ejpam-6275	757	15	is	be	AUX
ejpam-6275	757	16	a	a	DET
ejpam-6275	757	17	codeword	codeword	NOUN
ejpam-6275	757	18	from	from	ADP
ejpam-6275	757	19	a	a	DET
ejpam-6275	757	20	narrow	narrow	ADJ
ejpam-6275	757	21	-	-	PUNCT
ejpam-6275	757	22	sense	sense	NOUN
ejpam-6275	757	23	shortened	shorten	VERB
ejpam-6275	757	24	bch	bch	PROPN
ejpam-6275	757	25	code	code	PROPN
ejpam-6275	757	26	(	(	PUNCT
ejpam-6275	757	27	n	n	CCONJ
ejpam-6275	757	28	,	,	PUNCT
ejpam-6275	757	29	k	k	NOUN
ejpam-6275	757	30	,	,	PUNCT
ejpam-6275	757	31	d	d	NOUN
ejpam-6275	757	32	)	)	PUNCT
ejpam-6275	757	33	over	over	ADP
ejpam-6275	757	34	an	an	DET
ejpam-6275	757	35	eisenstein	eisenstein	NOUN
ejpam-6275	757	36	field	field	NOUN
ejpam-6275	757	37	,	,	PUNCT
ejpam-6275	757	38	and	and	CCONJ
ejpam-6275	757	39	the	the	DET
ejpam-6275	757	40	received	receive	VERB
ejpam-6275	757	41	word	word	NOUN
ejpam-6275	757	42	is	be	AUX
ejpam-6275	757	43	r.	r.	NOUN
ejpam-6275	757	44	the	the	DET
ejpam-6275	757	45	designed	design	VERB
ejpam-6275	757	46	distance	distance	NOUN
ejpam-6275	757	47	is	be	AUX
ejpam-6275	757	48	d.	d.	PROPN
ejpam-6275	757	49	the	the	DET
ejpam-6275	757	50	following	follow	VERB
ejpam-6275	757	51	steps	step	NOUN
ejpam-6275	757	52	outline	outline	VERB
ejpam-6275	757	53	the	the	DET
ejpam-6275	757	54	decoding	decode	VERB
ejpam-6275	757	55	process	process	NOUN
ejpam-6275	757	56	using	use	VERB
ejpam-6275	757	57	the	the	DET
ejpam-6275	757	58	modified	modify	VERB
ejpam-6275	757	59	berlekamp	berlekamp	NOUN
ejpam-6275	757	60	–	–	PUNCT
ejpam-6275	757	61	massey	massey	NOUN
ejpam-6275	757	62	algorithm	algorithm	NOUN
ejpam-6275	757	63	(	(	PUNCT
ejpam-6275	757	64	bma	bma	PROPN
ejpam-6275	757	65	)	)	PUNCT
ejpam-6275	757	66	.	.	PUNCT
ejpam-6275	758	1	step	step	NOUN
ejpam-6275	758	2	1	1	NUM
ejpam-6275	758	3	:	:	PUNCT
ejpam-6275	758	4	syndrome	syndrome	NOUN
ejpam-6275	758	5	calculation	calculation	NOUN
ejpam-6275	758	6	let	let	VERB
ejpam-6275	758	7	si	si	PROPN
ejpam-6275	758	8	for	for	ADP
ejpam-6275	758	9	i	i	PRON
ejpam-6275	758	10	=	=	SYM
ejpam-6275	758	11	c	c	X
ejpam-6275	758	12	,	,	PUNCT
ejpam-6275	758	13	c+1	c+1	VERB
ejpam-6275	758	14	,	,	PUNCT
ejpam-6275	758	15	.	.	PUNCT
ejpam-6275	758	16	.	.	PUNCT
ejpam-6275	759	1	.	.	PUNCT
ejpam-6275	760	1	,	,	PUNCT
ejpam-6275	760	2	c+	c+	VERB
ejpam-6275	760	3	d−	d−	PROPN
ejpam-6275	760	4	2	2	NUM
ejpam-6275	760	5	denote	denote	VERB
ejpam-6275	760	6	the	the	DET
ejpam-6275	760	7	syndromes	syndrome	NOUN
ejpam-6275	760	8	computed	compute	VERB
ejpam-6275	760	9	from	from	ADP
ejpam-6275	760	10	the	the	DET
ejpam-6275	760	11	received	receive	VERB
ejpam-6275	760	12	word	word	NOUN
ejpam-6275	760	13	r	r	NOUN
ejpam-6275	760	14	and	and	CCONJ
ejpam-6275	760	15	the	the	DET
ejpam-6275	760	16	matrix	matrix	NOUN
ejpam-6275	760	17	n	n	NOUN
ejpam-6275	760	18	:	:	PUNCT
ejpam-6275	760	19	si	si	X
ejpam-6275	760	20	=	=	PUNCT
ejpam-6275	760	21	rnt	rnt	VERB
ejpam-6275	760	22	mod	mod	PROPN
ejpam-6275	760	23	p	p	PROPN
ejpam-6275	760	24	=	=	X
ejpam-6275	760	25	(	(	PUNCT
ejpam-6275	760	26	sc	sc	PROPN
ejpam-6275	760	27	,	,	PUNCT
ejpam-6275	760	28	sc+1	sc+1	PROPN
ejpam-6275	760	29	,	,	PUNCT
ejpam-6275	760	30	.	.	PUNCT
ejpam-6275	760	31	.	.	PUNCT
ejpam-6275	761	1	.	.	PUNCT
ejpam-6275	762	1	,	,	PUNCT
ejpam-6275	762	2	sc+d−2	sc+d−2	PROPN
ejpam-6275	762	3	)	)	PUNCT
ejpam-6275	762	4	.	.	PUNCT
ejpam-6275	763	1	alternatively	alternatively	ADV
ejpam-6275	763	2	,	,	PUNCT
ejpam-6275	763	3	each	each	DET
ejpam-6275	763	4	syndrome	syndrome	NOUN
ejpam-6275	763	5	can	can	AUX
ejpam-6275	763	6	be	be	AUX
ejpam-6275	763	7	computed	compute	VERB
ejpam-6275	763	8	using	use	VERB
ejpam-6275	763	9	:	:	PUNCT
ejpam-6275	763	10	si	si	PROPN
ejpam-6275	763	11	=	=	PUNCT
ejpam-6275	763	12	rβi	rβi	PROPN
ejpam-6275	763	13	=	=	PUNCT
ejpam-6275	763	14	b0	b0	PROPN
ejpam-6275	763	15	+	+	CCONJ
ejpam-6275	763	16	b1β	b1β	PROPN
ejpam-6275	763	17	i	i	PRON
ejpam-6275	763	18	+	+	X
ejpam-6275	763	19	·	·	PUNCT
ejpam-6275	763	20	·	·	PUNCT
ejpam-6275	763	21	·	·	PUNCT
ejpam-6275	764	1	+	+	CCONJ
ejpam-6275	764	2	bn−1β	bn−1β	PROPN
ejpam-6275	764	3	(	(	PUNCT
ejpam-6275	764	4	n−1)i	n−1)i	NOUN
ejpam-6275	764	5	,	,	PUNCT
ejpam-6275	764	6	for	for	ADP
ejpam-6275	764	7	i	i	PRON
ejpam-6275	764	8	=	=	SYM
ejpam-6275	764	9	c	c	X
ejpam-6275	764	10	,	,	PUNCT
ejpam-6275	764	11	c+	c+	VERB
ejpam-6275	764	12	1	1	NUM
ejpam-6275	764	13	,	,	PUNCT
ejpam-6275	764	14	.	.	PUNCT
ejpam-6275	764	15	.	.	PUNCT
ejpam-6275	765	1	.	.	PUNCT
ejpam-6275	766	1	,	,	PUNCT
ejpam-6275	766	2	c+	c+	VERB
ejpam-6275	766	3	d−	d−	PROPN
ejpam-6275	766	4	2	2	NUM
ejpam-6275	766	5	.	.	PUNCT
ejpam-6275	767	1	if	if	SCONJ
ejpam-6275	767	2	all	all	DET
ejpam-6275	767	3	si	si	NOUN
ejpam-6275	767	4	=	=	SYM
ejpam-6275	767	5	0	0	NUM
ejpam-6275	767	6	,	,	PUNCT
ejpam-6275	767	7	then	then	ADV
ejpam-6275	767	8	c	c	NOUN
ejpam-6275	767	9	=	=	SYM
ejpam-6275	767	10	r	r	NOUN
ejpam-6275	767	11	,	,	PUNCT
ejpam-6275	767	12	indicating	indicate	VERB
ejpam-6275	767	13	no	no	DET
ejpam-6275	767	14	error	error	NOUN
ejpam-6275	767	15	.	.	PUNCT
ejpam-6275	768	1	otherwise	otherwise	ADV
ejpam-6275	768	2	,	,	PUNCT
ejpam-6275	768	3	proceed	proceed	VERB
ejpam-6275	768	4	to	to	ADP
ejpam-6275	768	5	the	the	DET
ejpam-6275	768	6	next	next	ADJ
ejpam-6275	768	7	step	step	NOUN
ejpam-6275	768	8	.	.	PUNCT
ejpam-6275	769	1	step	step	NOUN
ejpam-6275	769	2	2	2	NUM
ejpam-6275	769	3	:	:	PUNCT
ejpam-6275	769	4	compute	compute	VERB
ejpam-6275	769	5	the	the	DET
ejpam-6275	769	6	error	error	NOUN
ejpam-6275	769	7	locator	locator	NOUN
ejpam-6275	769	8	polynomial	polynomial	NOUN
ejpam-6275	769	9	∆n(y	∆n(y	ADV
ejpam-6275	769	10	)	)	PUNCT
ejpam-6275	769	11	apply	apply	VERB
ejpam-6275	769	12	the	the	DET
ejpam-6275	769	13	modified	modified	ADJ
ejpam-6275	769	14	bma	bma	PROPN
ejpam-6275	769	15	to	to	PART
ejpam-6275	769	16	find	find	VERB
ejpam-6275	769	17	∆n(y	∆n(y	NUM
ejpam-6275	769	18	)	)	PUNCT
ejpam-6275	769	19	,	,	PUNCT
ejpam-6275	769	20	the	the	DET
ejpam-6275	769	21	error	error	NOUN
ejpam-6275	769	22	locator	locator	NOUN
ejpam-6275	769	23	polynomial	polynomial	NOUN
ejpam-6275	769	24	.	.	PUNCT
ejpam-6275	770	1	let	let	VERB
ejpam-6275	770	2	ϑn	ϑn	NOUN
ejpam-6275	770	3	be	be	AUX
ejpam-6275	770	4	the	the	DET
ejpam-6275	770	5	discrepancy	discrepancy	NOUN
ejpam-6275	770	6	,	,	PUNCT
ejpam-6275	770	7	and	and	CCONJ
ejpam-6275	770	8	un	un	PROPN
ejpam-6275	770	9	=	=	PROPN
ejpam-6275	770	10	deg(∆n(y	deg(∆n(y	PROPN
ejpam-6275	770	11	)	)	PUNCT
ejpam-6275	770	12	)	)	PUNCT
ejpam-6275	770	13	.	.	PUNCT
ejpam-6275	771	1	the	the	DET
ejpam-6275	771	2	maximum	maximum	ADJ
ejpam-6275	771	3	number	number	NOUN
ejpam-6275	771	4	of	of	ADP
ejpam-6275	771	5	correctable	correctable	ADJ
ejpam-6275	771	6	errors	error	NOUN
ejpam-6275	771	7	is	be	AUX
ejpam-6275	771	8	t.	t.	NOUN
ejpam-6275	771	9	table	table	NOUN
ejpam-6275	771	10	4	4	NUM
ejpam-6275	771	11	shows	show	VERB
ejpam-6275	771	12	the	the	DET
ejpam-6275	771	13	initialization	initialization	NOUN
ejpam-6275	771	14	values	value	NOUN
ejpam-6275	771	15	.	.	PUNCT
ejpam-6275	772	1	table	table	NOUN
ejpam-6275	772	2	4	4	NUM
ejpam-6275	772	3	:	:	PUNCT
ejpam-6275	772	4	error	error	NOUN
ejpam-6275	772	5	correcting	correcting	NOUN
ejpam-6275	772	6	polynomials	polynomial	NOUN
ejpam-6275	772	7	by	by	ADP
ejpam-6275	772	8	modified	modify	VERB
ejpam-6275	772	9	bma	bma	PROPN
ejpam-6275	772	10	iterations	iteration	NOUN
ejpam-6275	772	11	n	n	CCONJ
ejpam-6275	772	12	∆n(y	∆n(y	PRON
ejpam-6275	772	13	)	)	PUNCT
ejpam-6275	772	14	ϑn	ϑn	PROPN
ejpam-6275	772	15	un	un	PROPN
ejpam-6275	772	16	n−	n−	PROPN
ejpam-6275	772	17	un	un	PROPN
ejpam-6275	772	18	-1	-1	PROPN
ejpam-6275	772	19	1	1	NUM
ejpam-6275	772	20	1	1	NUM
ejpam-6275	772	21	0	0	NUM
ejpam-6275	772	22	-1	-1	SYM
ejpam-6275	772	23	0	0	NUM
ejpam-6275	772	24	1	1	NUM
ejpam-6275	772	25	first	first	ADJ
ejpam-6275	772	26	non	non	ADJ
ejpam-6275	772	27	-	-	ADJ
ejpam-6275	772	28	zero	zero	NUM
ejpam-6275	772	29	syndrome	syndrome	NOUN
ejpam-6275	772	30	0	0	NUM
ejpam-6275	772	31	0	0	NUM
ejpam-6275	772	32	1	1	NUM
ejpam-6275	772	33	...	...	PUNCT
ejpam-6275	772	34	...	...	PUNCT
ejpam-6275	772	35	...	...	PUNCT
ejpam-6275	772	36	...	...	PUNCT
ejpam-6275	773	1	...	...	PUNCT
ejpam-6275	773	2	2	2	NUM
ejpam-6275	773	3	t	t	NOUN
ejpam-6275	773	4	m.	m.	NOUN
ejpam-6275	773	5	sajjad	sajjad	PROPN
ejpam-6275	773	6	et	et	PROPN
ejpam-6275	773	7	al	al	PROPN
ejpam-6275	773	8	.	.	PUNCT
ejpam-6275	773	9	/	/	SYM
ejpam-6275	773	10	eur	eur	PROPN
ejpam-6275	773	11	.	.	PUNCT
ejpam-6275	774	1	j.	j.	PROPN
ejpam-6275	774	2	pure	pure	PROPN
ejpam-6275	774	3	appl	appl	PROPN
ejpam-6275	774	4	.	.	PROPN
ejpam-6275	774	5	math	math	PROPN
ejpam-6275	774	6	,	,	PUNCT
ejpam-6275	774	7	18	18	NUM
ejpam-6275	774	8	(	(	PUNCT
ejpam-6275	774	9	3	3	NUM
ejpam-6275	774	10	)	)	PUNCT
ejpam-6275	774	11	(	(	PUNCT
ejpam-6275	774	12	2025	2025	NUM
ejpam-6275	774	13	)	)	PUNCT
ejpam-6275	774	14	,	,	PUNCT
ejpam-6275	774	15	6275	6275	NUM
ejpam-6275	774	16	28	28	NUM
ejpam-6275	774	17	of	of	ADP
ejpam-6275	774	18	36	36	NUM
ejpam-6275	774	19	case	case	NOUN
ejpam-6275	774	20	1	1	NUM
ejpam-6275	774	21	:	:	PUNCT
ejpam-6275	774	22	if	if	SCONJ
ejpam-6275	774	23	ϑn	ϑn	NOUN
ejpam-6275	774	24	=	=	SYM
ejpam-6275	774	25	0	0	NUM
ejpam-6275	774	26	,	,	PUNCT
ejpam-6275	774	27	then	then	ADV
ejpam-6275	774	28	:	:	PUNCT
ejpam-6275	774	29	∆n+1(y	∆n+1(y	X
ejpam-6275	774	30	)	)	PUNCT
ejpam-6275	774	31	=	=	SYM
ejpam-6275	774	32	∆n(y	∆n(y	ADV
ejpam-6275	774	33	)	)	PUNCT
ejpam-6275	774	34	,	,	PUNCT
ejpam-6275	774	35	un+1	un+1	NOUN
ejpam-6275	774	36	=	=	SYM
ejpam-6275	774	37	un	un	PROPN
ejpam-6275	774	38	.	.	PROPN
ejpam-6275	774	39	case	case	NOUN
ejpam-6275	774	40	2	2	NUM
ejpam-6275	774	41	:	:	PUNCT
ejpam-6275	774	42	if	if	SCONJ
ejpam-6275	774	43	ϑn	ϑn	NOUN
ejpam-6275	774	44	̸=	̸=	PROPN
ejpam-6275	774	45	0	0	NUM
ejpam-6275	774	46	,	,	PUNCT
ejpam-6275	774	47	choose	choose	VERB
ejpam-6275	774	48	m	m	PRON
ejpam-6275	774	49	≤	≤	ADJ
ejpam-6275	774	50	n−	n−	NOUN
ejpam-6275	774	51	1	1	NUM
ejpam-6275	774	52	such	such	ADJ
ejpam-6275	774	53	that	that	PRON
ejpam-6275	774	54	n−	n−	NOUN
ejpam-6275	774	55	um	um	INTJ
ejpam-6275	774	56	is	be	AUX
ejpam-6275	774	57	maximal	maximal	ADJ
ejpam-6275	774	58	or	or	CCONJ
ejpam-6275	774	59	equal	equal	ADJ
ejpam-6275	774	60	to	to	ADP
ejpam-6275	774	61	n−	n−	PROPN
ejpam-6275	774	62	un	un	PROPN
ejpam-6275	774	63	.	.	PUNCT
ejpam-6275	775	1	then	then	ADV
ejpam-6275	775	2	,	,	PUNCT
ejpam-6275	775	3	using	use	VERB
ejpam-6275	775	4	ϑn	ϑn	NOUN
ejpam-6275	775	5	−	−	PROPN
ejpam-6275	775	6	zϑm	zϑm	PROPN
ejpam-6275	775	7	=	=	SYM
ejpam-6275	775	8	0	0	NUM
ejpam-6275	775	9	,	,	PUNCT
ejpam-6275	775	10	solve	solve	VERB
ejpam-6275	775	11	for	for	ADP
ejpam-6275	775	12	z	z	NOUN
ejpam-6275	775	13	and	and	CCONJ
ejpam-6275	775	14	compute	compute	PROPN
ejpam-6275	775	15	:	:	PUNCT
ejpam-6275	775	16	∆n+1(y	∆n+1(y	PROPN
ejpam-6275	775	17	)	)	PUNCT
ejpam-6275	775	18	=	=	SYM
ejpam-6275	775	19	∆n(y)−	∆n(y)−	PROPN
ejpam-6275	775	20	zyn−m∆m(y	zyn−m∆m(y	NOUN
ejpam-6275	775	21	)	)	PUNCT
ejpam-6275	775	22	,	,	PUNCT
ejpam-6275	775	23	and	and	CCONJ
ejpam-6275	775	24	update	update	VERB
ejpam-6275	775	25	the	the	DET
ejpam-6275	775	26	discrepancy	discrepancy	NOUN
ejpam-6275	775	27	as	as	ADP
ejpam-6275	775	28	:	:	PUNCT
ejpam-6275	775	29	ϑn+1	ϑn+1	X
ejpam-6275	775	30	=	=	SYM
ejpam-6275	775	31	sn+2	sn+2	PROPN
ejpam-6275	775	32	+	+	NOUN
ejpam-6275	775	33	∆	∆	PROPN
ejpam-6275	775	34	(	(	PUNCT
ejpam-6275	775	35	n+1	n+1	NOUN
ejpam-6275	775	36	)	)	PUNCT
ejpam-6275	775	37	1	1	NUM
ejpam-6275	775	38	(	(	PUNCT
ejpam-6275	775	39	y)sn+1	y)sn+1	NOUN
ejpam-6275	776	1	+	+	PROPN
ejpam-6275	776	2	∆	∆	PROPN
ejpam-6275	776	3	(	(	PUNCT
ejpam-6275	776	4	n+1	n+1	NOUN
ejpam-6275	776	5	)	)	PUNCT
ejpam-6275	776	6	2	2	NUM
ejpam-6275	776	7	(	(	PUNCT
ejpam-6275	776	8	y)sn	y)sn	PROPN
ejpam-6275	776	9	+	+	PROPN
ejpam-6275	776	10	·	·	PUNCT
ejpam-6275	776	11	·	·	PUNCT
ejpam-6275	776	12	·	·	PUNCT
ejpam-6275	776	13	+	+	NOUN
ejpam-6275	776	14	∆(n+1	∆(n+1	NOUN
ejpam-6275	776	15	)	)	PUNCT
ejpam-6275	776	16	un+1	un+1	NOUN
ejpam-6275	776	17	(	(	PUNCT
ejpam-6275	776	18	y)sn+2−un+1	y)sn+2−un+1	PROPN
ejpam-6275	776	19	.	.	PUNCT
ejpam-6275	777	1	step	step	NOUN
ejpam-6275	777	2	3	3	NUM
ejpam-6275	777	3	:	:	PUNCT
ejpam-6275	777	4	find	find	VERB
ejpam-6275	777	5	the	the	DET
ejpam-6275	777	6	reciprocal	reciprocal	ADJ
ejpam-6275	777	7	function	function	NOUN
ejpam-6275	777	8	from	from	ADP
ejpam-6275	777	9	∆n(y	∆n(y	ADV
ejpam-6275	777	10	)	)	PUNCT
ejpam-6275	777	11	,	,	PUNCT
ejpam-6275	777	12	compute	compute	VERB
ejpam-6275	777	13	its	its	PRON
ejpam-6275	777	14	reciprocal	reciprocal	ADJ
ejpam-6275	777	15	g(y	g(y	NOUN
ejpam-6275	777	16	)	)	PUNCT
ejpam-6275	777	17	.	.	PUNCT
ejpam-6275	778	1	let	let	VERB
ejpam-6275	778	2	yj	yj	PROPN
ejpam-6275	778	3	be	be	AUX
ejpam-6275	778	4	the	the	DET
ejpam-6275	778	5	roots	root	NOUN
ejpam-6275	778	6	of	of	ADP
ejpam-6275	778	7	g(y	g(y	NOUN
ejpam-6275	778	8	)	)	PUNCT
ejpam-6275	778	9	,	,	PUNCT
ejpam-6275	778	10	and	and	CCONJ
ejpam-6275	778	11	suppose	suppose	VERB
ejpam-6275	778	12	xj	xj	PROPN
ejpam-6275	778	13	=	=	PROPN
ejpam-6275	778	14	pj	pj	PROPN
ejpam-6275	778	15	are	be	AUX
ejpam-6275	778	16	error	error	NOUN
ejpam-6275	778	17	positions	position	NOUN
ejpam-6275	778	18	if	if	SCONJ
ejpam-6275	778	19	xj	xj	PROPN
ejpam-6275	778	20	−	−	PROPN
ejpam-6275	778	21	yj	yj	PROPN
ejpam-6275	778	22	=	=	SYM
ejpam-6275	778	23	0	0	PROPN
ejpam-6275	778	24	,	,	PUNCT
ejpam-6275	778	25	for	for	ADP
ejpam-6275	778	26	1	1	NUM
ejpam-6275	778	27	≤	≤	NUM
ejpam-6275	778	28	j	j	PROPN
ejpam-6275	778	29	≤	≤	PROPN
ejpam-6275	778	30	n−	n−	PROPN
ejpam-6275	778	31	1	1	NUM
ejpam-6275	778	32	.	.	PUNCT
ejpam-6275	779	1	step	step	NOUN
ejpam-6275	779	2	4	4	NUM
ejpam-6275	779	3	:	:	PUNCT
ejpam-6275	779	4	symmetric	symmetric	ADJ
ejpam-6275	779	5	function	function	NOUN
ejpam-6275	779	6	determine	determine	VERB
ejpam-6275	779	7	the	the	DET
ejpam-6275	779	8	elementary	elementary	ADJ
ejpam-6275	779	9	symmetric	symmetric	ADJ
ejpam-6275	779	10	function	function	NOUN
ejpam-6275	779	11	of	of	ADP
ejpam-6275	779	12	the	the	DET
ejpam-6275	779	13	error	error	NOUN
ejpam-6275	779	14	locations	location	NOUN
ejpam-6275	779	15	:	:	PUNCT
ejpam-6275	779	16	(	(	PUNCT
ejpam-6275	779	17	y	y	PROPN
ejpam-6275	779	18	−	−	PROPN
ejpam-6275	779	19	x1)(y	x1)(y	PROPN
ejpam-6275	780	1	−	−	PROPN
ejpam-6275	780	2	x2	x2	PROPN
ejpam-6275	780	3	)	)	PUNCT
ejpam-6275	780	4	·	·	PUNCT
ejpam-6275	780	5	·	·	PUNCT
ejpam-6275	780	6	·	·	PUNCT
ejpam-6275	780	7	(	(	PUNCT
ejpam-6275	780	8	y	y	PROPN
ejpam-6275	780	9	−	−	PROPN
ejpam-6275	780	10	xv	xv	PROPN
ejpam-6275	780	11	)	)	PUNCT
ejpam-6275	780	12	=	=	PUNCT
ejpam-6275	780	13	∆0y	∆0y	NOUN
ejpam-6275	780	14	v	v	ADP
ejpam-6275	780	15	+	+	NOUN
ejpam-6275	780	16	∆1y	∆1y	NOUN
ejpam-6275	780	17	v−1	v−1	X
ejpam-6275	780	18	+	+	X
ejpam-6275	780	19	·	·	PUNCT
ejpam-6275	780	20	·	·	PUNCT
ejpam-6275	780	21	·	·	PUNCT
ejpam-6275	781	1	+	+	NUM
ejpam-6275	781	2	∆v	∆v	PROPN
ejpam-6275	781	3	,	,	PUNCT
ejpam-6275	781	4	where	where	SCONJ
ejpam-6275	781	5	v	v	NOUN
ejpam-6275	781	6	is	be	AUX
ejpam-6275	781	7	the	the	DET
ejpam-6275	781	8	total	total	ADJ
ejpam-6275	781	9	number	number	NOUN
ejpam-6275	781	10	of	of	ADP
