id	sid	tid	token	lemma	pos
ejpam-6409	1	1	european	european	PROPN
ejpam-6409	1	2	journal	journal	PROPN
ejpam-6409	1	3	of	of	ADP
ejpam-6409	1	4	pure	pure	ADJ
ejpam-6409	1	5	and	and	CCONJ
ejpam-6409	1	6	applied	applied	ADJ
ejpam-6409	1	7	mathematics	mathematic	NOUN
ejpam-6409	1	8	2025	2025	NUM
ejpam-6409	1	9	,	,	PUNCT
ejpam-6409	1	10	vol	vol	NOUN
ejpam-6409	1	11	.	.	PROPN
ejpam-6409	1	12	18	18	NUM
ejpam-6409	1	13	,	,	PUNCT
ejpam-6409	1	14	issue	issue	NOUN
ejpam-6409	1	15	3	3	NUM
ejpam-6409	1	16	,	,	PUNCT
ejpam-6409	1	17	article	article	NOUN
ejpam-6409	1	18	number	number	NOUN
ejpam-6409	1	19	6409	6409	NUM
ejpam-6409	1	20	issn	issn	PROPN
ejpam-6409	1	21	1307	1307	NUM
ejpam-6409	1	22	-	-	SYM
ejpam-6409	1	23	5543	5543	NUM
ejpam-6409	1	24	–	–	PUNCT
ejpam-6409	1	25	ejpam.com	ejpam.com	X
ejpam-6409	1	26	published	publish	VERB
ejpam-6409	1	27	by	by	ADP
ejpam-6409	1	28	new	new	PROPN
ejpam-6409	1	29	york	york	PROPN
ejpam-6409	1	30	business	business	PROPN
ejpam-6409	1	31	global	global	ADJ
ejpam-6409	1	32	construction	construction	NOUN
ejpam-6409	1	33	and	and	CCONJ
ejpam-6409	1	34	classification	classification	NOUN
ejpam-6409	1	35	of	of	ADP
ejpam-6409	1	36	generalized	generalized	ADJ
ejpam-6409	1	37	hadamard	hadamard	ADJ
ejpam-6409	1	38	codes	code	NOUN
ejpam-6409	1	39	over	over	ADP
ejpam-6409	1	40	eisenstein	eisenstein	PROPN
ejpam-6409	1	41	local	local	ADJ
ejpam-6409	1	42	rings	ring	NOUN
ejpam-6409	1	43	z2s[ω	z2s[ω	PROPN
ejpam-6409	1	44	]	]	X
ejpam-6409	1	45	muhammad	muhammad	PROPN
ejpam-6409	1	46	sajjad1,∗	sajjad1,∗	PROPN
ejpam-6409	1	47	,	,	PUNCT
ejpam-6409	1	48	muhammad	muhammad	PROPN
ejpam-6409	1	49	farhan	farhan	PROPN
ejpam-6409	1	50	ali	ali	PROPN
ejpam-6409	1	51	khan2	khan2	PROPN
ejpam-6409	1	52	,	,	PUNCT
ejpam-6409	1	53	maha	maha	PROPN
ejpam-6409	1	54	alammari3	alammari3	PROPN
ejpam-6409	1	55	,	,	PUNCT
ejpam-6409	1	56	robinson	robinson	PROPN
ejpam-6409	1	57	-	-	PUNCT
ejpam-6409	1	58	julian	julian	PROPN
ejpam-6409	1	59	serna4	serna4	PROPN
ejpam-6409	1	60	1	1	NUM
ejpam-6409	1	61	nutech	nutech	NOUN
ejpam-6409	1	62	school	school	NOUN
ejpam-6409	1	63	of	of	ADP
ejpam-6409	1	64	applied	apply	VERB
ejpam-6409	1	65	science	science	NOUN
ejpam-6409	1	66	and	and	CCONJ
ejpam-6409	1	67	humanities	humanity	NOUN
ejpam-6409	1	68	,	,	PUNCT
ejpam-6409	1	69	national	national	ADJ
ejpam-6409	1	70	university	university	PROPN
ejpam-6409	1	71	of	of	ADP
ejpam-6409	1	72	technology	technology	NOUN
ejpam-6409	1	73	,	,	PUNCT
ejpam-6409	1	74	islamabad	islamabad	PROPN
ejpam-6409	1	75	,	,	PUNCT
ejpam-6409	1	76	44000	44000	NUM
ejpam-6409	1	77	,	,	PUNCT
ejpam-6409	1	78	pakistan	pakistan	PROPN
ejpam-6409	1	79	2	2	NUM
ejpam-6409	1	80	department	department	NOUN
ejpam-6409	1	81	of	of	ADP
ejpam-6409	1	82	mathematics	mathematic	NOUN
ejpam-6409	1	83	,	,	PUNCT
ejpam-6409	1	84	quaid	quaid	PROPN
ejpam-6409	1	85	-	-	PUNCT
ejpam-6409	1	86	i	i	PROPN
ejpam-6409	1	87	-	-	PUNCT
ejpam-6409	1	88	azam	azam	PROPN
ejpam-6409	1	89	university	university	PROPN
ejpam-6409	1	90	,	,	PUNCT
ejpam-6409	1	91	islamabad	islamabad	PROPN
ejpam-6409	1	92	,	,	PUNCT
ejpam-6409	1	93	pakistan	pakistan	PROPN
ejpam-6409	1	94	3	3	NUM
ejpam-6409	1	95	department	department	NOUN
ejpam-6409	1	96	of	of	ADP
ejpam-6409	1	97	mathematics	mathematics	PROPN
ejpam-6409	1	98	,	,	PUNCT
ejpam-6409	1	99	college	college	NOUN
ejpam-6409	1	100	of	of	ADP
ejpam-6409	1	101	science	science	NOUN
ejpam-6409	1	102	,	,	PUNCT
ejpam-6409	1	103	king	king	PROPN
ejpam-6409	1	104	saud	saud	PROPN
ejpam-6409	1	105	university	university	PROPN
ejpam-6409	1	106	,	,	PUNCT
ejpam-6409	1	107	p.o	p.o	PROPN
ejpam-6409	1	108	.	.	PROPN
ejpam-6409	1	109	box	box	PROPN
ejpam-6409	1	110	22452	22452	NUM
ejpam-6409	1	111	riyadh	riyadh	PROPN
ejpam-6409	1	112	11495	11495	NUM
ejpam-6409	1	113	,	,	PUNCT
ejpam-6409	1	114	saudi	saudi	PROPN
ejpam-6409	1	115	arabia	arabia	PROPN
ejpam-6409	1	116	4	4	NUM
ejpam-6409	1	117	escuela	escuela	NOUN
ejpam-6409	1	118	de	de	X
ejpam-6409	1	119	matemáticas	matemáticas	PROPN
ejpam-6409	1	120	y	y	PROPN
ejpam-6409	1	121	estad́ıstica	estad́ıstica	PROPN
ejpam-6409	1	122	,	,	PUNCT
ejpam-6409	1	123	universidad	universidad	PROPN
ejpam-6409	1	124	pedagógica	pedagógica	PROPN
ejpam-6409	1	125	y	y	PROPN
ejpam-6409	1	126	tecnológica	tecnológica	PROPN
ejpam-6409	1	127	de	de	PROPN
ejpam-6409	1	128	colombia	colombia	PROPN
ejpam-6409	1	129	,	,	PUNCT
ejpam-6409	1	130	tunja	tunja	NUM
ejpam-6409	1	131	,	,	PUNCT
ejpam-6409	1	132	colombia	colombia	PROPN
ejpam-6409	1	133	abstract	abstract	NOUN
ejpam-6409	1	134	.	.	PUNCT
ejpam-6409	2	1	the	the	DET
ejpam-6409	2	2	research	research	NOUN
ejpam-6409	2	3	paper	paper	NOUN
ejpam-6409	2	4	examines	examine	VERB
ejpam-6409	2	5	the	the	DET
ejpam-6409	2	6	design	design	NOUN
ejpam-6409	2	7	principles	principle	NOUN
ejpam-6409	2	8	and	and	CCONJ
ejpam-6409	2	9	structural	structural	ADJ
ejpam-6409	2	10	features	feature	NOUN
ejpam-6409	2	11	of	of	ADP
ejpam-6409	2	12	generalized	generalized	ADJ
ejpam-6409	2	13	hadamard	hadamard	NOUN
ejpam-6409	2	14	(	(	PUNCT
ejpam-6409	2	15	gh	gh	PROPN
ejpam-6409	2	16	)	)	PUNCT
ejpam-6409	2	17	codes	code	NOUN
ejpam-6409	2	18	that	that	PRON
ejpam-6409	2	19	operate	operate	VERB
ejpam-6409	2	20	within	within	ADP
ejpam-6409	2	21	eisenstein	eisenstein	PROPN
ejpam-6409	2	22	local	local	ADJ
ejpam-6409	2	23	rings	ring	NOUN
ejpam-6409	2	24	z2s	z2s	PROPN
ejpam-6409	3	1	[	[	X
ejpam-6409	3	2	ω	ω	X
ejpam-6409	3	3	]	]	X
ejpam-6409	3	4	,	,	PUNCT
ejpam-6409	3	5	utilizing	utilize	VERB
ejpam-6409	3	6	a	a	DET
ejpam-6409	3	7	primitive	primitive	ADJ
ejpam-6409	3	8	cube	cube	NOUN
ejpam-6409	3	9	root	root	NOUN
ejpam-6409	3	10	of	of	ADP
ejpam-6409	3	11	unity	unity	NOUN
ejpam-6409	3	12	ω	ω	NUM
ejpam-6409	3	13	that	that	PRON
ejpam-6409	3	14	satisfies	satisfy	VERB
ejpam-6409	3	15	the	the	DET
ejpam-6409	3	16	relation	relation	NOUN
ejpam-6409	3	17	ω2	ω2	PROPN
ejpam-6409	4	1	+	+	CCONJ
ejpam-6409	4	2	ω	ω	NUM
ejpam-6409	4	3	+	+	CCONJ
ejpam-6409	4	4	1	1	NUM
ejpam-6409	4	5	=	=	SYM
ejpam-6409	4	6	0	0	NUM
ejpam-6409	4	7	.	.	PUNCT
ejpam-6409	5	1	the	the	DET
ejpam-6409	5	2	paper	paper	NOUN
ejpam-6409	5	3	first	first	ADV
ejpam-6409	5	4	introduces	introduce	VERB
ejpam-6409	5	5	an	an	DET
ejpam-6409	5	6	algebraic	algebraic	ADJ
ejpam-6409	5	7	eisenstein	eisenstein	NOUN
ejpam-6409	5	8	integer	integer	NOUN
ejpam-6409	5	9	framework	framework	NOUN
ejpam-6409	5	10	before	before	ADP
ejpam-6409	5	11	developing	develop	VERB
ejpam-6409	5	12	an	an	DET
ejpam-6409	5	13	appropriate	appropriate	ADJ
ejpam-6409	5	14	gray	gray	ADJ
ejpam-6409	5	15	mapping	mapping	NOUN
ejpam-6409	5	16	to	to	PART
ejpam-6409	5	17	examine	examine	VERB
ejpam-6409	5	18	binary	binary	ADJ
ejpam-6409	5	19	-	-	PUNCT
ejpam-6409	5	20	domain	domain	NOUN
ejpam-6409	5	21	representations	representation	NOUN
ejpam-6409	5	22	of	of	ADP
ejpam-6409	5	23	these	these	DET
ejpam-6409	5	24	codes	code	NOUN
ejpam-6409	5	25	.	.	PUNCT
ejpam-6409	6	1	we	we	PRON
ejpam-6409	6	2	establish	establish	VERB
ejpam-6409	6	3	the	the	DET
ejpam-6409	6	4	essential	essential	ADJ
ejpam-6409	6	5	criteria	criterion	NOUN
ejpam-6409	6	6	and	and	CCONJ
ejpam-6409	6	7	necessary	necessary	ADJ
ejpam-6409	6	8	checks	check	NOUN
ejpam-6409	6	9	for	for	ADP
ejpam-6409	6	10	determining	determine	VERB
ejpam-6409	6	11	the	the	DET
ejpam-6409	6	12	linear	linear	ADJ
ejpam-6409	6	13	properties	property	NOUN
ejpam-6409	6	14	of	of	ADP
ejpam-6409	6	15	gh	gh	PROPN
ejpam-6409	6	16	codes	code	NOUN
ejpam-6409	6	17	based	base	VERB
ejpam-6409	6	18	on	on	ADP
ejpam-6409	6	19	z2s	z2s	PROPN
ejpam-6409	6	20	[	[	X
ejpam-6409	6	21	ω	ω	X
ejpam-6409	6	22	]	]	X
ejpam-6409	6	23	structures	structure	NOUN
ejpam-6409	6	24	.	.	PUNCT
ejpam-6409	7	1	this	this	DET
ejpam-6409	7	2	research	research	NOUN
ejpam-6409	7	3	defines	define	VERB
ejpam-6409	7	4	the	the	DET
ejpam-6409	7	5	kernel	kernel	PROPN
ejpam-6409	7	6	structure	structure	NOUN
ejpam-6409	7	7	of	of	ADP
ejpam-6409	7	8	these	these	DET
ejpam-6409	7	9	codes	code	NOUN
ejpam-6409	7	10	together	together	ADV
ejpam-6409	7	11	with	with	ADP
ejpam-6409	7	12	their	their	PRON
ejpam-6409	7	13	rank	rank	NOUN
ejpam-6409	7	14	specification	specification	NOUN
ejpam-6409	7	15	and	and	CCONJ
ejpam-6409	7	16	an	an	DET
ejpam-6409	7	17	evaluation	evaluation	NOUN
ejpam-6409	7	18	of	of	ADP
ejpam-6409	7	19	their	their	PRON
ejpam-6409	7	20	structural	structural	ADJ
ejpam-6409	7	21	properties	property	NOUN
ejpam-6409	7	22	.	.	PUNCT
ejpam-6409	8	1	a	a	DET
ejpam-6409	8	2	classification	classification	NOUN
ejpam-6409	8	3	system	system	NOUN
ejpam-6409	8	4	for	for	ADP
ejpam-6409	8	5	z2s	z2s	PROPN
ejpam-6409	8	6	[	[	X
ejpam-6409	8	7	ω]-linear	ω]-linear	ADJ
ejpam-6409	8	8	hadamard	hadamard	ADJ
ejpam-6409	8	9	codes	code	NOUN
ejpam-6409	8	10	is	be	AUX
ejpam-6409	8	11	presented	present	VERB
ejpam-6409	8	12	in	in	ADP
ejpam-6409	8	13	the	the	DET
ejpam-6409	8	14	final	final	ADJ
ejpam-6409	8	15	part	part	NOUN
ejpam-6409	8	16	of	of	ADP
ejpam-6409	8	17	the	the	DET
ejpam-6409	8	18	paper	paper	NOUN
ejpam-6409	8	19	,	,	PUNCT
ejpam-6409	8	20	based	base	VERB
ejpam-6409	8	21	on	on	ADP
ejpam-6409	8	22	their	their	PRON
ejpam-6409	8	23	algebraic	algebraic	ADJ
ejpam-6409	8	24	and	and	CCONJ
ejpam-6409	8	25	combinatorial	combinatorial	ADJ
ejpam-6409	8	26	characteristics	characteristic	NOUN
ejpam-6409	8	27	.	.	PUNCT
ejpam-6409	9	1	future	future	ADJ
ejpam-6409	9	2	studies	study	NOUN
ejpam-6409	9	3	on	on	ADP
ejpam-6409	9	4	coding	code	VERB
ejpam-6409	9	5	techniques	technique	NOUN
ejpam-6409	9	6	within	within	ADP
ejpam-6409	9	7	algebraic	algebraic	ADJ
ejpam-6409	9	8	integer	integer	NOUN
ejpam-6409	9	9	rings	ring	NOUN
ejpam-6409	9	10	can	can	AUX
ejpam-6409	9	11	build	build	VERB
ejpam-6409	9	12	upon	upon	SCONJ
ejpam-6409	9	13	this	this	DET
ejpam-6409	9	14	work	work	NOUN
ejpam-6409	9	15	,	,	PUNCT
ejpam-6409	9	16	as	as	SCONJ
ejpam-6409	9	17	our	our	PRON
ejpam-6409	9	18	research	research	NOUN
ejpam-6409	9	19	expands	expand	VERB
ejpam-6409	9	20	the	the	DET
ejpam-6409	9	21	understanding	understanding	NOUN
ejpam-6409	9	22	of	of	ADP
ejpam-6409	9	23	code	code	NOUN
ejpam-6409	9	24	theory	theory	NOUN
ejpam-6409	9	25	over	over	ADP
ejpam-6409	9	26	non	non	ADJ
ejpam-6409	9	27	-	-	ADJ
ejpam-6409	9	28	traditional	traditional	ADJ
ejpam-6409	9	29	rings	ring	NOUN
ejpam-6409	9	30	.	.	PUNCT
ejpam-6409	10	1	2020	2020	NUM
ejpam-6409	10	2	mathematics	mathematics	PROPN
ejpam-6409	10	3	subject	subject	NOUN
ejpam-6409	10	4	classifications	classification	NOUN
ejpam-6409	10	5	:	:	PUNCT
ejpam-6409	10	6	94b75	94b75	NUM
ejpam-6409	10	7	,	,	PUNCT
ejpam-6409	10	8	11t71	11t71	NUM
ejpam-6409	10	9	,	,	PUNCT
ejpam-6409	10	10	94a24	94a24	NUM
ejpam-6409	10	11	,	,	PUNCT
ejpam-6409	10	12	68p30	68p30	NUM
ejpam-6409	10	13	,	,	PUNCT
ejpam-6409	10	14	14g50	14g50	NUM
ejpam-6409	10	15	,	,	PUNCT
ejpam-6409	10	16	94a05	94a05	NUM
ejpam-6409	10	17	key	key	ADJ
ejpam-6409	10	18	words	word	NOUN
ejpam-6409	10	19	and	and	CCONJ
ejpam-6409	10	20	phrases	phrase	NOUN
ejpam-6409	10	21	:	:	PUNCT
ejpam-6409	10	22	generalized	generalized	ADJ
ejpam-6409	10	23	hadamard	hadamard	ADJ
ejpam-6409	10	24	codes	code	NOUN
ejpam-6409	10	25	,	,	PUNCT
ejpam-6409	10	26	eisenstein	eisenstein	NOUN
ejpam-6409	10	27	integers	integer	NOUN
ejpam-6409	10	28	,	,	PUNCT
ejpam-6409	10	29	local	local	ADJ
ejpam-6409	10	30	rings	ring	NOUN
ejpam-6409	10	31	,	,	PUNCT
ejpam-6409	10	32	gray	gray	ADJ
ejpam-6409	10	33	map	map	NOUN
ejpam-6409	10	34	,	,	PUNCT
ejpam-6409	10	35	code	code	NOUN
ejpam-6409	10	36	linearity	linearity	NOUN
ejpam-6409	10	37	,	,	PUNCT
ejpam-6409	10	38	code	code	NOUN
ejpam-6409	10	39	kernel	kernel	NOUN
ejpam-6409	10	40	1	1	NUM
ejpam-6409	10	41	.	.	PUNCT
ejpam-6409	10	42	introduction	introduction	NOUN
ejpam-6409	10	43	modulator	modulator	NOUN
ejpam-6409	10	44	–	–	PUNCT
ejpam-6409	10	45	demodulator	demodulator	NOUN
ejpam-6409	10	46	,	,	PUNCT
ejpam-6409	10	47	as	as	SCONJ
ejpam-6409	10	48	we	we	PRON
ejpam-6409	10	49	are	be	AUX
ejpam-6409	10	50	all	all	ADV
ejpam-6409	10	51	accustomed	accustomed	ADJ
ejpam-6409	10	52	to	to	ADP
ejpam-6409	10	53	it	it	PRON
ejpam-6409	10	54	,	,	PUNCT
ejpam-6409	10	55	is	be	AUX
ejpam-6409	10	56	one	one	NUM
ejpam-6409	10	57	of	of	ADP
ejpam-6409	10	58	the	the	DET
ejpam-6409	10	59	cornerstones	cornerstone	NOUN
ejpam-6409	10	60	of	of	ADP
ejpam-6409	10	61	contemporary	contemporary	ADJ
ejpam-6409	10	62	digital	digital	ADJ
ejpam-6409	10	63	communication	communication	NOUN
ejpam-6409	10	64	,	,	PUNCT
ejpam-6409	10	65	thanks	thank	NOUN
ejpam-6409	10	66	to	to	ADP
ejpam-6409	10	67	which	which	PRON
ejpam-6409	10	68	we	we	PRON
ejpam-6409	10	69	are	be	AUX
ejpam-6409	10	70	able	able	ADJ
ejpam-6409	10	71	to	to	PART
ejpam-6409	10	72	detect	detect	VERB
ejpam-6409	10	73	and	and	CCONJ
ejpam-6409	10	74	correct	correct	ADJ
ejpam-6409	10	75	errors	error	NOUN
ejpam-6409	10	76	in	in	ADP
ejpam-6409	10	77	transmitted	transmit	VERB
ejpam-6409	10	78	data	datum	NOUN
ejpam-6409	10	79	.	.	PUNCT
ejpam-6409	11	1	it	it	PRON
ejpam-6409	11	2	is	be	AUX
ejpam-6409	11	3	mainly	mainly	ADV
ejpam-6409	11	4	a	a	DET
ejpam-6409	11	5	theory	theory	NOUN
ejpam-6409	11	6	for	for	ADP
ejpam-6409	11	7	the	the	DET
ejpam-6409	11	8	construction	construction	NOUN
ejpam-6409	11	9	of	of	ADP
ejpam-6409	11	10	structured	structured	ADJ
ejpam-6409	11	11	codes	code	NOUN
ejpam-6409	11	12	that	that	PRON
ejpam-6409	11	13	can	can	AUX
ejpam-6409	11	14	efficiently	efficiently	ADV
ejpam-6409	11	15	handle	handle	VERB
ejpam-6409	11	16	errors	error	NOUN
ejpam-6409	11	17	while	while	SCONJ
ejpam-6409	11	18	remaining	remain	VERB
ejpam-6409	11	19	data	datum	NOUN
ejpam-6409	11	20	-	-	PUNCT
ejpam-6409	11	21	intuitive	intuitive	ADJ
ejpam-6409	11	22	.	.	PUNCT
ejpam-6409	12	1	it	it	PRON
ejpam-6409	12	2	laid	lay	VERB
ejpam-6409	12	3	the	the	DET
ejpam-6409	12	4	foundations	foundation	NOUN
ejpam-6409	12	5	∗corresponding	∗corresponde	VERB
ejpam-6409	12	6	author	author	NOUN
ejpam-6409	12	7	.	.	PUNCT
ejpam-6409	13	1	doi	doi	NOUN
ejpam-6409	13	2	:	:	PUNCT
ejpam-6409	13	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6409	https://doi.org/10.29020/nybg.ejpam.v18i3.6409	DET
ejpam-6409	13	4	email	email	NOUN
ejpam-6409	13	5	addresses	address	NOUN
ejpam-6409	13	6	:	:	PUNCT
ejpam-6409	13	7	muhammad.sajjad@nutech.edu.pk	muhammad.sajjad@nutech.edu.pk	PROPN
ejpam-6409	13	8	(	(	PUNCT
ejpam-6409	13	9	m.	m.	PROPN
ejpam-6409	13	10	sajjad	sajjad	PROPN
ejpam-6409	13	11	)	)	PUNCT
ejpam-6409	13	12	,	,	PUNCT
ejpam-6409	13	13	muhammadfarhan20117@gmail.com	muhammadfarhan20117@gmail.com	X
ejpam-6409	14	1	(	(	PUNCT
ejpam-6409	14	2	m.	m.	PROPN
ejpam-6409	14	3	f.	f.	PROPN
ejpam-6409	14	4	a.	a.	PROPN
ejpam-6409	14	5	khan	khan	PROPN
ejpam-6409	14	6	)	)	PUNCT
ejpam-6409	14	7	,	,	PUNCT
ejpam-6409	14	8	malammari@ksu.edu.sa	malammari@ksu.edu.sa	PROPN
ejpam-6409	14	9	(	(	PUNCT
ejpam-6409	14	10	m.	m.	NOUN
ejpam-6409	14	11	alammari	alammari	PROPN
ejpam-6409	14	12	)	)	PUNCT
ejpam-6409	14	13	,	,	PUNCT
ejpam-6409	14	14	robinson.serna@uptc.edu.co	robinson.serna@uptc.edu.co	ADP
ejpam-6409	14	15	(	(	PUNCT
ejpam-6409	14	16	r.	r.	PROPN
ejpam-6409	14	17	j.	j.	PROPN
ejpam-6409	14	18	serna	serna	PROPN
ejpam-6409	14	19	)	)	PUNCT
ejpam-6409	14	20	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6409	15	1	1	1	NUM
ejpam-6409	15	2	copyright	copyright	NOUN
ejpam-6409	15	3	:	:	PUNCT
ejpam-6409	15	4	©	©	PROPN
ejpam-6409	15	5	2025	2025	NUM
ejpam-6409	15	6	the	the	DET
ejpam-6409	15	7	author(s	author(s	NOUN
ejpam-6409	15	8	)	)	PUNCT
ejpam-6409	15	9	.	.	PUNCT
ejpam-6409	16	1	(	(	PUNCT
ejpam-6409	16	2	cc	cc	NOUN
ejpam-6409	16	3	by	by	ADP
ejpam-6409	16	4	-	-	PUNCT
ejpam-6409	16	5	nc	nc	PROPN
ejpam-6409	16	6	4.0	4.0	NUM
ejpam-6409	16	7	)	)	PUNCT
ejpam-6409	16	8	muhammad	muhammad	PROPN
ejpam-6409	16	9	sajjad	sajjad	PROPN
ejpam-6409	16	10	et	et	PROPN
ejpam-6409	16	11	al	al	PROPN
ejpam-6409	16	12	.	.	PUNCT
ejpam-6409	16	13	/	/	SYM
ejpam-6409	16	14	eur	eur	PROPN
ejpam-6409	16	15	.	.	PUNCT
ejpam-6409	17	1	j.	j.	PROPN
ejpam-6409	17	2	pure	pure	PROPN
ejpam-6409	17	3	appl	appl	PROPN
ejpam-6409	17	4	.	.	PROPN
ejpam-6409	17	5	math	math	PROPN
ejpam-6409	17	6	,	,	PUNCT
ejpam-6409	17	7	18	18	NUM
ejpam-6409	17	8	(	(	PUNCT
ejpam-6409	17	9	3	3	NUM
ejpam-6409	17	10	)	)	PUNCT
ejpam-6409	17	11	(	(	PUNCT
ejpam-6409	17	12	2025	2025	NUM
ejpam-6409	17	13	)	)	PUNCT
ejpam-6409	17	14	,	,	PUNCT
ejpam-6409	17	15	6409	6409	NUM
ejpam-6409	17	16	2	2	NUM
ejpam-6409	17	17	of	of	ADP
ejpam-6409	17	18	32	32	NUM
ejpam-6409	17	19	for	for	ADP
ejpam-6409	17	20	several	several	ADJ
ejpam-6409	17	21	critical	critical	ADJ
ejpam-6409	17	22	concepts	concept	NOUN
ejpam-6409	17	23	in	in	ADP
ejpam-6409	17	24	this	this	DET
ejpam-6409	17	25	area	area	NOUN
ejpam-6409	17	26	,	,	PUNCT
ejpam-6409	17	27	including	include	VERB
ejpam-6409	17	28	linear	linear	PROPN
ejpam-6409	17	29	codes	code	NOUN
ejpam-6409	17	30	,	,	PUNCT
ejpam-6409	17	31	cyclic	cyclic	ADJ
ejpam-6409	17	32	codes	code	NOUN
ejpam-6409	17	33	,	,	PUNCT
ejpam-6409	17	34	and	and	CCONJ
ejpam-6409	17	35	bch	bch	PROPN
ejpam-6409	17	36	codes	code	NOUN
ejpam-6409	17	37	,	,	PUNCT
ejpam-6409	17	38	which	which	PRON
ejpam-6409	17	39	have	have	AUX
ejpam-6409	17	40	seen	see	VERB
ejpam-6409	17	41	practical	practical	ADJ
ejpam-6409	17	42	applications	application	NOUN
ejpam-6409	17	43	in	in	ADP
ejpam-6409	17	44	data	datum	NOUN
ejpam-6409	17	45	storage	storage	NOUN
ejpam-6409	17	46	,	,	PUNCT
ejpam-6409	17	47	satellite	satellite	NOUN
ejpam-6409	17	48	communication	communication	NOUN
ejpam-6409	17	49	,	,	PUNCT
ejpam-6409	17	50	and	and	CCONJ
ejpam-6409	17	51	cybersecurity	cybersecurity	NOUN
ejpam-6409	17	52	[	[	X
ejpam-6409	17	53	1–4	1–4	PROPN
ejpam-6409	17	54	]	]	X
ejpam-6409	17	55	.	.	PUNCT
ejpam-6409	18	1	the	the	DET
ejpam-6409	18	2	use	use	NOUN
ejpam-6409	18	3	of	of	ADP
ejpam-6409	18	4	algebraic	algebraic	ADJ
ejpam-6409	18	5	structures	structure	NOUN
ejpam-6409	18	6	,	,	PUNCT
ejpam-6409	18	7	such	such	ADJ
ejpam-6409	18	8	as	as	ADP
ejpam-6409	18	9	rings	ring	NOUN
ejpam-6409	18	10	and	and	CCONJ
ejpam-6409	18	11	fields	field	NOUN
ejpam-6409	18	12	,	,	PUNCT
ejpam-6409	18	13	to	to	PART
ejpam-6409	18	14	define	define	VERB
ejpam-6409	18	15	codes	code	NOUN
ejpam-6409	18	16	with	with	ADP
ejpam-6409	18	17	robust	robust	ADJ
ejpam-6409	18	18	or	or	CCONJ
ejpam-6409	18	19	stable	stable	ADJ
ejpam-6409	18	20	properties	property	NOUN
ejpam-6409	18	21	has	have	AUX
ejpam-6409	18	22	led	lead	VERB
ejpam-6409	18	23	to	to	ADP
ejpam-6409	18	24	the	the	DET
ejpam-6409	18	25	development	development	NOUN
ejpam-6409	18	26	of	of	ADP
ejpam-6409	18	27	more	more	ADV
ejpam-6409	18	28	general	general	ADJ
ejpam-6409	18	29	and	and	CCONJ
ejpam-6409	18	30	efficient	efficient	ADJ
ejpam-6409	18	31	error	error	NOUN
ejpam-6409	18	32	-	-	PUNCT
ejpam-6409	18	33	correcting	correct	VERB
ejpam-6409	18	34	techniques	technique	NOUN
ejpam-6409	18	35	[	[	X
ejpam-6409	18	36	5–7	5–7	X
ejpam-6409	18	37	]	]	PUNCT
ejpam-6409	18	38	.	.	PUNCT
ejpam-6409	19	1	these	these	DET
ejpam-6409	19	2	advances	advance	NOUN
ejpam-6409	19	3	were	be	AUX
ejpam-6409	19	4	complemented	complement	VERB
ejpam-6409	19	5	by	by	ADP
ejpam-6409	19	6	gray	gray	ADJ
ejpam-6409	19	7	maps	map	NOUN
ejpam-6409	19	8	,	,	PUNCT
ejpam-6409	19	9	isometries	isometry	NOUN
ejpam-6409	19	10	,	,	PUNCT
ejpam-6409	19	11	and	and	CCONJ
ejpam-6409	19	12	duality	duality	NOUN
ejpam-6409	19	13	concepts	concept	NOUN
ejpam-6409	19	14	,	,	PUNCT
ejpam-6409	19	15	linking	link	VERB
ejpam-6409	19	16	binary	binary	ADJ
ejpam-6409	19	17	code	code	NOUN
ejpam-6409	19	18	representations	representation	NOUN
ejpam-6409	19	19	with	with	ADP
ejpam-6409	19	20	algebraic	algebraic	ADJ
ejpam-6409	19	21	tools	tool	NOUN
ejpam-6409	19	22	[	[	X
ejpam-6409	19	23	7–9	7–9	X
ejpam-6409	19	24	]	]	X
ejpam-6409	19	25	.	.	PUNCT
ejpam-6409	20	1	in	in	ADP
ejpam-6409	20	2	recent	recent	ADJ
ejpam-6409	20	3	years	year	NOUN
ejpam-6409	20	4	,	,	PUNCT
ejpam-6409	20	5	this	this	DET
ejpam-6409	20	6	algebraic	algebraic	ADJ
ejpam-6409	20	7	framework	framework	NOUN
ejpam-6409	20	8	has	have	AUX
ejpam-6409	20	9	been	be	AUX
ejpam-6409	20	10	extended	extend	VERB
ejpam-6409	20	11	to	to	ADP
ejpam-6409	20	12	non	non	ADJ
ejpam-6409	20	13	-	-	ADJ
ejpam-6409	20	14	field	field	ADJ
ejpam-6409	20	15	structures	structure	NOUN
ejpam-6409	20	16	,	,	PUNCT
ejpam-6409	20	17	such	such	ADJ
ejpam-6409	20	18	as	as	ADP
ejpam-6409	20	19	finite	finite	ADJ
ejpam-6409	20	20	rings	ring	NOUN
ejpam-6409	20	21	,	,	PUNCT
ejpam-6409	20	22	group	group	NOUN
ejpam-6409	20	23	rings	ring	NOUN
ejpam-6409	20	24	,	,	PUNCT
ejpam-6409	20	25	and	and	CCONJ
ejpam-6409	20	26	number	number	NOUN
ejpam-6409	20	27	-	-	PUNCT
ejpam-6409	20	28	theoretic	theoretic	NOUN
ejpam-6409	20	29	rings	ring	NOUN
ejpam-6409	20	30	like	like	ADP
ejpam-6409	20	31	gaussian	gaussian	NOUN
ejpam-6409	20	32	and	and	CCONJ
ejpam-6409	20	33	eisenstein	eisenstein	NOUN
ejpam-6409	20	34	integers	integer	NOUN
ejpam-6409	20	35	[	[	X
ejpam-6409	20	36	4	4	NUM
ejpam-6409	20	37	,	,	PUNCT
ejpam-6409	20	38	10–12	10–12	NUM
ejpam-6409	20	39	]	]	PUNCT
ejpam-6409	20	40	.	.	PUNCT
ejpam-6409	21	1	many	many	ADJ
ejpam-6409	21	2	algebraic	algebraic	ADJ
ejpam-6409	21	3	domains	domain	NOUN
ejpam-6409	21	4	have	have	AUX
ejpam-6409	21	5	studied	study	VERB
ejpam-6409	21	6	hadamard	hadamard	ADJ
ejpam-6409	21	7	codes	code	NOUN
ejpam-6409	21	8	—	—	PUNCT
ejpam-6409	21	9	a	a	DET
ejpam-6409	21	10	class	class	NOUN
ejpam-6409	21	11	of	of	ADP
ejpam-6409	21	12	codes	code	NOUN
ejpam-6409	21	13	with	with	ADP
ejpam-6409	21	14	a	a	DET
ejpam-6409	21	15	simple	simple	ADJ
ejpam-6409	21	16	structure	structure	NOUN
ejpam-6409	21	17	and	and	CCONJ
ejpam-6409	21	18	an	an	DET
ejpam-6409	21	19	optimal	optimal	ADJ
ejpam-6409	21	20	minimum	minimum	ADJ
ejpam-6409	21	21	distance	distance	NOUN
ejpam-6409	21	22	.	.	PUNCT
ejpam-6409	22	1	these	these	DET
ejpam-6409	22	2	codes	code	NOUN
ejpam-6409	22	3	,	,	PUNCT
ejpam-6409	22	4	originally	originally	ADV
ejpam-6409	22	5	built	build	VERB
ejpam-6409	22	6	from	from	ADP
ejpam-6409	22	7	hadamard	hadamard	ADJ
ejpam-6409	22	8	matrices	matrix	NOUN
ejpam-6409	22	9	,	,	PUNCT
ejpam-6409	22	10	are	be	AUX
ejpam-6409	22	11	a	a	DET
ejpam-6409	22	12	backbone	backbone	NOUN
ejpam-6409	22	13	of	of	ADP
ejpam-6409	22	14	many	many	ADJ
ejpam-6409	22	15	applications	application	NOUN
ejpam-6409	22	16	due	due	ADP
ejpam-6409	22	17	to	to	ADP
ejpam-6409	22	18	their	their	PRON
ejpam-6409	22	19	strong	strong	ADJ
ejpam-6409	22	20	error	error	NOUN
ejpam-6409	22	21	detection	detection	NOUN
ejpam-6409	22	22	and	and	CCONJ
ejpam-6409	22	23	simple	simple	ADJ
ejpam-6409	22	24	structure	structure	NOUN
ejpam-6409	22	25	.	.	PUNCT
ejpam-6409	23	1	researchers	researcher	NOUN
ejpam-6409	23	2	have	have	AUX
ejpam-6409	23	3	explored	explore	VERB
ejpam-6409	23	4	the	the	DET
ejpam-6409	23	5	rank	rank	NOUN
ejpam-6409	23	6	and	and	CCONJ
ejpam-6409	23	7	kernel	kernel	NOUN
ejpam-6409	23	8	properties	property	NOUN
ejpam-6409	23	9	in	in	ADP
ejpam-6409	23	10	the	the	DET
ejpam-6409	23	11	binary	binary	ADJ
ejpam-6409	23	12	and	and	CCONJ
ejpam-6409	23	13	z4	z4	PROPN
ejpam-6409	23	14	-	-	PUNCT
ejpam-6409	23	15	linear	linear	NOUN
ejpam-6409	23	16	settings	setting	NOUN
ejpam-6409	23	17	to	to	PART
ejpam-6409	23	18	understand	understand	VERB
ejpam-6409	23	19	whether	whether	SCONJ
ejpam-6409	23	20	the	the	DET
ejpam-6409	23	21	algebraic	algebraic	ADJ
ejpam-6409	23	22	complexity	complexity	NOUN
ejpam-6409	23	23	of	of	ADP
ejpam-6409	23	24	these	these	DET
ejpam-6409	23	25	problems	problem	NOUN
ejpam-6409	23	26	impacts	impact	VERB
ejpam-6409	23	27	code	code	NOUN
ejpam-6409	23	28	dimension	dimension	NOUN
ejpam-6409	23	29	.	.	PUNCT
ejpam-6409	24	1	the	the	DET
ejpam-6409	24	2	study	study	NOUN
ejpam-6409	24	3	of	of	ADP