ejpam-6275	781	11	error	error	NOUN
ejpam-6275	781	12	locations	location	NOUN
ejpam-6275	781	13	.	.	PUNCT
ejpam-6275	782	1	step	step	NOUN
ejpam-6275	782	2	5	5	NUM
ejpam-6275	782	3	:	:	PUNCT
ejpam-6275	782	4	error	error	NOUN
ejpam-6275	782	5	magnitudes	magnitude	NOUN
ejpam-6275	782	6	by	by	ADP
ejpam-6275	782	7	forney	forney	PROPN
ejpam-6275	782	8	’s	’s	PART
ejpam-6275	782	9	formula	formula	NOUN
ejpam-6275	783	1	[	[	X
ejpam-6275	783	2	27	27	NUM
ejpam-6275	783	3	]	]	PUNCT
ejpam-6275	783	4	the	the	DET
ejpam-6275	783	5	magnitude	magnitude	NOUN
ejpam-6275	783	6	zi	zi	NOUN
ejpam-6275	783	7	of	of	ADP
ejpam-6275	783	8	the	the	DET
ejpam-6275	783	9	error	error	NOUN
ejpam-6275	783	10	at	at	ADP
ejpam-6275	783	11	location	location	NOUN
ejpam-6275	783	12	yi	yi	PROPN
ejpam-6275	783	13	is	be	AUX
ejpam-6275	783	14	given	give	VERB
ejpam-6275	783	15	by	by	ADP
ejpam-6275	783	16	:	:	PUNCT
ejpam-6275	783	17	zi	zi	NOUN
ejpam-6275	783	18	=	=	PUNCT
ejpam-6275	783	19	∑v−1	∑v−1	PROPN
ejpam-6275	783	20	l=0	l=0	PROPN
ejpam-6275	783	21	∆(i	∆(i	PROPN
ejpam-6275	783	22	,	,	PUNCT
ejpam-6275	783	23	l)sv−l∑v−1	l)sv−l∑v−1	PROPN
ejpam-6275	783	24	l=0	l=0	PROPN
ejpam-6275	783	25	∆(i	∆(i	PROPN
ejpam-6275	783	26	,	,	PUNCT
ejpam-6275	783	27	l)y	l)y	VERB
ejpam-6275	783	28	v−l	v−l	NOUN
ejpam-6275	783	29	i	i	PRON
ejpam-6275	783	30	,	,	PUNCT
ejpam-6275	783	31	with	with	ADP
ejpam-6275	783	32	initialization	initialization	NOUN
ejpam-6275	783	33	∆0	∆0	NOUN
ejpam-6275	783	34	=	=	SYM
ejpam-6275	783	35	∆(i,0	∆(i,0	PROPN
ejpam-6275	783	36	)	)	PUNCT
ejpam-6275	783	37	=	=	SYM
ejpam-6275	783	38	1	1	NUM
ejpam-6275	783	39	,	,	PUNCT
ejpam-6275	783	40	and	and	CCONJ
ejpam-6275	783	41	recurrence	recurrence	NOUN
ejpam-6275	783	42	:	:	PUNCT
ejpam-6275	783	43	∆(i	∆(i	PROPN
ejpam-6275	783	44	,	,	PUNCT
ejpam-6275	783	45	j	j	PROPN
ejpam-6275	783	46	)	)	PUNCT
ejpam-6275	783	47	=	=	SYM
ejpam-6275	783	48	∆j	∆j	PROPN
ejpam-6275	783	49	+	+	CCONJ
ejpam-6275	783	50	xi∆(i	xi∆(i	PROPN
ejpam-6275	783	51	,	,	PUNCT
ejpam-6275	783	52	j−1	j−1	PROPN
ejpam-6275	783	53	)	)	PUNCT
ejpam-6275	783	54	,	,	PUNCT
ejpam-6275	783	55	j	j	PROPN
ejpam-6275	783	56	=	=	SYM
ejpam-6275	783	57	1	1	NUM
ejpam-6275	783	58	,	,	PUNCT
ejpam-6275	783	59	2	2	NUM
ejpam-6275	783	60	,	,	PUNCT
ejpam-6275	783	61	.	.	PUNCT
ejpam-6275	783	62	.	.	PUNCT
ejpam-6275	784	1	.	.	PUNCT
ejpam-6275	785	1	,	,	PUNCT
ejpam-6275	785	2	v	v	ADP
ejpam-6275	785	3	−	−	PROPN
ejpam-6275	785	4	1	1	NUM
ejpam-6275	785	5	,	,	PUNCT
ejpam-6275	785	6	i	i	PRON
ejpam-6275	785	7	=	=	NOUN
ejpam-6275	785	8	1	1	NUM
ejpam-6275	785	9	,	,	PUNCT
ejpam-6275	785	10	2	2	NUM
ejpam-6275	785	11	,	,	PUNCT
ejpam-6275	785	12	.	.	PUNCT
ejpam-6275	785	13	.	.	PUNCT
ejpam-6275	785	14	.	.	PUNCT
ejpam-6275	786	1	,	,	PUNCT
ejpam-6275	786	2	v.	v.	ADP
ejpam-6275	786	3	step	step	NOUN
ejpam-6275	786	4	6	6	NUM
ejpam-6275	786	5	:	:	PUNCT
ejpam-6275	786	6	recover	recover	VERB
ejpam-6275	786	7	the	the	DET
ejpam-6275	786	8	codeword	codeword	NOUN
ejpam-6275	786	9	finally	finally	ADV
ejpam-6275	786	10	,	,	PUNCT
ejpam-6275	786	11	compute	compute	VERB
ejpam-6275	786	12	the	the	DET
ejpam-6275	786	13	corrected	correct	VERB
ejpam-6275	786	14	codeword	codeword	NOUN
ejpam-6275	786	15	:	:	PUNCT
ejpam-6275	786	16	c	c	NOUN
ejpam-6275	786	17	=	=	SYM
ejpam-6275	786	18	r−	r−	PROPN
ejpam-6275	786	19	e	e	NOUN
ejpam-6275	786	20	,	,	PUNCT
ejpam-6275	786	21	where	where	SCONJ
ejpam-6275	786	22	e	e	NOUN
ejpam-6275	786	23	is	be	AUX
ejpam-6275	786	24	the	the	DET
ejpam-6275	786	25	error	error	NOUN
ejpam-6275	786	26	vector	vector	NOUN
ejpam-6275	786	27	.	.	PUNCT
ejpam-6275	787	1	the	the	DET
ejpam-6275	787	2	pseudocode	pseudocode	NOUN
ejpam-6275	787	3	of	of	ADP
ejpam-6275	787	4	this	this	DET
ejpam-6275	787	5	decoding	decode	VERB
ejpam-6275	787	6	process	process	NOUN
ejpam-6275	787	7	is	be	AUX
ejpam-6275	787	8	given	give	VERB
ejpam-6275	787	9	in	in	ADP
ejpam-6275	787	10	algorithm	algorithm	NOUN
ejpam-6275	787	11	4.1	4.1	NUM
ejpam-6275	787	12	.	.	PUNCT
ejpam-6275	788	1	illustration	illustration	NOUN
ejpam-6275	788	2	4.1	4.1	NUM
ejpam-6275	788	3	:	:	PUNCT
ejpam-6275	788	4	assume	assume	VERB
ejpam-6275	788	5	that	that	SCONJ
ejpam-6275	788	6	[	[	X
ejpam-6275	788	7	3	3	NUM
ejpam-6275	788	8	,	,	PUNCT
ejpam-6275	788	9	1	1	NUM
ejpam-6275	788	10	,	,	PUNCT
ejpam-6275	788	11	3	3	NUM
ejpam-6275	788	12	]	]	ADJ
ejpam-6275	788	13	narrow	narrow	ADJ
ejpam-6275	788	14	-	-	PUNCT
ejpam-6275	788	15	sense	sense	NOUN
ejpam-6275	788	16	shortened	shorten	VERB
ejpam-6275	788	17	bch	bch	PROPN
ejpam-6275	788	18	code	code	NOUN
ejpam-6275	788	19	over	over	ADP
ejpam-6275	788	20	the	the	DET
ejpam-6275	788	21	eisenstein	eisenstein	PROPN
ejpam-6275	788	22	field	field	PROPN
ejpam-6275	788	23	z2[ω	z2[ω	NOUN
ejpam-6275	788	24	]	]	X
ejpam-6275	788	25	2	2	NUM
ejpam-6275	788	26	and	and	CCONJ
ejpam-6275	788	27	the	the	DET
ejpam-6275	788	28	received	receive	VERB
ejpam-6275	788	29	vector	vector	NOUN
ejpam-6275	788	30	r	r	NOUN
ejpam-6275	788	31	=	=	PUNCT
ejpam-6275	788	32	(	(	PUNCT
ejpam-6275	788	33	1	1	NUM
ejpam-6275	788	34	,	,	PUNCT
ejpam-6275	788	35	1	1	NUM
ejpam-6275	788	36	,	,	PUNCT
ejpam-6275	788	37	ω	ω	NUM
ejpam-6275	788	38	)	)	PUNCT
ejpam-6275	788	39	.	.	PUNCT
ejpam-6275	789	1	determine	determine	VERB
ejpam-6275	789	2	the	the	DET
ejpam-6275	789	3	corrected	correct	VERB
ejpam-6275	789	4	codeword	codeword	NOUN
ejpam-6275	789	5	(	(	PUNCT
ejpam-6275	789	6	if	if	SCONJ
ejpam-6275	789	7	possible	possible	ADJ
ejpam-6275	789	8	)	)	PUNCT
ejpam-6275	789	9	.	.	PUNCT
ejpam-6275	790	1	m.	m.	PROPN
ejpam-6275	790	2	sajjad	sajjad	PROPN
ejpam-6275	790	3	et	et	PROPN
ejpam-6275	790	4	al	al	PROPN
ejpam-6275	790	5	.	.	PUNCT
ejpam-6275	790	6	/	/	SYM
ejpam-6275	790	7	eur	eur	PROPN
ejpam-6275	790	8	.	.	PUNCT
ejpam-6275	791	1	j.	j.	PROPN
ejpam-6275	791	2	pure	pure	PROPN
ejpam-6275	791	3	appl	appl	PROPN
ejpam-6275	791	4	.	.	PROPN
ejpam-6275	791	5	math	math	PROPN
ejpam-6275	791	6	,	,	PUNCT
ejpam-6275	791	7	18	18	NUM
ejpam-6275	791	8	(	(	PUNCT
ejpam-6275	791	9	3	3	NUM
ejpam-6275	791	10	)	)	PUNCT
ejpam-6275	791	11	(	(	PUNCT
ejpam-6275	791	12	2025	2025	NUM
ejpam-6275	791	13	)	)	PUNCT
ejpam-6275	791	14	,	,	PUNCT
ejpam-6275	791	15	6275	6275	NUM
ejpam-6275	791	16	29	29	NUM
ejpam-6275	791	17	of	of	ADP
ejpam-6275	791	18	36	36	NUM
ejpam-6275	791	19	as	as	ADP
ejpam-6275	791	20	t	t	NOUN
ejpam-6275	791	21	=	=	PUNCT
ejpam-6275	791	22	⌊	⌊	PROPN
ejpam-6275	791	23	d−1	d−1	PROPN
ejpam-6275	791	24	2	2	NUM
ejpam-6275	791	25	⌋	⌋	NOUN
ejpam-6275	791	26	=	=	PUNCT
ejpam-6275	792	1	⌊	⌊	VERB
ejpam-6275	792	2	3−1	3−1	NUM
ejpam-6275	792	3	2	2	NUM
ejpam-6275	792	4	⌋	⌋	NOUN
ejpam-6275	792	5	=	=	SYM
ejpam-6275	792	6	1	1	NUM
ejpam-6275	792	7	,	,	PUNCT
ejpam-6275	792	8	the	the	DET
ejpam-6275	792	9	number	number	NOUN
ejpam-6275	792	10	of	of	ADP
ejpam-6275	792	11	iterations	iteration	NOUN
ejpam-6275	792	12	will	will	AUX
ejpam-6275	792	13	be	be	AUX
ejpam-6275	792	14	2	2	NUM
ejpam-6275	792	15	t	t	NOUN
ejpam-6275	792	16	=	=	SYM
ejpam-6275	792	17	2	2	X
ejpam-6275	792	18	.	.	PUNCT
ejpam-6275	793	1	let	let	VERB
ejpam-6275	793	2	s	s	PRON
ejpam-6275	793	3	=	=	NOUN
ejpam-6275	793	4	rht	rht	X
ejpam-6275	793	5	=	=	SYM
ejpam-6275	793	6	(	(	PUNCT
ejpam-6275	793	7	1	1	NUM
ejpam-6275	793	8	,	,	PUNCT
ejpam-6275	793	9	1	1	NUM
ejpam-6275	793	10	,	,	PUNCT
ejpam-6275	793	11	ω	ω	NOUN
ejpam-6275	793	12	)	)	PUNCT
ejpam-6275	793	13			PROPN
ejpam-6275	793	14	1	1	NUM
ejpam-6275	793	15	1	1	NUM
ejpam-6275	793	16	β	β	NOUN
ejpam-6275	793	17	β2	β2	NOUN
ejpam-6275	793	18	β2	β2	PROPN
ejpam-6275	793	19	β4	β4	PROPN
ejpam-6275	793	20	t	t	PUNCT
ejpam-6275	794	1	=	=	PRON
ejpam-6275	794	2	(	(	PUNCT
ejpam-6275	794	3	1	1	NUM
ejpam-6275	794	4	ω	ω	NUM
ejpam-6275	794	5	)	)	PUNCT
ejpam-6275	794	6	,	,	PUNCT
ejpam-6275	794	7	where	where	SCONJ
ejpam-6275	794	8	s1	s1	NOUN
ejpam-6275	794	9	=	=	SYM
ejpam-6275	794	10	β3	β3	VERB
ejpam-6275	794	11	and	and	CCONJ
ejpam-6275	794	12	s2	s2	VERB
ejpam-6275	794	13	=	=	PUNCT
ejpam-6275	794	14	β	β	NOUN
ejpam-6275	794	15	are	be	AUX
ejpam-6275	794	16	the	the	DET
ejpam-6275	794	17	syndromes	syndrome	NOUN
ejpam-6275	794	18	.	.	PUNCT
ejpam-6275	795	1	next	next	ADV
ejpam-6275	795	2	,	,	PUNCT
ejpam-6275	795	3	we	we	PRON
ejpam-6275	795	4	find	find	VERB
ejpam-6275	795	5	∆2(y	∆2(y	NOUN
ejpam-6275	795	6	)	)	PUNCT
ejpam-6275	795	7	using	use	VERB
ejpam-6275	795	8	the	the	DET
ejpam-6275	795	9	modified	modify	VERB
ejpam-6275	795	10	berlekamp	berlekamp	NOUN
ejpam-6275	795	11	–	–	PUNCT
ejpam-6275	795	12	massey	massey	NOUN
ejpam-6275	795	13	algorithm	algorithm	NOUN
ejpam-6275	795	14	(	(	PUNCT
ejpam-6275	795	15	bma	bma	PROPN
ejpam-6275	795	16	)	)	PUNCT
ejpam-6275	795	17	,	,	PUNCT
ejpam-6275	795	18	summarized	summarize	VERB
ejpam-6275	795	19	in	in	ADP
ejpam-6275	795	20	table	table	NOUN
ejpam-6275	795	21	5	5	NUM
ejpam-6275	795	22	.	.	PUNCT
ejpam-6275	795	23	iteration	iteration	NOUN
ejpam-6275	795	24	1	1	NUM
ejpam-6275	795	25	:	:	PUNCT
ejpam-6275	795	26	the	the	DET
ejpam-6275	795	27	initial	initial	ADJ
ejpam-6275	795	28	non	non	ADJ
ejpam-6275	795	29	-	-	ADJ
ejpam-6275	795	30	zero	zero	NUM
ejpam-6275	795	31	syndrome	syndrome	NOUN
ejpam-6275	795	32	is	be	AUX
ejpam-6275	795	33	1	1	NUM
ejpam-6275	795	34	.	.	PUNCT
ejpam-6275	795	35	applying	apply	VERB
ejpam-6275	795	36	case	case	NOUN
ejpam-6275	795	37	2	2	NUM
ejpam-6275	795	38	of	of	ADP
ejpam-6275	795	39	step	step	NOUN
ejpam-6275	795	40	2	2	NUM
ejpam-6275	795	41	from	from	ADP
ejpam-6275	795	42	the	the	DET
ejpam-6275	795	43	modified	modified	PROPN
ejpam-6275	795	44	bma	bma	PROPN
ejpam-6275	795	45	,	,	PUNCT
ejpam-6275	795	46	since	since	SCONJ
ejpam-6275	795	47	ϑ0	ϑ0	PROPN
ejpam-6275	795	48	̸=	̸=	PROPN
ejpam-6275	795	49	0	0	NUM
ejpam-6275	795	50	,	,	PUNCT
ejpam-6275	795	51	−1	−1	NOUN
ejpam-6275	795	52	≤	≤	NOUN
ejpam-6275	795	53	−1	−1	NOUN
ejpam-6275	795	54	,	,	PUNCT
ejpam-6275	795	55	and	and	CCONJ
ejpam-6275	795	56	0−	0−	NUM
ejpam-6275	795	57	u−1	u−1	PROPN
ejpam-6275	795	58	=	=	SYM
ejpam-6275	795	59	0	0	NUM
ejpam-6275	795	60	is	be	AUX
ejpam-6275	795	61	the	the	DET
ejpam-6275	795	62	maximum	maximum	ADJ
ejpam-6275	795	63	value	value	NOUN
ejpam-6275	795	64	of	of	ADP
ejpam-6275	795	65	the	the	DET
ejpam-6275	795	66	last	last	ADJ
ejpam-6275	795	67	column	column	NOUN
ejpam-6275	795	68	,	,	PUNCT
ejpam-6275	795	69	we	we	PRON
ejpam-6275	795	70	get	get	VERB
ejpam-6275	795	71	ϑ0	ϑ0	NOUN
ejpam-6275	795	72	−	−	PROPN
ejpam-6275	795	73	zϑ−1	zϑ−1	NOUN
ejpam-6275	795	74	=	=	SYM
ejpam-6275	795	75	0	0	NUM
ejpam-6275	796	1	⇒	⇒	NOUN
ejpam-6275	797	1	z	z	NOUN
ejpam-6275	798	1	=	=	PUNCT
ejpam-6275	799	1	ϑ0	ϑ0	PROPN
ejpam-6275	799	2	ϑ−1	ϑ−1	NOUN
ejpam-6275	799	3	=	=	NOUN
ejpam-6275	799	4	1	1	NUM
ejpam-6275	799	5	1	1	NUM
ejpam-6275	799	6	=	=	SYM
ejpam-6275	799	7	1	1	X
ejpam-6275	799	8	.	.	PUNCT
ejpam-6275	799	9	m.	m.	PROPN
ejpam-6275	799	10	sajjad	sajjad	PROPN
ejpam-6275	799	11	et	et	PROPN
ejpam-6275	799	12	al	al	PROPN
ejpam-6275	799	13	.	.	PUNCT
ejpam-6275	799	14	/	/	SYM
ejpam-6275	799	15	eur	eur	PROPN
ejpam-6275	799	16	.	.	PUNCT
ejpam-6275	800	1	j.	j.	PROPN
ejpam-6275	800	2	pure	pure	PROPN
ejpam-6275	800	3	appl	appl	PROPN
ejpam-6275	800	4	.	.	PROPN
ejpam-6275	800	5	math	math	PROPN
ejpam-6275	800	6	,	,	PUNCT
ejpam-6275	800	7	18	18	NUM
ejpam-6275	800	8	(	(	PUNCT
ejpam-6275	800	9	3	3	NUM
ejpam-6275	800	10	)	)	PUNCT
ejpam-6275	800	11	(	(	PUNCT
ejpam-6275	800	12	2025	2025	NUM
ejpam-6275	800	13	)	)	PUNCT
ejpam-6275	800	14	,	,	PUNCT
ejpam-6275	800	15	6275	6275	NUM
ejpam-6275	800	16	30	30	NUM
ejpam-6275	800	17	of	of	ADP
ejpam-6275	800	18	36	36	NUM
ejpam-6275	800	19	table	table	NOUN
ejpam-6275	800	20	5	5	NUM
ejpam-6275	800	21	:	:	PUNCT
ejpam-6275	800	22	single	single	ADJ
ejpam-6275	800	23	error	error	NOUN
ejpam-6275	800	24	locating	locating	NOUN
ejpam-6275	800	25	polynomials	polynomial	NOUN
ejpam-6275	800	26	n	n	PRON
ejpam-6275	800	27	∆n(y	∆n(y	PRON
ejpam-6275	800	28	)	)	PUNCT
ejpam-6275	800	29	ϑn	ϑn	PROPN
ejpam-6275	800	30	un	un	PROPN
ejpam-6275	801	1	n−	n−	PROPN
ejpam-6275	801	2	un	un	PROPN
ejpam-6275	801	3	−1	−1	NOUN
ejpam-6275	801	4	1	1	NUM
ejpam-6275	801	5	1	1	NUM
ejpam-6275	801	6	0	0	NUM
ejpam-6275	801	7	−1	−1	NOUN
ejpam-6275	801	8	0	0	NUM
ejpam-6275	801	9	1	1	NUM
ejpam-6275	801	10	1	1	NUM
ejpam-6275	801	11	0	0	NUM
ejpam-6275	801	12	0	0	NUM
ejpam-6275	801	13	1	1	NUM
ejpam-6275	801	14	1	1	NUM
ejpam-6275	801	15	+	+	NUM
ejpam-6275	801	16	y	y	PROPN
ejpam-6275	801	17	1	1	NUM
ejpam-6275	801	18	+	+	NUM
ejpam-6275	801	19	ω	ω	NUM
ejpam-6275	801	20	1	1	NUM
ejpam-6275	801	21	0	0	NUM
ejpam-6275	801	22	2	2	NUM
ejpam-6275	801	23	1	1	NUM
ejpam-6275	801	24	+	+	CCONJ
ejpam-6275	801	25	ωy	ωy	PROPN
ejpam-6275	801	26	thus	thus	ADV
ejpam-6275	801	27	,	,	PUNCT
ejpam-6275	801	28	∆1(y	∆1(y	PROPN
ejpam-6275	801	29	)	)	PUNCT
ejpam-6275	801	30	=	=	SYM
ejpam-6275	801	31	∆0(y)−	∆0(y)−	PROPN
ejpam-6275	801	32	(	(	PUNCT
ejpam-6275	801	33	1)y0	1)y0	NUM
ejpam-6275	801	34	+	+	NOUN
ejpam-6275	801	35	1∆−1(y	1∆−1(y	NUM
ejpam-6275	801	36	)	)	PUNCT
ejpam-6275	801	37	=	=	SYM
ejpam-6275	801	38	1	1	NUM
ejpam-6275	802	1	+	+	CCONJ
ejpam-6275	802	2	y.	y.	NOUN
ejpam-6275	802	3	now	now	ADV
ejpam-6275	802	4	compute	compute	VERB
ejpam-6275	802	5	ϑ1	ϑ1	NOUN
ejpam-6275	802	6	=	=	NOUN
ejpam-6275	802	7	s2	s2	PROPN
ejpam-6275	802	8	+	+	NOUN
ejpam-6275	802	9	∆	∆	X
ejpam-6275	802	10	(	(	PUNCT
ejpam-6275	802	11	1	1	NUM
ejpam-6275	802	12	)	)	SYM
ejpam-6275	802	13	1	1	NUM
ejpam-6275	802	14	s1	s1	NOUN
ejpam-6275	802	15	=	=	SYM
ejpam-6275	802	16	ω	ω	PROPN
ejpam-6275	802	17	+	+	CCONJ
ejpam-6275	802	18	(	(	PUNCT
ejpam-6275	802	19	1)(1	1)(1	NUM
ejpam-6275	802	20	)	)	PUNCT
ejpam-6275	802	21	=	=	SYM
ejpam-6275	803	1	1	1	NUM
ejpam-6275	803	2	+	+	NUM
ejpam-6275	803	3	ω	ω	X
ejpam-6275	803	4	.	.	PUNCT
ejpam-6275	803	5	iteration	iteration	NOUN
ejpam-6275	803	6	2	2	NUM
ejpam-6275	803	7	:	:	PUNCT
ejpam-6275	803	8	since	since	SCONJ
ejpam-6275	803	9	ϑ1	ϑ1	PROPN
ejpam-6275	803	10	̸=	̸=	PROPN
ejpam-6275	803	11	0	0	NUM
ejpam-6275	803	12	,	,	PUNCT
ejpam-6275	803	13	and	and	CCONJ
ejpam-6275	803	14	1	1	NUM
ejpam-6275	803	15	−	−	NOUN
ejpam-6275	803	16	u0	u0	ADJ
ejpam-6275	803	17	=	=	NOUN
ejpam-6275	803	18	1	1	NUM
ejpam-6275	803	19	−	−	NOUN
ejpam-6275	803	20	0	0	NUM
ejpam-6275	804	1	=	=	SYM
ejpam-6275	804	2	1	1	NUM
ejpam-6275	804	3	is	be	AUX
ejpam-6275	804	4	the	the	DET
ejpam-6275	804	5	highest	high	ADJ
ejpam-6275	804	6	value	value	NOUN
ejpam-6275	804	7	of	of	ADP
ejpam-6275	804	8	the	the	DET
ejpam-6275	804	9	last	last	ADJ
ejpam-6275	804	10	column	column	NOUN
ejpam-6275	804	11	,	,	PUNCT
ejpam-6275	804	12	we	we	PRON
ejpam-6275	804	13	have	have	VERB
ejpam-6275	804	14	:	:	PUNCT
ejpam-6275	804	15	ϑ1	ϑ1	PROPN
ejpam-6275	804	16	−	−	PROPN
ejpam-6275	804	17	zϑ0	zϑ0	X
ejpam-6275	804	18	=	=	SYM
ejpam-6275	804	19	0	0	NUM
ejpam-6275	804	20	⇒	⇒	NOUN
ejpam-6275	804	21	z	z	NOUN
ejpam-6275	805	1	=	=	SYM
ejpam-6275	805	2	ϑ1	ϑ1	NOUN
ejpam-6275	805	3	ϑ0	ϑ0	NOUN
ejpam-6275	805	4	=	=	SYM
ejpam-6275	805	5	1	1	NUM
ejpam-6275	805	6	+	+	NUM
ejpam-6275	805	7	ω	ω	NUM
ejpam-6275	805	8	1	1	NUM
ejpam-6275	805	9	=	=	SYM
ejpam-6275	805	10	1	1	NUM
ejpam-6275	805	11	+	+	NUM
ejpam-6275	805	12	ω	ω	X
ejpam-6275	805	13	.	.	PUNCT
ejpam-6275	806	1	hence	hence	ADV
ejpam-6275	806	2	,	,	PUNCT
ejpam-6275	806	3	∆2(y	∆2(y	PROPN