ejpam-6409	24	4	hadamard	hadamard	ADJ
ejpam-6409	24	5	codes	code	NOUN
ejpam-6409	24	6	over	over	ADP
ejpam-6409	24	7	z2s	z2s	PROPN
ejpam-6409	24	8	,	,	PUNCT
ejpam-6409	24	9	zps	zps	PROPN
ejpam-6409	24	10	,	,	PUNCT
ejpam-6409	24	11	and	and	CCONJ
ejpam-6409	24	12	mixed	mixed	ADJ
ejpam-6409	24	13	modules	module	NOUN
ejpam-6409	24	14	has	have	AUX
ejpam-6409	24	15	allowed	allow	VERB
ejpam-6409	24	16	a	a	DET
ejpam-6409	24	17	deeper	deep	ADJ
ejpam-6409	24	18	examination	examination	NOUN
ejpam-6409	24	19	of	of	ADP
ejpam-6409	24	20	structural	structural	ADJ
ejpam-6409	24	21	invariants	invariant	NOUN
ejpam-6409	24	22	and	and	CCONJ
ejpam-6409	24	23	equivalence	equivalence	NOUN
ejpam-6409	24	24	classes	class	NOUN
ejpam-6409	24	25	of	of	ADP
ejpam-6409	24	26	such	such	ADJ
ejpam-6409	24	27	codes	code	NOUN
ejpam-6409	24	28	[	[	X
ejpam-6409	24	29	4	4	NUM
ejpam-6409	24	30	,	,	PUNCT
ejpam-6409	24	31	7	7	NUM
ejpam-6409	24	32	,	,	PUNCT
ejpam-6409	24	33	9	9	NUM
ejpam-6409	24	34	]	]	PUNCT
ejpam-6409	24	35	.	.	PUNCT
ejpam-6409	25	1	these	these	DET
ejpam-6409	25	2	studies	study	NOUN
ejpam-6409	25	3	highlight	highlight	VERB
ejpam-6409	25	4	the	the	DET
ejpam-6409	25	5	influence	influence	NOUN
ejpam-6409	25	6	of	of	ADP
ejpam-6409	25	7	the	the	DET
ejpam-6409	25	8	underlying	underlie	VERB
ejpam-6409	25	9	ring	ring	NOUN
ejpam-6409	25	10	on	on	ADP
ejpam-6409	25	11	linearity	linearity	NOUN
ejpam-6409	25	12	,	,	PUNCT
ejpam-6409	25	13	decoding	decoding	NOUN
ejpam-6409	25	14	,	,	PUNCT
ejpam-6409	25	15	and	and	CCONJ
ejpam-6409	25	16	code	code	NOUN
ejpam-6409	25	17	equivalence	equivalence	NOUN
ejpam-6409	25	18	,	,	PUNCT
ejpam-6409	25	19	motivating	motivating	NOUN
ejpam-6409	25	20	extensions	extension	NOUN
ejpam-6409	25	21	to	to	PART
ejpam-6409	25	22	code	code	VERB
ejpam-6409	25	23	constructions	construction	NOUN
ejpam-6409	25	24	over	over	ADP
ejpam-6409	25	25	eisenstein	eisenstein	PROPN
ejpam-6409	25	26	and	and	CCONJ
ejpam-6409	25	27	quaternion	quaternion	NOUN
ejpam-6409	25	28	integers	integer	NOUN
ejpam-6409	25	29	[	[	X
ejpam-6409	25	30	9	9	NUM
ejpam-6409	25	31	,	,	PUNCT
ejpam-6409	25	32	13	13	NUM
ejpam-6409	25	33	,	,	PUNCT
ejpam-6409	25	34	14	14	NUM
ejpam-6409	25	35	]	]	PUNCT
ejpam-6409	25	36	.	.	PUNCT
ejpam-6409	26	1	recently	recently	ADV
ejpam-6409	26	2	,	,	PUNCT
ejpam-6409	26	3	much	much	ADJ
ejpam-6409	26	4	effort	effort	NOUN
ejpam-6409	26	5	has	have	AUX
ejpam-6409	26	6	been	be	AUX
ejpam-6409	26	7	directed	direct	VERB
ejpam-6409	26	8	toward	toward	ADP
ejpam-6409	26	9	classifying	classify	VERB
ejpam-6409	26	10	and	and	CCONJ
ejpam-6409	26	11	constructing	construct	VERB
ejpam-6409	26	12	generalized	generalized	ADJ
ejpam-6409	26	13	hadamard	hadamard	NOUN
ejpam-6409	26	14	(	(	PUNCT
ejpam-6409	26	15	gh	gh	PROPN
ejpam-6409	26	16	)	)	PUNCT
ejpam-6409	26	17	codes	code	NOUN
ejpam-6409	26	18	over	over	ADP
ejpam-6409	26	19	various	various	ADJ
ejpam-6409	26	20	algebraic	algebraic	ADJ
ejpam-6409	26	21	structures	structure	NOUN
ejpam-6409	26	22	.	.	PUNCT
ejpam-6409	27	1	bhunia	bhunia	NOUN
ejpam-6409	27	2	et	et	PROPN
ejpam-6409	27	3	al	al	PROPN
ejpam-6409	27	4	.	.	PUNCT
ejpam-6409	28	1	[	[	X
ejpam-6409	28	2	15	15	NUM
ejpam-6409	28	3	,	,	PUNCT
ejpam-6409	28	4	16	16	NUM
ejpam-6409	28	5	]	]	PUNCT
ejpam-6409	28	6	developed	develop	VERB
ejpam-6409	28	7	a	a	DET
ejpam-6409	28	8	framework	framework	NOUN
ejpam-6409	28	9	for	for	ADP
ejpam-6409	28	10	zps	zps	PROPN
ejpam-6409	28	11	-	-	PUNCT
ejpam-6409	28	12	linear	linear	PROPN
ejpam-6409	28	13	gh	gh	PROPN
ejpam-6409	28	14	codes	code	NOUN
ejpam-6409	28	15	in	in	ADP
ejpam-6409	28	16	terms	term	NOUN
ejpam-6409	28	17	of	of	ADP
ejpam-6409	28	18	kernel	kernel	NOUN
ejpam-6409	28	19	,	,	PUNCT
ejpam-6409	28	20	linearity	linearity	NOUN
ejpam-6409	28	21	,	,	PUNCT
ejpam-6409	28	22	and	and	CCONJ
ejpam-6409	28	23	equivalence	equivalence	NOUN
ejpam-6409	28	24	.	.	PUNCT
ejpam-6409	29	1	this	this	PRON
ejpam-6409	29	2	builds	build	VERB
ejpam-6409	29	3	upon	upon	SCONJ
ejpam-6409	29	4	previous	previous	ADJ
ejpam-6409	29	5	work	work	NOUN
ejpam-6409	29	6	by	by	ADP
ejpam-6409	29	7	dougherty	dougherty	PROPN
ejpam-6409	29	8	,	,	PUNCT
ejpam-6409	29	9	villanueva	villanueva	PROPN
ejpam-6409	29	10	,	,	PUNCT
ejpam-6409	29	11	and	and	CCONJ
ejpam-6409	29	12	rifà	rifà	NOUN
ejpam-6409	30	1	[	[	X
ejpam-6409	30	2	6	6	NUM
ejpam-6409	30	3	,	,	PUNCT
ejpam-6409	30	4	17	17	NUM
ejpam-6409	30	5	,	,	PUNCT
ejpam-6409	30	6	18	18	NUM
ejpam-6409	30	7	]	]	PUNCT
ejpam-6409	30	8	,	,	PUNCT
ejpam-6409	30	9	and	and	CCONJ
ejpam-6409	30	10	other	other	ADJ
ejpam-6409	30	11	contributions	contribution	NOUN
ejpam-6409	30	12	[	[	X
ejpam-6409	30	13	5	5	NUM
ejpam-6409	30	14	,	,	PUNCT
ejpam-6409	30	15	19	19	NUM
ejpam-6409	30	16	,	,	PUNCT
ejpam-6409	30	17	20	20	NUM
ejpam-6409	30	18	]	]	PUNCT
ejpam-6409	30	19	that	that	PRON
ejpam-6409	30	20	focused	focus	VERB
ejpam-6409	30	21	on	on	ADP
ejpam-6409	30	22	rank	rank	NOUN
ejpam-6409	30	23	and	and	CCONJ
ejpam-6409	30	24	kernel	kernel	PROPN
ejpam-6409	30	25	of	of	ADP
ejpam-6409	30	26	codes	code	NOUN
ejpam-6409	30	27	over	over	ADP
ejpam-6409	30	28	z2s	z2s	PROPN
ejpam-6409	30	29	and	and	CCONJ
ejpam-6409	30	30	related	related	ADJ
ejpam-6409	30	31	rings	ring	NOUN
ejpam-6409	30	32	.	.	PUNCT
ejpam-6409	31	1	these	these	DET
ejpam-6409	31	2	investigations	investigation	NOUN
ejpam-6409	31	3	have	have	AUX
ejpam-6409	31	4	greatly	greatly	ADV
ejpam-6409	31	5	enhanced	enhance	VERB
ejpam-6409	31	6	the	the	DET
ejpam-6409	31	7	understanding	understanding	NOUN
ejpam-6409	31	8	of	of	ADP
ejpam-6409	31	9	code	code	NOUN
ejpam-6409	31	10	structure	structure	NOUN
ejpam-6409	31	11	,	,	PUNCT
ejpam-6409	31	12	gray	gray	ADJ
ejpam-6409	31	13	maps	map	NOUN
ejpam-6409	31	14	,	,	PUNCT
ejpam-6409	31	15	and	and	CCONJ
ejpam-6409	31	16	their	their	PRON
ejpam-6409	31	17	connection	connection	NOUN
ejpam-6409	31	18	to	to	ADP
ejpam-6409	31	19	classification	classification	NOUN
ejpam-6409	31	20	theory	theory	NOUN
ejpam-6409	31	21	.	.	PUNCT
ejpam-6409	32	1	the	the	DET
ejpam-6409	32	2	significance	significance	NOUN
ejpam-6409	32	3	of	of	ADP
ejpam-6409	32	4	z4	z4	PROPN
ejpam-6409	32	5	-	-	PUNCT
ejpam-6409	32	6	linear	linear	NOUN
ejpam-6409	32	7	codes	code	NOUN
ejpam-6409	32	8	was	be	AUX
ejpam-6409	32	9	demonstrated	demonstrate	VERB
ejpam-6409	32	10	by	by	ADP
ejpam-6409	32	11	carlet	carlet	NOUN
ejpam-6409	32	12	[	[	X
ejpam-6409	32	13	5	5	NUM
ejpam-6409	32	14	]	]	PUNCT
ejpam-6409	32	15	and	and	CCONJ
ejpam-6409	32	16	hammons	hammon	NOUN
ejpam-6409	32	17	et	et	PROPN
ejpam-6409	32	18	al	al	PROPN
ejpam-6409	32	19	.	.	PUNCT
ejpam-6409	33	1	[	[	X
ejpam-6409	33	2	7	7	NUM
ejpam-6409	33	3	]	]	PUNCT
ejpam-6409	33	4	,	,	PUNCT
ejpam-6409	33	5	who	who	PRON
ejpam-6409	33	6	revealed	reveal	VERB
ejpam-6409	33	7	that	that	SCONJ
ejpam-6409	33	8	z4linear	z4linear	PROPN
ejpam-6409	33	9	codes	code	NOUN
ejpam-6409	33	10	underpin	underpin	VERB
ejpam-6409	33	11	other	other	ADJ
ejpam-6409	33	12	well	well	ADV
ejpam-6409	33	13	-	-	PUNCT
ejpam-6409	33	14	known	know	VERB
ejpam-6409	33	15	nonlinear	nonlinear	ADJ
ejpam-6409	33	16	codes	code	NOUN
ejpam-6409	33	17	such	such	ADJ
ejpam-6409	33	18	as	as	ADP
ejpam-6409	33	19	kerdock	kerdock	NOUN
ejpam-6409	33	20	and	and	CCONJ
ejpam-6409	33	21	preparata	preparata	NOUN
ejpam-6409	33	22	codes	code	NOUN
ejpam-6409	33	23	.	.	PUNCT
ejpam-6409	34	1	further	further	ADJ
ejpam-6409	34	2	studies	study	NOUN
ejpam-6409	34	3	into	into	ADP
ejpam-6409	34	4	other	other	ADJ
ejpam-6409	34	5	ring	ring	NOUN
ejpam-6409	34	6	-	-	PUNCT
ejpam-6409	34	7	based	base	VERB
ejpam-6409	34	8	codes	code	NOUN
ejpam-6409	34	9	include	include	VERB
ejpam-6409	34	10	extended	extend	VERB
ejpam-6409	34	11	perfect	perfect	ADJ
ejpam-6409	34	12	codes	code	NOUN
ejpam-6409	34	13	and	and	CCONJ
ejpam-6409	34	14	duality	duality	NOUN
ejpam-6409	34	15	over	over	ADP
ejpam-6409	34	16	z2k	z2k	NOUN
ejpam-6409	34	17	,	,	PUNCT
ejpam-6409	34	18	as	as	SCONJ
ejpam-6409	34	19	explored	explore	VERB
ejpam-6409	34	20	by	by	ADP
ejpam-6409	34	21	krotov	krotov	ADJ
ejpam-6409	34	22	[	[	X
ejpam-6409	34	23	21	21	NUM
ejpam-6409	34	24	,	,	PUNCT
ejpam-6409	34	25	22	22	NUM
ejpam-6409	34	26	]	]	PUNCT
ejpam-6409	34	27	.	.	PUNCT
ejpam-6409	35	1	more	more	ADV
ejpam-6409	35	2	recent	recent	ADJ
ejpam-6409	35	3	work	work	NOUN
ejpam-6409	35	4	by	by	ADP
ejpam-6409	35	5	shi	shi	PROPN
ejpam-6409	35	6	et	et	PROPN
ejpam-6409	35	7	al	al	PROPN
ejpam-6409	35	8	.	.	PUNCT
ejpam-6409	36	1	[	[	X
ejpam-6409	36	2	9	9	NUM
ejpam-6409	36	3	,	,	PUNCT
ejpam-6409	36	4	23	23	NUM
ejpam-6409	36	5	]	]	PUNCT
ejpam-6409	36	6	has	have	AUX
ejpam-6409	36	7	focused	focus	VERB
ejpam-6409	36	8	on	on	ADP
ejpam-6409	36	9	additive	additive	ADJ
ejpam-6409	36	10	codes	code	NOUN
ejpam-6409	36	11	over	over	ADP
ejpam-6409	36	12	mixed	mixed	ADJ
ejpam-6409	36	13	rings	ring	NOUN
ejpam-6409	36	14	,	,	PUNCT
ejpam-6409	36	15	duality	duality	NOUN
ejpam-6409	36	16	principles	principle	NOUN
ejpam-6409	36	17	,	,	PUNCT
ejpam-6409	36	18	and	and	CCONJ
ejpam-6409	36	19	classification	classification	NOUN
ejpam-6409	36	20	criteria	criterion	NOUN
ejpam-6409	36	21	,	,	PUNCT
ejpam-6409	36	22	demonstrating	demonstrate	VERB
ejpam-6409	36	23	the	the	DET
ejpam-6409	36	24	algebraic	algebraic	ADJ
ejpam-6409	36	25	depth	depth	NOUN
ejpam-6409	36	26	and	and	CCONJ
ejpam-6409	36	27	practical	practical	ADJ
ejpam-6409	36	28	relevance	relevance	NOUN
ejpam-6409	36	29	of	of	ADP
ejpam-6409	36	30	such	such	ADJ
ejpam-6409	36	31	constructions	construction	NOUN
ejpam-6409	36	32	.	.	PUNCT
ejpam-6409	37	1	the	the	DET
ejpam-6409	37	2	other	other	ADJ
ejpam-6409	37	3	significant	significant	ADJ
ejpam-6409	37	4	direction	direction	NOUN
ejpam-6409	37	5	has	have	AUX
ejpam-6409	37	6	been	be	AUX
ejpam-6409	37	7	the	the	DET
ejpam-6409	37	8	exploration	exploration	NOUN
ejpam-6409	37	9	of	of	ADP
ejpam-6409	37	10	hadamard	hadamard	ADJ
ejpam-6409	37	11	and	and	CCONJ
ejpam-6409	37	12	generalized	generalized	ADJ
ejpam-6409	37	13	hadamard	hadamard	NOUN
ejpam-6409	37	14	(	(	PUNCT
ejpam-6409	37	15	gh	gh	PROPN
ejpam-6409	37	16	)	)	PUNCT
ejpam-6409	37	17	codes	code	NOUN
ejpam-6409	37	18	over	over	ADP
ejpam-6409	37	19	number	number	NOUN
ejpam-6409	37	20	-	-	PUNCT
ejpam-6409	37	21	theoretic	theoretic	NOUN
ejpam-6409	37	22	rings	ring	NOUN
ejpam-6409	37	23	.	.	PUNCT
ejpam-6409	38	1	in	in	ADP
ejpam-6409	38	2	particular	particular	ADJ
ejpam-6409	38	3	,	,	PUNCT
ejpam-6409	38	4	sajjad	sajjad	PROPN
ejpam-6409	38	5	et	et	PROPN
ejpam-6409	38	6	al	al	PROPN
ejpam-6409	38	7	.	.	PROPN
ejpam-6409	38	8	have	have	AUX
ejpam-6409	38	9	made	make	VERB
ejpam-6409	38	10	substantial	substantial	ADJ
ejpam-6409	38	11	contributions	contribution	NOUN
ejpam-6409	38	12	to	to	ADP
ejpam-6409	38	13	coding	code	VERB
ejpam-6409	38	14	theory	theory	NOUN
ejpam-6409	38	15	in	in	ADP
ejpam-6409	38	16	the	the	DET
ejpam-6409	38	17	context	context	NOUN
ejpam-6409	38	18	of	of	ADP
ejpam-6409	38	19	gaussian	gaussian	ADJ
ejpam-6409	38	20	and	and	CCONJ
ejpam-6409	38	21	eisenstein	eisenstein	NOUN
ejpam-6409	38	22	integers	integer	NOUN
ejpam-6409	38	23	[	[	X
ejpam-6409	38	24	10	10	NUM
ejpam-6409	38	25	,	,	PUNCT
ejpam-6409	38	26	12	12	NUM
ejpam-6409	38	27	]	]	PUNCT
ejpam-6409	38	28	,	,	PUNCT
ejpam-6409	38	29	including	include	VERB
ejpam-6409	38	30	bch	bch	PROPN
ejpam-6409	38	31	code	code	NOUN
ejpam-6409	38	32	constructions	construction	NOUN
ejpam-6409	38	33	and	and	CCONJ
ejpam-6409	38	34	alternant	alternant	ADJ
ejpam-6409	38	35	codes	code	NOUN
ejpam-6409	38	36	with	with	ADP
ejpam-6409	38	37	applications	application	NOUN
ejpam-6409	38	38	in	in	ADP
ejpam-6409	38	39	modern	modern	ADJ
ejpam-6409	38	40	technology	technology	NOUN
ejpam-6409	38	41	[	[	X
ejpam-6409	38	42	24	24	NUM
ejpam-6409	38	43	]	]	PUNCT
ejpam-6409	38	44	.	.	PUNCT
ejpam-6409	39	1	modified	modify	VERB
ejpam-6409	39	2	berlekamp	berlekamp	PROPN
ejpam-6409	39	3	–	–	PUNCT
ejpam-6409	39	4	massey	massey	NOUN
ejpam-6409	39	5	algorithms	algorithm	NOUN
ejpam-6409	39	6	,	,	PUNCT
ejpam-6409	39	7	along	along	ADP
ejpam-6409	39	8	with	with	ADP
ejpam-6409	39	9	included	include	VERB
ejpam-6409	39	10	algebraic	algebraic	ADJ
ejpam-6409	39	11	tools	tool	NOUN
ejpam-6409	39	12	,	,	PUNCT
ejpam-6409	39	13	have	have	AUX
ejpam-6409	39	14	been	be	AUX
ejpam-6409	39	15	employed	employ	VERB
ejpam-6409	39	16	in	in	ADP
ejpam-6409	39	17	their	their	PRON
ejpam-6409	39	18	decoding	decode	VERB
ejpam-6409	39	19	frameworks	framework	NOUN
ejpam-6409	39	20	,	,	PUNCT
ejpam-6409	39	21	which	which	PRON
ejpam-6409	39	22	generalize	generalize	VERB
ejpam-6409	39	23	classical	classical	ADJ
ejpam-6409	39	24	coding	code	VERB
ejpam-6409	39	25	theory	theory	NOUN
ejpam-6409	39	26	into	into	ADP
ejpam-6409	39	27	broader	broad	ADJ
ejpam-6409	39	28	algebraic	algebraic	ADJ
ejpam-6409	39	29	domains	domain	NOUN
ejpam-6409	39	30	[	[	X
ejpam-6409	39	31	4	4	NUM
ejpam-6409	39	32	,	,	PUNCT
ejpam-6409	39	33	11	11	NUM
ejpam-6409	39	34	]	]	PUNCT
ejpam-6409	39	35	.	.	PUNCT
ejpam-6409	40	1	additionally	additionally	ADV
ejpam-6409	40	2	,	,	PUNCT
ejpam-6409	40	3	the	the	DET
ejpam-6409	40	4	work	work	NOUN
ejpam-6409	40	5	of	of	ADP
ejpam-6409	40	6	villanueva	villanueva	PROPN
ejpam-6409	40	7	and	and	CCONJ
ejpam-6409	40	8	zinoviev	zinoviev	PROPN
ejpam-6409	40	9	[	[	X
ejpam-6409	40	10	25	25	NUM
ejpam-6409	40	11	,	,	PUNCT
ejpam-6409	40	12	26	26	NUM
ejpam-6409	40	13	]	]	PUNCT
ejpam-6409	40	14	on	on	ADP
ejpam-6409	40	15	hadamard	hadamard	ADJ
ejpam-6409	40	16	matrix	matrix	NOUN
ejpam-6409	40	17	construction	construction	NOUN
ejpam-6409	40	18	has	have	AUX
ejpam-6409	40	19	influenced	influence	VERB
ejpam-6409	40	20	generalized	generalized	ADJ
ejpam-6409	40	21	code	code	NOUN
ejpam-6409	40	22	design	design	NOUN
ejpam-6409	40	23	across	across	ADP
ejpam-6409	40	24	diverse	diverse	ADJ
ejpam-6409	40	25	metrics	metric	NOUN
ejpam-6409	40	26	and	and	CCONJ
ejpam-6409	40	27	algebraic	algebraic	ADJ
ejpam-6409	40	28	rings	ring	NOUN
ejpam-6409	40	29	.	.	PUNCT
ejpam-6409	41	1	despite	despite	SCONJ
ejpam-6409	41	2	the	the	DET
ejpam-6409	41	3	fact	fact	NOUN
ejpam-6409	41	4	that	that	SCONJ
ejpam-6409	41	5	the	the	DET
ejpam-6409	41	6	theory	theory	NOUN
ejpam-6409	41	7	of	of	ADP
ejpam-6409	41	8	gh	gh	PROPN
ejpam-6409	41	9	codes	code	NOUN
ejpam-6409	41	10	over	over	ADP
ejpam-6409	41	11	classical	classical	ADJ
ejpam-6409	41	12	rings	ring	NOUN
ejpam-6409	41	13	like	like	SCONJ
ejpam-6409	41	14	z2s	z2s	PROPN
ejpam-6409	41	15	is	be	AUX
ejpam-6409	41	16	fairly	fairly	ADV
ejpam-6409	41	17	well	well	ADV
ejpam-6409	41	18	established	establish	VERB
ejpam-6409	41	19	,	,	PUNCT
ejpam-6409	41	20	there	there	PRON
ejpam-6409	41	21	exists	exist	VERB
ejpam-6409	41	22	a	a	DET
ejpam-6409	41	23	significant	significant	ADJ
ejpam-6409	41	24	gap	gap	NOUN
ejpam-6409	41	25	in	in	ADP
ejpam-6409	41	26	the	the	DET
ejpam-6409	41	27	literature	literature	NOUN
ejpam-6409	41	28	concerning	concern	VERB
ejpam-6409	41	29	their	their	PRON
ejpam-6409	41	30	generalization	generalization	NOUN
ejpam-6409	41	31	muhammad	muhammad	PROPN
ejpam-6409	41	32	sajjad	sajjad	PROPN
ejpam-6409	41	33	et	et	PROPN
ejpam-6409	41	34	al	al	PROPN
ejpam-6409	41	35	.	.	PUNCT
ejpam-6409	41	36	/	/	SYM
ejpam-6409	41	37	eur	eur	PROPN
ejpam-6409	41	38	.	.	PUNCT
ejpam-6409	42	1	j.	j.	PROPN
ejpam-6409	42	2	pure	pure	PROPN
ejpam-6409	42	3	appl	appl	PROPN
ejpam-6409	42	4	.	.	PROPN
ejpam-6409	42	5	math	math	PROPN
ejpam-6409	42	6	,	,	PUNCT
ejpam-6409	42	7	18	18	NUM
ejpam-6409	42	8	(	(	PUNCT
ejpam-6409	42	9	3	3	NUM
ejpam-6409	42	10	)	)	PUNCT
ejpam-6409	42	11	(	(	PUNCT
ejpam-6409	42	12	2025	2025	NUM
ejpam-6409	42	13	)	)	PUNCT
ejpam-6409	42	14	,	,	PUNCT
ejpam-6409	42	15	6409	6409	NUM
ejpam-6409	42	16	3	3	NUM
ejpam-6409	42	17	of	of	ADP
ejpam-6409	42	18	32	32	NUM
ejpam-6409	42	19	over	over	ADP
ejpam-6409	42	20	eisenstein	eisenstein	NOUN
ejpam-6409	42	21	integers	integer	NOUN
ejpam-6409	42	22	and	and	CCONJ
ejpam-6409	42	23	corresponding	correspond	VERB
ejpam-6409	42	24	rational	rational	ADJ
ejpam-6409	42	25	domains	domain	NOUN
ejpam-6409	42	26	.	.	PUNCT
ejpam-6409	43	1	eisenstein	eisenstein	PROPN
ejpam-6409	43	2	integers	integer	NOUN
ejpam-6409	43	3	offer	offer	VERB
ejpam-6409	43	4	a	a	DET
ejpam-6409	43	5	rich	rich	ADJ
ejpam-6409	43	6	algebraic	algebraic	ADJ
ejpam-6409	43	7	structure	structure	NOUN
ejpam-6409	43	8	with	with	ADP
ejpam-6409	43	9	complex	complex	ADJ
ejpam-6409	43	10	arithmetic	arithmetic	NOUN
ejpam-6409	43	11	,	,	PUNCT
ejpam-6409	43	12	which	which	PRON
ejpam-6409	43	13	has	have	AUX
ejpam-6409	43	14	already	already	ADV
ejpam-6409	43	15	proven	prove	VERB
ejpam-6409	43	16	useful	useful	ADJ
ejpam-6409	43	17	in	in	ADP
ejpam-6409	43	18	bch	bch	PROPN
ejpam-6409	43	19	and	and	CCONJ
ejpam-6409	43	20	alternant	alternant	ADJ
ejpam-6409	43	21	code	code	NOUN
ejpam-6409	43	22	design	design	NOUN
ejpam-6409	43	23	.	.	PUNCT
ejpam-6409	44	1	it	it	PRON
ejpam-6409	44	2	is	be	AUX
ejpam-6409	44	3	anticipated	anticipate	VERB
ejpam-6409	44	4	that	that	SCONJ
ejpam-6409	44	5	the	the	DET
ejpam-6409	44	6	promising	promising	ADJ
ejpam-6409	44	7	results	result	NOUN
ejpam-6409	44	8	of	of	ADP
ejpam-6409	44	9	sajjad	sajjad	PROPN
ejpam-6409	44	10	et	et	PROPN
ejpam-6409	44	11	al	al	PROPN
ejpam-6409	44	12	.	.	PUNCT
ejpam-6409	45	1	[	[	X
ejpam-6409	45	2	10	10	NUM
ejpam-6409	45	3	,	,	PUNCT
ejpam-6409	45	4	12	12	NUM
ejpam-6409	45	5	,	,	PUNCT
ejpam-6409	45	6	24	24	NUM
ejpam-6409	45	7	]	]	PUNCT
ejpam-6409	45	8	in	in	ADP
ejpam-6409	45	9	robust	robust	ADJ
ejpam-6409	45	10	code	code	NOUN
ejpam-6409	45	11	construction	construction	NOUN
ejpam-6409	45	12	and	and	CCONJ
ejpam-6409	45	13	error	error	NOUN
ejpam-6409	45	14	correction	correction	NOUN
ejpam-6409	45	15	using	use	VERB
ejpam-6409	45	16	eisenstein	eisenstein	NOUN
ejpam-6409	45	17	integers	integer	NOUN
ejpam-6409	45	18	will	will	AUX
ejpam-6409	45	19	yield	yield	VERB
ejpam-6409	45	20	analogous	analogous	ADJ
ejpam-6409	45	21	benefits	benefit	NOUN
ejpam-6409	45	22	in	in	ADP
ejpam-6409	45	23	the	the	DET
ejpam-6409	45	24	context	context	NOUN
ejpam-6409	45	25	of	of	ADP
ejpam-6409	45	26	gh	gh	PROPN
ejpam-6409	45	27	codes	code	NOUN
ejpam-6409	45	28	.	.	PUNCT
ejpam-6409	46	1	furthermore	furthermore	ADV
ejpam-6409	46	2	,	,	PUNCT
ejpam-6409	46	3	intellectual	intellectual	ADJ
ejpam-6409	46	4	stimulation	stimulation	NOUN
ejpam-6409	46	5	arises	arise	VERB
ejpam-6409	46	6	from	from	ADP
ejpam-6409	46	7	the	the	DET
ejpam-6409	46	8	complexity	complexity	NOUN
ejpam-6409	46	9	of	of	ADP
ejpam-6409	46	10	such	such	DET
ejpam-6409	46	11	a	a	DET
ejpam-6409	46	12	non	non	ADJ
ejpam-6409	46	13	-	-	ADJ
ejpam-6409	46	14	trivial	trivial	ADJ
ejpam-6409	46	15	ring	ring	NOUN
ejpam-6409	46	16	system	system	NOUN
ejpam-6409	46	17	,	,	PUNCT
ejpam-6409	46	18	where	where	SCONJ
ejpam-6409	46	19	understanding	understand	VERB
ejpam-6409	46	20	linearity	linearity	NOUN
ejpam-6409	46	21	,	,	PUNCT
ejpam-6409	46	22	the	the	DET
ejpam-6409	46	23	behaviour	behaviour	NOUN
ejpam-6409	46	24	of	of	ADP
ejpam-6409	46	25	the	the	DET
ejpam-6409	46	26	gray	gray	ADJ
ejpam-6409	46	27	map	map	NOUN
ejpam-6409	46	28	,	,	PUNCT
ejpam-6409	46	29	and	and	CCONJ
ejpam-6409	46	30	the	the	DET
ejpam-6409	46	31	structure	structure	NOUN
ejpam-6409	46	32	of	of	ADP
ejpam-6409	46	33	the	the	DET
ejpam-6409	46	34	kernel	kernel	NOUN
ejpam-6409	46	35	becomes	become	VERB
ejpam-6409	46	36	crucial	crucial	ADJ
ejpam-6409	46	37	.	.	PUNCT
ejpam-6409	47	1	as	as	SCONJ
ejpam-6409	47	2	the	the	DET
ejpam-6409	47	3	foundational	foundational	ADJ
ejpam-6409	47	4	works	work	NOUN
ejpam-6409	47	5	on	on	ADP
ejpam-6409	47	6	ring	ring	NOUN
ejpam-6409	47	7	-	-	PUNCT
ejpam-6409	47	8	based	base	VERB
ejpam-6409	47	9	and	and	CCONJ
ejpam-6409	47	10	non	non	ADJ
ejpam-6409	47	11	-	-	ADJ
ejpam-6409	47	12	binary	binary	ADJ
ejpam-6409	47	13	hadamard	hadamard	NOUN
ejpam-6409	47	14	codes	code	NOUN
ejpam-6409	47	15	[	[	X
ejpam-6409	47	16	6	6	NUM
ejpam-6409	47	17	,	,	PUNCT
ejpam-6409	47	18	15	15	NUM
ejpam-6409	47	19	,	,	PUNCT
ejpam-6409	47	20	17	17	NUM
ejpam-6409	47	21	,	,	PUNCT
ejpam-6409	47	22	27	27	NUM
ejpam-6409	47	23	]	]	PUNCT
ejpam-6409	47	24	already	already	ADV
ejpam-6409	47	25	offer	offer	VERB
ejpam-6409	47	26	natural	natural	ADJ
ejpam-6409	47	27	starting	starting	NOUN
ejpam-6409	47	28	points	point	NOUN
ejpam-6409	47	29	for	for	ADP
ejpam-6409	47	30	generalization	generalization	NOUN
ejpam-6409	47	31	,	,	PUNCT
ejpam-6409	47	32	the	the	DET
ejpam-6409	47	33	eisenstein	eisenstein	PROPN
ejpam-6409	47	34	local	local	ADJ
ejpam-6409	47	35	ring	ring	NOUN
ejpam-6409	47	36	z2s	z2s	PROPN
ejpam-6409	48	1	[	[	X
ejpam-6409	48	2	ω	ω	X
ejpam-6409	48	3	]	]	X
ejpam-6409	48	4	,	,	PUNCT
ejpam-6409	48	5	where	where	SCONJ
ejpam-6409	48	6	ω2	ω2	ADJ
ejpam-6409	48	7	+	+	CCONJ
ejpam-6409	48	8	ω	ω	PROPN
ejpam-6409	49	1	+	+	CCONJ
ejpam-6409	49	2	1	1	NUM
ejpam-6409	49	3	=	=	SYM
ejpam-6409	49	4	0	0	NUM
ejpam-6409	49	5	,	,	PUNCT
ejpam-6409	49	6	is	be	AUX
ejpam-6409	49	7	chosen	choose	VERB
ejpam-6409	49	8	as	as	ADP
ejpam-6409	49	9	the	the	DET
ejpam-6409	49	10	natural	natural	ADJ
ejpam-6409	49	11	candidate	candidate	NOUN
ejpam-6409	49	12	for	for	ADP
ejpam-6409	49	13	extension	extension	NOUN
ejpam-6409	49	14	.	.	PUNCT
ejpam-6409	50	1	finally	finally	ADV
ejpam-6409	50	2	,	,	PUNCT
ejpam-6409	50	3	this	this	DET
ejpam-6409	50	4	study	study	NOUN
ejpam-6409	50	5	helps	help	VERB
ejpam-6409	50	6	to	to	PART
ejpam-6409	50	7	bridge	bridge	VERB
ejpam-6409	50	8	a	a	DET
ejpam-6409	50	9	critical	critical	ADJ
ejpam-6409	50	10	gap	gap	NOUN
ejpam-6409	50	11	between	between	ADP
ejpam-6409	50	12	classical	classical	ADJ
ejpam-6409	50	13	gh	gh	PROPN
ejpam-6409	50	14	code	code	PROPN
ejpam-6409	50	15	theory	theory	NOUN
ejpam-6409	50	16	and	and	CCONJ
ejpam-6409	50	17	a	a	DET
ejpam-6409	50	18	novel	novel	ADJ
ejpam-6409	50	19	algebraic	algebraic	ADJ
ejpam-6409	50	20	domain	domain	NOUN
ejpam-6409	50	21	,	,	PUNCT
ejpam-6409	50	22	with	with	ADP
ejpam-6409	50	23	potential	potential	ADJ
ejpam-6409	50	24	applications	application	NOUN
ejpam-6409	50	25	in	in	ADP
ejpam-6409	50	26	secure	secure	ADJ
ejpam-6409	50	27	communications	communication	NOUN
ejpam-6409	50	28	and	and	CCONJ
ejpam-6409	50	29	high	high	ADJ
ejpam-6409	50	30	-	-	PUNCT
ejpam-6409	50	31	reliability	reliability	NOUN
ejpam-6409	50	32	systems	system	NOUN
ejpam-6409	50	33	.	.	PUNCT
ejpam-6409	51	1	this	this	DET
ejpam-6409	51	2	article	article	NOUN
ejpam-6409	51	3	contributes	contribute	VERB
ejpam-6409	51	4	the	the	DET
ejpam-6409	51	5	following	following	NOUN
ejpam-6409	51	6	:	:	PUNCT
ejpam-6409	51	7	•	•	NUM
ejpam-6409	51	8	it	it	PRON
ejpam-6409	51	9	demonstrates	demonstrate	VERB
ejpam-6409	51	10	how	how	SCONJ
ejpam-6409	51	11	gh	gh	PROPN
ejpam-6409	51	12	codes	code	NOUN
ejpam-6409	51	13	can	can	AUX
ejpam-6409	51	14	be	be	AUX
ejpam-6409	51	15	constructed	construct	VERB
ejpam-6409	51	16	over	over	ADP
ejpam-6409	51	17	eisenstein	eisenstein	PROPN
ejpam-6409	51	18	local	local	ADJ
ejpam-6409	51	19	rings	ring	NOUN
ejpam-6409	51	20	z2s	z2s	PROPN
ejpam-6409	52	1	[	[	X
ejpam-6409	52	2	ω	ω	X
ejpam-6409	52	3	]	]	X
ejpam-6409	52	4	,	,	PUNCT
ejpam-6409	52	5	where	where	SCONJ
ejpam-6409	52	6	ω2	ω2	ADJ
ejpam-6409	52	7	+	+	CCONJ
ejpam-6409	52	8	ω	ω	PROPN
ejpam-6409	53	1	+	+	CCONJ
ejpam-6409	53	2	1	1	NUM
ejpam-6409	53	3	=	=	SYM
ejpam-6409	53	4	0	0	NUM
ejpam-6409	53	5	.	.	NOUN
ejpam-6409	53	6	•	•	NUM
ejpam-6409	53	7	it	it	PRON
ejpam-6409	53	8	defines	define	VERB
ejpam-6409	53	9	and	and	CCONJ
ejpam-6409	53	10	analyzes	analyze	VERB
ejpam-6409	53	11	a	a	DET
ejpam-6409	53	12	gray	gray	ADJ
ejpam-6409	53	13	map	map	NOUN
ejpam-6409	53	14	suitable	suitable	ADJ
ejpam-6409	53	15	for	for	ADP
ejpam-6409	53	16	eisenstein	eisenstein	PROPN
ejpam-6409	53	17	local	local	ADJ