ejpam-6275	806	4	)	)	PUNCT
ejpam-6275	806	5	=	=	PUNCT
ejpam-6275	807	1	∆1(y)−	∆1(y)−	X
ejpam-6275	807	2	(	(	PUNCT
ejpam-6275	807	3	1	1	NUM
ejpam-6275	807	4	+	+	NUM
ejpam-6275	807	5	ω)y1−0∆0(y	ω)y1−0∆0(y	NOUN
ejpam-6275	807	6	)	)	PUNCT
ejpam-6275	807	7	=	=	SYM
ejpam-6275	807	8	1	1	NUM
ejpam-6275	807	9	+	+	NUM
ejpam-6275	807	10	ωy	ωy	PROPN
ejpam-6275	807	11	.	.	PUNCT
ejpam-6275	808	1	the	the	DET
ejpam-6275	808	2	reciprocal	reciprocal	ADJ
ejpam-6275	808	3	polynomial	polynomial	NOUN
ejpam-6275	808	4	is	be	AUX
ejpam-6275	808	5	g(y	g(y	NOUN
ejpam-6275	808	6	)	)	PUNCT
ejpam-6275	809	1	=	=	SYM
ejpam-6275	809	2	y	y	PROPN
ejpam-6275	810	1	+	+	PROPN
ejpam-6275	810	2	ω	ω	PROPN
ejpam-6275	810	3	.	.	PUNCT
ejpam-6275	811	1	the	the	DET
ejpam-6275	811	2	root	root	NOUN
ejpam-6275	811	3	of	of	ADP
ejpam-6275	811	4	g(y	g(y	NOUN
ejpam-6275	811	5	)	)	PUNCT
ejpam-6275	811	6	is	be	AUX
ejpam-6275	811	7	ω	ω	NUM
ejpam-6275	811	8	,	,	PUNCT
ejpam-6275	811	9	i.e.	i.e.	X
ejpam-6275	811	10	,	,	PUNCT
ejpam-6275	811	11	y1	y1	INTJ
ejpam-6275	811	12	=	=	SYM
ejpam-6275	811	13	ω	ω	PROPN
ejpam-6275	812	1	=	=	SYM
ejpam-6275	812	2	β2	β2	PROPN
ejpam-6275	812	3	,	,	PUNCT
ejpam-6275	812	4	indicating	indicate	VERB
ejpam-6275	812	5	that	that	SCONJ
ejpam-6275	812	6	the	the	DET
ejpam-6275	812	7	error	error	NOUN
ejpam-6275	812	8	occurred	occur	VERB
ejpam-6275	812	9	at	at	ADP
ejpam-6275	812	10	the	the	DET
ejpam-6275	812	11	second	second	ADJ
ejpam-6275	812	12	position	position	NOUN
ejpam-6275	812	13	of	of	ADP
ejpam-6275	812	14	r.	r.	PROPN
ejpam-6275	812	15	the	the	DET
ejpam-6275	812	16	elementary	elementary	ADJ
ejpam-6275	812	17	symmetric	symmetric	ADJ
ejpam-6275	812	18	function	function	NOUN
ejpam-6275	812	19	(	(	PUNCT
ejpam-6275	812	20	esf	esf	PROPN
ejpam-6275	812	21	)	)	PUNCT
ejpam-6275	812	22	is	be	AUX
ejpam-6275	812	23	given	give	VERB
ejpam-6275	812	24	by	by	ADP
ejpam-6275	812	25	:	:	PUNCT
ejpam-6275	812	26	∆0y	∆0y	PROPN
ejpam-6275	812	27	v	v	ADP
ejpam-6275	812	28	+	+	NOUN
ejpam-6275	812	29	∆1	∆1	NOUN
ejpam-6275	812	30	=	=	SYM
ejpam-6275	812	31	y	y	PROPN
ejpam-6275	812	32	−	−	PROPN
ejpam-6275	812	33	β2	β2	PROPN
ejpam-6275	812	34	.	.	PUNCT
ejpam-6275	813	1	error	error	NOUN
ejpam-6275	813	2	magnitude	magnitude	NOUN
ejpam-6275	813	3	:	:	PUNCT
ejpam-6275	813	4	z1	z1	PROPN
ejpam-6275	813	5	=	=	SYM
ejpam-6275	813	6	∆1,0s1	∆1,0s1	NOUN
ejpam-6275	813	7	∆1,0y1	∆1,0y1	NOUN
ejpam-6275	813	8	=	=	SYM
ejpam-6275	813	9	β3	β3	ADJ
ejpam-6275	813	10	β2	β2	NOUN
ejpam-6275	813	11	=	=	SYM
ejpam-6275	813	12	β	β	NOUN
ejpam-6275	813	13	=	=	SYM
ejpam-6275	813	14	1	1	NUM
ejpam-6275	813	15	+	+	NUM
ejpam-6275	813	16	ω	ω	X
ejpam-6275	813	17	.	.	PUNCT
ejpam-6275	814	1	where	where	SCONJ
ejpam-6275	814	2	∆0	∆0	ADV
ejpam-6275	814	3	=	=	SYM
ejpam-6275	814	4	1	1	NUM
ejpam-6275	814	5	,	,	PUNCT
ejpam-6275	814	6	∆1	∆1	PROPN
ejpam-6275	814	7	=	=	SYM
ejpam-6275	814	8	β2	β2	VERB
ejpam-6275	814	9	,	,	PUNCT
ejpam-6275	814	10	and	and	CCONJ
ejpam-6275	814	11	v	v	NOUN
ejpam-6275	814	12	=	=	SYM
ejpam-6275	814	13	1	1	X
ejpam-6275	814	14	.	.	NUM
ejpam-6275	814	15	corrected	correct	VERB
ejpam-6275	814	16	codeword	codeword	NOUN
ejpam-6275	814	17	:	:	PUNCT
ejpam-6275	815	1	c	c	NOUN
ejpam-6275	815	2	=	=	SYM
ejpam-6275	816	1	r	r	NOUN
ejpam-6275	816	2	−	−	NOUN
ejpam-6275	816	3	e	e	NOUN
ejpam-6275	816	4	=	=	PUNCT
ejpam-6275	816	5	(	(	PUNCT
ejpam-6275	816	6	1	1	NUM
ejpam-6275	816	7	,	,	PUNCT
ejpam-6275	816	8	1	1	NUM
ejpam-6275	816	9	,	,	PUNCT
ejpam-6275	816	10	ω)−	ω)−	PROPN
ejpam-6275	816	11	(	(	PUNCT
ejpam-6275	816	12	0	0	NUM
ejpam-6275	816	13	,	,	PUNCT
ejpam-6275	816	14	1	1	NUM
ejpam-6275	816	15	+	+	NUM
ejpam-6275	816	16	ω	ω	NUM
ejpam-6275	816	17	,	,	PUNCT
ejpam-6275	816	18	0	0	NUM
ejpam-6275	816	19	)	)	PUNCT
ejpam-6275	816	20	=	=	NOUN
ejpam-6275	816	21	(	(	PUNCT
ejpam-6275	816	22	1	1	NUM
ejpam-6275	816	23	,	,	PUNCT
ejpam-6275	816	24	1	1	NUM
ejpam-6275	816	25	,	,	PUNCT
ejpam-6275	816	26	1	1	NUM
ejpam-6275	816	27	)	)	PUNCT
ejpam-6275	816	28	.	.	PUNCT
ejpam-6275	817	1	thus	thus	ADV
ejpam-6275	817	2	,	,	PUNCT
ejpam-6275	817	3	the	the	DET
ejpam-6275	817	4	corrected	correct	VERB
ejpam-6275	817	5	codeword	codeword	NOUN
ejpam-6275	817	6	is	be	AUX
ejpam-6275	817	7	c	c	NOUN
ejpam-6275	817	8	=	=	SYM
ejpam-6275	817	9	(	(	PUNCT
ejpam-6275	817	10	1	1	NUM
ejpam-6275	817	11	,	,	PUNCT
ejpam-6275	817	12	1	1	NUM
ejpam-6275	817	13	,	,	PUNCT
ejpam-6275	817	14	1	1	NUM
ejpam-6275	817	15	)	)	PUNCT
ejpam-6275	817	16	,	,	PUNCT
ejpam-6275	817	17	which	which	PRON
ejpam-6275	817	18	is	be	AUX
ejpam-6275	817	19	a	a	DET
ejpam-6275	817	20	valid	valid	ADJ
ejpam-6275	817	21	codeword	codeword	NOUN
ejpam-6275	817	22	of	of	ADP
ejpam-6275	817	23	the	the	DET
ejpam-6275	817	24	(	(	PUNCT
ejpam-6275	817	25	3	3	NUM
ejpam-6275	817	26	,	,	PUNCT
ejpam-6275	817	27	1	1	NUM
ejpam-6275	817	28	,	,	PUNCT
ejpam-6275	817	29	3	3	X
ejpam-6275	817	30	)	)	PUNCT
ejpam-6275	817	31	bch	bch	PROPN
ejpam-6275	817	32	code	code	NOUN
ejpam-6275	817	33	.	.	PUNCT
ejpam-6275	818	1	illustration	illustration	NOUN
ejpam-6275	818	2	4.2	4.2	NUM
ejpam-6275	818	3	:	:	PUNCT
ejpam-6275	818	4	assume	assume	VERB
ejpam-6275	818	5	that	that	SCONJ
ejpam-6275	818	6	[	[	X
ejpam-6275	818	7	9	9	NUM
ejpam-6275	818	8	,	,	PUNCT
ejpam-6275	818	9	3	3	NUM
ejpam-6275	818	10	,	,	PUNCT
ejpam-6275	818	11	3	3	NUM
ejpam-6275	818	12	]	]	X
ejpam-6275	818	13	narrow	narrow	ADJ
ejpam-6275	818	14	sense	sense	NOUN
ejpam-6275	818	15	shortened	shorten	VERB
ejpam-6275	818	16	bch	bch	PROPN
ejpam-6275	818	17	code	code	NOUN
ejpam-6275	818	18	over	over	ADP
ejpam-6275	818	19	the	the	DET
ejpam-6275	818	20	eisenstein	eisenstein	PROPN
ejpam-6275	818	21	field	field	PROPN
ejpam-6275	818	22	z2[ω	z2[ω	NOUN
ejpam-6275	818	23	]	]	X
ejpam-6275	818	24	2	2	NUM
ejpam-6275	818	25	and	and	CCONJ
ejpam-6275	818	26	the	the	DET
ejpam-6275	818	27	received	receive	VERB
ejpam-6275	818	28	vector	vector	NOUN
ejpam-6275	818	29	r	r	NOUN
ejpam-6275	818	30	=	=	PUNCT
ejpam-6275	818	31	(	(	PUNCT
ejpam-6275	818	32	1	1	NUM
ejpam-6275	818	33	,	,	PUNCT
ejpam-6275	818	34	0	0	NUM
ejpam-6275	818	35	,	,	PUNCT
ejpam-6275	818	36	ω	ω	NOUN
ejpam-6275	818	37	,	,	PUNCT
ejpam-6275	818	38	1	1	NUM
ejpam-6275	818	39	,	,	PUNCT
ejpam-6275	818	40	0	0	NUM
ejpam-6275	818	41	,	,	PUNCT
ejpam-6275	818	42	0	0	NUM
ejpam-6275	818	43	,	,	PUNCT
ejpam-6275	818	44	1	1	NUM
ejpam-6275	818	45	,	,	PUNCT
ejpam-6275	818	46	0	0	NUM
ejpam-6275	818	47	,	,	PUNCT
ejpam-6275	818	48	0	0	NUM
ejpam-6275	818	49	)	)	PUNCT
ejpam-6275	818	50	,	,	PUNCT
ejpam-6275	818	51	determine	determine	VERB
ejpam-6275	818	52	the	the	DET
ejpam-6275	818	53	corrected	correct	VERB
ejpam-6275	818	54	code	code	NOUN
ejpam-6275	818	55	word	word	NOUN
ejpam-6275	818	56	(	(	PUNCT
ejpam-6275	818	57	if	if	SCONJ
ejpam-6275	818	58	possible	possible	ADJ
ejpam-6275	818	59	)	)	PUNCT
ejpam-6275	818	60	.	.	PUNCT
ejpam-6275	819	1	as	as	ADP
ejpam-6275	819	2	t	t	PROPN
ejpam-6275	819	3	=	=	PUNCT
ejpam-6275	819	4	⌊	⌊	PROPN
ejpam-6275	819	5	d−1	d−1	PROPN
ejpam-6275	819	6	2	2	NUM
ejpam-6275	819	7	⌋	⌋	NOUN
ejpam-6275	819	8	=	=	PUNCT
ejpam-6275	819	9	⌊	⌊	VERB
ejpam-6275	819	10	3−1	3−1	NUM
ejpam-6275	819	11	2	2	NUM
ejpam-6275	819	12	⌋	⌋	NOUN
ejpam-6275	819	13	=	=	PUNCT
ejpam-6275	820	1	1	1	X
ejpam-6275	820	2	.	.	PUNCT
ejpam-6275	821	1	so	so	ADV
ejpam-6275	821	2	,	,	PUNCT
ejpam-6275	821	3	the	the	DET
ejpam-6275	821	4	number	number	NOUN
ejpam-6275	821	5	of	of	ADP
ejpam-6275	821	6	iterations	iteration	NOUN
ejpam-6275	821	7	will	will	AUX
ejpam-6275	821	8	also	also	ADV
ejpam-6275	821	9	be	be	AUX
ejpam-6275	821	10	2	2	NUM
ejpam-6275	821	11	t	t	NOUN
ejpam-6275	821	12	=	=	SYM
ejpam-6275	821	13	2	2	X
ejpam-6275	821	14	.	.	PUNCT
ejpam-6275	822	1	let	let	VERB
ejpam-6275	822	2	s	s	PRON
ejpam-6275	822	3	=	=	NOUN
ejpam-6275	822	4	rht	rht	X
ejpam-6275	822	5	=	=	SYM
ejpam-6275	822	6	(	(	PUNCT
ejpam-6275	822	7	1	1	NUM
ejpam-6275	822	8	,	,	PUNCT
ejpam-6275	822	9	0	0	NUM
ejpam-6275	822	10	,	,	PUNCT
ejpam-6275	822	11	ω	ω	NOUN
ejpam-6275	822	12	,	,	PUNCT
ejpam-6275	822	13	1	1	NUM
ejpam-6275	822	14	,	,	PUNCT
ejpam-6275	822	15	0	0	NUM
ejpam-6275	822	16	,	,	PUNCT
ejpam-6275	822	17	0	0	NUM
ejpam-6275	822	18	,	,	PUNCT
ejpam-6275	822	19	1	1	NUM
ejpam-6275	822	20	,	,	PUNCT
ejpam-6275	822	21	0	0	NUM
ejpam-6275	822	22	,	,	PUNCT
ejpam-6275	822	23	0	0	NUM
ejpam-6275	822	24	)	)	PUNCT
ejpam-6275	822	25	(	(	PUNCT
ejpam-6275	822	26	1	1	NUM
ejpam-6275	822	27	β	β	X
ejpam-6275	822	28	β2	β2	NOUN
ejpam-6275	822	29	·	·	PUNCT
ejpam-6275	822	30	·	·	PUNCT
ejpam-6275	822	31	·	·	PUNCT
ejpam-6275	823	1	β8	β8	NOUN
ejpam-6275	823	2	1	1	NUM
ejpam-6275	823	3	β2	β2	NOUN
ejpam-6275	823	4	β4	β4	PROPN
ejpam-6275	823	5	·	·	PUNCT
ejpam-6275	823	6	·	·	PUNCT
ejpam-6275	823	7	·	·	PUNCT
ejpam-6275	823	8	β16	β16	NUM
ejpam-6275	823	9	)	)	PUNCT
ejpam-6275	823	10	t	t	NOUN
ejpam-6275	823	11	=	=	PUNCT
ejpam-6275	823	12	(	(	PUNCT
ejpam-6275	823	13	β8	β8	NOUN
ejpam-6275	823	14	β	β	X
ejpam-6275	823	15	)	)	PUNCT
ejpam-6275	823	16	,	,	PUNCT
ejpam-6275	823	17	m.	m.	NOUN
ejpam-6275	823	18	sajjad	sajjad	PROPN
ejpam-6275	823	19	et	et	PROPN
ejpam-6275	823	20	al	al	PROPN
ejpam-6275	823	21	.	.	PUNCT
ejpam-6275	823	22	/	/	SYM
ejpam-6275	823	23	eur	eur	PROPN
ejpam-6275	823	24	.	.	PUNCT
ejpam-6275	824	1	j.	j.	PROPN
ejpam-6275	824	2	pure	pure	PROPN
ejpam-6275	824	3	appl	appl	PROPN
ejpam-6275	824	4	.	.	PROPN
ejpam-6275	824	5	math	math	PROPN
ejpam-6275	824	6	,	,	PUNCT
ejpam-6275	824	7	18	18	NUM
ejpam-6275	824	8	(	(	PUNCT
ejpam-6275	824	9	3	3	NUM
ejpam-6275	824	10	)	)	PUNCT
ejpam-6275	824	11	(	(	PUNCT
ejpam-6275	824	12	2025	2025	NUM
ejpam-6275	824	13	)	)	PUNCT
ejpam-6275	824	14	,	,	PUNCT
ejpam-6275	824	15	6275	6275	NUM
ejpam-6275	824	16	31	31	NUM
ejpam-6275	824	17	of	of	ADP
ejpam-6275	824	18	36	36	NUM
ejpam-6275	824	19	table	table	NOUN
ejpam-6275	824	20	6	6	NUM
ejpam-6275	824	21	:	:	PUNCT
ejpam-6275	824	22	error	error	NOUN
ejpam-6275	824	23	locating	locating	NOUN
ejpam-6275	824	24	polynomials	polynomial	NOUN
ejpam-6275	824	25	n	n	PRON
ejpam-6275	824	26	∆n(y	∆n(y	PRON
ejpam-6275	824	27	)	)	PUNCT
ejpam-6275	824	28	ϑn	ϑn	PROPN
ejpam-6275	824	29	un	un	PROPN
ejpam-6275	825	1	n−	n−	PROPN
ejpam-6275	825	2	un	un	PROPN
ejpam-6275	825	3	−1	−1	NOUN
ejpam-6275	825	4	1	1	NUM
ejpam-6275	825	5	1	1	NUM
ejpam-6275	825	6	0	0	NUM
ejpam-6275	825	7	−1	−1	NOUN
ejpam-6275	825	8	0	0	NUM
ejpam-6275	825	9	1	1	NUM
ejpam-6275	825	10	β8	β8	NOUN
ejpam-6275	825	11	0	0	NUM
ejpam-6275	825	12	0	0	NUM
ejpam-6275	825	13	1	1	NUM
ejpam-6275	825	14	1	1	NUM
ejpam-6275	825	15	+	+	NOUN
ejpam-6275	825	16	β8y	β8y	PUNCT
ejpam-6275	825	17	β4	β4	PROPN
ejpam-6275	825	18	1	1	NUM
ejpam-6275	825	19	0	0	NUM
ejpam-6275	825	20	2	2	NUM
ejpam-6275	825	21	1	1	NUM
ejpam-6275	825	22	+	+	CCONJ
ejpam-6275	825	23	ωy	ωy	VERB
ejpam-6275	825	24	where	where	SCONJ
ejpam-6275	825	25	s1	s1	NOUN
ejpam-6275	825	26	=	=	SYM
ejpam-6275	825	27	β8	β8	NOUN
ejpam-6275	825	28	and	and	CCONJ
ejpam-6275	825	29	s2	s2	VERB
ejpam-6275	825	30	=	=	PUNCT
ejpam-6275	825	31	β	β	NOUN
ejpam-6275	825	32	are	be	AUX
ejpam-6275	825	33	the	the	DET
ejpam-6275	825	34	syndromes	syndrome	NOUN
ejpam-6275	825	35	.	.	PUNCT
ejpam-6275	826	1	next	next	ADV
ejpam-6275	826	2	,	,	PUNCT
ejpam-6275	826	3	we	we	PRON
ejpam-6275	826	4	will	will	AUX
ejpam-6275	826	5	find	find	VERB
ejpam-6275	826	6	∆2(y	∆2(y	NOUN
ejpam-6275	826	7	)	)	PUNCT
ejpam-6275	826	8	(	(	PUNCT
ejpam-6275	826	9	see	see	VERB
ejpam-6275	826	10	table	table	NOUN
ejpam-6275	826	11	2	2	NUM
ejpam-6275	826	12	)	)	PUNCT
ejpam-6275	826	13	by	by	ADP
ejpam-6275	826	14	applying	apply	VERB
ejpam-6275	826	15	the	the	DET
ejpam-6275	826	16	modified	modify	VERB
ejpam-6275	826	17	bma	bma	NOUN
ejpam-6275	826	18	shown	show	VERB
ejpam-6275	826	19	in	in	ADP
ejpam-6275	826	20	table	table	NOUN
ejpam-6275	826	21	6	6	NUM
ejpam-6275	826	22	.	.	PUNCT
ejpam-6275	826	23	iteration	iteration	NOUN
ejpam-6275	826	24	1	1	NUM
ejpam-6275	826	25	:	:	PUNCT
ejpam-6275	826	26	the	the	DET
ejpam-6275	826	27	initial	initial	ADJ
ejpam-6275	826	28	non	non	ADJ
ejpam-6275	826	29	-	-	ADJ
ejpam-6275	826	30	zero	zero	NUM
ejpam-6275	826	31	syndrome	syndrome	NOUN
ejpam-6275	826	32	is	be	AUX
ejpam-6275	826	33	β8	β8	ADJ
ejpam-6275	826	34	.	.	PUNCT
ejpam-6275	827	1	we	we	PRON
ejpam-6275	827	2	apply	apply	VERB
ejpam-6275	827	3	case	case	NOUN
ejpam-6275	827	4	2	2	NUM
ejpam-6275	827	5	of	of	ADP
ejpam-6275	827	6	step	step	NOUN
ejpam-6275	827	7	2	2	NUM
ejpam-6275	827	8	from	from	ADP
ejpam-6275	827	9	the	the	DET
ejpam-6275	827	10	modified	modified	PROPN
ejpam-6275	827	11	bma	bma	PROPN
ejpam-6275	827	12	as	as	ADP
ejpam-6275	827	13	ϑ0	ϑ0	PROPN
ejpam-6275	827	14	̸=	̸=	PROPN
ejpam-6275	827	15	0	0	NUM
ejpam-6275	827	16	,	,	PUNCT
ejpam-6275	827	17	−1	−1	NOUN
ejpam-6275	827	18	≤	≤	NOUN
ejpam-6275	827	19	−1	−1	NOUN
ejpam-6275	827	20	,	,	PUNCT
ejpam-6275	827	21	and	and	CCONJ
ejpam-6275	827	22	0−	0−	NUM
ejpam-6275	827	23	u−1	u−1	PROPN
ejpam-6275	827	24	=	=	SYM
ejpam-6275	827	25	0	0	NUM
ejpam-6275	827	26	is	be	AUX
ejpam-6275	827	27	the	the	DET
ejpam-6275	827	28	extreme	extreme	ADJ
ejpam-6275	827	29	value	value	NOUN
ejpam-6275	827	30	of	of	ADP
ejpam-6275	827	31	the	the	DET
ejpam-6275	827	32	end	end	NOUN
ejpam-6275	827	33	column	column	NOUN
ejpam-6275	827	34	.	.	PUNCT
ejpam-6275	828	1	then	then	ADV
ejpam-6275	828	2	,	,	PUNCT
ejpam-6275	828	3	ϑ0	ϑ0	PROPN
ejpam-6275	828	4	−	−	PROPN
ejpam-6275	828	5	zϑ−1	zϑ−1	NOUN
ejpam-6275	828	6	=	=	SYM
ejpam-6275	828	7	0	0	NUM
ejpam-6275	828	8	⇒	⇒	NOUN
ejpam-6275	828	9	z	z	NOUN
ejpam-6275	829	1	=	=	PUNCT
ejpam-6275	829	2	ϑ0	ϑ0	PROPN
ejpam-6275	829	3	ϑ−1	ϑ−1	NOUN
ejpam-6275	829	4	=	=	NOUN
ejpam-6275	829	5	β8	β8	NOUN
ejpam-6275	829	6	1	1	NUM
ejpam-6275	829	7	=	=	SYM
ejpam-6275	829	8	β8	β8	NOUN
ejpam-6275	829	9	.	.	PUNCT
ejpam-6275	830	1	so	so	ADV
ejpam-6275	830	2	,	,	PUNCT
ejpam-6275	830	3	the	the	DET
ejpam-6275	830	4	polynomial	polynomial	ADJ
ejpam-6275	830	5	:	:	PUNCT
ejpam-6275	830	6	∆1(y	∆1(y	ADJ
ejpam-6275	830	7	)	)	PUNCT
ejpam-6275	830	8	=	=	PUNCT
ejpam-6275	830	9	∆0(y)−	∆0(y)−	PROPN
ejpam-6275	830	10	β8y0	β8y0	PROPN
ejpam-6275	830	11	+	+	NOUN
ejpam-6275	830	12	1∆−1(y	1∆−1(y	NUM
ejpam-6275	830	13	)	)	PUNCT
ejpam-6275	830	14	=	=	SYM
ejpam-6275	830	15	1	1	NUM
ejpam-6275	830	16	+	+	NUM
ejpam-6275	830	17	β8y	β8y	PROPN
ejpam-6275	830	18	.	.	PUNCT
ejpam-6275	831	1	ϑ1	ϑ1	NOUN
ejpam-6275	831	2	=	=	NOUN
ejpam-6275	831	3	s2	s2	VERB
ejpam-6275	831	4	+	+	NOUN
ejpam-6275	831	5	∆1	∆1	PROPN
ejpam-6275	831	6	1(y)s1	1(y)s1	NUM
ejpam-6275	831	7	=	=	SYM
ejpam-6275	831	8	β	β	X
ejpam-6275	831	9	+	+	CCONJ
ejpam-6275	831	10	β8	β8	NOUN
ejpam-6275	831	11	·	·	PUNCT
ejpam-6275	831	12	β8	β8	NOUN
ejpam-6275	831	13	=	=	PUNCT
ejpam-6275	831	14	β	β	X
ejpam-6275	831	15	+	+	CCONJ
ejpam-6275	831	16	β7	β7	ADJ
ejpam-6275	831	17	=	=	SYM
ejpam-6275	831	18	β4	β4	PROPN
ejpam-6275	831	19	.	.	PUNCT
ejpam-6275	831	20	iteration	iteration	NOUN
ejpam-6275	831	21	2	2	NUM
ejpam-6275	831	22	:	:	PUNCT
ejpam-6275	831	23	let	let	VERB
ejpam-6275	831	24	ϑ1	ϑ1	PROPN
ejpam-6275	831	25	̸=	̸=	PROPN
ejpam-6275	831	26	0	0	NUM
ejpam-6275	832	1	in	in	ADP
ejpam-6275	832	2	iteration	iteration	NOUN
ejpam-6275	832	3	one	one	NUM
ejpam-6275	832	4	,	,	PUNCT
ejpam-6275	832	5	0	0	NUM
ejpam-6275	832	6	=	=	SYM
ejpam-6275	832	7	1	1	NUM
ejpam-6275	832	8	−	−	NUM
ejpam-6275	832	9	1	1	NUM
ejpam-6275	832	10	,	,	PUNCT
ejpam-6275	832	11	and	and	CCONJ
ejpam-6275	832	12	1	1	NUM
ejpam-6275	832	13	−	−	NOUN
ejpam-6275	832	14	u0	u0	ADJ
ejpam-6275	832	15	=	=	NOUN
ejpam-6275	832	16	1	1	NUM
ejpam-6275	832	17	−	−	NOUN
ejpam-6275	832	18	0	0	NUM
ejpam-6275	833	1	=	=	SYM
ejpam-6275	833	2	1	1	NUM
ejpam-6275	833	3	is	be	AUX
ejpam-6275	833	4	the	the	DET
ejpam-6275	833	5	highest	high	ADJ
ejpam-6275	833	6	value	value	NOUN
ejpam-6275	833	7	of	of	ADP
ejpam-6275	833	8	the	the	DET
ejpam-6275	833	9	last	last	ADJ
ejpam-6275	833	10	column	column	NOUN
ejpam-6275	833	11	.	.	PUNCT
ejpam-6275	834	1	thus	thus	ADV
ejpam-6275	834	2	,	,	PUNCT
ejpam-6275	834	3	ϑ1	ϑ1	PROPN
ejpam-6275	834	4	−	−	PROPN
ejpam-6275	834	5	zϑ0	zϑ0	X
ejpam-6275	834	6	=	=	SYM
ejpam-6275	834	7	0	0	NUM
ejpam-6275	834	8	⇒	⇒	NOUN
ejpam-6275	834	9	z	z	NOUN
ejpam-6275	834	10	=	=	SYM
ejpam-6275	834	11	ϑ1	ϑ1	NOUN
ejpam-6275	834	12	ϑ0	ϑ0	NOUN
ejpam-6275	834	13	=	=	SYM
ejpam-6275	834	14	β4	β4	PROPN
ejpam-6275	834	15	β8	β8	NOUN
ejpam-6275	834	16	=	=	PUNCT
ejpam-6275	834	17	β5	β5	PROPN
ejpam-6275	834	18	.	.	PUNCT
ejpam-6275	835	1	so	so	ADV
ejpam-6275	835	2	,	,	PUNCT
ejpam-6275	835	3	the	the	DET
ejpam-6275	835	4	polynomial	polynomial	ADJ
ejpam-6275	835	5	:	:	PUNCT
ejpam-6275	835	6	∆2(y	∆2(y	X
ejpam-6275	835	7	)	)	PUNCT
ejpam-6275	835	8	=	=	PUNCT
ejpam-6275	836	1	∆1(y)−	∆1(y)−	VERB
ejpam-6275	836	2	β5y1−0∆0(y	β5y1−0∆0(y	NOUN
ejpam-6275	836	3	)	)	PUNCT
ejpam-6275	836	4	=	=	SYM
ejpam-6275	836	5	1	1	NUM
ejpam-6275	836	6	+	+	CCONJ
ejpam-6275	836	7	(	(	PUNCT
ejpam-6275	836	8	β5	β5	NOUN
ejpam-6275	836	9	+	+	CCONJ
ejpam-6275	836	10	β8)y	β8)y	ADJ
ejpam-6275	836	11	=	=	SYM
ejpam-6275	836	12	1	1	NUM
ejpam-6275	836	13	+	+	CCONJ
ejpam-6275	836	14	β2y	β2y	PRON
ejpam-6275	836	15	.	.	PUNCT
ejpam-6275	837	1	now	now	ADV
ejpam-6275	837	2	,	,	PUNCT
ejpam-6275	837	3	the	the	DET
ejpam-6275	837	4	reciprocal	reciprocal	ADJ
ejpam-6275	837	5	function	function	NOUN
ejpam-6275	837	6	of	of	ADP
ejpam-6275	837	7	∆2(y	∆2(y	PROPN
ejpam-6275	837	8	)	)	PUNCT
ejpam-6275	837	9	=	=	SYM
ejpam-6275	837	10	1	1	NUM
ejpam-6275	837	11	+	+	CCONJ
ejpam-6275	837	12	β2y	β2y	PRON
ejpam-6275	837	13	is	be	AUX
ejpam-6275	837	14	g(y	g(y	NOUN
ejpam-6275	837	15	)	)	PUNCT
ejpam-6275	838	1	=	=	SYM
ejpam-6275	838	2	y	y	PROPN
ejpam-6275	839	1	+	+	CCONJ
ejpam-6275	839	2	β2	β2	ADJ
ejpam-6275	839	3	.	.	PUNCT
ejpam-6275	840	1	so	so	ADV
ejpam-6275	840	2	β2	β2	PROPN
ejpam-6275	840	3	is	be	AUX
ejpam-6275	840	4	the	the	DET
ejpam-6275	840	5	only	only	ADJ
ejpam-6275	840	6	root	root	NOUN
ejpam-6275	840	7	of	of	ADP
ejpam-6275	840	8	g(y	g(y	NOUN
ejpam-6275	840	9	)	)	PUNCT
ejpam-6275	840	10	.	.	PUNCT
ejpam-6275	841	1	hence	hence	ADV
ejpam-6275	841	2	y1	y1	ADV
ejpam-6275	841	3	=	=	PUNCT
ejpam-6275	841	4	β2	β2	VERB
ejpam-6275	841	5	,	,	PUNCT
ejpam-6275	841	6	and	and	CCONJ
ejpam-6275	841	7	the	the	DET
ejpam-6275	841	8	error	error	NOUN
ejpam-6275	841	9	takes	take	VERB
ejpam-6275	841	10	place	place	NOUN
ejpam-6275	841	11	in	in	ADP
ejpam-6275	841	12	the	the	DET
ejpam-6275	841	13	third	third	ADJ
ejpam-6275	841	14	position	position	NOUN
ejpam-6275	841	15	of	of	ADP
ejpam-6275	841	16	r.	r.	PROPN
ejpam-6275	841	17	∆0y	∆0y	PROPN
ejpam-6275	841	18	v	v	ADP
ejpam-6275	841	19	+	+	NOUN
ejpam-6275	841	20	∆1	∆1	NOUN
ejpam-6275	841	21	=	=	SYM
ejpam-6275	841	22	y	y	PROPN
ejpam-6275	841	23	−	−	PROPN
ejpam-6275	841	24	β2	β2	PROPN
ejpam-6275	841	25	is	be	AUX
ejpam-6275	841	26	an	an	DET
ejpam-6275	841	27	error	error	NOUN
ejpam-6275	841	28	-	-	PUNCT
ejpam-6275	841	29	locator	locator	NOUN
ejpam-6275	841	30	polynomial	polynomial	NOUN
ejpam-6275	841	31	(	(	PUNCT
ejpam-6275	841	32	esf	esf	PROPN
ejpam-6275	841	33	)	)	PUNCT
ejpam-6275	841	34	.	.	PUNCT
ejpam-6275	842	1	the	the	DET
ejpam-6275	842	2	error	error	NOUN
ejpam-6275	842	3	magnitude	magnitude	NOUN
ejpam-6275	842	4	is	be	AUX
ejpam-6275	842	5	:	:	PUNCT
ejpam-6275	842	6	z1	z1	PROPN
ejpam-6275	842	7	=	=	PUNCT
ejpam-6275	842	8	∆1,0s1	∆1,0s1	NOUN
ejpam-6275	842	9	∆1,0y1	∆1,0y1	NOUN
ejpam-6275	842	10	=	=	SYM
ejpam-6275	842	11	β8	β8	NOUN
ejpam-6275	842	12	β2	β2	NOUN
ejpam-6275	842	13	=	=	SYM
ejpam-6275	842	14	β6	β6	PROPN
ejpam-6275	842	15	=	=	SYM
ejpam-6275	842	16	ω	ω	PROPN
ejpam-6275	842	17	,	,	PUNCT
ejpam-6275	842	18	where	where	SCONJ
ejpam-6275	842	19	∆0	∆0	ADV
ejpam-6275	842	20	=	=	SYM
ejpam-6275	842	21	1	1	NUM
ejpam-6275	842	22	,	,	PUNCT
ejpam-6275	842	23	∆1	∆1	PROPN
ejpam-6275	842	24	=	=	SYM
ejpam-6275	842	25	β2	β2	VERB
ejpam-6275	842	26	,	,	PUNCT
ejpam-6275	842	27	and	and	CCONJ
ejpam-6275	842	28	v	v	X
ejpam-6275	842	29	=	=	SYM
ejpam-6275	842	30	1	1	NUM
ejpam-6275	842	31	.	.	PUNCT
ejpam-6275	843	1	the	the	DET
ejpam-6275	843	2	revised	revise	VERB
ejpam-6275	843	3	code	code	NOUN
ejpam-6275	843	4	word	word	NOUN
ejpam-6275	843	5	is	be	AUX
ejpam-6275	843	6	:	:	PUNCT
ejpam-6275	843	7	c	c	X
ejpam-6275	843	8	=	=	SYM
ejpam-6275	844	1	r	r	NOUN
ejpam-6275	844	2	−	−	NOUN
ejpam-6275	844	3	e	e	NOUN
ejpam-6275	844	4	=	=	PUNCT
ejpam-6275	844	5	(	(	PUNCT
ejpam-6275	844	6	1	1	NUM
ejpam-6275	844	7	,	,	PUNCT
ejpam-6275	844	8	0	0	NUM
ejpam-6275	844	9	,	,	PUNCT
ejpam-6275	844	10	ω	ω	NOUN
ejpam-6275	844	11	,	,	PUNCT
ejpam-6275	844	12	1	1	NUM
ejpam-6275	844	13	,	,	PUNCT
ejpam-6275	844	14	0	0	NUM
ejpam-6275	844	15	,	,	PUNCT
ejpam-6275	844	16	0	0	NUM
ejpam-6275	844	17	,	,	PUNCT
ejpam-6275	844	18	1	1	NUM
ejpam-6275	844	19	,	,	PUNCT
ejpam-6275	844	20	0	0	NUM
ejpam-6275	844	21	,	,	PUNCT
ejpam-6275	844	22	0)−	0)−	NUM
ejpam-6275	844	23	(	(	PUNCT
ejpam-6275	844	24	0	0	NUM
ejpam-6275	844	25	,	,	PUNCT
ejpam-6275	844	26	0	0	NUM
ejpam-6275	844	27	,	,	PUNCT
ejpam-6275	844	28	ω	ω	PROPN
ejpam-6275	844	29	,	,	PUNCT
ejpam-6275	844	30	0	0	NUM
ejpam-6275	844	31	,	,	PUNCT
ejpam-6275	844	32	0	0	NUM
ejpam-6275	844	33	,	,	PUNCT
ejpam-6275	844	34	0	0	NUM
ejpam-6275	844	35	,	,	PUNCT
ejpam-6275	844	36	0	0	NUM
ejpam-6275	844	37	,	,	PUNCT
ejpam-6275	844	38	0	0	NUM
ejpam-6275	844	39	,	,	PUNCT
ejpam-6275	844	40	0	0	NUM
ejpam-6275	844	41	)	)	PUNCT
ejpam-6275	844	42	=	=	NOUN
ejpam-6275	844	43	(	(	PUNCT
ejpam-6275	844	44	1	1	NUM
ejpam-6275	844	45	,	,	PUNCT
ejpam-6275	844	46	0	0	NUM
ejpam-6275	844	47	,	,	PUNCT
ejpam-6275	844	48	0	0	NUM
ejpam-6275	844	49	,	,	PUNCT
ejpam-6275	844	50	1	1	NUM
ejpam-6275	844	51	,	,	PUNCT
ejpam-6275	844	52	0	0	NUM
ejpam-6275	844	53	,	,	PUNCT
ejpam-6275	844	54	0	0	NUM
ejpam-6275	844	55	,	,	PUNCT
ejpam-6275	844	56	1	1	NUM
ejpam-6275	844	57	,	,	PUNCT
ejpam-6275	844	58	0	0	NUM
ejpam-6275	844	59	,	,	PUNCT
ejpam-6275	844	60	0	0	NUM
ejpam-6275	844	61	)	)	PUNCT
ejpam-6275	844	62	.	.	PUNCT
ejpam-6275	845	1	thus	thus	ADV
ejpam-6275	845	2	,	,	PUNCT
ejpam-6275	845	3	c	c	PROPN
ejpam-6275	845	4	is	be	AUX
ejpam-6275	845	5	the	the	DET
ejpam-6275	845	6	corrected	correct	VERB
ejpam-6275	845	7	narrow	narrow	ADJ
ejpam-6275	845	8	sense	sense	NOUN
ejpam-6275	845	9	shortened	shorten	VERB
ejpam-6275	845	10	(	(	PUNCT
ejpam-6275	845	11	9	9	NUM
ejpam-6275	845	12	,	,	PUNCT
ejpam-6275	845	13	3	3	NUM
ejpam-6275	845	14	,	,	PUNCT
ejpam-6275	845	15	3	3	X
ejpam-6275	845	16	)	)	PUNCT
ejpam-6275	845	17	bch	bch	PROPN
ejpam-6275	845	18	code	code	PROPN
ejpam-6275	845	19	.	.	PUNCT
ejpam-6275	846	1	m.	m.	PROPN
ejpam-6275	846	2	sajjad	sajjad	PROPN
ejpam-6275	846	3	et	et	PROPN
ejpam-6275	846	4	al	al	PROPN
ejpam-6275	846	5	.	.	PUNCT
ejpam-6275	846	6	/	/	SYM
ejpam-6275	846	7	eur	eur	PROPN
ejpam-6275	846	8	.	.	PUNCT
ejpam-6275	847	1	j.	j.	PROPN
ejpam-6275	847	2	pure	pure	PROPN
ejpam-6275	847	3	appl	appl	PROPN
ejpam-6275	847	4	.	.	PROPN
ejpam-6275	847	5	math	math	PROPN
ejpam-6275	847	6	,	,	PUNCT
ejpam-6275	847	7	18	18	NUM
ejpam-6275	847	8	(	(	PUNCT
ejpam-6275	847	9	3	3	NUM
ejpam-6275	847	10	)	)	PUNCT
ejpam-6275	847	11	(	(	PUNCT
ejpam-6275	847	12	2025	2025	NUM
ejpam-6275	847	13	)	)	PUNCT
ejpam-6275	847	14	,	,	PUNCT
ejpam-6275	847	15	6275	6275	NUM
ejpam-6275	847	16	32	32	NUM
ejpam-6275	847	17	of	of	ADP
ejpam-6275	847	18	36	36	NUM
ejpam-6275	847	19	5	5	NUM
ejpam-6275	847	20	.	.	PUNCT
ejpam-6275	848	1	the	the	DET
ejpam-6275	848	2	importance	importance	NOUN
ejpam-6275	848	3	of	of	ADP
ejpam-6275	848	4	shortened	shorten	VERB
ejpam-6275	848	5	codes	code	NOUN
ejpam-6275	848	6	over	over	ADP
ejpam-6275	848	7	ef	ef	PROPN
ejpam-6275	848	8	in	in	ADP
ejpam-6275	848	9	modern	modern	ADJ
ejpam-6275	848	10	-	-	PUNCT
ejpam-6275	848	11	day	day	NOUN
ejpam-6275	848	12	transmission	transmission	NOUN
ejpam-6275	848	13	being	be	AUX
ejpam-6275	848	14	able	able	ADJ
ejpam-6275	848	15	to	to	PART
ejpam-6275	848	16	code	code	VERB
ejpam-6275	848	17	and	and	CCONJ
ejpam-6275	848	18	decode	decode	VERB
ejpam-6275	848	19	shortened	shorten	VERB
ejpam-6275	848	20	bch	bch	PROPN
ejpam-6275	848	21	codes	code	NOUN
ejpam-6275	848	22	within	within	ADP
ejpam-6275	848	23	the	the	DET
ejpam-6275	848	24	eisenstein	eisenstein	PROPN
ejpam-6275	848	25	field	field	NOUN
ejpam-6275	848	26	zp[ω	zp[ω	PROPN
ejpam-6275	848	27	]	]	X
ejpam-6275	848	28	,	,	PUNCT
ejpam-6275	848	29	where	where	SCONJ
ejpam-6275	848	30	p	p	PRON
ejpam-6275	848	31	≡	≡	PROPN
ejpam-6275	848	32	2	2	NUM
ejpam-6275	848	33	(	(	PUNCT
ejpam-6275	848	34	mod	mod	NOUN
ejpam-6275	848	35	3	3	NUM
ejpam-6275	848	36	)	)	PUNCT
ejpam-6275	848	37	,	,	PUNCT
ejpam-6275	848	38	is	be	AUX
ejpam-6275	848	39	highly	highly	ADV
ejpam-6275	848	40	beneficial	beneficial	ADJ
ejpam-6275	848	41	in	in	ADP
ejpam-6275	848	42	modern	modern	ADJ
ejpam-6275	848	43	data	datum	NOUN
ejpam-6275	848	44	transmission	transmission	NOUN
ejpam-6275	848	45	systems	system	NOUN
ejpam-6275	848	46	.	.	PUNCT
ejpam-6275	849	1	defining	define	VERB
ejpam-6275	849	2	codes	code	NOUN
ejpam-6275	849	3	over	over	ADP
ejpam-6275	849	4	zp[ω	zp[ω	PROPN
ejpam-6275	849	5	]	]	PUNCT
ejpam-6275	849	6	provides	provide	VERB
ejpam-6275	849	7	the	the	DET
ejpam-6275	849	8	codes	code	NOUN
ejpam-6275	849	9	with	with	ADP
ejpam-6275	849	10	symmetry	symmetry	NOUN
ejpam-6275	849	11	and	and	CCONJ
ejpam-6275	849	12	enables	enable	VERB
ejpam-6275	849	13	shorter	short	ADJ
ejpam-6275	849	14	distances	distance	NOUN
ejpam-6275	849	15	due	due	ADP
ejpam-6275	849	16	to	to	ADP
ejpam-6275	849	17	the	the	DET
ejpam-6275	849	18	identity	identity	NOUN
ejpam-6275	849	19	ω3	ω3	NOUN
ejpam-6275	849	20	=	=	PUNCT
ejpam-6275	850	1	1	1	X
ejpam-6275	850	2	.	.	PUNCT
ejpam-6275	850	3	when	when	SCONJ
ejpam-6275	850	4	p	p	PRON
ejpam-6275	850	5	≡	≡	PROPN
ejpam-6275	850	6	2	2	NUM
ejpam-6275	850	7	(	(	PUNCT
ejpam-6275	850	8	mod	mod	NOUN
ejpam-6275	850	9	3	3	NUM
ejpam-6275	850	10	)	)	PUNCT
ejpam-6275	850	11	is	be	AUX
ejpam-6275	850	12	selected	select	VERB
ejpam-6275	850	13	,	,	PUNCT
ejpam-6275	850	14	the	the	DET
ejpam-6275	850	15	corresponding	corresponding	ADJ
ejpam-6275	850	16	polynomial	polynomial	NOUN
ejpam-6275	850	17	becomes	become	VERB
ejpam-6275	850	18	irreducible	irreducible	ADJ
ejpam-6275	850	19	in	in	ADP
ejpam-6275	850	20	zp[ω	zp[ω	PROPN
ejpam-6275	850	21	]	]	PUNCT
ejpam-6275	850	22	,	,	PUNCT
ejpam-6275	850	23	thereby	thereby	ADV
ejpam-6275	850	24	preserving	preserve	VERB
ejpam-6275	850	25	the	the	DET
ejpam-6275	850	26	field	field	NOUN
ejpam-6275	850	27	extension	extension	NOUN
ejpam-6275	850	28	and	and	CCONJ
ejpam-6275	850	29	making	make	VERB
ejpam-6275	850	30	the	the	DET
ejpam-6275	850	31	construction	construction	NOUN
ejpam-6275	850	32	of	of	ADP
ejpam-6275	850	33	codes	code	NOUN
ejpam-6275	850	34	using	use	VERB
ejpam-6275	850	35	eisenstein	eisenstein	PROPN
ejpam-6275	850	36	arithmetic	arithmetic	ADJ
ejpam-6275	850	37	more	more	ADV
ejpam-6275	850	38	convenient	convenient	ADJ
ejpam-6275	850	39	.	.	PUNCT
ejpam-6275	851	1	depending	depend	VERB
ejpam-6275	851	2	on	on	ADP
ejpam-6275	851	3	the	the	DET
ejpam-6275	851	4	required	require	VERB
ejpam-6275	851	5	block	block	NOUN
ejpam-6275	851	6	length	length	NOUN
ejpam-6275	851	7	and	and	CCONJ
ejpam-6275	851	8	level	level	NOUN
ejpam-6275	851	9	of	of	ADP
ejpam-6275	851	10	error	error	NOUN
ejpam-6275	851	11	correction	correction	NOUN
ejpam-6275	851	12	,	,	PUNCT
ejpam-6275	851	13	shortened	shorten	VERB
ejpam-6275	851	14	bch	bch	PROPN
ejpam-6275	851	15	codes	code	NOUN
ejpam-6275	851	16	can	can	AUX
ejpam-6275	851	17	be	be	AUX
ejpam-6275	851	18	designed	design	VERB
ejpam-6275	851	19	in	in	ADP
ejpam-6275	851	20	this	this	DET
ejpam-6275	851	21	setting	setting	NOUN
ejpam-6275	851	22	.	.	PUNCT
ejpam-6275	852	1	these	these	DET
ejpam-6275	852	2	codes	code	NOUN
ejpam-6275	852	3	are	be	AUX
ejpam-6275	852	4	particularly	particularly	ADV
ejpam-6275	852	5	suitable	suitable	ADJ
ejpam-6275	852	6	for	for	ADP
ejpam-6275	852	7	applications	application	NOUN
ejpam-6275	852	8	such	such	ADJ
ejpam-6275	852	9	as	as	ADP
ejpam-6275	852	10	internet	internet	NOUN
ejpam-6275	852	11	of	of	ADP
ejpam-6275	852	12	things	thing	NOUN
ejpam-6275	852	13	(	(	PUNCT
ejpam-6275	852	14	iot	iot	NOUN
ejpam-6275	852	15	)	)	PUNCT
ejpam-6275	852	16	devices	device	NOUN
ejpam-6275	852	17	,	,	PUNCT
ejpam-6275	852	18	satellite	satellite	NOUN
ejpam-6275	852	19	communication	communication	NOUN
ejpam-6275	852	20	,	,	PUNCT
ejpam-6275	852	21	and	and	CCONJ
ejpam-6275	852	22	mobile	mobile	NOUN
ejpam-6275	852	23	networks	network	NOUN
ejpam-6275	852	24	where	where	SCONJ
ejpam-6275	852	25	strict	strict	ADJ
ejpam-6275	852	26	power	power	NOUN
ejpam-6275	852	27	and	and	CCONJ
ejpam-6275	852	28	bandwidth	bandwidth	ADJ
ejpam-6275	852	29	limitations	limitation	NOUN
ejpam-6275	852	30	exist	exist	VERB
ejpam-6275	852	31	.	.	PUNCT
ejpam-6275	853	1	the	the	DET
ejpam-6275	853	2	anti	anti	ADJ
ejpam-6275	853	3	-	-	ADJ
ejpam-6275	853	4	noise	noise	ADJ
ejpam-6275	853	5	efficiency	efficiency	NOUN
ejpam-6275	853	6	of	of	ADP
ejpam-6275	853	7	these	these	DET
ejpam-6275	853	8	codes	code	NOUN
ejpam-6275	853	9	is	be	AUX
ejpam-6275	853	10	enhanced	enhance	VERB
ejpam-6275	853	11	owing	owe	VERB
ejpam-6275	853	12	to	to	ADP
ejpam-6275	853	13	dense	dense	ADJ
ejpam-6275	853	14	encoding	encoding	NOUN
ejpam-6275	853	15	schemes	scheme	NOUN
ejpam-6275	853	16	and	and	CCONJ
ejpam-6275	853	17	improved	improved	ADJ
ejpam-6275	853	18	distances	distance	NOUN
ejpam-6275	853	19	measured	measure	VERB
ejpam-6275	853	20	via	via	ADP
ejpam-6275	853	21	both	both	CCONJ
ejpam-6275	853	22	euclidean	euclidean	ADJ
ejpam-6275	853	23	and	and	CCONJ
ejpam-6275	853	24	eisenstein	eisenstein	PROPN
ejpam-6275	853	25	weights	weight	NOUN
ejpam-6275	853	26	.	.	PUNCT
ejpam-6275	854	1	in	in	ADP
ejpam-6275	854	2	this	this	DET
ejpam-6275	854	3	context	context	NOUN
ejpam-6275	854	4	,	,	PUNCT
ejpam-6275	854	5	coding	code	VERB
ejpam-6275	854	6	algorithms	algorithm	NOUN
ejpam-6275	854	7	,	,	PUNCT
ejpam-6275	854	8	especially	especially	ADV
ejpam-6275	854	9	those	those	PRON
ejpam-6275	854	10	based	base	VERB
ejpam-6275	854	11	on	on	ADP
ejpam-6275	854	12	syndrome	syndrome	NOUN
ejpam-6275	854	13	decoding	decoding	NOUN
ejpam-6275	854	14	and	and	CCONJ
ejpam-6275	854	15	the	the	DET
ejpam-6275	854	16	berlekamp	berlekamp	NOUN
ejpam-6275	854	17	–	–	PUNCT
ejpam-6275	854	18	massey	massey	NOUN
ejpam-6275	854	19	algorithm	algorithm	NOUN
ejpam-6275	854	20	,	,	PUNCT
ejpam-6275	854	21	exploit	exploit	VERB
ejpam-6275	854	22	the	the	DET
ejpam-6275	854	23	algebraic	algebraic	ADJ
ejpam-6275	854	24	structure	structure	NOUN