ejpam-6409	53	18	rings	ring	NOUN
ejpam-6409	53	19	with	with	ADP
ejpam-6409	53	20	binary	binary	ADJ
ejpam-6409	53	21	image	image	NOUN
ejpam-6409	53	22	representation	representation	NOUN
ejpam-6409	53	23	.	.	PUNCT
ejpam-6409	54	1	•	•	INTJ
ejpam-6409	54	2	it	it	PRON
ejpam-6409	54	3	obtains	obtain	VERB
ejpam-6409	54	4	linearity	linearity	NOUN
ejpam-6409	54	5	conditions	condition	NOUN
ejpam-6409	54	6	for	for	ADP
ejpam-6409	54	7	gh	gh	PROPN
ejpam-6409	54	8	codes	code	NOUN
ejpam-6409	54	9	with	with	ADP
ejpam-6409	54	10	respect	respect	NOUN
ejpam-6409	54	11	to	to	ADP
ejpam-6409	54	12	elements	element	NOUN
ejpam-6409	54	13	in	in	ADP
ejpam-6409	54	14	z2s	z2s	PROPN
ejpam-6409	54	15	[	[	X
ejpam-6409	54	16	ω	ω	X
ejpam-6409	54	17	]	]	X
ejpam-6409	54	18	.	.	PUNCT
ejpam-6409	55	1	•	•	INTJ
ejpam-6409	55	2	it	it	PRON
ejpam-6409	55	3	investigates	investigate	VERB
ejpam-6409	55	4	the	the	DET
ejpam-6409	55	5	kernel	kernel	NOUN
ejpam-6409	55	6	and	and	CCONJ
ejpam-6409	55	7	rank	rank	NOUN
ejpam-6409	55	8	structures	structure	NOUN
ejpam-6409	55	9	of	of	ADP
ejpam-6409	55	10	these	these	DET
ejpam-6409	55	11	codes	code	NOUN
ejpam-6409	55	12	,	,	PUNCT
ejpam-6409	55	13	leading	lead	VERB
ejpam-6409	55	14	to	to	ADP
ejpam-6409	55	15	the	the	DET
ejpam-6409	55	16	determination	determination	NOUN
ejpam-6409	55	17	of	of	ADP
ejpam-6409	55	18	some	some	PRON
ejpam-6409	55	19	of	of	ADP
ejpam-6409	55	20	their	their	PRON
ejpam-6409	55	21	algebraic	algebraic	ADJ
ejpam-6409	55	22	invariants	invariant	NOUN
ejpam-6409	55	23	.	.	PUNCT
ejpam-6409	56	1	•	•	INTJ
ejpam-6409	56	2	it	it	PRON
ejpam-6409	56	3	partially	partially	ADV
ejpam-6409	56	4	classifies	classify	VERB
ejpam-6409	56	5	z2s	z2s	PROPN
ejpam-6409	56	6	[	[	X
ejpam-6409	56	7	ω]-linear	ω]-linear	ADJ
ejpam-6409	56	8	hadamard	hadamard	ADJ
ejpam-6409	56	9	codes	code	NOUN
ejpam-6409	56	10	in	in	ADP
ejpam-6409	56	11	terms	term	NOUN
ejpam-6409	56	12	of	of	ADP
ejpam-6409	56	13	their	their	PRON
ejpam-6409	56	14	structural	structural	ADJ
ejpam-6409	56	15	and	and	CCONJ
ejpam-6409	56	16	combinatorial	combinatorial	ADJ
ejpam-6409	56	17	properties	property	NOUN
ejpam-6409	56	18	.	.	PUNCT
ejpam-6409	57	1	•	•	NUM
ejpam-6409	57	2	it	it	PRON
ejpam-6409	57	3	distinguishes	distinguish	VERB
ejpam-6409	57	4	classical	classical	ADJ
ejpam-6409	57	5	gh	gh	PROPN
ejpam-6409	57	6	code	code	PROPN
ejpam-6409	57	7	theory	theory	NOUN
ejpam-6409	57	8	from	from	ADP
ejpam-6409	57	9	modern	modern	ADJ
ejpam-6409	57	10	algebraic	algebraic	ADJ
ejpam-6409	57	11	settings	setting	NOUN
ejpam-6409	57	12	such	such	ADJ
ejpam-6409	57	13	as	as	ADP
ejpam-6409	57	14	eisenstein	eisenstein	NOUN
ejpam-6409	57	15	rings	ring	NOUN
ejpam-6409	57	16	.	.	PUNCT
ejpam-6409	58	1	2	2	X
ejpam-6409	58	2	.	.	X
ejpam-6409	58	3	eisenstein	eisenstein	NOUN
ejpam-6409	58	4	integers	integer	VERB
ejpam-6409	58	5	[	[	X
ejpam-6409	58	6	12	12	NUM
ejpam-6409	58	7	,	,	PUNCT
ejpam-6409	58	8	24	24	NUM
ejpam-6409	58	9	]	]	PUNCT
ejpam-6409	58	10	let	let	VERB
ejpam-6409	58	11	ω	ω	NUM
ejpam-6409	58	12	=	=	SYM
ejpam-6409	58	13	−1+i	−1+i	NOUN
ejpam-6409	58	14	√	√	NOUN
ejpam-6409	58	15	3	3	NUM
ejpam-6409	58	16	2	2	NUM
ejpam-6409	58	17	be	be	AUX
ejpam-6409	58	18	a	a	DET
ejpam-6409	58	19	primitive	primitive	ADJ
ejpam-6409	58	20	cube	cube	NOUN
ejpam-6409	58	21	root	root	NOUN
ejpam-6409	58	22	of	of	ADP
ejpam-6409	58	23	unity	unity	NOUN
ejpam-6409	58	24	.	.	PUNCT
ejpam-6409	59	1	then	then	ADV
ejpam-6409	59	2	the	the	DET
ejpam-6409	59	3	identity	identity	NOUN
ejpam-6409	59	4	1	1	NUM
ejpam-6409	59	5	+	+	NUM
ejpam-6409	59	6	ω	ω	NUM
ejpam-6409	59	7	+	+	CCONJ
ejpam-6409	59	8	ω2	ω2	ADJ
ejpam-6409	59	9	=	=	SYM
ejpam-6409	59	10	0	0	NUM
ejpam-6409	59	11	implies	imply	VERB
ejpam-6409	59	12	that	that	SCONJ
ejpam-6409	59	13	ω2	ω2	ADJ
ejpam-6409	59	14	=	=	PUNCT
ejpam-6409	59	15	−ω	−ω	NOUN
ejpam-6409	59	16	−	−	NOUN
ejpam-6409	59	17	1	1	NUM
ejpam-6409	59	18	.	.	PUNCT
ejpam-6409	60	1	the	the	DET
ejpam-6409	60	2	set	set	NOUN
ejpam-6409	60	3	of	of	ADP
ejpam-6409	60	4	eisenstein	eisenstein	NOUN
ejpam-6409	60	5	integers	integer	NOUN
ejpam-6409	60	6	consists	consist	VERB
ejpam-6409	60	7	of	of	ADP
ejpam-6409	60	8	complex	complex	ADJ
ejpam-6409	60	9	numbers	number	NOUN
ejpam-6409	60	10	that	that	PRON
ejpam-6409	60	11	can	can	AUX
ejpam-6409	60	12	be	be	AUX
ejpam-6409	60	13	written	write	VERB
ejpam-6409	60	14	in	in	ADP
ejpam-6409	60	15	the	the	DET
ejpam-6409	60	16	form	form	NOUN
ejpam-6409	60	17	a+	a+	PUNCT
ejpam-6409	60	18	ωb	ωb	NOUN
ejpam-6409	60	19	,	,	PUNCT
ejpam-6409	60	20	where	where	SCONJ
ejpam-6409	60	21	a	a	PRON
ejpam-6409	60	22	,	,	PUNCT
ejpam-6409	60	23	b	b	PROPN
ejpam-6409	60	24	∈	∈	PROPN
ejpam-6409	60	25	z.	z.	PROPN
ejpam-6409	60	26	mathematically	mathematically	ADV
ejpam-6409	60	27	,	,	PUNCT
ejpam-6409	60	28	this	this	DET
ejpam-6409	60	29	set	set	NOUN
ejpam-6409	60	30	is	be	AUX
ejpam-6409	60	31	defined	define	VERB
ejpam-6409	60	32	as	as	ADP
ejpam-6409	60	33	e	e	X
ejpam-6409	60	34	=	=	PUNCT
ejpam-6409	60	35	{	{	PUNCT
ejpam-6409	60	36	a+	a+	SYM
ejpam-6409	60	37	ωb	ωb	NOUN
ejpam-6409	60	38	|	|	ADV
ejpam-6409	60	39	a	a	X
ejpam-6409	60	40	,	,	PUNCT
ejpam-6409	60	41	b	b	X
ejpam-6409	60	42	∈	∈	PROPN
ejpam-6409	60	43	z	z	NOUN
ejpam-6409	60	44	}	}	PUNCT
ejpam-6409	60	45	.	.	PUNCT
ejpam-6409	61	1	the	the	DET
ejpam-6409	61	2	set	set	NOUN
ejpam-6409	61	3	e	e	NOUN
ejpam-6409	61	4	forms	form	VERB
ejpam-6409	61	5	a	a	DET
ejpam-6409	61	6	ring	ring	NOUN
ejpam-6409	61	7	.	.	PUNCT
ejpam-6409	62	1	muhammad	muhammad	PROPN
ejpam-6409	62	2	sajjad	sajjad	PROPN
ejpam-6409	62	3	et	et	PROPN
ejpam-6409	62	4	al	al	PROPN
ejpam-6409	62	5	.	.	PUNCT
ejpam-6409	62	6	/	/	SYM
ejpam-6409	62	7	eur	eur	PROPN
ejpam-6409	62	8	.	.	PUNCT
ejpam-6409	63	1	j.	j.	PROPN
ejpam-6409	63	2	pure	pure	PROPN
ejpam-6409	63	3	appl	appl	PROPN
ejpam-6409	63	4	.	.	PROPN
ejpam-6409	63	5	math	math	PROPN
ejpam-6409	63	6	,	,	PUNCT
ejpam-6409	63	7	18	18	NUM
ejpam-6409	63	8	(	(	PUNCT
ejpam-6409	63	9	3	3	NUM
ejpam-6409	63	10	)	)	PUNCT
ejpam-6409	63	11	(	(	PUNCT
ejpam-6409	63	12	2025	2025	NUM
ejpam-6409	63	13	)	)	PUNCT
ejpam-6409	63	14	,	,	PUNCT
ejpam-6409	63	15	6409	6409	NUM
ejpam-6409	63	16	4	4	NUM
ejpam-6409	63	17	of	of	ADP
ejpam-6409	63	18	32	32	NUM
ejpam-6409	63	19	the	the	DET
ejpam-6409	63	20	conjugate	conjugate	NOUN
ejpam-6409	63	21	of	of	ADP
ejpam-6409	63	22	z	z	NOUN
ejpam-6409	63	23	=	=	SYM
ejpam-6409	63	24	a+	a+	PUNCT
ejpam-6409	63	25	ωb	ωb	NOUN
ejpam-6409	63	26	∈	∈	NOUN
ejpam-6409	63	27	e	e	NOUN
ejpam-6409	63	28	is	be	AUX
ejpam-6409	63	29	given	give	VERB
ejpam-6409	63	30	by	by	ADP
ejpam-6409	63	31	z∗	z∗	NOUN
ejpam-6409	63	32	=	=	SYM
ejpam-6409	63	33	a+	a+	PUNCT
ejpam-6409	63	34	ω2b	ω2b	PROPN
ejpam-6409	63	35	.	.	PUNCT
ejpam-6409	64	1	for	for	ADP
ejpam-6409	64	2	example	example	NOUN
ejpam-6409	64	3	,	,	PUNCT
ejpam-6409	64	4	z∗	z∗	NOUN
ejpam-6409	64	5	=	=	SYM
ejpam-6409	64	6	1	1	NUM
ejpam-6409	64	7	+	+	NUM
ejpam-6409	64	8	2ω2	2ω2	NUM
ejpam-6409	64	9	is	be	AUX
ejpam-6409	64	10	the	the	DET
ejpam-6409	64	11	conjugate	conjugate	NOUN
ejpam-6409	64	12	of	of	ADP
ejpam-6409	64	13	z	z	NOUN
ejpam-6409	64	14	=	=	SYM
ejpam-6409	64	15	a+	a+	PUNCT
ejpam-6409	64	16	bω	bω	NOUN
ejpam-6409	64	17	in	in	ADP
ejpam-6409	64	18	the	the	DET
ejpam-6409	64	19	set	set	NOUN
ejpam-6409	64	20	e.	e.	PROPN
ejpam-6409	64	21	every	every	DET
ejpam-6409	64	22	eisenstein	eisenstein	PROPN
ejpam-6409	64	23	integer	integer	PROPN
ejpam-6409	64	24	z	z	PROPN
ejpam-6409	64	25	has	have	VERB
ejpam-6409	64	26	a	a	DET
ejpam-6409	64	27	norm	norm	NOUN
ejpam-6409	64	28	denoted	denote	VERB
ejpam-6409	64	29	by	by	ADP
ejpam-6409	64	30	n(z	n(z	NOUN
ejpam-6409	64	31	)	)	PUNCT
ejpam-6409	64	32	,	,	PUNCT
ejpam-6409	64	33	which	which	PRON
ejpam-6409	64	34	is	be	AUX
ejpam-6409	64	35	defined	define	VERB
ejpam-6409	64	36	as	as	ADP
ejpam-6409	64	37	n(z	n(z	NOUN
ejpam-6409	64	38	)	)	PUNCT
ejpam-6409	64	39	=	=	SYM
ejpam-6409	64	40	zz∗	zz∗	NOUN
ejpam-6409	64	41	=	=	SYM
ejpam-6409	64	42	a2	a2	PROPN
ejpam-6409	64	43	−	−	PROPN
ejpam-6409	64	44	ab+	ab+	NOUN
ejpam-6409	64	45	b2	b2	NOUN
ejpam-6409	64	46	.	.	PUNCT
ejpam-6409	65	1	moreover	moreover	ADV
ejpam-6409	65	2	,	,	PUNCT
ejpam-6409	65	3	the	the	DET
ejpam-6409	65	4	norm	norm	NOUN
ejpam-6409	65	5	is	be	AUX
ejpam-6409	65	6	multiplicative	multiplicative	ADJ
ejpam-6409	65	7	:	:	PUNCT
ejpam-6409	65	8	n(z1z2	n(z1z2	NUM
ejpam-6409	65	9	)	)	PUNCT
ejpam-6409	65	10	=	=	SYM
ejpam-6409	65	11	n(z1)n(z2	n(z1)n(z2	PROPN
ejpam-6409	65	12	)	)	PUNCT
ejpam-6409	65	13	,	,	PUNCT
ejpam-6409	65	14	for	for	ADP
ejpam-6409	65	15	all	all	DET
ejpam-6409	65	16	z1	z1	VERB
ejpam-6409	65	17	,	,	PUNCT
ejpam-6409	65	18	z2	z2	PROPN
ejpam-6409	65	19	∈	∈	PROPN
ejpam-6409	65	20	e.	e.	PROPN
ejpam-6409	65	21	proposition	proposition	NOUN
ejpam-6409	65	22	2.1	2.1	NUM
ejpam-6409	65	23	[	[	X
ejpam-6409	65	24	12	12	NUM
ejpam-6409	65	25	,	,	PUNCT
ejpam-6409	65	26	24	24	NUM
ejpam-6409	65	27	]	]	PUNCT
ejpam-6409	65	28	:	:	PUNCT
ejpam-6409	65	29	for	for	ADP
ejpam-6409	65	30	ω	ω	NUM
ejpam-6409	65	31	=	=	SYM
ejpam-6409	65	32	−1+i	−1+i	PROPN
ejpam-6409	65	33	√	√	NOUN
ejpam-6409	65	34	3	3	NUM
ejpam-6409	65	35	2	2	NUM
ejpam-6409	65	36	,	,	PUNCT
ejpam-6409	65	37	the	the	DET
ejpam-6409	65	38	ring	ring	NOUN
ejpam-6409	65	39	e	e	NOUN
ejpam-6409	65	40	is	be	AUX
ejpam-6409	65	41	a	a	DET
ejpam-6409	65	42	euclidean	euclidean	ADJ
ejpam-6409	65	43	domain	domain	NOUN
ejpam-6409	65	44	.	.	PUNCT
ejpam-6409	66	1	the	the	DET
ejpam-6409	66	2	units	unit	NOUN
ejpam-6409	66	3	in	in	ADP
ejpam-6409	66	4	the	the	DET
ejpam-6409	66	5	ring	ring	NOUN
ejpam-6409	66	6	e	e	PROPN
ejpam-6409	66	7	of	of	ADP
ejpam-6409	66	8	eisenstein	eisenstein	NOUN
ejpam-6409	66	9	integers	integer	NOUN
ejpam-6409	66	10	are	be	AUX
ejpam-6409	66	11	±1,±ω,±ω2	±1,±ω,±ω2	NOUN
ejpam-6409	66	12	.	.	PUNCT
ejpam-6409	67	1	the	the	DET
ejpam-6409	67	2	primes	prime	NOUN
ejpam-6409	67	3	in	in	ADP
ejpam-6409	67	4	e	e	NOUN
ejpam-6409	67	5	include	include	VERB
ejpam-6409	67	6	:	:	PUNCT
ejpam-6409	67	7	•	•	X
ejpam-6409	67	8	rational	rational	ADJ
ejpam-6409	67	9	primes	prime	NOUN
ejpam-6409	68	1	p	p	NOUN
ejpam-6409	68	2	satisfying	satisfy	VERB
ejpam-6409	68	3	p	p	X
ejpam-6409	68	4	≡	≡	PROPN
ejpam-6409	68	5	2	2	NUM
ejpam-6409	68	6	(	(	PUNCT
ejpam-6409	68	7	mod	mod	NOUN
ejpam-6409	68	8	3	3	NUM
ejpam-6409	68	9	)	)	PUNCT
ejpam-6409	68	10	,	,	PUNCT
ejpam-6409	68	11	•	•	NUM
ejpam-6409	68	12	eisenstein	eisenstein	PROPN
ejpam-6409	68	13	integers	integer	VERB
ejpam-6409	68	14	z	z	NOUN
ejpam-6409	68	15	such	such	ADJ
ejpam-6409	68	16	that	that	SCONJ
ejpam-6409	68	17	n(z	n(z	NOUN
ejpam-6409	68	18	)	)	PUNCT
ejpam-6409	69	1	=	=	SYM
ejpam-6409	70	1	p	p	X
ejpam-6409	70	2	,	,	PUNCT
ejpam-6409	70	3	where	where	SCONJ
ejpam-6409	70	4	p	p	NOUN
ejpam-6409	70	5	is	be	AUX
ejpam-6409	70	6	a	a	DET
ejpam-6409	70	7	prime	prime	NOUN
ejpam-6409	70	8	.	.	PUNCT
ejpam-6409	71	1	the	the	DET
ejpam-6409	71	2	quotient	quotient	NOUN
ejpam-6409	71	3	ring	ring	NOUN
ejpam-6409	71	4	e	e	PROPN
ejpam-6409	71	5	/	/	SYM
ejpam-6409	71	6	ne	ne	PROPN
ejpam-6409	71	7	is	be	AUX
ejpam-6409	71	8	canonically	canonically	ADV
ejpam-6409	71	9	isomorphic	isomorphic	ADJ
ejpam-6409	71	10	to	to	ADP
ejpam-6409	71	11	the	the	DET
ejpam-6409	71	12	ring	ring	NOUN
ejpam-6409	71	13	en	en	X
ejpam-6409	71	14	=	=	X
ejpam-6409	71	15	{	{	PUNCT
ejpam-6409	71	16	a+	a+	PUNCT
ejpam-6409	71	17	bω	bω	NOUN
ejpam-6409	71	18	|	|	ADV
ejpam-6409	71	19	a	a	PRON
ejpam-6409	71	20	,	,	PUNCT
ejpam-6409	71	21	b	b	PROPN
ejpam-6409	71	22	∈	∈	PROPN
ejpam-6409	71	23	zn	zn	PROPN
ejpam-6409	71	24	}	}	PUNCT
ejpam-6409	71	25	,	,	PUNCT
ejpam-6409	71	26	which	which	PRON
ejpam-6409	71	27	is	be	AUX
ejpam-6409	71	28	the	the	DET
ejpam-6409	71	29	ring	ring	NOUN
ejpam-6409	71	30	of	of	ADP
ejpam-6409	71	31	eisenstein	eisenstein	PROPN
ejpam-6409	71	32	integers	integer	NOUN
ejpam-6409	71	33	modulo	modulo	PROPN
ejpam-6409	71	34	n.	n.	PROPN