ejpam-6275	854	25	of	of	ADP
ejpam-6275	854	26	the	the	DET
ejpam-6275	854	27	field	field	NOUN
ejpam-6275	854	28	to	to	PART
ejpam-6275	854	29	efficiently	efficiently	ADV
ejpam-6275	854	30	correct	correct	ADJ
ejpam-6275	854	31	errors	error	NOUN
ejpam-6275	854	32	.	.	PUNCT
ejpam-6275	855	1	overall	overall	ADJ
ejpam-6275	855	2	,	,	PUNCT
ejpam-6275	855	3	utilizing	utilize	VERB
ejpam-6275	855	4	bch	bch	PROPN
ejpam-6275	855	5	codes	code	NOUN
ejpam-6275	855	6	over	over	ADP
ejpam-6275	855	7	zp[ω	zp[ω	PROPN
ejpam-6275	855	8	]	]	PUNCT
ejpam-6275	855	9	presents	present	VERB
ejpam-6275	855	10	a	a	DET
ejpam-6275	855	11	promising	promising	ADJ
ejpam-6275	855	12	approach	approach	NOUN
ejpam-6275	855	13	to	to	ADP
ejpam-6275	855	14	enhancing	enhance	VERB
ejpam-6275	855	15	the	the	DET
ejpam-6275	855	16	robustness	robustness	NOUN
ejpam-6275	855	17	and	and	CCONJ
ejpam-6275	855	18	reliability	reliability	NOUN
ejpam-6275	855	19	of	of	ADP
ejpam-6275	855	20	modern	modern	ADJ
ejpam-6275	855	21	digital	digital	ADJ
ejpam-6275	855	22	communication	communication	NOUN
ejpam-6275	855	23	infrastructures	infrastructure	NOUN
ejpam-6275	855	24	.	.	PUNCT
ejpam-6275	856	1	6	6	X
ejpam-6275	856	2	.	.	X
ejpam-6275	856	3	comparative	comparative	ADJ
ejpam-6275	856	4	analysis	analysis	NOUN
ejpam-6275	856	5	in	in	ADP
ejpam-6275	856	6	this	this	DET
ejpam-6275	856	7	section	section	NOUN
ejpam-6275	856	8	,	,	PUNCT
ejpam-6275	856	9	we	we	PRON
ejpam-6275	856	10	compare	compare	VERB
ejpam-6275	856	11	the	the	DET
ejpam-6275	856	12	performance	performance	NOUN
ejpam-6275	856	13	of	of	ADP
ejpam-6275	856	14	narrow	narrow	ADJ
ejpam-6275	856	15	-	-	PUNCT
ejpam-6275	856	16	sense	sense	NOUN
ejpam-6275	856	17	shortened	shorten	VERB
ejpam-6275	856	18	bch	bch	PROPN
ejpam-6275	856	19	codes	code	NOUN
ejpam-6275	856	20	over	over	ADP
ejpam-6275	856	21	the	the	DET
ejpam-6275	856	22	finite	finite	ADJ
ejpam-6275	856	23	field	field	NOUN
ejpam-6275	856	24	gf(pm	gf(pm	NOUN
ejpam-6275	856	25	)	)	PUNCT
ejpam-6275	856	26	and	and	CCONJ
ejpam-6275	856	27	the	the	DET
ejpam-6275	856	28	eisenstein	eisenstein	PROPN
ejpam-6275	856	29	field	field	NOUN
ejpam-6275	856	30	zp[ω	zp[ω	PROPN
ejpam-6275	856	31	]	]	X
ejpam-6275	856	32	m/2	m/2	NUM
ejpam-6275	856	33	,	,	PUNCT
ejpam-6275	856	34	where	where	SCONJ
ejpam-6275	856	35	p	p	PRON
ejpam-6275	856	36	≡	≡	PROPN
ejpam-6275	856	37	2	2	NUM
ejpam-6275	856	38	(	(	PUNCT
ejpam-6275	856	39	mod	mod	NOUN
ejpam-6275	856	40	3	3	NUM
ejpam-6275	856	41	)	)	PUNCT
ejpam-6275	856	42	.	.	PUNCT
ejpam-6275	857	1	the	the	DET
ejpam-6275	857	2	comparison	comparison	NOUN
ejpam-6275	857	3	is	be	AUX
ejpam-6275	857	4	made	make	VERB
ejpam-6275	857	5	by	by	ADP
ejpam-6275	857	6	identifying	identify	VERB
ejpam-6275	857	7	key	key	ADJ
ejpam-6275	857	8	parameters	parameter	NOUN
ejpam-6275	857	9	,	,	PUNCT
ejpam-6275	857	10	including	include	VERB
ejpam-6275	857	11	the	the	DET
ejpam-6275	857	12	code	code	NOUN
ejpam-6275	857	13	length	length	NOUN
ejpam-6275	857	14	(	(	PUNCT
ejpam-6275	857	15	n	n	CCONJ
ejpam-6275	857	16	)	)	PUNCT
ejpam-6275	857	17	,	,	PUNCT
ejpam-6275	857	18	minimum	minimum	ADJ
ejpam-6275	857	19	distance	distance	NOUN
ejpam-6275	857	20	(	(	PUNCT
ejpam-6275	857	21	d	d	NOUN
ejpam-6275	857	22	)	)	PUNCT
ejpam-6275	857	23	,	,	PUNCT
ejpam-6275	857	24	code	code	NOUN
ejpam-6275	857	25	dimension	dimension	NOUN
ejpam-6275	857	26	(	(	PUNCT
ejpam-6275	857	27	k	k	NOUN
ejpam-6275	857	28	)	)	PUNCT
ejpam-6275	857	29	,	,	PUNCT
ejpam-6275	857	30	number	number	NOUN
ejpam-6275	857	31	of	of	ADP
ejpam-6275	857	32	codewords	codeword	NOUN
ejpam-6275	857	33	(	(	PUNCT
ejpam-6275	857	34	|c|	|c|	PROPN
ejpam-6275	857	35	)	)	PUNCT
ejpam-6275	857	36	,	,	PUNCT
ejpam-6275	857	37	and	and	CCONJ
ejpam-6275	857	38	code	code	NOUN
ejpam-6275	857	39	rate	rate	NOUN
ejpam-6275	857	40	(	(	PUNCT
ejpam-6275	857	41	r	r	NOUN
ejpam-6275	857	42	)	)	PUNCT
ejpam-6275	857	43	.	.	PUNCT
ejpam-6275	858	1	the	the	DET
ejpam-6275	858	2	analysis	analysis	NOUN
ejpam-6275	858	3	draws	draw	VERB
ejpam-6275	858	4	on	on	ADP
ejpam-6275	858	5	prior	prior	ADJ
ejpam-6275	858	6	research	research	NOUN
ejpam-6275	858	7	findings	finding	NOUN
ejpam-6275	858	8	,	,	PUNCT
ejpam-6275	858	9	particularly	particularly	ADV
ejpam-6275	858	10	[	[	X
ejpam-6275	858	11	6	6	NUM
ejpam-6275	858	12	]	]	PUNCT
ejpam-6275	858	13	and	and	CCONJ
ejpam-6275	858	14	sections	section	NOUN
ejpam-6275	858	15	3	3	NUM
ejpam-6275	858	16	and	and	CCONJ
ejpam-6275	858	17	4	4	NUM
ejpam-6275	858	18	of	of	ADP
ejpam-6275	858	19	this	this	DET
ejpam-6275	858	20	article	article	NOUN
ejpam-6275	858	21	.	.	PUNCT
ejpam-6275	859	1	we	we	PRON
ejpam-6275	859	2	provide	provide	VERB
ejpam-6275	859	3	a	a	DET
ejpam-6275	859	4	comparison	comparison	NOUN
ejpam-6275	859	5	based	base	VERB
ejpam-6275	859	6	on	on	ADP
ejpam-6275	859	7	the	the	DET
ejpam-6275	859	8	finite	finite	ADJ
ejpam-6275	859	9	field	field	NOUN
ejpam-6275	859	10	gf(26	gf(26	PROPN
ejpam-6275	859	11	)	)	PUNCT
ejpam-6275	859	12	and	and	CCONJ
ejpam-6275	859	13	the	the	DET
ejpam-6275	859	14	eisenstein	eisenstein	PROPN
ejpam-6275	859	15	field	field	PROPN
ejpam-6275	859	16	z2[ω	z2[ω	NOUN
ejpam-6275	859	17	]	]	X
ejpam-6275	859	18	3	3	NUM
ejpam-6275	859	19	,	,	PUNCT
ejpam-6275	859	20	as	as	SCONJ
ejpam-6275	859	21	represented	represent	VERB
ejpam-6275	859	22	in	in	ADP
ejpam-6275	859	23	tables	table	NOUN
ejpam-6275	859	24	7	7	NUM
ejpam-6275	859	25	and	and	CCONJ
ejpam-6275	859	26	8	8	NUM
ejpam-6275	859	27	.	.	PUNCT
ejpam-6275	859	28	from	from	ADP
ejpam-6275	859	29	tables	table	NOUN
ejpam-6275	859	30	7	7	NUM
ejpam-6275	859	31	and	and	CCONJ
ejpam-6275	859	32	8	8	NUM
ejpam-6275	859	33	,	,	PUNCT
ejpam-6275	859	34	both	both	DET
ejpam-6275	859	35	gf(26	gf(26	NOUN
ejpam-6275	859	36	)	)	PUNCT
ejpam-6275	859	37	and	and	CCONJ
ejpam-6275	859	38	z2[ω	z2[ω	NOUN
ejpam-6275	859	39	]	]	X
ejpam-6275	859	40	3	3	NUM
ejpam-6275	859	41	yield	yield	NOUN
ejpam-6275	859	42	shortened	shorten	VERB
ejpam-6275	859	43	bch	bch	PROPN
ejpam-6275	859	44	codes	code	NOUN
ejpam-6275	859	45	with	with	ADP
ejpam-6275	859	46	comparable	comparable	ADJ
ejpam-6275	859	47	lengths	length	NOUN
ejpam-6275	859	48	and	and	CCONJ
ejpam-6275	859	49	distances	distance	NOUN
ejpam-6275	859	50	.	.	PUNCT
ejpam-6275	860	1	the	the	DET
ejpam-6275	860	2	comparative	comparative	ADJ
ejpam-6275	860	3	outcomes	outcome	NOUN
ejpam-6275	860	4	are	be	AUX
ejpam-6275	860	5	summarized	summarize	VERB
ejpam-6275	860	6	below	below	ADP
ejpam-6275	860	7	:	:	PUNCT
ejpam-6275	860	8	dimension	dimension	NOUN
ejpam-6275	860	9	:	:	PUNCT
ejpam-6275	860	10	the	the	DET
ejpam-6275	860	11	dimensions	dimension	NOUN
ejpam-6275	860	12	of	of	ADP
ejpam-6275	860	13	codes	code	NOUN
ejpam-6275	860	14	over	over	ADP
ejpam-6275	860	15	z2[ω	z2[ω	NOUN
ejpam-6275	860	16	]	]	X
ejpam-6275	860	17	3	3	NUM
ejpam-6275	860	18	often	often	ADV
ejpam-6275	860	19	match	match	VERB
ejpam-6275	860	20	or	or	CCONJ
ejpam-6275	860	21	surpass	surpass	VERB
ejpam-6275	860	22	their	their	PRON
ejpam-6275	860	23	gf(26	gf(26	NOUN
ejpam-6275	860	24	)	)	PUNCT
ejpam-6275	860	25	counterparts	counterpart	NOUN
ejpam-6275	860	26	.	.	PUNCT
ejpam-6275	861	1	for	for	ADP
ejpam-6275	861	2	instance	instance	NOUN
ejpam-6275	861	3	,	,	PUNCT
ejpam-6275	861	4	for	for	ADP
ejpam-6275	861	5	n	n	NOUN
ejpam-6275	861	6	=	=	SYM
ejpam-6275	861	7	21	21	NUM
ejpam-6275	861	8	,	,	PUNCT
ejpam-6275	861	9	both	both	DET
ejpam-6275	861	10	fields	field	NOUN
ejpam-6275	861	11	provide	provide	VERB
ejpam-6275	861	12	k	k	PROPN
ejpam-6275	861	13	=	=	PUNCT
ejpam-6275	861	14	15	15	NUM
ejpam-6275	861	15	at	at	ADP
ejpam-6275	861	16	d	d	PROPN
ejpam-6275	861	17	=	=	SYM
ejpam-6275	861	18	3	3	NUM
ejpam-6275	861	19	,	,	PUNCT
ejpam-6275	861	20	indicating	indicate	VERB
ejpam-6275	861	21	equivalent	equivalent	ADJ
ejpam-6275	861	22	capacity	capacity	NOUN
ejpam-6275	861	23	.	.	PUNCT
ejpam-6275	862	1	however	however	ADV
ejpam-6275	862	2	,	,	PUNCT
ejpam-6275	862	3	z2[ω	z2[ω	NOUN
ejpam-6275	862	4	]	]	X
ejpam-6275	862	5	3	3	NUM
ejpam-6275	862	6	offers	offer	VERB
ejpam-6275	862	7	greater	great	ADJ
ejpam-6275	862	8	dimensions	dimension	NOUN
ejpam-6275	862	9	at	at	ADP
ejpam-6275	862	10	higher	high	ADJ
ejpam-6275	862	11	distances	distance	NOUN
ejpam-6275	862	12	,	,	PUNCT
ejpam-6275	862	13	indicating	indicate	VERB
ejpam-6275	862	14	better	well	ADJ
ejpam-6275	862	15	payload	payload	NOUN
ejpam-6275	862	16	accommodation	accommodation	NOUN
ejpam-6275	862	17	.	.	PUNCT
ejpam-6275	863	1	number	number	NOUN
ejpam-6275	863	2	of	of	ADP
ejpam-6275	863	3	codewords	codeword	NOUN
ejpam-6275	863	4	:	:	PUNCT
ejpam-6275	863	5	due	due	ADP
ejpam-6275	863	6	to	to	ADP
ejpam-6275	863	7	the	the	DET
ejpam-6275	863	8	quadratic	quadratic	ADJ
ejpam-6275	863	9	growth	growth	NOUN
ejpam-6275	863	10	in	in	ADP
ejpam-6275	863	11	the	the	DET
ejpam-6275	863	12	symbol	symbol	NOUN
ejpam-6275	863	13	set	set	NOUN
ejpam-6275	863	14	,	,	PUNCT
ejpam-6275	863	15	codes	code	NOUN
ejpam-6275	863	16	over	over	ADP
ejpam-6275	863	17	z2[ω	z2[ω	NOUN
ejpam-6275	863	18	]	]	X
ejpam-6275	863	19	3	3	NUM
ejpam-6275	863	20	provide	provide	VERB
ejpam-6275	863	21	significantly	significantly	ADV
ejpam-6275	863	22	more	more	ADJ
ejpam-6275	863	23	codewords	codeword	NOUN
ejpam-6275	863	24	for	for	ADP
ejpam-6275	863	25	the	the	DET
ejpam-6275	863	26	same	same	ADJ
ejpam-6275	863	27	dimension	dimension	NOUN
ejpam-6275	863	28	k.	k.	PROPN
ejpam-6275	864	1	for	for	ADP
ejpam-6275	864	2	example	example	NOUN
ejpam-6275	864	3	,	,	PUNCT
ejpam-6275	864	4	at	at	ADP
ejpam-6275	864	5	n	n	NOUN
ejpam-6275	864	6	=	=	SYM
ejpam-6275	864	7	21	21	NUM
ejpam-6275	864	8	and	and	CCONJ
ejpam-6275	864	9	d	d	NOUN
ejpam-6275	864	10	=	=	SYM
ejpam-6275	864	11	3	3	NUM
ejpam-6275	864	12	,	,	PUNCT
ejpam-6275	864	13	we	we	PRON
ejpam-6275	864	14	obtain	obtain	VERB
ejpam-6275	864	15	|c|	|c|	PROPN
ejpam-6275	864	16	=	=	SYM
ejpam-6275	864	17	1073741824	1073741824	NUM
ejpam-6275	864	18	for	for	ADP
ejpam-6275	864	19	z2[ω	z2[ω	NOUN
ejpam-6275	864	20	]	]	X
ejpam-6275	864	21	3	3	NUM
ejpam-6275	864	22	,	,	PUNCT
ejpam-6275	864	23	compared	compare	VERB
ejpam-6275	864	24	to	to	ADP
ejpam-6275	864	25	only	only	ADV
ejpam-6275	864	26	|c|	|c|	PROPN
ejpam-6275	864	27	=	=	NUM
ejpam-6275	864	28	32768	32768	NUM
ejpam-6275	864	29	in	in	ADP
ejpam-6275	864	30	gf(26	gf(26	NOUN
ejpam-6275	864	31	)	)	PUNCT
ejpam-6275	864	32	.	.	PUNCT
ejpam-6275	865	1	this	this	PRON
ejpam-6275	865	2	suggests	suggest	VERB
ejpam-6275	865	3	an	an	DET
ejpam-6275	865	4	advantage	advantage	NOUN
ejpam-6275	865	5	in	in	ADP
ejpam-6275	865	6	applications	application	NOUN
ejpam-6275	865	7	requiring	require	VERB
ejpam-6275	865	8	high	high	ADJ
ejpam-6275	865	9	codeword	codeword	NOUN
ejpam-6275	865	10	dispersion	dispersion	NOUN
ejpam-6275	865	11	.	.	PUNCT
ejpam-6275	866	1	code	code	NOUN
ejpam-6275	866	2	rate	rate	NOUN
ejpam-6275	866	3	:	:	PUNCT
ejpam-6275	866	4	the	the	DET
ejpam-6275	866	5	code	code	NOUN
ejpam-6275	866	6	rate	rate	NOUN
ejpam-6275	866	7	is	be	AUX
ejpam-6275	866	8	equivalent	equivalent	ADJ
ejpam-6275	866	9	in	in	ADP
ejpam-6275	866	10	some	some	DET
ejpam-6275	866	11	cases	case	NOUN
ejpam-6275	866	12	.	.	PUNCT
ejpam-6275	867	1	for	for	ADP
ejpam-6275	867	2	n	n	NOUN
ejpam-6275	867	3	=	=	SYM
ejpam-6275	867	4	21	21	NUM
ejpam-6275	867	5	and	and	CCONJ
ejpam-6275	867	6	d	d	NOUN
ejpam-6275	867	7	=	=	SYM
ejpam-6275	867	8	3	3	NUM
ejpam-6275	867	9	,	,	PUNCT
ejpam-6275	867	10	both	both	DET
ejpam-6275	867	11	fields	field	NOUN
ejpam-6275	867	12	yield	yield	VERB
ejpam-6275	867	13	r	r	NOUN
ejpam-6275	867	14	=	=	SYM
ejpam-6275	867	15	0.7143	0.7143	NUM
ejpam-6275	867	16	.	.	PUNCT
ejpam-6275	868	1	yet	yet	ADV
ejpam-6275	868	2	,	,	PUNCT
ejpam-6275	868	3	for	for	ADP
ejpam-6275	868	4	other	other	ADJ
ejpam-6275	868	5	configurations	configuration	NOUN
ejpam-6275	868	6	,	,	PUNCT
ejpam-6275	868	7	z2[ω	z2[ω	NOUN
ejpam-6275	868	8	]	]	X
ejpam-6275	868	9	3	3	NUM
ejpam-6275	868	10	offers	offer	VERB
ejpam-6275	868	11	better	well	ADJ
ejpam-6275	868	12	performance	performance	NOUN
ejpam-6275	868	13	,	,	PUNCT
ejpam-6275	868	14	m.	m.	NOUN
ejpam-6275	868	15	sajjad	sajjad	PROPN
ejpam-6275	868	16	et	et	PROPN
ejpam-6275	868	17	al	al	PROPN
ejpam-6275	868	18	.	.	PUNCT
ejpam-6275	868	19	/	/	SYM
ejpam-6275	868	20	eur	eur	PROPN
ejpam-6275	868	21	.	.	PUNCT
ejpam-6275	869	1	j.	j.	PROPN
ejpam-6275	869	2	pure	pure	PROPN
ejpam-6275	869	3	appl	appl	PROPN
ejpam-6275	869	4	.	.	PROPN
ejpam-6275	869	5	math	math	PROPN
ejpam-6275	869	6	,	,	PUNCT
ejpam-6275	869	7	18	18	NUM
ejpam-6275	869	8	(	(	PUNCT
ejpam-6275	869	9	3	3	NUM
ejpam-6275	869	10	)	)	PUNCT
ejpam-6275	869	11	(	(	PUNCT
ejpam-6275	869	12	2025	2025	NUM
ejpam-6275	869	13	)	)	PUNCT
ejpam-6275	869	14	,	,	PUNCT
ejpam-6275	869	15	6275	6275	NUM
ejpam-6275	869	16	33	33	NUM
ejpam-6275	869	17	of	of	ADP
ejpam-6275	869	18	36	36	NUM
ejpam-6275	869	19	table	table	NOUN
ejpam-6275	869	20	7	7	NUM
ejpam-6275	869	21	:	:	PUNCT
ejpam-6275	869	22	analysis	analysis	NOUN
ejpam-6275	869	23	of	of	ADP
ejpam-6275	869	24	shortened	shorten	VERB
ejpam-6275	869	25	bch	bch	PROPN
ejpam-6275	869	26	codes	code	NOUN
ejpam-6275	869	27	over	over	ADP
ejpam-6275	869	28	the	the	DET
ejpam-6275	869	29	galois	galois	PROPN
ejpam-6275	869	30	field	field	NOUN
ejpam-6275	869	31	gf(26	gf(26	PROPN
ejpam-6275	869	32	)	)	PUNCT
ejpam-6275	869	33	n	n	PROPN
ejpam-6275	870	1	d	d	PROPN
ejpam-6275	870	2	k	k	PROPN
ejpam-6275	870	3	|c|	|c|	PROPN
ejpam-6275	870	4	=	=	SYM
ejpam-6275	870	5	2k	2k	NUM
ejpam-6275	870	6	r	r	NOUN
ejpam-6275	870	7	=	=	SYM
ejpam-6275	870	8	k	k	NOUN
ejpam-6275	870	9	/	/	SYM
ejpam-6275	870	10	n	n	PROPN
ejpam-6275	870	11	3	3	NUM
ejpam-6275	870	12	3	3	NUM
ejpam-6275	870	13	1	1	NUM
ejpam-6275	870	14	2	2	NUM
ejpam-6275	870	15	0.3333	0.3333	NUM
ejpam-6275	870	16	7	7	NUM
ejpam-6275	870	17	3	3	NUM
ejpam-6275	870	18	4	4	NUM
ejpam-6275	870	19	16	16	NUM
ejpam-6275	870	20	0.5714	0.5714	NUM
ejpam-6275	870	21	5	5	NUM
ejpam-6275	870	22	1	1	NUM
ejpam-6275	870	23	2	2	NUM
ejpam-6275	870	24	0.1429	0.1429	NUM
ejpam-6275	870	25	7	7	NUM
ejpam-6275	870	26	1	1	NUM
ejpam-6275	870	27	2	2	NUM
ejpam-6275	870	28	0.1429	0.1429	NUM
ejpam-6275	870	29	9	9	NUM
ejpam-6275	870	30	3	3	NUM
ejpam-6275	870	31	3	3	NUM
ejpam-6275	870	32	8	8	NUM
ejpam-6275	870	33	0.3333	0.3333	NUM
ejpam-6275	870	34	5	5	NUM
ejpam-6275	870	35	1	1	NUM
ejpam-6275	870	36	2	2	NUM
ejpam-6275	870	37	0.1111	0.1111	NUM
ejpam-6275	870	38	7	7	NUM
ejpam-6275	870	39	1	1	NUM
ejpam-6275	870	40	2	2	NUM
ejpam-6275	870	41	0.1111	0.1111	NUM
ejpam-6275	870	42	9	9	NUM
ejpam-6275	870	43	1	1	NUM
ejpam-6275	870	44	2	2	NUM
ejpam-6275	870	45	0.1111	0.1111	NUM
ejpam-6275	870	46	21	21	NUM
ejpam-6275	870	47	3	3	NUM
ejpam-6275	870	48	15	15	NUM
ejpam-6275	870	49	32768	32768	NUM
ejpam-6275	870	50	0.7143	0.7143	NOUN
ejpam-6275	870	51	5	5	NUM
ejpam-6275	870	52	12	12	NUM
ejpam-6275	870	53	4096	4096	NUM
ejpam-6275	870	54	0.5714	0.5714	NUM
ejpam-6275	870	55	7	7	NUM
ejpam-6275	870	56	6	6	NUM
ejpam-6275	870	57	64	64	NUM
ejpam-6275	870	58	0.2857	0.2857	NUM
ejpam-6275	870	59	9	9	NUM
ejpam-6275	870	60	4	4	NUM
ejpam-6275	870	61	16	16	NUM
ejpam-6275	870	62	0.1905	0.1905	NUM
ejpam-6275	870	63	11	11	NUM
ejpam-6275	870	64	1	1	NUM
ejpam-6275	870	65	2	2	NUM
ejpam-6275	870	66	0.0476	0.0476	NUM
ejpam-6275	870	67	13	13	NUM
ejpam-6275	870	68	na	na	NOUN
ejpam-6275	870	69	na	na	NOUN
ejpam-6275	870	70	na	na	SYM
ejpam-6275	870	71	15	15	NUM
ejpam-6275	870	72	na	na	NOUN
ejpam-6275	870	73	na	na	NOUN
ejpam-6275	870	74	na	na	PROPN
ejpam-6275	870	75	17	17	NUM
ejpam-6275	870	76	na	na	PART
ejpam-6275	870	77	na	na	NOUN
ejpam-6275	870	78	na	na	PROPN
ejpam-6275	870	79	19	19	NUM
ejpam-6275	870	80	na	na	PART
ejpam-6275	870	81	na	na	NOUN
ejpam-6275	870	82	na	na	PART
ejpam-6275	870	83	21	21	NUM
ejpam-6275	870	84	na	na	NOUN
ejpam-6275	870	85	na	na	NOUN
ejpam-6275	870	86	na	na	ADV
ejpam-6275	870	87	such	such	ADJ
ejpam-6275	870	88	as	as	ADP
ejpam-6275	870	89	r	r	NOUN
ejpam-6275	870	90	=	=	SYM
ejpam-6275	870	91	0.3900	0.3900	NUM
ejpam-6275	870	92	at	at	ADP
ejpam-6275	870	93	n	n	NOUN
ejpam-6275	870	94	=	=	NUM
ejpam-6275	870	95	21	21	NUM
ejpam-6275	870	96	,	,	PUNCT
ejpam-6275	870	97	d	d	NOUN
ejpam-6275	870	98	=	=	SYM
ejpam-6275	870	99	9	9	NUM
ejpam-6275	870	100	,	,	PUNCT
ejpam-6275	870	101	compared	compare	VERB
ejpam-6275	870	102	to	to	ADP
ejpam-6275	870	103	r	r	NOUN
ejpam-6275	870	104	=	=	PUNCT
ejpam-6275	870	105	0.2857	0.2857	NUM
ejpam-6275	870	106	in	in	ADP
ejpam-6275	870	107	gf(26	gf(26	NOUN
ejpam-6275	870	108	)	)	PUNCT
ejpam-6275	870	109	.	.	PUNCT
ejpam-6275	871	1	in	in	ADP
ejpam-6275	871	2	summary	summary	NOUN
ejpam-6275	871	3	,	,	PUNCT
ejpam-6275	871	4	the	the	DET
ejpam-6275	871	5	higher	high	ADJ
ejpam-6275	871	6	codeword	codeword	NOUN
ejpam-6275	871	7	diversity	diversity	NOUN
ejpam-6275	871	8	and	and	CCONJ
ejpam-6275	871	9	greater	great	ADJ
ejpam-6275	871	10	dimensions	dimension	NOUN
ejpam-6275	871	11	offered	offer	VERB
ejpam-6275	871	12	by	by	ADP
ejpam-6275	871	13	zp[ω	zp[ω	PROPN
ejpam-6275	871	14	]	]	PUNCT
ejpam-6275	871	15	m/2	m/2	NUM
ejpam-6275	871	16	,	,	PUNCT
ejpam-6275	871	17	where	where	SCONJ
ejpam-6275	871	18	p	p	PRON
ejpam-6275	871	19	≡	≡	PROPN
ejpam-6275	871	20	2	2	NUM
ejpam-6275	871	21	(	(	PUNCT
ejpam-6275	871	22	mod	mod	NOUN
ejpam-6275	871	23	3	3	NUM
ejpam-6275	871	24	)	)	PUNCT
ejpam-6275	871	25	,	,	PUNCT
ejpam-6275	871	26	make	make	VERB
ejpam-6275	871	27	it	it	PRON
ejpam-6275	871	28	particularly	particularly	ADV
ejpam-6275	871	29	suitable	suitable	ADJ
ejpam-6275	871	30	for	for	ADP
ejpam-6275	871	31	applications	application	NOUN
ejpam-6275	871	32	demanding	demand	VERB
ejpam-6275	871	33	robust	robust	ADJ
ejpam-6275	871	34	error	error	NOUN
ejpam-6275	871	35	correction	correction	NOUN
ejpam-6275	871	36	.	.	PUNCT
ejpam-6275	872	1	these	these	DET
ejpam-6275	872	2	benefits	benefit	NOUN
ejpam-6275	872	3	come	come	VERB
ejpam-6275	872	4	at	at	ADP
ejpam-6275	872	5	the	the	DET
ejpam-6275	872	6	cost	cost	NOUN
ejpam-6275	872	7	of	of	ADP
ejpam-6275	872	8	potentially	potentially	ADV
ejpam-6275	872	9	lower	low	ADJ
ejpam-6275	872	10	data	datum	NOUN
ejpam-6275	872	11	rates	rate	NOUN
ejpam-6275	872	12	.	.	PUNCT
ejpam-6275	873	1	furthermore	furthermore	ADV
ejpam-6275	873	2	,	,	PUNCT
ejpam-6275	873	3	the	the	PRON
ejpam-6275	873	4	demonstrated	demonstrate	VERB
ejpam-6275	873	5	higher	high	ADJ
ejpam-6275	873	6	-	-	PUNCT
ejpam-6275	873	7	order	order	NOUN
ejpam-6275	873	8	minimum	minimum	ADJ
ejpam-6275	873	9	distances	distance	NOUN
ejpam-6275	873	10	in	in	ADP
ejpam-6275	873	11	the	the	DET
ejpam-6275	873	12	eisenstein	eisenstein	NOUN
ejpam-6275	873	13	field	field	NOUN
ejpam-6275	873	14	indicate	indicate	VERB
ejpam-6275	873	15	enhanced	enhanced	ADJ
ejpam-6275	873	16	error	error	NOUN
ejpam-6275	873	17	resilience	resilience	NOUN
ejpam-6275	873	18	,	,	PUNCT
ejpam-6275	873	19	positioning	position	VERB
ejpam-6275	873	20	such	such	ADJ
ejpam-6275	873	21	codes	code	NOUN
ejpam-6275	873	22	well	well	ADV
ejpam-6275	873	23	for	for	ADP
ejpam-6275	873	24	next	next	ADJ
ejpam-6275	873	25	-	-	PUNCT
ejpam-6275	873	26	generation	generation	NOUN
ejpam-6275	873	27	communication	communication	NOUN
ejpam-6275	873	28	systems	system	NOUN
ejpam-6275	873	29	and	and	CCONJ
ejpam-6275	873	30	storage	storage	NOUN
ejpam-6275	873	31	solutions	solution	NOUN
ejpam-6275	873	32	requiring	require	VERB
ejpam-6275	873	33	high	high	ADJ
ejpam-6275	873	34	reliability	reliability	NOUN
ejpam-6275	873	35	and	and	CCONJ
ejpam-6275	873	36	minimal	minimal	ADJ
ejpam-6275	873	37	redundancy	redundancy	NOUN
ejpam-6275	873	38	.	.	PUNCT
ejpam-6275	874	1	7	7	X
ejpam-6275	874	2	.	.	X
ejpam-6275	874	3	conclusion	conclusion	NOUN
ejpam-6275	874	4	and	and	CCONJ
ejpam-6275	874	5	future	future	ADJ
ejpam-6275	874	6	directions	direction	NOUN
ejpam-6275	874	7	this	this	DET
ejpam-6275	874	8	article	article	NOUN
ejpam-6275	874	9	proposed	propose	VERB
ejpam-6275	874	10	a	a	DET
ejpam-6275	874	11	new	new	ADJ
ejpam-6275	874	12	approach	approach	NOUN
ejpam-6275	874	13	in	in	ADP
ejpam-6275	874	14	shortened	shorten	VERB
ejpam-6275	874	15	bch	bch	PROPN
ejpam-6275	874	16	codes	code	NOUN
ejpam-6275	874	17	over	over	ADP
ejpam-6275	874	18	the	the	DET
ejpam-6275	874	19	eisenstein	eisenstein	NOUN
ejpam-6275	874	20	fields	field	NOUN
ejpam-6275	874	21	and	and	CCONJ
ejpam-6275	874	22	established	establish	VERB
ejpam-6275	874	23	its	its	PRON
ejpam-6275	874	24	improved	improved	ADJ
ejpam-6275	874	25	error	error	NOUN
ejpam-6275	874	26	correction	correction	NOUN
ejpam-6275	874	27	performance	performance	NOUN
ejpam-6275	874	28	through	through	ADP
ejpam-6275	874	29	modified	modified	PROPN
ejpam-6275	874	30	bma	bma	PROPN
ejpam-6275	874	31	.	.	PUNCT
ejpam-6275	875	1	due	due	ADP
ejpam-6275	875	2	to	to	ADP
ejpam-6275	875	3	the	the	DET
ejpam-6275	875	4	natural	natural	ADJ
ejpam-6275	875	5	extension	extension	NOUN
ejpam-6275	875	6	of	of	ADP
ejpam-6275	875	7	gaussian	gaussian	ADJ
ejpam-6275	875	8	fields	field	NOUN
ejpam-6275	875	9	,	,	PUNCT
ejpam-6275	875	10	the	the	DET
ejpam-6275	875	11	algebraic	algebraic	ADJ
ejpam-6275	875	12	characteristics	characteristic	NOUN
ejpam-6275	875	13	of	of	ADP
ejpam-6275	875	14	eisenstein	eisenstein	PROPN
ejpam-6275	875	15	fields	field	NOUN
ejpam-6275	875	16	turn	turn	VERB
ejpam-6275	875	17	out	out	ADP
ejpam-6275	875	18	to	to	PART
ejpam-6275	875	19	be	be	AUX
ejpam-6275	875	20	suitable	suitable	ADJ
ejpam-6275	875	21	for	for	ADP
ejpam-6275	875	22	the	the	DET
ejpam-6275	875	23	development	development	NOUN
ejpam-6275	875	24	of	of	ADP
ejpam-6275	875	25	efficient	efficient	ADJ
ejpam-6275	875	26	codes	code	NOUN
ejpam-6275	875	27	where	where	SCONJ
ejpam-6275	875	28	high	high	ADJ
ejpam-6275	875	29	reliability	reliability	NOUN
ejpam-6275	875	30	and	and	CCONJ
ejpam-6275	875	31	low	low	ADJ
ejpam-6275	875	32	latency	latency	NOUN
ejpam-6275	875	33	are	be	AUX
ejpam-6275	875	34	desired	desire	VERB
ejpam-6275	875	35	in	in	ADP
ejpam-6275	875	36	noisy	noisy	ADJ
ejpam-6275	875	37	channels	channel	NOUN
ejpam-6275	875	38	.	.	PUNCT
ejpam-6275	876	1	many	many	ADJ
ejpam-6275	876	2	of	of	ADP
ejpam-6275	876	3	the	the	DET
ejpam-6275	876	4	changes	change	NOUN
ejpam-6275	876	5	that	that	PRON
ejpam-6275	876	6	have	have	AUX
ejpam-6275	876	7	been	be	AUX
ejpam-6275	876	8	proposed	propose	VERB
ejpam-6275	876	9	for	for	ADP
ejpam-6275	876	10	the	the	DET
ejpam-6275	876	11	bma	bma	PROPN
ejpam-6275	876	12	strongly	strongly	ADV
ejpam-6275	876	13	contribute	contribute	VERB
ejpam-6275	876	14	to	to	ADP
ejpam-6275	876	15	the	the	DET
ejpam-6275	876	16	growth	growth	NOUN
ejpam-6275	876	17	of	of	ADP
ejpam-6275	876	18	the	the	DET
ejpam-6275	876	19	decoding	decode	VERB
ejpam-6275	876	20	efficiency	efficiency	NOUN
ejpam-6275	876	21	by	by	ADP
ejpam-6275	876	22	taking	take	VERB
ejpam-6275	876	23	into	into	ADP
ejpam-6275	876	24	account	account	NOUN
ejpam-6275	876	25	the	the	DET
ejpam-6275	876	26	computational	computational	ADJ
ejpam-6275	876	27	concerns	concern	NOUN
ejpam-6275	876	28	of	of	ADP
ejpam-6275	876	29	using	use	VERB
ejpam-6275	876	30	eisenstein	eisenstein	PROPN
ejpam-6275	876	31	field	field	PROPN
ejpam-6275	876	32	arithmetic	arithmetic	ADJ
ejpam-6275	876	33	.	.	PUNCT
ejpam-6275	877	1	therefore	therefore	ADV
ejpam-6275	877	2	,	,	PUNCT
ejpam-6275	877	3	this	this	DET
ejpam-6275	877	4	study	study	NOUN
ejpam-6275	877	5	adds	add	VERB
ejpam-6275	877	6	value	value	NOUN
ejpam-6275	877	7	to	to	ADP
ejpam-6275	877	8	the	the	DET
ejpam-6275	877	9	development	development	NOUN
ejpam-6275	877	10	of	of	ADP
ejpam-6275	877	11	the	the	DET
ejpam-6275	877	12	coding	code	VERB
ejpam-6275	877	13	theory	theory	NOUN
ejpam-6275	877	14	by	by	ADP
ejpam-6275	877	15	bringing	bring	VERB
ejpam-6275	877	16	conceptuality	conceptuality	NOUN
ejpam-6275	877	17	in	in	ADP
ejpam-6275	877	18	theory	theory	NOUN
ejpam-6275	877	19	and	and	CCONJ
ejpam-6275	877	20	idea	idea	NOUN
ejpam-6275	877	21	in	in	ADP
ejpam-6275	877	22	application	application	NOUN
ejpam-6275	877	23	in	in	ADP
ejpam-6275	877	24	order	order	NOUN
ejpam-6275	877	25	to	to	PART
ejpam-6275	877	26	accommodate	accommodate	VERB
ejpam-6275	877	27	future	future	ADJ
ejpam-6275	877	28	issues	issue	NOUN
ejpam-6275	877	29	with	with	ADP
ejpam-6275	877	30	respect	respect	NOUN
ejpam-6275	877	31	to	to	ADP
ejpam-6275	877	32	reliable	reliable	ADJ
ejpam-6275	877	33	data	data	NOUN
ejpam-6275	877	34	transmission	transmission	NOUN
ejpam-6275	877	35	.	.	PUNCT
ejpam-6275	878	1	efficient	efficient	ADJ
ejpam-6275	878	2	implementations	implementation	NOUN
ejpam-6275	878	3	of	of	ADP
ejpam-6275	878	4	the	the	DET
ejpam-6275	878	5	proposed	propose	VERB
ejpam-6275	878	6	codes	code	NOUN
ejpam-6275	878	7	and	and	CCONJ
ejpam-6275	878	8	decoding	decode	VERB
ejpam-6275	878	9	algorithms	algorithm	NOUN
ejpam-6275	878	10	in	in	ADP
ejpam-6275	878	11	communication	communication	NOUN
ejpam-6275	878	12	structures	structure	NOUN
ejpam-6275	878	13	are	be	AUX
ejpam-6275	878	14	possible	possible	ADJ
ejpam-6275	878	15	if	if	SCONJ
ejpam-6275	878	16	the	the	DET
ejpam-6275	878	17	suggested	suggest	VERB
ejpam-6275	878	18	structures	structure	NOUN
ejpam-6275	878	19	are	be	AUX
ejpam-6275	878	20	used	use	VERB
ejpam-6275	878	21	.	.	PUNCT
ejpam-6275	879	1	the	the	DET
ejpam-6275	879	2	extension	extension	NOUN
ejpam-6275	879	3	m.	m.	NOUN
ejpam-6275	879	4	sajjad	sajjad	PROPN
ejpam-6275	879	5	et	et	PROPN
ejpam-6275	879	6	al	al	PROPN
ejpam-6275	879	7	.	.	PUNCT
ejpam-6275	879	8	/	/	SYM
ejpam-6275	879	9	eur	eur	PROPN
ejpam-6275	879	10	.	.	PUNCT
ejpam-6275	880	1	j.	j.	PROPN
ejpam-6275	880	2	pure	pure	PROPN
ejpam-6275	880	3	appl	appl	PROPN
ejpam-6275	880	4	.	.	PROPN
ejpam-6275	880	5	math	math	PROPN
ejpam-6275	880	6	,	,	PUNCT
ejpam-6275	880	7	18	18	NUM
ejpam-6275	880	8	(	(	PUNCT
ejpam-6275	880	9	3	3	NUM
ejpam-6275	880	10	)	)	PUNCT
ejpam-6275	880	11	(	(	PUNCT
ejpam-6275	880	12	2025	2025	NUM
ejpam-6275	880	13	)	)	PUNCT
ejpam-6275	880	14	,	,	PUNCT
ejpam-6275	880	15	6275	6275	NUM
ejpam-6275	880	16	34	34	NUM
ejpam-6275	880	17	of	of	ADP
ejpam-6275	880	18	36	36	NUM
ejpam-6275	880	19	table	table	NOUN
ejpam-6275	880	20	8	8	NUM
ejpam-6275	880	21	:	:	PUNCT
ejpam-6275	880	22	analysis	analysis	NOUN
ejpam-6275	880	23	of	of	ADP
ejpam-6275	880	24	shortened	shorten	VERB
ejpam-6275	880	25	bch	bch	PROPN
ejpam-6275	880	26	codes	code	NOUN
ejpam-6275	880	27	over	over	ADP
ejpam-6275	880	28	the	the	DET
ejpam-6275	880	29	eisenstein	eisenstein	PROPN
ejpam-6275	880	30	field	field	PROPN
ejpam-6275	880	31	z2[ω	z2[ω	NOUN
ejpam-6275	880	32	]	]	X
ejpam-6275	881	1	3	3	NUM
ejpam-6275	881	2	n	n	PROPN
ejpam-6275	881	3	d	d	PROPN
ejpam-6275	881	4	k	k	PROPN
ejpam-6275	881	5	|c|	|c|	PROPN
ejpam-6275	881	6	=	=	SYM
ejpam-6275	881	7	22k	22k	NOUN
ejpam-6275	881	8	r	r	NOUN
ejpam-6275	881	9	=	=	SYM
ejpam-6275	881	10	k	k	NOUN
ejpam-6275	881	11	/	/	SYM
ejpam-6275	881	12	n	n	PROPN
ejpam-6275	881	13	3	3	NUM
ejpam-6275	881	14	3	3	NUM
ejpam-6275	881	15	1	1	NUM
ejpam-6275	881	16	4	4	NUM
ejpam-6275	881	17	0.3333	0.3333	NUM
ejpam-6275	881	18	7	7	NUM
ejpam-6275	881	19	3	3	NUM
ejpam-6275	881	20	1	1	NUM
ejpam-6275	881	21	4	4	NUM
ejpam-6275	881	22	0.1429	0.1429	NUM
ejpam-6275	881	23	5	5	NUM
ejpam-6275	881	24	2	2	NUM
ejpam-6275	881	25	16	16	NUM
ejpam-6275	881	26	0.2857	0.2857	NUM
ejpam-6275	881	27	7	7	NUM
ejpam-6275	881	28	1	1	NUM
ejpam-6275	881	29	4	4	NUM
ejpam-6275	881	30	0.1429	0.1429	NUM
ejpam-6275	881	31	9	9	NUM
ejpam-6275	881	32	3	3	NUM
ejpam-6275	881	33	1	1	NUM
ejpam-6275	881	34	4	4	NUM
ejpam-6275	881	35	0.1111	0.1111	NUM
ejpam-6275	881	36	5	5	NUM
ejpam-6275	881	37	2	2	NUM
ejpam-6275	881	38	16	16	NUM
ejpam-6275	881	39	0.2222	0.2222	NUM
ejpam-6275	881	40	7	7	NUM
ejpam-6275	881	41	1	1	NUM
ejpam-6275	881	42	4	4	NUM
ejpam-6275	881	43	0.1111	0.1111	NUM
ejpam-6275	881	44	9	9	NUM
ejpam-6275	881	45	1	1	NUM
ejpam-6275	881	46	4	4	NUM
ejpam-6275	881	47	0.1111	0.1111	NUM
ejpam-6275	881	48	21	21	NUM
ejpam-6275	881	49	3	3	NUM
ejpam-6275	881	50	15	15	NUM
ejpam-6275	881	51	1073741824	1073741824	NUM
ejpam-6275	881	52	0.7143	0.7143	NOUN
ejpam-6275	881	53	5	5	NUM
ejpam-6275	881	54	12	12	NUM
ejpam-6275	881	55	16777216	16777216	NUM
ejpam-6275	881	56	0.5714	0.5714	NUM
ejpam-6275	881	57	7	7	NUM
ejpam-6275	881	58	9	9	NUM
ejpam-6275	881	59	262144	262144	NUM
ejpam-6275	881	60	0.4286	0.4286	NUM
ejpam-6275	881	61	9	9	NUM
ejpam-6275	881	62	8	8	NUM
ejpam-6275	881	63	65536	65536	NUM
ejpam-6275	881	64	0.3900	0.3900	NUM
ejpam-6275	881	65	11	11	NUM
ejpam-6275	881	66	2	2	NUM
ejpam-6275	881	67	16	16	NUM
ejpam-6275	881	68	0.0952	0.0952	NUM
ejpam-6275	881	69	13	13	NUM
ejpam-6275	881	70	1	1	NUM
ejpam-6275	881	71	2	2	NUM
ejpam-6275	881	72	0.0476	0.0476	NUM
ejpam-6275	881	73	15	15	NUM
ejpam-6275	881	74	1	1	NUM
ejpam-6275	881	75	2	2	NUM
ejpam-6275	881	76	0.0476	0.0476	NUM
ejpam-6275	881	77	17	17	NUM
ejpam-6275	881	78	1	1	NUM
ejpam-6275	881	79	2	2	NUM
ejpam-6275	881	80	0.0476	0.0476	NUM
ejpam-6275	881	81	19	19	NUM
ejpam-6275	881	82	1	1	NUM
ejpam-6275	881	83	2	2	NUM
ejpam-6275	881	84	0.0476	0.0476	NUM
ejpam-6275	881	85	21	21	NUM
ejpam-6275	881	86	1	1	NUM
ejpam-6275	881	87	2	2	NUM
ejpam-6275	881	88	0.0476	0.0476	NUM
ejpam-6275	881	89	of	of	ADP
ejpam-6275	881	90	these	these	DET
ejpam-6275	881	91	methods	method	NOUN
ejpam-6275	881	92	into	into	ADP
ejpam-6275	881	93	multidimensional	multidimensional	ADJ
ejpam-6275	881	94	and	and	CCONJ
ejpam-6275	881	95	spatially	spatially	ADV
ejpam-6275	881	96	coupled	couple	VERB
ejpam-6275	881	97	codes	code	NOUN
ejpam-6275	881	98	holds	hold	VERB
ejpam-6275	881	99	great	great	ADJ
ejpam-6275	881	100	potential	potential	NOUN
ejpam-6275	881	101	to	to	PART
ejpam-6275	881	102	be	be	AUX
ejpam-6275	881	103	useful	useful	ADJ
ejpam-6275	881	104	in	in	ADP
ejpam-6275	881	105	new	new	ADJ
ejpam-6275	881	106	fields	field	NOUN
ejpam-6275	881	107	such	such	ADJ
ejpam-6275	881	108	as	as	ADP
ejpam-6275	881	109	storage	storage	NOUN
ejpam-6275	881	110	systems	system	NOUN
ejpam-6275	881	111	and	and	CCONJ
ejpam-6275	881	112	sensor	sensor	NOUN
ejpam-6275	881	113	networks	network	NOUN
ejpam-6275	881	114	.	.	PUNCT
ejpam-6275	882	1	likely	likely	ADV
ejpam-6275	882	2	expanding	expand	VERB
ejpam-6275	882	3	the	the	DET
ejpam-6275	882	4	areas	area	NOUN
ejpam-6275	882	5	of	of	ADP
ejpam-6275	882	6	application	application	NOUN
ejpam-6275	882	7	in	in	ADP
ejpam-6275	882	8	qec	qec	NOUN
ejpam-6275	882	9	and	and	CCONJ
ejpam-6275	882	10	in	in	ADP
ejpam-6275	882	11	the	the	DET
ejpam-6275	882	12	further	further	ADJ