ejpam-6409	71	35	this	this	DET
ejpam-6409	71	36	ring	ring	NOUN
ejpam-6409	71	37	is	be	AUX
ejpam-6409	71	38	also	also	ADV
ejpam-6409	71	39	a	a	DET
ejpam-6409	71	40	principal	principal	ADJ
ejpam-6409	71	41	ideal	ideal	ADJ
ejpam-6409	71	42	ring	ring	NOUN
ejpam-6409	71	43	.	.	PUNCT
ejpam-6409	72	1	lemma	lemma	PROPN
ejpam-6409	72	2	2.1	2.1	NUM
ejpam-6409	73	1	[	[	X
ejpam-6409	73	2	24	24	NUM
ejpam-6409	73	3	]	]	PUNCT
ejpam-6409	73	4	:	:	PUNCT
ejpam-6409	73	5	let	let	VERB
ejpam-6409	73	6	z	z	NOUN
ejpam-6409	73	7	=	=	SYM
ejpam-6409	73	8	a+	a+	PUNCT
ejpam-6409	73	9	ωb	ωb	NOUN
ejpam-6409	73	10	∈	∈	PROPN
ejpam-6409	73	11	en	en	PROPN
ejpam-6409	73	12	.	.	PUNCT
ejpam-6409	74	1	then	then	ADV
ejpam-6409	74	2	z	z	PROPN
ejpam-6409	74	3	is	be	AUX
ejpam-6409	74	4	a	a	DET
ejpam-6409	74	5	unit	unit	NOUN
ejpam-6409	74	6	in	in	ADP
ejpam-6409	74	7	en	en	ADV
ejpam-6409	74	8	if	if	SCONJ
ejpam-6409	74	9	and	and	CCONJ
ejpam-6409	74	10	only	only	ADV
ejpam-6409	74	11	if	if	SCONJ
ejpam-6409	74	12	n(z	n(z	NOUN
ejpam-6409	74	13	)	)	PUNCT
ejpam-6409	74	14	is	be	AUX
ejpam-6409	74	15	a	a	DET
ejpam-6409	74	16	unit	unit	NOUN
ejpam-6409	74	17	in	in	ADP
ejpam-6409	74	18	zn	zn	PROPN
ejpam-6409	74	19	.	.	PUNCT
ejpam-6409	75	1	since	since	SCONJ
ejpam-6409	75	2	our	our	PRON
ejpam-6409	75	3	main	main	ADJ
ejpam-6409	75	4	focus	focus	NOUN
ejpam-6409	75	5	is	be	AUX
ejpam-6409	75	6	on	on	ADP
ejpam-6409	75	7	local	local	ADJ
ejpam-6409	75	8	rings	ring	NOUN
ejpam-6409	75	9	of	of	ADP
ejpam-6409	75	10	eisenstein	eisenstein	NOUN
ejpam-6409	75	11	integers	integer	NOUN
ejpam-6409	75	12	,	,	PUNCT
ejpam-6409	75	13	we	we	PRON
ejpam-6409	75	14	consider	consider	VERB
ejpam-6409	75	15	n	n	PRON
ejpam-6409	75	16	=	=	SYM
ejpam-6409	75	17	ps	ps	PROPN
ejpam-6409	75	18	,	,	PUNCT
ejpam-6409	75	19	where	where	SCONJ
ejpam-6409	75	20	p	p	NOUN
ejpam-6409	75	21	is	be	AUX
ejpam-6409	75	22	a	a	DET
ejpam-6409	75	23	prime	prime	ADJ
ejpam-6409	75	24	integer	integer	NOUN
ejpam-6409	75	25	and	and	CCONJ
ejpam-6409	75	26	s	s	NOUN
ejpam-6409	75	27	is	be	AUX
ejpam-6409	75	28	a	a	DET
ejpam-6409	75	29	positive	positive	ADJ
ejpam-6409	75	30	integer	integer	NOUN
ejpam-6409	75	31	.	.	PUNCT
ejpam-6409	76	1	note	note	VERB
ejpam-6409	76	2	that	that	SCONJ
ejpam-6409	76	3	eps	eps	PROPN
ejpam-6409	76	4	is	be	AUX
ejpam-6409	76	5	not	not	PART
ejpam-6409	76	6	always	always	ADV
ejpam-6409	76	7	a	a	DET
ejpam-6409	76	8	local	local	ADJ
ejpam-6409	76	9	ring	ring	NOUN
ejpam-6409	76	10	,	,	PUNCT
ejpam-6409	76	11	unlike	unlike	ADP
ejpam-6409	76	12	zps	zps	PROPN
ejpam-6409	76	13	.	.	PUNCT
ejpam-6409	77	1	similarly	similarly	ADV
ejpam-6409	77	2	,	,	PUNCT
ejpam-6409	77	3	it	it	PRON
ejpam-6409	77	4	ep	ep	PROPN
ejpam-6409	77	5	is	be	AUX
ejpam-6409	77	6	not	not	PART
ejpam-6409	77	7	always	always	ADV
ejpam-6409	77	8	a	a	DET
ejpam-6409	77	9	field	field	NOUN
ejpam-6409	77	10	even	even	ADV
ejpam-6409	77	11	when	when	SCONJ
ejpam-6409	77	12	p	p	NOUN
ejpam-6409	77	13	is	be	AUX
ejpam-6409	77	14	prime	prime	ADJ
ejpam-6409	77	15	.	.	PUNCT
ejpam-6409	78	1	theorem	theorem	VERB
ejpam-6409	78	2	2.1	2.1	NUM
ejpam-6409	78	3	[	[	SYM
ejpam-6409	78	4	12	12	NUM
ejpam-6409	78	5	,	,	PUNCT
ejpam-6409	78	6	24	24	NUM
ejpam-6409	78	7	]	]	PUNCT
ejpam-6409	78	8	:	:	PUNCT
ejpam-6409	78	9	for	for	ADP
ejpam-6409	78	10	p	p	PRON
ejpam-6409	78	11	≥	≥	NUM
ejpam-6409	78	12	3	3	NUM
ejpam-6409	78	13	,	,	PUNCT
ejpam-6409	78	14	the	the	DET
ejpam-6409	78	15	ring	ring	NOUN
ejpam-6409	78	16	eps	eps	PROPN
ejpam-6409	78	17	is	be	AUX
ejpam-6409	78	18	local	local	ADJ
ejpam-6409	78	19	if	if	SCONJ
ejpam-6409	78	20	and	and	CCONJ
ejpam-6409	78	21	only	only	ADV
ejpam-6409	78	22	if	if	SCONJ
ejpam-6409	78	23	p	p	PRON
ejpam-6409	78	24	≡	≡	PROPN
ejpam-6409	78	25	2	2	NUM
ejpam-6409	78	26	(	(	PUNCT
ejpam-6409	78	27	mod	mod	NOUN
ejpam-6409	78	28	3	3	NUM
ejpam-6409	78	29	)	)	PUNCT
ejpam-6409	78	30	or	or	CCONJ
ejpam-6409	78	31	p	p	X
ejpam-6409	78	32	=	=	ADJ
ejpam-6409	78	33	2	2	NUM
ejpam-6409	78	34	.	.	NOUN
ejpam-6409	78	35	3	3	NUM
ejpam-6409	78	36	.	.	X
ejpam-6409	78	37	gray	gray	ADJ
ejpam-6409	78	38	map	map	NOUN
ejpam-6409	78	39	and	and	CCONJ
ejpam-6409	78	40	related	related	ADJ
ejpam-6409	78	41	results	result	NOUN
ejpam-6409	78	42	[	[	X
ejpam-6409	78	43	6	6	NUM
ejpam-6409	78	44	,	,	PUNCT
ejpam-6409	78	45	16	16	NUM
ejpam-6409	78	46	,	,	PUNCT
ejpam-6409	78	47	17	17	NUM
ejpam-6409	78	48	]	]	PUNCT
ejpam-6409	78	49	in	in	ADP
ejpam-6409	78	50	this	this	DET
ejpam-6409	78	51	section	section	NOUN
ejpam-6409	78	52	,	,	PUNCT
ejpam-6409	78	53	we	we	PRON
ejpam-6409	78	54	provide	provide	VERB
ejpam-6409	78	55	the	the	DET
ejpam-6409	78	56	definition	definition	NOUN
ejpam-6409	78	57	of	of	ADP
ejpam-6409	78	58	the	the	DET
ejpam-6409	78	59	generalized	generalize	VERB
ejpam-6409	78	60	gray	gray	ADJ
ejpam-6409	78	61	map	map	NOUN
ejpam-6409	78	62	for	for	ADP
ejpam-6409	78	63	z2s	z2s	PROPN
ejpam-6409	78	64	[	[	X
ejpam-6409	78	65	ω	ω	X
ejpam-6409	78	66	]	]	X
ejpam-6409	78	67	gh	gh	PROPN
ejpam-6409	78	68	codes	code	NOUN
ejpam-6409	78	69	.	.	PUNCT
ejpam-6409	79	1	then	then	ADV
ejpam-6409	79	2	we	we	PRON
ejpam-6409	79	3	establish	establish	VERB
ejpam-6409	79	4	some	some	DET
ejpam-6409	79	5	properties	property	NOUN
ejpam-6409	79	6	of	of	ADP
ejpam-6409	79	7	the	the	DET
ejpam-6409	79	8	gray	gray	ADJ
ejpam-6409	79	9	map	map	NOUN
ejpam-6409	79	10	for	for	ADP
ejpam-6409	79	11	z2s	z2s	PROPN
ejpam-6409	79	12	[	[	X
ejpam-6409	79	13	ω	ω	X
ejpam-6409	79	14	]	]	X
ejpam-6409	79	15	,	,	PUNCT
ejpam-6409	79	16	based	base	VERB
ejpam-6409	79	17	on	on	ADP
ejpam-6409	79	18	the	the	DET
ejpam-6409	79	19	results	result	NOUN
ejpam-6409	79	20	given	give	VERB
ejpam-6409	79	21	in	in	ADP
ejpam-6409	79	22	section	section	NOUN
ejpam-6409	79	23	2	2	NUM
ejpam-6409	79	24	of	of	ADP
ejpam-6409	79	25	[	[	X
ejpam-6409	79	26	6	6	NUM
ejpam-6409	79	27	,	,	PUNCT
ejpam-6409	79	28	16	16	NUM
ejpam-6409	79	29	,	,	PUNCT
ejpam-6409	79	30	17	17	NUM
ejpam-6409	79	31	]	]	PUNCT
ejpam-6409	79	32	.	.	PUNCT
ejpam-6409	80	1	let	let	VERB
ejpam-6409	80	2	ϕs	ϕs	AUX
ejpam-6409	80	3	be	be	AUX
ejpam-6409	80	4	carlet	carlet	NOUN
ejpam-6409	80	5	’s	’s	PART
ejpam-6409	80	6	gray	gray	ADJ
ejpam-6409	80	7	map	map	NOUN
ejpam-6409	80	8	from	from	ADP
ejpam-6409	80	9	z2s	z2s	PROPN
ejpam-6409	80	10	[	[	X
ejpam-6409	80	11	ω	ω	X
ejpam-6409	80	12	]	]	X
ejpam-6409	80	13	to	to	ADP
ejpam-6409	80	14	z22(s−1	z22(s−1	NOUN
ejpam-6409	80	15	)	)	PUNCT
ejpam-6409	80	16	2	2	NUM
ejpam-6409	81	1	[	[	X
ejpam-6409	81	2	ω	ω	X
ejpam-6409	81	3	]	]	X
ejpam-6409	81	4	,	,	PUNCT
ejpam-6409	81	5	defined	define	VERB
ejpam-6409	81	6	as	as	ADP
ejpam-6409	81	7	ϕs(h	ϕs(h	PRON
ejpam-6409	81	8	)	)	PUNCT
ejpam-6409	81	9	=	=	SYM
ejpam-6409	81	10	(	(	PUNCT
ejpam-6409	81	11	hs−1	hs−1	PROPN
ejpam-6409	81	12	,	,	PUNCT
ejpam-6409	81	13	hs−1	hs−1	PROPN
ejpam-6409	81	14	,	,	PUNCT
ejpam-6409	81	15	.	.	PUNCT
ejpam-6409	81	16	.	.	PUNCT
ejpam-6409	82	1	.	.	PUNCT
ejpam-6409	83	1	,	,	PUNCT
ejpam-6409	83	2	hs−1	hs−1	PROPN
ejpam-6409	83	3	)	)	PUNCT
ejpam-6409	83	4	+	+	CCONJ
ejpam-6409	83	5	(	(	PUNCT
ejpam-6409	83	6	h0	h0	PROPN
ejpam-6409	83	7	,	,	PUNCT
ejpam-6409	83	8	.	.	PUNCT
ejpam-6409	83	9	.	.	PUNCT
ejpam-6409	83	10	.	.	PUNCT
ejpam-6409	84	1	,	,	PUNCT
ejpam-6409	84	2	hs−2)ys−1	hs−2)ys−1	NOUN
ejpam-6409	84	3	,	,	PUNCT
ejpam-6409	84	4	where	where	SCONJ
ejpam-6409	84	5	h	h	NOUN
ejpam-6409	84	6	∈	∈	PROPN
ejpam-6409	84	7	z2s	z2s	X
ejpam-6409	85	1	[	[	X
ejpam-6409	85	2	ω	ω	X
ejpam-6409	85	3	]	]	X
ejpam-6409	85	4	,	,	PUNCT
ejpam-6409	85	5	and	and	CCONJ
ejpam-6409	85	6	[	[	X
ejpam-6409	85	7	h0	h0	PROPN
ejpam-6409	85	8	,	,	PUNCT
ejpam-6409	85	9	h1	h1	NOUN
ejpam-6409	85	10	,	,	PUNCT
ejpam-6409	85	11	.	.	PUNCT
ejpam-6409	85	12	.	.	PUNCT
ejpam-6409	85	13	.	.	PUNCT
ejpam-6409	86	1	,	,	PUNCT
ejpam-6409	86	2	hs−1]2	hs−1]2	X
ejpam-6409	86	3	is	be	AUX
ejpam-6409	86	4	the	the	DET
ejpam-6409	86	5	binary	binary	ADJ
ejpam-6409	86	6	(	(	PUNCT
ejpam-6409	86	7	2	2	NUM
ejpam-6409	86	8	-	-	PUNCT
ejpam-6409	86	9	ary	ary	NOUN
ejpam-6409	86	10	)	)	PUNCT
ejpam-6409	86	11	expansion	expansion	NOUN
ejpam-6409	86	12	of	of	ADP
ejpam-6409	86	13	h	h	NOUN
ejpam-6409	86	14	,	,	PUNCT
ejpam-6409	86	15	i.e.	i.e.	X
ejpam-6409	86	16	,	,	PUNCT
ejpam-6409	86	17	h	h	NOUN
ejpam-6409	86	18	=	=	SYM
ejpam-6409	86	19	s−1∑	s−1∑	NUM
ejpam-6409	86	20	i=0	i=0	PROPN
ejpam-6409	86	21	2ihi	2ihi	PROPN
ejpam-6409	86	22	,	,	PUNCT
ejpam-6409	86	23	muhammad	muhammad	PROPN
ejpam-6409	86	24	sajjad	sajjad	PROPN
ejpam-6409	86	25	et	et	PROPN
ejpam-6409	86	26	al	al	PROPN
ejpam-6409	86	27	.	.	PUNCT
ejpam-6409	86	28	/	/	SYM
ejpam-6409	86	29	eur	eur	PROPN
ejpam-6409	86	30	.	.	PUNCT
ejpam-6409	87	1	j.	j.	PROPN
ejpam-6409	87	2	pure	pure	PROPN
ejpam-6409	87	3	appl	appl	PROPN
ejpam-6409	87	4	.	.	PROPN
ejpam-6409	87	5	math	math	PROPN
ejpam-6409	87	6	,	,	PUNCT
ejpam-6409	87	7	18	18	NUM
ejpam-6409	87	8	(	(	PUNCT
ejpam-6409	87	9	3	3	NUM
ejpam-6409	87	10	)	)	PUNCT
ejpam-6409	87	11	(	(	PUNCT
ejpam-6409	87	12	2025	2025	NUM
ejpam-6409	87	13	)	)	PUNCT
ejpam-6409	87	14	,	,	PUNCT
ejpam-6409	87	15	6409	6409	NUM
ejpam-6409	87	16	5	5	NUM
ejpam-6409	87	17	of	of	ADP
ejpam-6409	87	18	32	32	NUM
ejpam-6409	87	19	with	with	ADP
ejpam-6409	87	20	hi	hi	PROPN
ejpam-6409	87	21	∈	∈	PROPN
ejpam-6409	87	22	z2[ω	z2[ω	NOUN
ejpam-6409	87	23	]	]	X
ejpam-6409	87	24	.	.	PUNCT
ejpam-6409	88	1	here	here	ADV
ejpam-6409	88	2	,	,	PUNCT
ejpam-6409	88	3	ys−1	ys−1	PROPN
ejpam-6409	88	4	is	be	AUX
ejpam-6409	88	5	an	an	DET
ejpam-6409	88	6	(	(	PUNCT
ejpam-6409	88	7	s	s	NOUN
ejpam-6409	88	8	−	−	NOUN
ejpam-6409	88	9	1	1	NUM
ejpam-6409	88	10	)	)	PUNCT
ejpam-6409	88	11	×	×	NOUN
ejpam-6409	88	12	22(s−1	22(s−1	NUM
ejpam-6409	88	13	)	)	PUNCT
ejpam-6409	88	14	matrix	matrix	NOUN
ejpam-6409	88	15	whose	whose	DET
ejpam-6409	88	16	columns	column	NOUN
ejpam-6409	88	17	are	be	AUX
ejpam-6409	88	18	all	all	PRON
ejpam-6409	88	19	distinct	distinct	ADJ
ejpam-6409	88	20	elements	element	NOUN
ejpam-6409	88	21	of	of	ADP
ejpam-6409	88	22	zs−1	zs−1	PROPN
ejpam-6409	88	23	2	2	PROPN
ejpam-6409	89	1	[	[	X
ejpam-6409	89	2	ω	ω	X
ejpam-6409	89	3	]	]	X
ejpam-6409	89	4	.	.	PUNCT
ejpam-6409	90	1	now	now	ADV
ejpam-6409	90	2	we	we	PRON
ejpam-6409	90	3	extend	extend	VERB
ejpam-6409	90	4	the	the	DET
ejpam-6409	90	5	map	map	NOUN
ejpam-6409	90	6	ϕs	ϕs	INTJ
ejpam-6409	90	7	to	to	ADP
ejpam-6409	90	8	a	a	DET
ejpam-6409	90	9	component	component	NOUN
ejpam-6409	90	10	-	-	PUNCT
ejpam-6409	90	11	wise	wise	ADJ
ejpam-6409	90	12	map	map	NOUN
ejpam-6409	90	13	:	:	PUNCT
ejpam-6409	90	14	φs	φs	ADJ
ejpam-6409	90	15	:	:	PUNCT
ejpam-6409	90	16	zn	zn	PROPN
ejpam-6409	90	17	2s	2s	X
ejpam-6409	91	1	[	[	X
ejpam-6409	91	2	ω	ω	X
ejpam-6409	91	3	]	]	X
ejpam-6409	91	4	−→	−→	NOUN
ejpam-6409	91	5	zn·22(s−1	zn·22(s−1	NUM
ejpam-6409	91	6	)	)	PUNCT
ejpam-6409	91	7	2	2	NUM
ejpam-6409	92	1	[	[	X
ejpam-6409	92	2	ω	ω	X
ejpam-6409	92	3	]	]	X
ejpam-6409	92	4	.	.	PUNCT
ejpam-6409	93	1	the	the	DET
ejpam-6409	93	2	matrix	matrix	NOUN
ejpam-6409	93	3	ys−1	ys−1	NOUN
ejpam-6409	93	4	can	can	AUX
ejpam-6409	93	5	be	be	AUX
ejpam-6409	93	6	obtained	obtain	VERB
ejpam-6409	93	7	recursively	recursively	ADV
ejpam-6409	93	8	,	,	PUNCT
ejpam-6409	93	9	starting	start	VERB
ejpam-6409	93	10	from	from	ADP
ejpam-6409	93	11	y1	y1	NOUN
ejpam-6409	93	12	=	=	PUNCT
ejpam-6409	94	1	[	[	PUNCT
ejpam-6409	94	2	00	00	NUM
ejpam-6409	94	3	01	01	NUM
ejpam-6409	94	4	10	10	NUM
ejpam-6409	94	5	11	11	NUM
ejpam-6409	94	6	]	]	PUNCT
ejpam-6409	94	7	,	,	PUNCT
ejpam-6409	94	8	and	and	CCONJ
ejpam-6409	94	9	for	for	ADP
ejpam-6409	94	10	s	s	PROPN
ejpam-6409	94	11	>	>	X
ejpam-6409	94	12	1	1	NUM
ejpam-6409	94	13	,	,	PUNCT
ejpam-6409	94	14	ys	ys	NOUN
ejpam-6409	94	15	=	=	SYM
ejpam-6409	94	16	(	(	PUNCT
ejpam-6409	94	17	ys−1	ys−1	ADV
ejpam-6409	94	18	ys−1	ys−1	PROPN
ejpam-6409	94	19	00	00	NUM
ejpam-6409	94	20	01	01	NUM
ejpam-6409	94	21	ys−1	ys−1	NOUN
ejpam-6409	94	22	ys−1	ys−1	PROPN