ejpam-6275	882	13	advancement	advancement	NOUN
ejpam-6275	882	14	of	of	ADP
ejpam-6275	882	15	next	next	ADJ
ejpam-6275	882	16	-	-	PUNCT
ejpam-6275	882	17	generation	generation	NOUN
ejpam-6275	882	18	wireless	wireless	NOUN
ejpam-6275	882	19	networks	network	NOUN
ejpam-6275	882	20	,	,	PUNCT
ejpam-6275	882	21	such	such	ADJ
ejpam-6275	882	22	as	as	ADP
ejpam-6275	882	23	6	6	NUM
ejpam-6275	882	24	g	g	NOUN
ejpam-6275	882	25	,	,	PUNCT
ejpam-6275	882	26	the	the	DET
ejpam-6275	882	27	techniques	technique	NOUN
ejpam-6275	882	28	based	base	VERB
ejpam-6275	882	29	on	on	ADP
ejpam-6275	882	30	the	the	DET
ejpam-6275	882	31	eisenstein	eisenstein	PROPN
ejpam-6275	882	32	field	field	NOUN
ejpam-6275	882	33	-	-	PUNCT
ejpam-6275	882	34	based	base	VERB
ejpam-6275	882	35	codes	code	NOUN
ejpam-6275	882	36	can	can	AUX
ejpam-6275	882	37	bring	bring	VERB
ejpam-6275	882	38	added	add	VERB
ejpam-6275	882	39	value	value	NOUN
ejpam-6275	882	40	to	to	ADP
ejpam-6275	882	41	the	the	DET
ejpam-6275	882	42	transformative	transformative	ADJ
ejpam-6275	882	43	technologies	technology	NOUN
ejpam-6275	882	44	.	.	PUNCT
ejpam-6275	883	1	acknowledgements	acknowledgement	NOUN
ejpam-6275	883	2	this	this	DET
ejpam-6275	883	3	research	research	NOUN
ejpam-6275	883	4	is	be	AUX
ejpam-6275	883	5	supported	support	VERB
ejpam-6275	883	6	by	by	ADP
ejpam-6275	883	7	the	the	DET
ejpam-6275	883	8	ongoing	ongoing	ADJ
ejpam-6275	883	9	research	research	NOUN
ejpam-6275	883	10	funding	funding	NOUN
ejpam-6275	883	11	program	program	NOUN
ejpam-6275	883	12	research	research	NOUN
ejpam-6275	883	13	chairs	chair	NOUN
ejpam-6275	883	14	(	(	PUNCT
ejpam-6275	883	15	orf	orf	ADJ
ejpam-6275	883	16	-	-	PUNCT
ejpam-6275	883	17	rc-2025	rc-2025	NOUN
ejpam-6275	883	18	-	-	PUNCT
ejpam-6275	883	19	5300	5300	NUM
ejpam-6275	883	20	)	)	PUNCT
ejpam-6275	883	21	,	,	PUNCT
ejpam-6275	883	22	king	king	PROPN
ejpam-6275	883	23	saud	saud	PROPN
ejpam-6275	883	24	university	university	PROPN
ejpam-6275	883	25	,	,	PUNCT
ejpam-6275	883	26	riyadh	riyadh	PROPN
ejpam-6275	883	27	,	,	PUNCT
ejpam-6275	883	28	saudi	saudi	PROPN
ejpam-6275	883	29	arabia	arabia	PROPN
ejpam-6275	883	30	.	.	PUNCT
ejpam-6275	884	1	tribute	tribute	NOUN
ejpam-6275	884	2	we	we	PRON
ejpam-6275	884	3	wish	wish	VERB
ejpam-6275	884	4	to	to	PART
ejpam-6275	884	5	express	express	VERB
ejpam-6275	884	6	our	our	PRON
ejpam-6275	884	7	heartfelt	heartfelt	ADJ
ejpam-6275	884	8	gratitude	gratitude	NOUN
ejpam-6275	884	9	to	to	ADP
ejpam-6275	884	10	our	our	PRON
ejpam-6275	884	11	beloved	beloved	ADJ
ejpam-6275	884	12	supervisor	supervisor	NOUN
ejpam-6275	884	13	,	,	PUNCT
ejpam-6275	884	14	professor	professor	PROPN
ejpam-6275	884	15	dr	dr	PROPN
ejpam-6275	884	16	.	.	PROPN
ejpam-6275	884	17	tariq	tariq	PROPN
ejpam-6275	884	18	shah	shah	PROPN
ejpam-6275	884	19	(	(	PUNCT
ejpam-6275	884	20	late	late	ADJ
ejpam-6275	884	21	)	)	PUNCT
ejpam-6275	884	22	,	,	PUNCT
ejpam-6275	884	23	whose	whose	DET
ejpam-6275	884	24	exceptional	exceptional	ADJ
ejpam-6275	884	25	guidance	guidance	NOUN
ejpam-6275	884	26	,	,	PUNCT
ejpam-6275	884	27	profound	profound	ADJ
ejpam-6275	884	28	knowledge	knowledge	NOUN
ejpam-6275	884	29	,	,	PUNCT
ejpam-6275	884	30	and	and	CCONJ
ejpam-6275	884	31	unwavering	unwavere	VERB
ejpam-6275	884	32	support	support	NOUN
ejpam-6275	884	33	profoundly	profoundly	ADV
ejpam-6275	884	34	shaped	shape	VERB
ejpam-6275	884	35	our	our	PRON
ejpam-6275	884	36	journey	journey	NOUN
ejpam-6275	884	37	as	as	ADP
ejpam-6275	884	38	researchers	researcher	NOUN
ejpam-6275	884	39	in	in	ADP
ejpam-6275	884	40	algebra	algebra	NOUN
ejpam-6275	884	41	,	,	PUNCT
ejpam-6275	884	42	number	number	NOUN
ejpam-6275	884	43	theory	theory	NOUN
ejpam-6275	884	44	,	,	PUNCT
ejpam-6275	884	45	coding	code	VERB
ejpam-6275	884	46	theory	theory	NOUN
ejpam-6275	884	47	,	,	PUNCT
ejpam-6275	884	48	and	and	CCONJ
ejpam-6275	884	49	cryptography	cryptography	NOUN
ejpam-6275	884	50	.	.	PUNCT
ejpam-6275	885	1	his	his	PRON
ejpam-6275	885	2	mentorship	mentorship	NOUN
ejpam-6275	885	3	was	be	AUX
ejpam-6275	885	4	a	a	DET
ejpam-6275	885	5	cornerstone	cornerstone	NOUN
ejpam-6275	885	6	of	of	ADP
ejpam-6275	885	7	our	our	PRON
ejpam-6275	885	8	academic	academic	ADJ
ejpam-6275	885	9	and	and	CCONJ
ejpam-6275	885	10	personal	personal	ADJ
ejpam-6275	885	11	growth	growth	NOUN
ejpam-6275	885	12	,	,	PUNCT
ejpam-6275	885	13	and	and	CCONJ
ejpam-6275	885	14	his	his	PRON
ejpam-6275	885	15	presence	presence	NOUN
ejpam-6275	885	16	continues	continue	VERB
ejpam-6275	885	17	to	to	PART
ejpam-6275	885	18	inspire	inspire	VERB
ejpam-6275	885	19	our	our	PRON
ejpam-6275	885	20	scholarly	scholarly	ADJ
ejpam-6275	885	21	endeavours	endeavour	NOUN
ejpam-6275	885	22	.	.	PUNCT
ejpam-6275	886	1	may	may	AUX
ejpam-6275	886	2	his	his	PRON
ejpam-6275	886	3	soul	soul	NOUN
ejpam-6275	886	4	rest	rest	VERB
ejpam-6275	886	5	in	in	ADP
ejpam-6275	886	6	eternal	eternal	ADJ
ejpam-6275	886	7	peace	peace	NOUN
ejpam-6275	886	8	.	.	PUNCT
ejpam-6275	887	1	m.	m.	NOUN
ejpam-6275	887	2	sajjad	sajjad	PROPN
ejpam-6275	887	3	et	et	PROPN
ejpam-6275	887	4	al	al	PROPN
ejpam-6275	887	5	.	.	PUNCT
ejpam-6275	887	6	/	/	SYM
ejpam-6275	887	7	eur	eur	PROPN
ejpam-6275	887	8	.	.	PUNCT
ejpam-6275	888	1	j.	j.	PROPN
ejpam-6275	888	2	pure	pure	PROPN
ejpam-6275	888	3	appl	appl	PROPN
ejpam-6275	888	4	.	.	PROPN
ejpam-6275	888	5	math	math	PROPN
ejpam-6275	888	6	,	,	PUNCT
ejpam-6275	888	7	18	18	NUM
ejpam-6275	888	8	(	(	PUNCT
ejpam-6275	888	9	3	3	NUM
ejpam-6275	888	10	)	)	PUNCT
ejpam-6275	888	11	(	(	PUNCT
ejpam-6275	888	12	2025	2025	NUM
ejpam-6275	888	13	)	)	PUNCT
ejpam-6275	888	14	,	,	PUNCT
ejpam-6275	888	15	6275	6275	NUM
ejpam-6275	888	16	35	35	NUM
ejpam-6275	888	17	of	of	ADP
ejpam-6275	888	18	36	36	NUM
ejpam-6275	888	19	data	datum	NOUN
ejpam-6275	888	20	availability	availability	NOUN
ejpam-6275	888	21	all	all	DET
ejpam-6275	888	22	the	the	DET
ejpam-6275	888	23	data	datum	NOUN
ejpam-6275	888	24	given	give	VERB
ejpam-6275	888	25	in	in	ADP
ejpam-6275	888	26	this	this	DET
ejpam-6275	888	27	article	article	NOUN
ejpam-6275	888	28	.	.	PUNCT
ejpam-6275	889	1	references	reference	NOUN
ejpam-6275	889	2	[	[	X
ejpam-6275	889	3	1	1	NUM
ejpam-6275	889	4	]	]	PUNCT
ejpam-6275	889	5	gilberto	gilberto	PROPN
ejpam-6275	889	6	bini	bini	PROPN
ejpam-6275	889	7	and	and	CCONJ
ejpam-6275	889	8	fabio	fabio	PROPN
ejpam-6275	889	9	flamini	flamini	PROPN
ejpam-6275	889	10	.	.	PUNCT
ejpam-6275	890	1	finite	finite	PROPN
ejpam-6275	890	2	commutative	commutative	ADJ
ejpam-6275	890	3	rings	ring	NOUN
ejpam-6275	890	4	and	and	CCONJ
ejpam-6275	890	5	their	their	PRON
ejpam-6275	890	6	applications	application	NOUN
ejpam-6275	890	7	,	,	PUNCT
ejpam-6275	890	8	volume	volume	NOUN
ejpam-6275	890	9	680	680	NUM
ejpam-6275	890	10	.	.	PUNCT
ejpam-6275	891	1	springer	springer	PROPN
ejpam-6275	891	2	science	science	PROPN
ejpam-6275	891	3	&	&	CCONJ
ejpam-6275	891	4	business	business	NOUN
ejpam-6275	891	5	media	medium	NOUN
ejpam-6275	891	6	,	,	PUNCT
ejpam-6275	891	7	2012	2012	NUM
ejpam-6275	891	8	.	.	PUNCT
ejpam-6275	892	1	[	[	X
ejpam-6275	892	2	2	2	X
ejpam-6275	892	3	]	]	X
ejpam-6275	892	4	robert	robert	PROPN
ejpam-6275	892	5	gilmer	gilmer	PROPN
ejpam-6275	892	6	.	.	PUNCT
ejpam-6275	893	1	commutative	commutative	PROPN
ejpam-6275	893	2	semigroup	semigroup	PROPN
ejpam-6275	893	3	rings	ring	NOUN
ejpam-6275	893	4	.	.	PUNCT
ejpam-6275	894	1	university	university	PROPN
ejpam-6275	894	2	of	of	ADP
ejpam-6275	894	3	chicago	chicago	PROPN
ejpam-6275	894	4	press	press	PROPN
ejpam-6275	894	5	,	,	PUNCT
ejpam-6275	894	6	1984	1984	NUM
ejpam-6275	894	7	.	.	PUNCT
ejpam-6275	895	1	[	[	X
ejpam-6275	895	2	3	3	X
ejpam-6275	895	3	]	]	X
ejpam-6275	895	4	anthony	anthony	PROPN
ejpam-6275	895	5	w	w	PROPN
ejpam-6275	895	6	knapp	knapp	PROPN
ejpam-6275	895	7	.	.	PUNCT
ejpam-6275	895	8	basic	basic	ADJ
ejpam-6275	895	9	real	real	ADJ
ejpam-6275	895	10	analysis	analysis	NOUN
ejpam-6275	895	11	.	.	PUNCT
ejpam-6275	896	1	springer	springer	NOUN
ejpam-6275	896	2	science	science	PROPN
ejpam-6275	896	3	&	&	CCONJ
ejpam-6275	896	4	business	business	NOUN
ejpam-6275	896	5	media	medium	NOUN
ejpam-6275	896	6	,	,	PUNCT
ejpam-6275	896	7	2007	2007	NUM
ejpam-6275	896	8	.	.	PUNCT
ejpam-6275	897	1	[	[	X
ejpam-6275	897	2	4	4	NUM
ejpam-6275	897	3	]	]	X
ejpam-6275	897	4	sanjay	sanjay	ADJ
ejpam-6275	897	5	r	r	NOUN
ejpam-6275	897	6	nagpaul	nagpaul	NOUN
ejpam-6275	897	7	.	.	PUNCT
ejpam-6275	898	1	topics	topic	NOUN
ejpam-6275	898	2	in	in	ADP
ejpam-6275	898	3	applied	applied	ADJ
ejpam-6275	898	4	abstract	abstract	ADJ
ejpam-6275	898	5	algebra	algebra	NOUN
ejpam-6275	898	6	,	,	PUNCT
ejpam-6275	898	7	volume	volume	NOUN
ejpam-6275	898	8	15	15	NUM
ejpam-6275	898	9	.	.	PUNCT
ejpam-6275	899	1	american	american	PROPN
ejpam-6275	899	2	mathematical	mathematical	PROPN
ejpam-6275	899	3	society	society	NOUN
ejpam-6275	899	4	,	,	PUNCT
ejpam-6275	899	5	2005	2005	NUM
ejpam-6275	899	6	.	.	PUNCT
ejpam-6275	900	1	[	[	X
ejpam-6275	900	2	5	5	NUM
ejpam-6275	900	3	]	]	X
ejpam-6275	900	4	tariq	tariq	NOUN
ejpam-6275	900	5	shah	shah	PROPN
ejpam-6275	900	6	,	,	PUNCT
ejpam-6275	900	7	sobia	sobia	NOUN
ejpam-6275	900	8	farwa	farwa	NOUN
ejpam-6275	900	9	,	,	PUNCT
ejpam-6275	900	10	and	and	CCONJ
ejpam-6275	900	11	rana	rana	PROPN
ejpam-6275	900	12	s	s	PROPN
ejpam-6275	900	13	badar	badar	PROPN
ejpam-6275	900	14	.	.	PUNCT
ejpam-6275	901	1	a	a	DET
ejpam-6275	901	2	generalization	generalization	NOUN
ejpam-6275	901	3	of	of	ADP
ejpam-6275	901	4	integral	integral	ADJ
ejpam-6275	901	5	dependence	dependence	NOUN
ejpam-6275	901	6	.	.	PUNCT
ejpam-6275	902	1	advances	advance	NOUN
ejpam-6275	902	2	in	in	ADP
ejpam-6275	902	3	algebra	algebra	NOUN
ejpam-6275	902	4	,	,	PUNCT
ejpam-6275	902	5	3(1):43–55	3(1):43–55	NUM
ejpam-6275	902	6	,	,	PUNCT
ejpam-6275	902	7	2010	2010	NUM
ejpam-6275	902	8	.	.	PUNCT
ejpam-6275	903	1	[	[	X
ejpam-6275	903	2	6	6	NUM
ejpam-6275	903	3	]	]	PUNCT
ejpam-6275	903	4	claude	claude	PROPN
ejpam-6275	903	5	e	e	PROPN
ejpam-6275	903	6	shannon	shannon	PROPN
ejpam-6275	903	7	.	.	PUNCT
ejpam-6275	904	1	a	a	DET
ejpam-6275	904	2	mathematical	mathematical	ADJ
ejpam-6275	904	3	theory	theory	NOUN
ejpam-6275	904	4	of	of	ADP
ejpam-6275	904	5	communication	communication	NOUN
ejpam-6275	904	6	.	.	PUNCT
ejpam-6275	905	1	the	the	DET
ejpam-6275	905	2	bell	bell	PROPN
ejpam-6275	905	3	system	system	PROPN
ejpam-6275	905	4	technical	technical	ADJ
ejpam-6275	905	5	journal	journal	NOUN
ejpam-6275	905	6	,	,	PUNCT
ejpam-6275	905	7	27(3):379–423	27(3):379–423	PROPN
ejpam-6275	905	8	,	,	PUNCT
ejpam-6275	905	9	1948	1948	NUM
ejpam-6275	905	10	.	.	PUNCT
ejpam-6275	906	1	[	[	X
ejpam-6275	906	2	7	7	X
ejpam-6275	906	3	]	]	X
ejpam-6275	906	4	hsien	hsien	PROPN
ejpam-6275	906	5	t	t	PROPN
ejpam-6275	906	6	hsu	hsu	PROPN
ejpam-6275	906	7	.	.	PUNCT
ejpam-6275	907	1	a	a	DET
ejpam-6275	907	2	class	class	NOUN
ejpam-6275	907	3	of	of	ADP
ejpam-6275	907	4	binary	binary	NOUN
ejpam-6275	907	5	shortened	shorten	VERB
ejpam-6275	907	6	cyclic	cyclic	NOUN
ejpam-6275	907	7	codes	code	NOUN
ejpam-6275	907	8	for	for	ADP
ejpam-6275	907	9	a	a	DET
ejpam-6275	907	10	compound	compound	NOUN
ejpam-6275	907	11	channel	channel	NOUN
ejpam-6275	907	12	.	.	PUNCT
ejpam-6275	908	1	information	information	NOUN
ejpam-6275	908	2	and	and	CCONJ
ejpam-6275	908	3	control	control	NOUN
ejpam-6275	908	4	,	,	PUNCT
ejpam-6275	908	5	18(2):126–139	18(2):126–139	PROPN
ejpam-6275	908	6	,	,	PUNCT
ejpam-6275	908	7	1971	1971	NUM
ejpam-6275	908	8	.	.	PUNCT
ejpam-6275	909	1	[	[	X
ejpam-6275	909	2	8	8	NUM
ejpam-6275	909	3	]	]	PUNCT
ejpam-6275	909	4	tadao	tadao	PROPN
ejpam-6275	909	5	kasami	kasami	PROPN
ejpam-6275	909	6	.	.	PUNCT
ejpam-6275	910	1	optimum	optimum	ADJ
ejpam-6275	910	2	shortened	shorten	VERB
ejpam-6275	910	3	cyclic	cyclic	ADJ
ejpam-6275	910	4	codes	code	NOUN
ejpam-6275	910	5	for	for	ADP
ejpam-6275	910	6	burst	burst	NOUN
ejpam-6275	910	7	-	-	PUNCT
ejpam-6275	910	8	error	error	NOUN
ejpam-6275	910	9	correction	correction	NOUN
ejpam-6275	910	10	.	.	PUNCT
ejpam-6275	911	1	ieee	ieee	NOUN
ejpam-6275	911	2	transactions	transaction	NOUN
ejpam-6275	911	3	on	on	ADP
ejpam-6275	911	4	information	information	NOUN
ejpam-6275	911	5	theory	theory	NOUN
ejpam-6275	911	6	,	,	PUNCT
ejpam-6275	911	7	9(2):105–109	9(2):105–109	NUM
ejpam-6275	911	8	,	,	PUNCT
ejpam-6275	911	9	1963	1963	NUM
ejpam-6275	911	10	.	.	PUNCT
ejpam-6275	912	1	[	[	X
ejpam-6275	912	2	9	9	NUM
ejpam-6275	912	3	]	]	PUNCT
ejpam-6275	912	4	h	h	NOUN
ejpam-6275	912	5	helgert	helgert	NOUN
ejpam-6275	912	6	and	and	CCONJ
ejpam-6275	912	7	r	r	PROPN
ejpam-6275	912	8	stinaff	stinaff	PROPN
ejpam-6275	912	9	.	.	PUNCT
ejpam-6275	913	1	shortened	shorten	VERB
ejpam-6275	913	2	bch	bch	PROPN
ejpam-6275	913	3	codes	code	NOUN
ejpam-6275	913	4	(	(	PUNCT
ejpam-6275	913	5	corresp	corresp	PROPN
ejpam-6275	913	6	.	.	PUNCT
ejpam-6275	913	7	)	)	PUNCT
ejpam-6275	913	8	.	.	PUNCT
ejpam-6275	914	1	ieee	ieee	NOUN
ejpam-6275	914	2	transactions	transaction	NOUN
ejpam-6275	914	3	on	on	ADP
ejpam-6275	914	4	information	information	NOUN
ejpam-6275	914	5	theory	theory	NOUN
ejpam-6275	914	6	,	,	PUNCT
ejpam-6275	914	7	19(6):818–820	19(6):818–820	PROPN
ejpam-6275	914	8	,	,	PUNCT
ejpam-6275	914	9	1973	1973	NUM
ejpam-6275	914	10	.	.	PUNCT
ejpam-6275	915	1	[	[	X
ejpam-6275	915	2	10	10	NUM
ejpam-6275	915	3	]	]	X
ejpam-6275	915	4	cunsheng	cunsheng	PROPN
ejpam-6275	915	5	ding	ding	PROPN
ejpam-6275	915	6	,	,	PUNCT
ejpam-6275	915	7	jinyuan	jinyuan	PROPN
ejpam-6275	915	8	luo	luo	PROPN
ejpam-6275	915	9	,	,	PUNCT
ejpam-6275	915	10	and	and	CCONJ
ejpam-6275	915	11	harald	harald	PROPN
ejpam-6275	915	12	niederreiter	niederreiter	PROPN
ejpam-6275	915	13	.	.	PUNCT
ejpam-6275	916	1	two	two	NUM
ejpam-6275	916	2	-	-	PUNCT
ejpam-6275	916	3	weight	weight	NOUN
ejpam-6275	916	4	codes	code	NOUN
ejpam-6275	916	5	punctured	puncture	VERB
ejpam-6275	916	6	from	from	ADP
ejpam-6275	916	7	irreducible	irreducible	ADJ
ejpam-6275	916	8	cyclic	cyclic	ADJ
ejpam-6275	916	9	codes	code	NOUN
ejpam-6275	916	10	.	.	PUNCT
ejpam-6275	917	1	in	in	ADP
ejpam-6275	917	2	coding	coding	NOUN
ejpam-6275	917	3	and	and	CCONJ
ejpam-6275	917	4	cryptology	cryptology	NOUN
ejpam-6275	917	5	,	,	PUNCT
ejpam-6275	917	6	pages	page	NOUN
ejpam-6275	917	7	119–124	119–124	NUM
ejpam-6275	917	8	,	,	PUNCT
ejpam-6275	917	9	2008	2008	NUM
ejpam-6275	917	10	.	.	PUNCT
ejpam-6275	918	1	[	[	X
ejpam-6275	918	2	11	11	NUM
ejpam-6275	918	3	]	]	X
ejpam-6275	918	4	cunsheng	cunsheng	PROPN
ejpam-6275	918	5	ding	ding	PROPN
ejpam-6275	918	6	and	and	CCONJ
ejpam-6275	918	7	harald	harald	PROPN
ejpam-6275	918	8	niederreiter	niederreiter	PROPN
ejpam-6275	918	9	.	.	PUNCT
ejpam-6275	919	1	cyclotomic	cyclotomic	ADJ
ejpam-6275	919	2	linear	linear	PROPN
ejpam-6275	919	3	codes	code	NOUN
ejpam-6275	919	4	of	of	ADP
ejpam-6275	919	5	order	order	NOUN
ejpam-6275	919	6	$	$	SYM
ejpam-6275	919	7	3$.	3$.	NUM
ejpam-6275	919	8	ieee	ieee	NOUN
ejpam-6275	919	9	transactions	transaction	NOUN
ejpam-6275	919	10	on	on	ADP
ejpam-6275	919	11	information	information	NOUN
ejpam-6275	919	12	theory	theory	NOUN
ejpam-6275	919	13	,	,	PUNCT
ejpam-6275	919	14	53(6):2274–2277	53(6):2274–2277	NUM
ejpam-6275	919	15	,	,	PUNCT
ejpam-6275	919	16	2007	2007	NUM
ejpam-6275	919	17	.	.	PUNCT