ejpam-6409	94	23	10	10	NUM
ejpam-6409	94	24	11	11	NUM
ejpam-6409	94	25	)	)	PUNCT
ejpam-6409	94	26	.	.	PUNCT
ejpam-6409	95	1	example	example	NOUN
ejpam-6409	96	1	3.1	3.1	NUM
ejpam-6409	96	2	.	.	PUNCT
ejpam-6409	97	1	let	let	VERB
ejpam-6409	97	2	s	s	NOUN
ejpam-6409	97	3	=	=	ADJ
ejpam-6409	97	4	2	2	X
ejpam-6409	97	5	.	.	X
ejpam-6409	97	6	take	take	VERB
ejpam-6409	97	7	h	h	NOUN
ejpam-6409	97	8	∈	∈	PROPN
ejpam-6409	97	9	z4[ω	z4[ω	NOUN
ejpam-6409	97	10	]	]	PUNCT
ejpam-6409	97	11	,	,	PUNCT
ejpam-6409	97	12	where	where	SCONJ
ejpam-6409	97	13	h	h	NOUN
ejpam-6409	97	14	=	=	PROPN
ejpam-6409	97	15	h0	h0	PROPN
ejpam-6409	97	16	+	+	CCONJ
ejpam-6409	97	17	2h1	2h1	NUM
ejpam-6409	97	18	is	be	AUX
ejpam-6409	97	19	the	the	DET
ejpam-6409	97	20	2	2	NUM
ejpam-6409	97	21	-	-	PUNCT
ejpam-6409	97	22	ary	ary	NOUN
ejpam-6409	97	23	expansion	expansion	NOUN
ejpam-6409	97	24	of	of	ADP
ejpam-6409	97	25	h.	h.	NOUN
ejpam-6409	97	26	then	then	ADV
ejpam-6409	97	27	ϕ2	ϕ2	ADV
ejpam-6409	97	28	is	be	AUX
ejpam-6409	97	29	a	a	DET
ejpam-6409	97	30	gray	gray	ADJ
ejpam-6409	97	31	map	map	NOUN
ejpam-6409	97	32	from	from	ADP
ejpam-6409	97	33	z4[ω	z4[ω	NOUN
ejpam-6409	97	34	]	]	PUNCT
ejpam-6409	97	35	to	to	ADP
ejpam-6409	97	36	z4	z4	PROPN
ejpam-6409	97	37	2[ω	2[ω	NUM
ejpam-6409	97	38	]	]	PUNCT
ejpam-6409	97	39	,	,	PUNCT
ejpam-6409	97	40	which	which	PRON
ejpam-6409	97	41	is	be	AUX
ejpam-6409	97	42	given	give	VERB
ejpam-6409	97	43	in	in	ADP
ejpam-6409	97	44	table	table	NOUN
ejpam-6409	97	45	1	1	NUM
ejpam-6409	97	46	.	.	PUNCT
ejpam-6409	98	1	as	as	SCONJ
ejpam-6409	98	2	h	h	PROPN
ejpam-6409	98	3	∈	∈	PROPN
ejpam-6409	98	4	z4[ω	z4[ω	NOUN
ejpam-6409	98	5	]	]	PUNCT
ejpam-6409	98	6	with	with	ADP
ejpam-6409	98	7	binary	binary	ADJ
ejpam-6409	98	8	representation	representation	PROPN
ejpam-6409	98	9	[	[	X
ejpam-6409	98	10	h0	h0	PROPN
ejpam-6409	98	11	,	,	PUNCT
ejpam-6409	98	12	h1]2	h1]2	ADJ
ejpam-6409	98	13	.	.	PUNCT
ejpam-6409	99	1	the	the	DET
ejpam-6409	99	2	carlet	carlet	NOUN
ejpam-6409	99	3	’s	’s	PART
ejpam-6409	99	4	generalized	generalize	VERB
ejpam-6409	99	5	gray	gray	ADJ
ejpam-6409	99	6	map	map	NOUN
ejpam-6409	99	7	ϕ2	ϕ2	ADV
ejpam-6409	99	8	is	be	AUX
ejpam-6409	99	9	defined	define	VERB
ejpam-6409	99	10	as	as	ADP
ejpam-6409	99	11	:	:	PUNCT
ejpam-6409	99	12	ϕ2(h	ϕ2(h	NUM
ejpam-6409	99	13	)	)	PUNCT
ejpam-6409	99	14	=	=	PRON
ejpam-6409	99	15	(	(	PUNCT
ejpam-6409	99	16	h1	h1	PROPN
ejpam-6409	99	17	,	,	PUNCT
ejpam-6409	99	18	h1	h1	PROPN
ejpam-6409	99	19	,	,	PUNCT
ejpam-6409	99	20	h1	h1	PROPN
ejpam-6409	99	21	,	,	PUNCT
ejpam-6409	99	22	h1	h1	PROPN
ejpam-6409	99	23	)	)	PUNCT
ejpam-6409	100	1	+	+	CCONJ
ejpam-6409	100	2	h0y1	h0y1	PUNCT
ejpam-6409	100	3	∈	∈	PROPN
ejpam-6409	100	4	z4	z4	PROPN
ejpam-6409	100	5	2[ω	2[ω	NUM
ejpam-6409	100	6	]	]	PUNCT
ejpam-6409	100	7	,	,	PUNCT
ejpam-6409	100	8	where	where	SCONJ
ejpam-6409	100	9	y1	y1	NOUN
ejpam-6409	100	10	=	=	SYM
ejpam-6409	100	11	(	(	PUNCT
ejpam-6409	100	12	00	00	NUM
ejpam-6409	100	13	,	,	PUNCT
ejpam-6409	100	14	01	01	NUM
ejpam-6409	100	15	,	,	PUNCT
ejpam-6409	100	16	10	10	NUM
ejpam-6409	100	17	,	,	PUNCT
ejpam-6409	100	18	11	11	NUM
ejpam-6409	100	19	)	)	PUNCT
ejpam-6409	100	20	.	.	PUNCT
ejpam-6409	101	1	table	table	NOUN
ejpam-6409	101	2	1	1	NUM
ejpam-6409	101	3	:	:	PUNCT
ejpam-6409	101	4	gray	gray	ADJ
ejpam-6409	101	5	map	map	NOUN
ejpam-6409	101	6	from	from	ADP
ejpam-6409	101	7	z4[ω	z4[ω	NOUN
ejpam-6409	101	8	]	]	PUNCT
ejpam-6409	101	9	to	to	ADP
ejpam-6409	101	10	z4	z4	PROPN
ejpam-6409	101	11	2[ω	2[ω	NUM
ejpam-6409	101	12	]	]	X
ejpam-6409	101	13	h	h	NOUN
ejpam-6409	101	14	∈	∈	PROPN
ejpam-6409	101	15	z4[ω	z4[ω	NOUN
ejpam-6409	101	16	]	]	PUNCT
ejpam-6409	102	1	[	[	X
ejpam-6409	102	2	h0	h0	PROPN
ejpam-6409	102	3	,	,	PUNCT
ejpam-6409	102	4	h1]2	h1]2	ADJ
ejpam-6409	102	5	ϕ2(h	ϕ2(h	NUM
ejpam-6409	102	6	)	)	PUNCT
ejpam-6409	102	7	h	h	NOUN
ejpam-6409	102	8	∈	∈	PROPN
ejpam-6409	102	9	z4[ω	z4[ω	NOUN
ejpam-6409	102	10	]	]	PUNCT
ejpam-6409	103	1	[	[	X
ejpam-6409	103	2	h0	h0	PROPN
ejpam-6409	103	3	,	,	PUNCT
ejpam-6409	103	4	h1]2	h1]2	ADJ
ejpam-6409	103	5	ϕ2(h	ϕ2(h	NUM
ejpam-6409	103	6	)	)	PUNCT
ejpam-6409	103	7	00	00	PUNCT
ejpam-6409	104	1	[	[	X
ejpam-6409	104	2	00	00	NUM
ejpam-6409	104	3	,	,	PUNCT
ejpam-6409	104	4	00]2	00]2	NUM
ejpam-6409	104	5	(	(	PUNCT
ejpam-6409	104	6	00	00	NUM
ejpam-6409	104	7	,	,	PUNCT
ejpam-6409	104	8	00	00	NUM
ejpam-6409	104	9	,	,	PUNCT
ejpam-6409	104	10	00	00	NUM
ejpam-6409	104	11	,	,	PUNCT
ejpam-6409	104	12	00	00	NUM
ejpam-6409	104	13	)	)	PUNCT
ejpam-6409	104	14	20	20	NUM
ejpam-6409	105	1	[	[	NOUN
ejpam-6409	105	2	00	00	NUM
ejpam-6409	105	3	,	,	PUNCT
ejpam-6409	105	4	10]2	10]2	NUM
ejpam-6409	105	5	(	(	PUNCT
ejpam-6409	105	6	10	10	NUM
ejpam-6409	105	7	,	,	PUNCT
ejpam-6409	105	8	10	10	NUM
ejpam-6409	105	9	,	,	PUNCT
ejpam-6409	105	10	10	10	NUM
ejpam-6409	105	11	,	,	PUNCT
ejpam-6409	105	12	10	10	NUM
ejpam-6409	105	13	)	)	PUNCT
ejpam-6409	105	14	01	01	NUM
ejpam-6409	106	1	[	[	X
ejpam-6409	106	2	01	01	NUM
ejpam-6409	106	3	,	,	PUNCT
ejpam-6409	106	4	00]2	00]2	NUM
ejpam-6409	106	5	(	(	PUNCT
ejpam-6409	106	6	00	00	NUM
ejpam-6409	106	7	,	,	PUNCT
ejpam-6409	106	8	11	11	NUM
ejpam-6409	106	9	,	,	PUNCT
ejpam-6409	106	10	01	01	NUM
ejpam-6409	106	11	,	,	PUNCT
ejpam-6409	106	12	10	10	NUM
ejpam-6409	106	13	)	)	PUNCT
ejpam-6409	106	14	21	21	NUM
ejpam-6409	107	1	[	[	X
ejpam-6409	107	2	01	01	NUM
ejpam-6409	107	3	,	,	PUNCT
ejpam-6409	107	4	10]2	10]2	NUM
ejpam-6409	107	5	(	(	PUNCT
ejpam-6409	107	6	10	10	NUM
ejpam-6409	107	7	,	,	PUNCT
ejpam-6409	107	8	01	01	NUM
ejpam-6409	107	9	,	,	PUNCT
ejpam-6409	107	10	11	11	NUM
ejpam-6409	107	11	,	,	PUNCT
ejpam-6409	107	12	00	00	NUM
ejpam-6409	107	13	)	)	PUNCT
ejpam-6409	107	14	02	02	NUM
ejpam-6409	108	1	[	[	X
ejpam-6409	108	2	00	00	NUM
ejpam-6409	108	3	,	,	PUNCT
ejpam-6409	108	4	01]2	01]2	NUM
ejpam-6409	108	5	(	(	PUNCT
ejpam-6409	108	6	01	01	NUM
ejpam-6409	108	7	,	,	PUNCT
ejpam-6409	108	8	01	01	NUM
ejpam-6409	108	9	,	,	PUNCT
ejpam-6409	108	10	01	01	NUM
ejpam-6409	108	11	,	,	PUNCT
ejpam-6409	108	12	01	01	NUM
ejpam-6409	108	13	)	)	PUNCT
ejpam-6409	108	14	22	22	NUM
ejpam-6409	109	1	[	[	X
ejpam-6409	109	2	00	00	NUM
ejpam-6409	109	3	,	,	PUNCT
ejpam-6409	109	4	11]2	11]2	NUM
ejpam-6409	109	5	(	(	PUNCT
ejpam-6409	109	6	11	11	NUM
ejpam-6409	109	7	,	,	PUNCT
ejpam-6409	109	8	11	11	NUM
ejpam-6409	109	9	,	,	PUNCT
ejpam-6409	109	10	11	11	NUM
ejpam-6409	109	11	,	,	PUNCT
ejpam-6409	109	12	11	11	NUM
ejpam-6409	109	13	)	)	PUNCT
ejpam-6409	109	14	03	03	NUM
ejpam-6409	110	1	[	[	X
ejpam-6409	110	2	01	01	NUM
ejpam-6409	110	3	,	,	PUNCT
ejpam-6409	110	4	01]2	01]2	NUM
ejpam-6409	110	5	(	(	PUNCT
ejpam-6409	110	6	01	01	NUM
ejpam-6409	110	7	,	,	PUNCT
ejpam-6409	110	8	10	10	NUM
ejpam-6409	110	9	,	,	PUNCT
ejpam-6409	110	10	00	00	NUM
ejpam-6409	110	11	,	,	PUNCT
ejpam-6409	110	12	11	11	NUM
ejpam-6409	110	13	)	)	PUNCT
ejpam-6409	110	14	23	23	NUM
ejpam-6409	111	1	[	[	X
ejpam-6409	111	2	01	01	NUM
ejpam-6409	111	3	,	,	PUNCT
ejpam-6409	111	4	11]2	11]2	NUM
ejpam-6409	111	5	(	(	PUNCT
ejpam-6409	111	6	11	11	NUM
ejpam-6409	111	7	,	,	PUNCT
ejpam-6409	111	8	00	00	NUM
ejpam-6409	111	9	,	,	PUNCT
ejpam-6409	111	10	10	10	NUM
ejpam-6409	111	11	,	,	PUNCT
ejpam-6409	111	12	01	01	NUM
ejpam-6409	111	13	)	)	PUNCT
ejpam-6409	111	14	10	10	NUM
ejpam-6409	112	1	[	[	X
ejpam-6409	112	2	10	10	NUM
ejpam-6409	112	3	,	,	PUNCT
ejpam-6409	112	4	00]2	00]2	PROPN
ejpam-6409	112	5	(	(	PUNCT
ejpam-6409	112	6	00	00	NUM
ejpam-6409	112	7	,	,	PUNCT
ejpam-6409	112	8	01	01	NUM
ejpam-6409	112	9	,	,	PUNCT
ejpam-6409	112	10	10	10	NUM
ejpam-6409	112	11	,	,	PUNCT
ejpam-6409	112	12	11	11	NUM
ejpam-6409	112	13	)	)	PUNCT
ejpam-6409	112	14	30	30	NUM
ejpam-6409	113	1	[	[	SYM
ejpam-6409	113	2	10	10	NUM
ejpam-6409	113	3	,	,	PUNCT
ejpam-6409	113	4	10]2	10]2	NUM
ejpam-6409	113	5	(	(	PUNCT
ejpam-6409	113	6	10	10	NUM
ejpam-6409	113	7	,	,	PUNCT
ejpam-6409	113	8	11	11	NUM
ejpam-6409	113	9	,	,	PUNCT
ejpam-6409	113	10	00	00	NUM
ejpam-6409	113	11	,	,	PUNCT
ejpam-6409	113	12	01	01	NUM
ejpam-6409	113	13	)	)	PUNCT
ejpam-6409	113	14	11	11	NUM
ejpam-6409	114	1	[	[	X
ejpam-6409	114	2	11	11	NUM
ejpam-6409	114	3	,	,	PUNCT
ejpam-6409	114	4	00]2	00]2	PROPN
ejpam-6409	114	5	(	(	PUNCT
ejpam-6409	114	6	00	00	NUM
ejpam-6409	114	7	,	,	PUNCT
ejpam-6409	114	8	10	10	NUM
ejpam-6409	114	9	,	,	PUNCT
ejpam-6409	114	10	11	11	NUM
ejpam-6409	114	11	,	,	PUNCT
ejpam-6409	114	12	01	01	NUM
ejpam-6409	114	13	)	)	PUNCT
ejpam-6409	114	14	31	31	NUM
ejpam-6409	115	1	[	[	X
ejpam-6409	115	2	11	11	NUM
ejpam-6409	115	3	,	,	PUNCT
ejpam-6409	115	4	10]2	10]2	NUM
ejpam-6409	115	5	(	(	PUNCT
ejpam-6409	115	6	10	10	NUM
ejpam-6409	115	7	,	,	PUNCT
ejpam-6409	115	8	00	00	NUM
ejpam-6409	115	9	,	,	PUNCT
ejpam-6409	115	10	01	01	NUM
ejpam-6409	115	11	,	,	PUNCT
ejpam-6409	115	12	11	11	NUM
ejpam-6409	115	13	)	)	PUNCT
ejpam-6409	115	14	12	12	NUM
ejpam-6409	116	1	[	[	X
ejpam-6409	116	2	10	10	NUM
ejpam-6409	116	3	,	,	PUNCT
ejpam-6409	116	4	01]2	01]2	NUM
ejpam-6409	116	5	(	(	PUNCT
ejpam-6409	116	6	01	01	NUM
ejpam-6409	116	7	,	,	PUNCT
ejpam-6409	116	8	00	00	NUM
ejpam-6409	116	9	,	,	PUNCT
ejpam-6409	116	10	11	11	NUM
ejpam-6409	116	11	,	,	PUNCT
ejpam-6409	116	12	10	10	NUM
ejpam-6409	116	13	)	)	PUNCT
ejpam-6409	116	14	32	32	NUM
ejpam-6409	117	1	[	[	SYM
ejpam-6409	117	2	10	10	NUM
ejpam-6409	117	3	,	,	PUNCT
ejpam-6409	117	4	11]2	11]2	NUM
ejpam-6409	117	5	(	(	PUNCT
ejpam-6409	117	6	11	11	NUM
ejpam-6409	117	7	,	,	PUNCT
ejpam-6409	117	8	10	10	NUM
ejpam-6409	117	9	,	,	PUNCT
ejpam-6409	117	10	01	01	NUM
ejpam-6409	117	11	,	,	PUNCT
ejpam-6409	117	12	00	00	NUM
ejpam-6409	117	13	)	)	PUNCT
ejpam-6409	117	14	13	13	NUM
ejpam-6409	118	1	[	[	X
ejpam-6409	118	2	11	11	NUM
ejpam-6409	118	3	,	,	PUNCT
ejpam-6409	118	4	01]2	01]2	NUM
ejpam-6409	118	5	(	(	PUNCT
ejpam-6409	118	6	01	01	NUM
ejpam-6409	118	7	,	,	PUNCT
ejpam-6409	118	8	11	11	NUM
ejpam-6409	118	9	,	,	PUNCT
ejpam-6409	118	10	10	10	NUM
ejpam-6409	118	11	,	,	PUNCT
ejpam-6409	118	12	00	00	NUM
ejpam-6409	118	13	)	)	PUNCT
ejpam-6409	118	14	33	33	NUM
ejpam-6409	119	1	[	[	SYM
ejpam-6409	119	2	11	11	NUM
ejpam-6409	119	3	,	,	PUNCT
ejpam-6409	119	4	11]2	11]2	NUM
ejpam-6409	119	5	(	(	PUNCT
ejpam-6409	119	6	11	11	NUM
ejpam-6409	119	7	,	,	PUNCT
ejpam-6409	119	8	01	01	NUM
ejpam-6409	119	9	,	,	PUNCT
ejpam-6409	119	10	00	00	NUM
ejpam-6409	119	11	,	,	PUNCT
ejpam-6409	119	12	10	10	NUM
ejpam-6409	119	13	)	)	PUNCT
ejpam-6409	119	14	now	now	ADV
ejpam-6409	119	15	some	some	DET
ejpam-6409	119	16	results	result	NOUN
ejpam-6409	119	17	for	for	ADP
ejpam-6409	119	18	the	the	DET
ejpam-6409	119	19	above	above	ADV
ejpam-6409	119	20	-	-	PUNCT
ejpam-6409	119	21	defined	define	VERB
ejpam-6409	119	22	carlet	carlet	NOUN
ejpam-6409	119	23	’s	’s	PART
ejpam-6409	119	24	generalized	generalize	VERB
ejpam-6409	119	25	gray	gray	ADJ
ejpam-6409	119	26	map	map	NOUN
ejpam-6409	119	27	are	be	AUX
ejpam-6409	119	28	presented	present	VERB
ejpam-6409	119	29	below	below	ADV
ejpam-6409	119	30	.	.	PUNCT
ejpam-6409	120	1	let	let	VERB
ejpam-6409	120	2	ek	ek	PRON
ejpam-6409	120	3	be	be	AUX
ejpam-6409	120	4	the	the	DET
ejpam-6409	120	5	vector	vector	NOUN
ejpam-6409	120	6	that	that	PRON
ejpam-6409	120	7	has	have	VERB
ejpam-6409	120	8	1	1	NUM
ejpam-6409	120	9	in	in	ADP
ejpam-6409	120	10	the	the	DET
ejpam-6409	120	11	kth	kth	PROPN
ejpam-6409	120	12	position	position	NOUN
ejpam-6409	120	13	and	and	CCONJ
ejpam-6409	120	14	0	0	NUM
ejpam-6409	120	15	elsewhere	elsewhere	ADV
ejpam-6409	120	16	.	.	PUNCT
ejpam-6409	121	1	let	let	VERB
ejpam-6409	121	2	u	u	NOUN
ejpam-6409	121	3	,	,	PUNCT
ejpam-6409	121	4	v	v	ADP
ejpam-6409	121	5	∈	∈	NOUN
ejpam-6409	121	6	z2s	z2s	X
ejpam-6409	122	1	[	[	X
ejpam-6409	122	2	ω	ω	X
ejpam-6409	122	3	]	]	PUNCT
ejpam-6409	122	4	and	and	CCONJ
ejpam-6409	122	5	[	[	X
ejpam-6409	122	6	u0	u0	ADJ
ejpam-6409	122	7	,	,	PUNCT
ejpam-6409	122	8	u1	u1	NOUN
ejpam-6409	122	9	,	,	PUNCT