ejpam-6275	920	1	[	[	X
ejpam-6275	920	2	12	12	NUM
ejpam-6275	920	3	]	]	X
ejpam-6275	920	4	tariq	tariq	NOUN
ejpam-6275	920	5	shah	shah	PROPN
ejpam-6275	920	6	,	,	PUNCT
ejpam-6275	920	7	asif	asif	PROPN
ejpam-6275	920	8	khan	khan	PROPN
ejpam-6275	920	9	,	,	PUNCT
ejpam-6275	920	10	and	and	CCONJ
ejpam-6275	920	11	aline	aline	PROPN
ejpam-6275	920	12	a	a	DET
ejpam-6275	920	13	andrade	andrade	PROPN
ejpam-6275	920	14	.	.	PUNCT
ejpam-6275	921	1	encoding	encode	VERB
ejpam-6275	921	2	through	through	ADP
ejpam-6275	921	3	generalized	generalized	ADJ
ejpam-6275	921	4	polynomial	polynomial	ADJ
ejpam-6275	921	5	codes	code	NOUN
ejpam-6275	921	6	.	.	PUNCT
ejpam-6275	922	1	computational	computational	ADJ
ejpam-6275	922	2	&	&	CCONJ
ejpam-6275	922	3	applied	applied	ADJ
ejpam-6275	922	4	mathematics	mathematic	NOUN
ejpam-6275	922	5	,	,	PUNCT
ejpam-6275	922	6	30:349–366	30:349–366	PROPN
ejpam-6275	922	7	,	,	PUNCT
ejpam-6275	922	8	2011	2011	NUM
ejpam-6275	922	9	.	.	PUNCT
ejpam-6275	923	1	[	[	X
ejpam-6275	923	2	13	13	NUM
ejpam-6275	923	3	]	]	X
ejpam-6275	923	4	tariq	tariq	NOUN
ejpam-6275	923	5	shah	shah	PROPN
ejpam-6275	923	6	,	,	PUNCT
ejpam-6275	923	7	awais	awais	PROPN
ejpam-6275	923	8	qamar	qamar	PROPN
ejpam-6275	923	9	,	,	PUNCT
ejpam-6275	923	10	and	and	CCONJ
ejpam-6275	923	11	aline	aline	PROPN
ejpam-6275	923	12	a	a	DET
ejpam-6275	923	13	de	de	X
ejpam-6275	923	14	andrade	andrade	PROPN
ejpam-6275	923	15	.	.	PUNCT
ejpam-6275	924	1	constructions	construction	NOUN
ejpam-6275	924	2	and	and	CCONJ
ejpam-6275	924	3	decoding	decode	VERB
ejpam-6275	924	4	of	of	ADP
ejpam-6275	924	5	a	a	DET
ejpam-6275	924	6	sequence	sequence	NOUN
ejpam-6275	924	7	of	of	ADP
ejpam-6275	924	8	bch	bch	PROPN
ejpam-6275	924	9	codes	code	NOUN
ejpam-6275	924	10	,	,	PUNCT
ejpam-6275	924	11	2012	2012	NUM
ejpam-6275	924	12	.	.	PUNCT
ejpam-6275	925	1	unpublished	unpublished	ADJ
ejpam-6275	925	2	manuscript	manuscript	NOUN
ejpam-6275	925	3	.	.	PUNCT
ejpam-6275	926	1	[	[	X
ejpam-6275	926	2	14	14	NUM
ejpam-6275	926	3	]	]	X
ejpam-6275	926	4	cunsheng	cunsheng	PROPN
ejpam-6275	926	5	ding	ding	PROPN
ejpam-6275	926	6	.	.	PUNCT
ejpam-6275	927	1	linear	linear	ADJ
ejpam-6275	927	2	codes	code	NOUN
ejpam-6275	927	3	from	from	ADP
ejpam-6275	927	4	some	some	DET
ejpam-6275	927	5	2	2	NUM
ejpam-6275	927	6	-	-	PUNCT
ejpam-6275	927	7	designs	design	NOUN
ejpam-6275	927	8	.	.	PUNCT
ejpam-6275	928	1	ieee	ieee	NOUN
ejpam-6275	928	2	transactions	transaction	NOUN
ejpam-6275	928	3	on	on	ADP
ejpam-6275	928	4	information	information	NOUN
ejpam-6275	928	5	theory	theory	NOUN
ejpam-6275	928	6	,	,	PUNCT
ejpam-6275	928	7	61(6):3265–3275	61(6):3265–3275	NUM
ejpam-6275	928	8	,	,	PUNCT
ejpam-6275	928	9	2015	2015	NUM
ejpam-6275	928	10	.	.	PUNCT
ejpam-6275	929	1	[	[	X
ejpam-6275	929	2	15	15	NUM
ejpam-6275	929	3	]	]	PUNCT
ejpam-6275	929	4	jay	jay	PROPN
ejpam-6275	929	5	l	l	PROPN
ejpam-6275	929	6	goldwasser	goldwasser	PROPN
ejpam-6275	929	7	.	.	PUNCT
ejpam-6275	930	1	shortened	shorten	VERB
ejpam-6275	930	2	and	and	CCONJ
ejpam-6275	930	3	punctured	punctured	ADJ
ejpam-6275	930	4	codes	code	NOUN
ejpam-6275	930	5	and	and	CCONJ
ejpam-6275	930	6	the	the	DET
ejpam-6275	930	7	macwilliams	macwilliam	NOUN
ejpam-6275	930	8	identities	identity	NOUN
ejpam-6275	930	9	.	.	PUNCT
ejpam-6275	931	1	linear	linear	ADJ
ejpam-6275	931	2	algebra	algebra	NOUN
ejpam-6275	931	3	and	and	CCONJ
ejpam-6275	931	4	its	its	PRON
ejpam-6275	931	5	applications	application	NOUN
ejpam-6275	931	6	,	,	PUNCT
ejpam-6275	931	7	253(1	253(1	PROPN
ejpam-6275	931	8	-	-	SYM
ejpam-6275	931	9	3):1–13	3):1–13	NUM
ejpam-6275	931	10	,	,	PUNCT
ejpam-6275	931	11	1997	1997	NUM
ejpam-6275	931	12	.	.	PUNCT
ejpam-6275	932	1	[	[	X
ejpam-6275	932	2	16	16	NUM
ejpam-6275	932	3	]	]	PUNCT
ejpam-6275	932	4	arpita	arpita	PROPN
ejpam-6275	932	5	yardi	yardi	PROPN
ejpam-6275	932	6	and	and	CCONJ
ejpam-6275	932	7	ruud	ruud	PROPN
ejpam-6275	932	8	pellikaan	pellikaan	PROPN
ejpam-6275	932	9	.	.	PUNCT
ejpam-6275	933	1	on	on	ADP
ejpam-6275	933	2	shortened	shorten	VERB
ejpam-6275	933	3	and	and	CCONJ
ejpam-6275	933	4	punctured	punctured	ADJ
ejpam-6275	933	5	cyclic	cyclic	ADJ
ejpam-6275	933	6	codes	code	NOUN
ejpam-6275	933	7	.	.	PUNCT
ejpam-6275	934	1	arxiv	arxiv	PROPN
ejpam-6275	934	2	preprint	preprint	VERB
ejpam-6275	934	3	arxiv:1705.09859	arxiv:1705.09859	PROPN
ejpam-6275	934	4	,	,	PUNCT
ejpam-6275	934	5	2017	2017	NUM
ejpam-6275	934	6	.	.	PUNCT
ejpam-6275	935	1	[	[	X
ejpam-6275	935	2	17	17	NUM
ejpam-6275	935	3	]	]	X
ejpam-6275	935	4	klaus	klaus	PROPN
ejpam-6275	935	5	huber	huber	PROPN
ejpam-6275	935	6	.	.	PUNCT
ejpam-6275	936	1	codes	code	NOUN
ejpam-6275	936	2	over	over	ADP
ejpam-6275	936	3	eisenstein	eisenstein	PROPN
ejpam-6275	936	4	-	-	PUNCT
ejpam-6275	936	5	jacobi	jacobi	PROPN
ejpam-6275	936	6	integers	integer	NOUN
ejpam-6275	936	7	.	.	PUNCT
ejpam-6275	937	1	contemporary	contemporary	ADJ
ejpam-6275	937	2	mathematics	mathematic	NOUN
ejpam-6275	937	3	,	,	PUNCT
ejpam-6275	937	4	168:165–165	168:165–165	NUM
ejpam-6275	937	5	,	,	PUNCT
ejpam-6275	937	6	1994	1994	NUM
ejpam-6275	937	7	.	.	PUNCT
ejpam-6275	938	1	[	[	X
ejpam-6275	938	2	18	18	NUM
ejpam-6275	938	3	]	]	X
ejpam-6275	938	4	mohammed	mohammed	PROPN
ejpam-6275	938	5	m	m	PROPN
ejpam-6275	938	6	hazzazi	hazzazi	PROPN
ejpam-6275	938	7	,	,	PUNCT
ejpam-6275	938	8	muhammad	muhammad	PROPN
ejpam-6275	938	9	sajjad	sajjad	PROPN
ejpam-6275	938	10	,	,	PUNCT
ejpam-6275	938	11	ziad	ziad	PROPN
ejpam-6275	938	12	bassfar	bassfar	PROPN
ejpam-6275	938	13	,	,	PUNCT
ejpam-6275	938	14	tariq	tariq	NOUN
ejpam-6275	938	15	shah	shah	NOUN
ejpam-6275	938	16	,	,	PUNCT
ejpam-6275	938	17	and	and	CCONJ
ejpam-6275	938	18	abdullah	abdullah	PROPN
ejpam-6275	938	19	albakri	albakri	PROPN
ejpam-6275	938	20	.	.	PUNCT
ejpam-6275	939	1	nonlinear	nonlinear	ADJ
ejpam-6275	939	2	components	component	NOUN
ejpam-6275	939	3	of	of	ADP
ejpam-6275	939	4	a	a	DET
ejpam-6275	939	5	block	block	NOUN
ejpam-6275	939	6	cipher	cipher	ADJ
ejpam-6275	939	7	over	over	ADP
ejpam-6275	939	8	eisenstein	eisenstein	NOUN
ejpam-6275	939	9	integers	integer	NOUN
ejpam-6275	939	10	.	.	PUNCT
ejpam-6275	940	1	cmccomputers	cmccomputer	NOUN
ejpam-6275	940	2	materials	material	NOUN
ejpam-6275	940	3	&	&	CCONJ
ejpam-6275	940	4	continua	continua	PROPN
ejpam-6275	940	5	,	,	PUNCT
ejpam-6275	940	6	77(3):3659–3675	77(3):3659–3675	NUM
ejpam-6275	940	7	,	,	PUNCT
ejpam-6275	940	8	2023	2023	NUM
ejpam-6275	940	9	.	.	PUNCT
ejpam-6275	941	1	[	[	X
ejpam-6275	941	2	19	19	NUM
ejpam-6275	941	3	]	]	X
ejpam-6275	941	4	muhammad	muhammad	PROPN
ejpam-6275	941	5	sajjad	sajjad	PROPN
ejpam-6275	941	6	,	,	PUNCT
ejpam-6275	941	7	tariq	tariq	PROPN
ejpam-6275	941	8	shah	shah	PROPN
ejpam-6275	941	9	,	,	PUNCT
ejpam-6275	941	10	mohammed	mohammed	PROPN
ejpam-6275	941	11	alammari	alammari	PROPN
ejpam-6275	941	12	,	,	PUNCT
ejpam-6275	941	13	and	and	CCONJ
ejpam-6275	941	14	hatem	hatem	PROPN
ejpam-6275	941	15	alsaud	alsaud	PROPN
ejpam-6275	941	16	.	.	PUNCT
ejpam-6275	942	1	construction	construction	NOUN
ejpam-6275	942	2	and	and	CCONJ
ejpam-6275	942	3	decoding	decoding	NOUN
ejpam-6275	942	4	of	of	ADP
ejpam-6275	942	5	bch	bch	NOUN
ejpam-6275	942	6	-	-	PUNCT
ejpam-6275	942	7	codes	code	NOUN
ejpam-6275	942	8	over	over	ADP
ejpam-6275	942	9	the	the	DET
ejpam-6275	942	10	gaussian	gaussian	ADJ
ejpam-6275	942	11	field	field	NOUN
ejpam-6275	942	12	.	.	PUNCT
ejpam-6275	943	1	ieee	ieee	NOUN
ejpam-6275	943	2	access	access	NOUN
ejpam-6275	943	3	,	,	PUNCT
ejpam-6275	943	4	2023	2023	NUM
ejpam-6275	943	5	.	.	PUNCT
ejpam-6275	944	1	m.	m.	NOUN
ejpam-6275	944	2	sajjad	sajjad	PROPN
ejpam-6275	944	3	et	et	PROPN
ejpam-6275	944	4	al	al	PROPN
ejpam-6275	944	5	.	.	PUNCT
ejpam-6275	944	6	/	/	SYM
ejpam-6275	944	7	eur	eur	PROPN
ejpam-6275	944	8	.	.	PUNCT
ejpam-6275	945	1	j.	j.	PROPN
ejpam-6275	945	2	pure	pure	PROPN
ejpam-6275	945	3	appl	appl	PROPN
ejpam-6275	945	4	.	.	PROPN
ejpam-6275	945	5	math	math	PROPN
ejpam-6275	945	6	,	,	PUNCT
ejpam-6275	945	7	18	18	NUM
ejpam-6275	945	8	(	(	PUNCT
ejpam-6275	945	9	3	3	NUM
ejpam-6275	945	10	)	)	PUNCT
ejpam-6275	945	11	(	(	PUNCT
ejpam-6275	945	12	2025	2025	NUM
ejpam-6275	945	13	)	)	PUNCT
ejpam-6275	945	14	,	,	PUNCT
ejpam-6275	945	15	6275	6275	NUM
ejpam-6275	945	16	36	36	NUM
ejpam-6275	945	17	of	of	ADP
ejpam-6275	945	18	36	36	NUM
ejpam-6275	945	19	[	[	SYM
ejpam-6275	945	20	20	20	NUM
ejpam-6275	945	21	]	]	X
ejpam-6275	945	22	cunsheng	cunsheng	PROPN
ejpam-6275	945	23	ding	ding	PROPN
ejpam-6275	945	24	.	.	PUNCT
ejpam-6275	946	1	designs	design	NOUN
ejpam-6275	946	2	from	from	ADP
ejpam-6275	946	3	linear	linear	ADJ
ejpam-6275	946	4	codes	code	NOUN
ejpam-6275	946	5	,	,	PUNCT
ejpam-6275	946	6	2022	2022	NUM
ejpam-6275	946	7	.	.	PUNCT
ejpam-6275	947	1	manuscript	manuscript	NOUN
ejpam-6275	947	2	.	.	PUNCT
ejpam-6275	948	1	[	[	X
ejpam-6275	948	2	21	21	NUM
ejpam-6275	948	3	]	]	X
ejpam-6275	948	4	shu	shu	PROPN
ejpam-6275	948	5	lin	lin	PROPN
ejpam-6275	948	6	.	.	PUNCT
ejpam-6275	949	1	shortened	shorten	VERB
ejpam-6275	949	2	finite	finite	ADJ
ejpam-6275	949	3	geometry	geometry	NOUN
ejpam-6275	949	4	codes	code	NOUN
ejpam-6275	949	5	(	(	PUNCT
ejpam-6275	949	6	corresp	corresp	PROPN
ejpam-6275	949	7	.	.	PUNCT
ejpam-6275	949	8	)	)	PUNCT
ejpam-6275	949	9	.	.	PUNCT
ejpam-6275	950	1	ieee	ieee	NOUN
ejpam-6275	950	2	transactions	transaction	NOUN
ejpam-6275	950	3	on	on	ADP
ejpam-6275	950	4	information	information	NOUN
ejpam-6275	950	5	theory	theory	NOUN
ejpam-6275	950	6	,	,	PUNCT
ejpam-6275	950	7	18(5):692–696	18(5):692–696	PROPN
ejpam-6275	950	8	,	,	PUNCT
ejpam-6275	950	9	1972	1972	NUM
ejpam-6275	950	10	.	.	PUNCT
ejpam-6275	951	1	[	[	X
ejpam-6275	951	2	22	22	NUM
ejpam-6275	951	3	]	]	X
ejpam-6275	951	4	cunsheng	cunsheng	PROPN
ejpam-6275	951	5	ding	ding	PROPN
ejpam-6275	951	6	and	and	CCONJ
ejpam-6275	951	7	chengju	chengju	PROPN
ejpam-6275	951	8	li	li	PROPN
ejpam-6275	951	9	.	.	PROPN
ejpam-6275	951	10	infinite	infinite	ADJ
ejpam-6275	951	11	families	family	NOUN
ejpam-6275	951	12	of	of	ADP
ejpam-6275	951	13	2	2	NUM
ejpam-6275	951	14	-	-	PUNCT
ejpam-6275	951	15	designs	design	NOUN
ejpam-6275	951	16	and	and	CCONJ
ejpam-6275	951	17	3	3	NUM
ejpam-6275	951	18	-	-	NOUN
ejpam-6275	951	19	designs	design	NOUN
ejpam-6275	951	20	from	from	ADP
ejpam-6275	951	21	linear	linear	ADJ
ejpam-6275	951	22	codes	code	NOUN
ejpam-6275	951	23	.	.	PUNCT
ejpam-6275	952	1	discrete	discrete	ADJ
ejpam-6275	952	2	mathematics	mathematic	NOUN
ejpam-6275	952	3	,	,	PUNCT
ejpam-6275	952	4	340(10):2415–2431	340(10):2415–2431	NUM
ejpam-6275	952	5	,	,	PUNCT
ejpam-6275	952	6	2017	2017	NUM
ejpam-6275	952	7	.	.	PUNCT
ejpam-6275	953	1	[	[	X
ejpam-6275	953	2	23	23	NUM
ejpam-6275	953	3	]	]	PUNCT
ejpam-6275	953	4	florence	florence	NOUN
ejpam-6275	953	5	jessie	jessie	PROPN
ejpam-6275	953	6	macwilliams	macwilliams	PROPN
ejpam-6275	953	7	.	.	PUNCT
ejpam-6275	954	1	the	the	DET
ejpam-6275	954	2	theory	theory	NOUN
ejpam-6275	954	3	of	of	ADP
ejpam-6275	954	4	error	error	NOUN
ejpam-6275	954	5	-	-	PUNCT
ejpam-6275	954	6	correcting	correct	VERB
ejpam-6275	954	7	codes	code	NOUN
ejpam-6275	954	8	,	,	PUNCT
ejpam-6275	954	9	volume	volume	NOUN
ejpam-6275	954	10	2	2	NUM
ejpam-6275	954	11	.	.	PUNCT
ejpam-6275	954	12	elsevier	elsevier	NOUN
ejpam-6275	954	13	,	,	PUNCT
ejpam-6275	954	14	1977	1977	NUM
ejpam-6275	954	15	.	.	PUNCT
ejpam-6275	955	1	[	[	X
ejpam-6275	955	2	24	24	NUM
ejpam-6275	955	3	]	]	PUNCT
ejpam-6275	955	4	muhammad	muhammad	PROPN
ejpam-6275	955	5	sajjad	sajjad	PROPN
ejpam-6275	955	6	,	,	PUNCT
ejpam-6275	955	7	tariq	tariq	PROPN
ejpam-6275	955	8	shah	shah	PROPN
ejpam-6275	955	9	,	,	PUNCT
ejpam-6275	955	10	muhammad	muhammad	PROPN
ejpam-6275	955	11	abbas	abbas	PROPN
ejpam-6275	955	12	,	,	PUNCT
ejpam-6275	955	13	mohammed	mohammed	PROPN
ejpam-6275	955	14	alammari	alammari	PROPN
ejpam-6275	955	15	,	,	PUNCT
ejpam-6275	955	16	and	and	CCONJ
ejpam-6275	955	17	julian	julian	PROPN
ejpam-6275	955	18	serna	serna	PROPN
ejpam-6275	955	19	.	.	PUNCT
ejpam-6275	956	1	the	the	DET
ejpam-6275	956	2	impact	impact	NOUN
ejpam-6275	956	3	of	of	ADP
ejpam-6275	956	4	alternant	alternant	ADJ
ejpam-6275	956	5	codes	code	NOUN
ejpam-6275	956	6	over	over	ADP
ejpam-6275	956	7	eisenstein	eisenstein	NOUN
ejpam-6275	956	8	integers	integer	NOUN
ejpam-6275	956	9	on	on	ADP
ejpam-6275	956	10	modern	modern	ADJ
ejpam-6275	956	11	technology	technology	NOUN
ejpam-6275	956	12	.	.	PUNCT
ejpam-6275	957	1	computational	computational	ADJ
ejpam-6275	957	2	and	and	CCONJ
ejpam-6275	957	3	applied	applied	ADJ
ejpam-6275	957	4	mathematics	mathematic	NOUN
ejpam-6275	957	5	,	,	PUNCT
ejpam-6275	957	6	44(1):95	44(1):95	NOUN
ejpam-6275	957	7	,	,	PUNCT
ejpam-6275	957	8	2025	2025	NUM
ejpam-6275	957	9	.	.	PUNCT
ejpam-6275	958	1	[	[	X
ejpam-6275	958	2	25	25	NUM
ejpam-6275	958	3	]	]	X
ejpam-6275	958	4	w	w	PROPN
ejpam-6275	958	5	cary	cary	PROPN
ejpam-6275	958	6	huffman	huffman	PROPN
ejpam-6275	958	7	and	and	CCONJ
ejpam-6275	958	8	vera	vera	PROPN
ejpam-6275	958	9	pless	pless	PROPN
ejpam-6275	958	10	.	.	PUNCT
ejpam-6275	959	1	fundamentals	fundamental	NOUN
ejpam-6275	959	2	of	of	ADP
ejpam-6275	959	3	error	error	NOUN
ejpam-6275	959	4	-	-	PUNCT
ejpam-6275	959	5	correcting	correct	VERB
ejpam-6275	959	6	codes	code	NOUN
ejpam-6275	959	7	.	.	PUNCT
ejpam-6275	960	1	cambridge	cambridge	PROPN
ejpam-6275	960	2	university	university	PROPN
ejpam-6275	960	3	press	press	NOUN
ejpam-6275	960	4	,	,	PUNCT
ejpam-6275	960	5	2010	2010	NUM
ejpam-6275	960	6	.	.	PUNCT
ejpam-6275	961	1	[	[	X
ejpam-6275	961	2	26	26	NUM
ejpam-6275	961	3	]	]	PUNCT
ejpam-6275	961	4	muhammad	muhammad	PROPN
ejpam-6275	961	5	sajjad	sajjad	PROPN
ejpam-6275	961	6	,	,	PUNCT
ejpam-6275	961	7	tariq	tariq	PROPN
ejpam-6275	961	8	shah	shah	PROPN
ejpam-6275	961	9	,	,	PUNCT
ejpam-6275	961	10	qiong	qiong	PROPN
ejpam-6275	961	11	xin	xin	PROPN
ejpam-6275	961	12	,	,	PUNCT
ejpam-6275	961	13	and	and	CCONJ
ejpam-6275	961	14	bader	bader	PROPN
ejpam-6275	961	15	almutairi	almutairi	PROPN
ejpam-6275	961	16	.	.	PUNCT
ejpam-6275	962	1	eisenstein	eisenstein	PROPN
ejpam-6275	962	2	field	field	PROPN
ejpam-6275	962	3	bch	bch	PROPN
ejpam-6275	962	4	codes	code	VERB
ejpam-6275	962	5	construction	construction	NOUN
ejpam-6275	962	6	and	and	CCONJ
ejpam-6275	962	7	decoding	decoding	NOUN
ejpam-6275	962	8	.	.	PUNCT
ejpam-6275	963	1	aims	aim	VERB
ejpam-6275	963	2	mathematics	mathematic	NOUN
ejpam-6275	963	3	,	,	PUNCT
ejpam-6275	963	4	8(12):29453–29473	8(12):29453–29473	NUM
ejpam-6275	963	5	,	,	PUNCT
ejpam-6275	963	6	2023	2023	NUM
ejpam-6275	963	7	.	.	PUNCT
ejpam-6275	964	1	[	[	X
ejpam-6275	964	2	27	27	NUM
ejpam-6275	964	3	]	]	X
ejpam-6275	964	4	g	g	PROPN
ejpam-6275	964	5	david	david	PROPN
ejpam-6275	964	6	forney	forney	PROPN
ejpam-6275	964	7	.	.	PUNCT
ejpam-6275	965	1	on	on	ADP
ejpam-6275	965	2	decoding	decode	VERB
ejpam-6275	965	3	bch	bch	PROPN
ejpam-6275	965	4	codes	code	NOUN
ejpam-6275	965	5	.	.	PUNCT
ejpam-6275	966	1	ieee	ieee	NOUN
ejpam-6275	966	2	transactions	transaction	NOUN
ejpam-6275	966	3	on	on	ADP
ejpam-6275	966	4	information	information	NOUN
ejpam-6275	966	5	theory	theory	NOUN
ejpam-6275	966	6	,	,	PUNCT
ejpam-6275	966	7	11(4):549–557	11(4):549–557	NUM
ejpam-6275	966	8	,	,	PUNCT
ejpam-6275	966	9	1965	1965	NUM
ejpam-6275	966	10	.	.	PUNCT