ejpam-6409	122	10	.	.	PUNCT
ejpam-6409	122	11	.	.	PUNCT
ejpam-6409	123	1	.	.	PUNCT
ejpam-6409	124	1	,	,	PUNCT
ejpam-6409	124	2	us−1]2	us−1]2	NOUN
ejpam-6409	124	3	,	,	PUNCT
ejpam-6409	124	4	[	[	X
ejpam-6409	124	5	v0	v0	NOUN
ejpam-6409	124	6	,	,	PUNCT
ejpam-6409	124	7	v1	v1	NOUN
ejpam-6409	124	8	,	,	PUNCT
ejpam-6409	124	9	.	.	PUNCT
ejpam-6409	124	10	.	.	PUNCT
ejpam-6409	124	11	.	.	PUNCT
ejpam-6409	125	1	,	,	PUNCT
ejpam-6409	125	2	vs−1]2	vs−1]2	ADV
ejpam-6409	125	3	be	be	AUX
ejpam-6409	125	4	the	the	DET
ejpam-6409	125	5	2	2	NUM
ejpam-6409	125	6	-	-	PUNCT
ejpam-6409	125	7	ary	ary	NOUN
ejpam-6409	125	8	expansions	expansion	NOUN
ejpam-6409	125	9	of	of	ADP
ejpam-6409	125	10	u	u	NOUN
ejpam-6409	125	11	and	and	CCONJ
ejpam-6409	125	12	v	v	NOUN
ejpam-6409	125	13	,	,	PUNCT
ejpam-6409	125	14	respectively	respectively	ADV
ejpam-6409	125	15	,	,	PUNCT
ejpam-6409	125	16	i.e.	i.e.	X
ejpam-6409	125	17	,	,	PUNCT
ejpam-6409	125	18	u	u	PROPN
ejpam-6409	125	19	=	=	SYM
ejpam-6409	125	20	s−1∑	s−1∑	NUM
ejpam-6409	125	21	i=0	i=0	PROPN
ejpam-6409	125	22	2iui	2iui	PROPN
ejpam-6409	125	23	,	,	PUNCT
ejpam-6409	125	24	v	v	NOUN
ejpam-6409	125	25	=	=	SYM
ejpam-6409	125	26	s−1∑	s−1∑	NUM
ejpam-6409	125	27	i=0	i=0	PROPN
ejpam-6409	125	28	2ivi	2ivi	PROPN
ejpam-6409	125	29	.	.	PUNCT
ejpam-6409	126	1	now	now	ADV
ejpam-6409	126	2	define	define	VERB
ejpam-6409	126	3	the	the	DET
ejpam-6409	126	4	operation	operation	NOUN
ejpam-6409	126	5	⊕2	⊕2	PROPN
ejpam-6409	126	6	for	for	ADP
ejpam-6409	126	7	elements	element	NOUN
ejpam-6409	126	8	of	of	ADP
ejpam-6409	126	9	z2s	z2s	PROPN
ejpam-6409	126	10	[	[	X
ejpam-6409	126	11	ω	ω	X
ejpam-6409	126	12	]	]	X
ejpam-6409	126	13	as	as	ADP
ejpam-6409	126	14	:	:	PUNCT
ejpam-6409	126	15	muhammad	muhammad	PROPN
ejpam-6409	126	16	sajjad	sajjad	PROPN
ejpam-6409	126	17	et	et	PROPN
ejpam-6409	126	18	al	al	PROPN
ejpam-6409	126	19	.	.	PUNCT
ejpam-6409	126	20	/	/	SYM
ejpam-6409	126	21	eur	eur	PROPN
ejpam-6409	126	22	.	.	PUNCT
ejpam-6409	127	1	j.	j.	PROPN
ejpam-6409	127	2	pure	pure	PROPN
ejpam-6409	127	3	appl	appl	PROPN
ejpam-6409	127	4	.	.	PROPN
ejpam-6409	127	5	math	math	PROPN
ejpam-6409	127	6	,	,	PUNCT
ejpam-6409	127	7	18	18	NUM
ejpam-6409	127	8	(	(	PUNCT
ejpam-6409	127	9	3	3	NUM
ejpam-6409	127	10	)	)	PUNCT
ejpam-6409	127	11	(	(	PUNCT
ejpam-6409	127	12	2025	2025	NUM
ejpam-6409	127	13	)	)	PUNCT
ejpam-6409	127	14	,	,	PUNCT
ejpam-6409	127	15	6409	6409	NUM
ejpam-6409	127	16	6	6	NUM
ejpam-6409	127	17	of	of	ADP
ejpam-6409	127	18	32	32	NUM
ejpam-6409	127	19	u⊕2	u⊕2	PROPN
ejpam-6409	127	20	v	v	NOUN
ejpam-6409	127	21	=	=	SYM
ejpam-6409	127	22	s−1∑	s−1∑	NUM
ejpam-6409	127	23	i=0	i=0	PROPN
ejpam-6409	127	24	ri2	ri2	X
ejpam-6409	128	1	i	i	PRON
ejpam-6409	128	2	,	,	PUNCT
ejpam-6409	128	3	where	where	SCONJ
ejpam-6409	128	4	ri	ri	PROPN
ejpam-6409	128	5	=	=	PUNCT
ejpam-6409	128	6	ui	ui	PROPN
ejpam-6409	128	7	+	+	CCONJ
ejpam-6409	128	8	vi	vi	X
ejpam-6409	128	9	mod	mod	NOUN
ejpam-6409	128	10	2	2	NUM
ejpam-6409	128	11	in	in	ADP
ejpam-6409	128	12	z2[ω	z2[ω	NOUN
ejpam-6409	128	13	]	]	PUNCT
ejpam-6409	128	14	,	,	PUNCT
ejpam-6409	128	15	and	and	CCONJ
ejpam-6409	128	16	the	the	DET
ejpam-6409	128	17	operation	operation	NOUN
ejpam-6409	128	18	⊙2	⊙2	NOUN
ejpam-6409	128	19	as	as	ADP
ejpam-6409	128	20	:	:	PUNCT
ejpam-6409	128	21	u⊙2	u⊙2	ADJ
ejpam-6409	128	22	v	v	NOUN
ejpam-6409	128	23	=	=	SYM
ejpam-6409	128	24	s−1∑	s−1∑	NUM
ejpam-6409	128	25	i=0	i=0	PROPN
ejpam-6409	128	26	ti2	ti2	PROPN
ejpam-6409	128	27	i	i	PRON
ejpam-6409	128	28	,	,	PUNCT
ejpam-6409	128	29	where	where	SCONJ
ejpam-6409	128	30	ti	ti	NOUN
ejpam-6409	128	31	=	=	SYM
ejpam-6409	128	32	{	{	PUNCT
ejpam-6409	128	33	10	10	NUM
ejpam-6409	128	34	if	if	SCONJ
ejpam-6409	128	35	ui	ui	PROPN
ejpam-6409	128	36	+	+	NOUN
ejpam-6409	128	37	vi	vi	NOUN
ejpam-6409	128	38	≥	≥	NOUN
ejpam-6409	128	39	2	2	NUM
ejpam-6409	128	40	,	,	PUNCT
ejpam-6409	128	41	00	00	PUNCT
ejpam-6409	128	42	otherwise	otherwise	ADV
ejpam-6409	128	43	.	.	PUNCT
ejpam-6409	129	1	the	the	DET
ejpam-6409	129	2	2	2	NUM
ejpam-6409	129	3	-	-	PUNCT
ejpam-6409	129	4	ary	ary	NOUN
ejpam-6409	129	5	expansion	expansion	NOUN
ejpam-6409	129	6	of	of	ADP
ejpam-6409	129	7	u⊙2	u⊙2	PROPN
ejpam-6409	129	8	v	v	NOUN
ejpam-6409	129	9	is	be	AUX
ejpam-6409	129	10	[	[	X
ejpam-6409	129	11	t0	t0	NOUN
ejpam-6409	129	12	,	,	PUNCT
ejpam-6409	129	13	t1	t1	PROPN
ejpam-6409	129	14	,	,	PUNCT
ejpam-6409	129	15	.	.	PUNCT
ejpam-6409	129	16	.	.	PUNCT
ejpam-6409	129	17	.	.	PUNCT
ejpam-6409	130	1	,	,	PUNCT
ejpam-6409	130	2	ts−1]2	ts−1]2	PRON
ejpam-6409	130	3	,	,	PUNCT
ejpam-6409	130	4	where	where	SCONJ
ejpam-6409	130	5	ti	ti	NOUN
ejpam-6409	130	6	∈	∈	PROPN
ejpam-6409	130	7	{	{	PUNCT
ejpam-6409	130	8	00	00	NUM
ejpam-6409	130	9	,	,	PUNCT
ejpam-6409	130	10	10	10	NUM
ejpam-6409	130	11	}	}	PUNCT
ejpam-6409	130	12	.	.	PUNCT
ejpam-6409	131	1	lemma	lemma	PROPN
ejpam-6409	131	2	3.1	3.1	NUM
ejpam-6409	131	3	:	:	PUNCT
ejpam-6409	131	4	let	let	VERB
ejpam-6409	131	5	u	u	PRON
ejpam-6409	131	6	∈	∈	PROPN
ejpam-6409	131	7	z2s	z2s	X
ejpam-6409	132	1	[	[	X
ejpam-6409	132	2	ω	ω	X
ejpam-6409	132	3	]	]	X
ejpam-6409	132	4	and	and	CCONJ
ejpam-6409	132	5	µ	µ	PRON
ejpam-6409	132	6	∈	∈	PROPN
ejpam-6409	132	7	z2[ω	z2[ω	NOUN
ejpam-6409	132	8	]	]	PUNCT
ejpam-6409	132	9	.	.	PUNCT
ejpam-6409	133	1	then	then	ADV
ejpam-6409	133	2	ϕs(u+	ϕs(u+	PROPN
ejpam-6409	133	3	µ2s−1	µ2s−1	PROPN
ejpam-6409	133	4	)	)	PUNCT
ejpam-6409	133	5	=	=	PUNCT
ejpam-6409	133	6	ϕs(u	ϕs(u	PUNCT
ejpam-6409	133	7	)	)	PUNCT
ejpam-6409	134	1	+	+	CCONJ
ejpam-6409	134	2	(	(	PUNCT
ejpam-6409	134	3	µ	µ	X
ejpam-6409	134	4	,	,	PUNCT
ejpam-6409	134	5	µ	µ	NOUN
ejpam-6409	134	6	,	,	PUNCT
ejpam-6409	134	7	.	.	PUNCT
ejpam-6409	134	8	.	.	PUNCT
ejpam-6409	135	1	.	.	PUNCT
ejpam-6409	136	1	,	,	PUNCT
ejpam-6409	136	2	µ	µ	NOUN
ejpam-6409	136	3	)	)	PUNCT
ejpam-6409	136	4	.	.	PUNCT
ejpam-6409	137	1	proof	proof	NOUN
ejpam-6409	137	2	:	:	PUNCT
ejpam-6409	137	3	since	since	SCONJ
ejpam-6409	137	4	u+	u+	NOUN
ejpam-6409	137	5	µ2s−1	µ2s−1	NOUN
ejpam-6409	137	6	=	=	SYM
ejpam-6409	137	7	u1	u1	NOUN
ejpam-6409	137	8	+	+	CCONJ
ejpam-6409	137	9	µ02	µ02	VERB
ejpam-6409	137	10	s−1	s−1	PROPN
ejpam-6409	137	11	+	+	CCONJ
ejpam-6409	137	12	µ2s−1	µ2s−1	NOUN
ejpam-6409	137	13	=	=	SYM
ejpam-6409	137	14	u1	u1	NOUN
ejpam-6409	137	15	+	+	CCONJ
ejpam-6409	137	16	(	(	PUNCT
ejpam-6409	137	17	µ0	µ0	NOUN
ejpam-6409	137	18	+	+	CCONJ
ejpam-6409	137	19	µ)2s−1	µ)2s−1	NOUN
ejpam-6409	137	20	,	,	PUNCT
ejpam-6409	137	21	where	where	SCONJ
ejpam-6409	137	22	let	let	VERB
ejpam-6409	137	23	u1	u1	PROPN
ejpam-6409	137	24	∈	∈	PROPN
ejpam-6409	137	25	{	{	PUNCT
ejpam-6409	137	26	00	00	NUM
ejpam-6409	137	27	,	,	PUNCT
ejpam-6409	137	28	.	.	PUNCT
ejpam-6409	137	29	.	.	PUNCT
ejpam-6409	137	30	.	.	PUNCT
ejpam-6409	138	1	,	,	PUNCT
ejpam-6409	138	2	0	0	X
ejpam-6409	138	3	·	·	SYM
ejpam-6409	138	4	2s−1	2s−1	NUM
ejpam-6409	138	5	−	−	NUM
ejpam-6409	138	6	1	1	NUM
ejpam-6409	138	7	,	,	PUNCT
ejpam-6409	138	8	.	.	PUNCT
ejpam-6409	138	9	.	.	PUNCT
ejpam-6409	138	10	.	.	PUNCT
ejpam-6409	139	1	,	,	PUNCT
ejpam-6409	139	2	2s−1	2s−1	NUM
ejpam-6409	139	3	−	−	NOUN
ejpam-6409	139	4	1	1	NUM
ejpam-6409	139	5	·	·	PUNCT
ejpam-6409	139	6	0	0	NUM
ejpam-6409	139	7	,	,	PUNCT
ejpam-6409	139	8	.	.	PUNCT
ejpam-6409	139	9	.	.	PUNCT
ejpam-6409	139	10	.	.	PUNCT
ejpam-6409	140	1	,	,	PUNCT
ejpam-6409	140	2	2s−1	2s−1	NUM
ejpam-6409	140	3	−	−	NOUN
ejpam-6409	140	4	1	1	NUM
ejpam-6409	140	5	·	·	SYM
ejpam-6409	140	6	2s−1	2s−1	NUM
ejpam-6409	140	7	−	−	NOUN
ejpam-6409	140	8	1	1	NUM
ejpam-6409	140	9	}	}	PUNCT
ejpam-6409	140	10	and	and	CCONJ
ejpam-6409	140	11	u0	u0	PROPN
ejpam-6409	140	12	∈	∈	PROPN
ejpam-6409	140	13	z2[ω	z2[ω	NOUN
ejpam-6409	140	14	]	]	PUNCT
ejpam-6409	140	15	.	.	PUNCT
ejpam-6409	141	1	then	then	ADV
ejpam-6409	141	2	,	,	PUNCT
ejpam-6409	141	3	by	by	ADP
ejpam-6409	141	4	the	the	DET
ejpam-6409	141	5	definition	definition	NOUN
ejpam-6409	141	6	of	of	ADP
ejpam-6409	141	7	the	the	DET
ejpam-6409	141	8	gray	gray	ADJ
ejpam-6409	141	9	map	map	NOUN
ejpam-6409	141	10	ϕs	ϕs	INTJ
ejpam-6409	141	11	,	,	PUNCT
ejpam-6409	141	12	we	we	PRON
ejpam-6409	141	13	have	have	VERB
ejpam-6409	141	14	:	:	PUNCT
ejpam-6409	141	15	ϕs(u+	ϕs(u+	NOUN
ejpam-6409	141	16	µ2s−1	µ2s−1	PROPN
ejpam-6409	141	17	)	)	PUNCT
ejpam-6409	141	18	=	=	PUNCT
ejpam-6409	141	19	ϕs(u1	ϕs(u1	X
ejpam-6409	141	20	)	)	PUNCT
ejpam-6409	142	1	+	+	CCONJ
ejpam-6409	142	2	(	(	PUNCT
ejpam-6409	142	3	µ0	µ0	NOUN
ejpam-6409	142	4	+	+	X
ejpam-6409	142	5	µ	µ	NOUN
ejpam-6409	142	6	,	,	PUNCT
ejpam-6409	142	7	.	.	PUNCT
ejpam-6409	142	8	.	.	PUNCT
ejpam-6409	142	9	.	.	PUNCT
ejpam-6409	143	1	,	,	PUNCT
ejpam-6409	143	2	µ0	µ0	NOUN
ejpam-6409	143	3	+	+	NOUN
ejpam-6409	143	4	µ	µ	X
ejpam-6409	143	5	)	)	PUNCT
ejpam-6409	143	6	=	=	PUNCT
ejpam-6409	143	7	ϕs(u1	ϕs(u1	X
ejpam-6409	143	8	)	)	PUNCT
ejpam-6409	144	1	+	+	CCONJ
ejpam-6409	144	2	(	(	PUNCT
ejpam-6409	144	3	µ0	µ0	NOUN
ejpam-6409	144	4	,	,	PUNCT
ejpam-6409	144	5	.	.	PUNCT
ejpam-6409	144	6	.	.	PUNCT
ejpam-6409	144	7	.	.	PUNCT
ejpam-6409	145	1	,	,	PUNCT
ejpam-6409	145	2	µ0	µ0	NOUN
ejpam-6409	145	3	)	)	PUNCT
ejpam-6409	146	1	+	+	CCONJ
ejpam-6409	146	2	(	(	PUNCT
ejpam-6409	146	3	µ	µ	NUM
ejpam-6409	146	4	,	,	PUNCT
ejpam-6409	146	5	.	.	PUNCT
ejpam-6409	146	6	.	.	PUNCT
ejpam-6409	147	1	.	.	PUNCT
ejpam-6409	148	1	,	,	PUNCT
ejpam-6409	148	2	µ	µ	X
ejpam-6409	148	3	)	)	PUNCT
ejpam-6409	148	4	=	=	PUNCT
ejpam-6409	148	5	ϕs(u1	ϕs(u1	X
ejpam-6409	148	6	)	)	PUNCT
ejpam-6409	149	1	+	+	CCONJ
ejpam-6409	149	2	(	(	PUNCT
ejpam-6409	149	3	µ	µ	NUM
ejpam-6409	149	4	,	,	PUNCT
ejpam-6409	149	5	.	.	PUNCT
ejpam-6409	149	6	.	.	PUNCT
ejpam-6409	150	1	.	.	PUNCT
ejpam-6409	151	1	,	,	PUNCT
ejpam-6409	151	2	µ	µ	NOUN
ejpam-6409	151	3	)	)	PUNCT
ejpam-6409	151	4	.	.	PUNCT
ejpam-6409	152	1	corollary	corollary	ADJ
ejpam-6409	152	2	3.1	3.1	NUM
ejpam-6409	152	3	:	:	PUNCT
ejpam-6409	152	4	let	let	VERB
ejpam-6409	152	5	λ	λ	PROPN
ejpam-6409	152	6	,	,	PUNCT
ejpam-6409	152	7	µ	µ	PRON
ejpam-6409	152	8	∈	∈	NOUN
ejpam-6409	152	9	z2[ω	z2[ω	NOUN
ejpam-6409	152	10	]	]	PUNCT
ejpam-6409	152	11	.	.	PUNCT
ejpam-6409	153	1	then	then	ADV
ejpam-6409	153	2	,	,	PUNCT
ejpam-6409	153	3	ϕs(λµ2	ϕs(λµ2	X
ejpam-6409	153	4	s−1	s−1	PROPN
ejpam-6409	153	5	)	)	PUNCT
ejpam-6409	153	6	=	=	PUNCT
ejpam-6409	153	7	λϕs(µ2	λϕs(µ2	ADJ
ejpam-6409	153	8	s−1	s−1	PROPN
ejpam-6409	153	9	)	)	PUNCT
ejpam-6409	153	10	=	=	PUNCT
ejpam-6409	154	1	λµϕs(2	λµϕs(2	NOUN
ejpam-6409	154	2	s−1	s−1	NOUN
ejpam-6409	154	3	)	)	PUNCT
ejpam-6409	154	4	.	.	PUNCT
ejpam-6409	155	1	proof	proof	NOUN
ejpam-6409	155	2	:	:	PUNCT
ejpam-6409	155	3	by	by	ADP
ejpam-6409	155	4	the	the	DET
ejpam-6409	155	5	definition	definition	NOUN
ejpam-6409	155	6	of	of	ADP
ejpam-6409	155	7	the	the	DET
ejpam-6409	155	8	gray	gray	ADJ
ejpam-6409	155	9	map	map	NOUN
ejpam-6409	155	10	ϕs	ϕs	INTJ
ejpam-6409	155	11	,	,	PUNCT
ejpam-6409	155	12	we	we	PRON
ejpam-6409	155	13	have	have	VERB
ejpam-6409	155	14	ϕs(µ2	ϕs(µ2	ADJ
ejpam-6409	155	15	s−1	s−1	PROPN
ejpam-6409	155	16	)	)	PUNCT
ejpam-6409	155	17	=	=	PUNCT
ejpam-6409	155	18	(	(	PUNCT
ejpam-6409	155	19	µ	µ	NOUN
ejpam-6409	155	20	,	,	PUNCT
ejpam-6409	155	21	.	.	PUNCT
ejpam-6409	155	22	.	.	PUNCT
ejpam-6409	156	1	.	.	PUNCT
ejpam-6409	157	1	,	,	PUNCT
ejpam-6409	157	2	µ	µ	NOUN
ejpam-6409	157	3	)	)	PUNCT
ejpam-6409	157	4	.	.	PUNCT
ejpam-6409	158	1	then	then	ADV
ejpam-6409	158	2	,	,	PUNCT
ejpam-6409	158	3	ϕs(λµ2	ϕs(λµ2	X
ejpam-6409	158	4	s−1	s−1	PROPN
ejpam-6409	158	5	)	)	PUNCT
ejpam-6409	158	6	=	=	PUNCT
ejpam-6409	158	7	(	(	PUNCT
ejpam-6409	158	8	λµ	λµ	INTJ
ejpam-6409	158	9	,	,	PUNCT
ejpam-6409	158	10	.	.	PUNCT
ejpam-6409	158	11	.	.	PUNCT
ejpam-6409	158	12	.	.	PUNCT
ejpam-6409	159	1	,	,	PUNCT
ejpam-6409	159	2	λµ	λµ	NOUN
ejpam-6409	159	3	)	)	PUNCT
ejpam-6409	159	4	=	=	SYM
ejpam-6409	159	5	λ(µ	λ(µ	NOUN
ejpam-6409	159	6	,	,	PUNCT
ejpam-6409	159	7	.	.	PUNCT
ejpam-6409	159	8	.	.	PUNCT
ejpam-6409	160	1	.	.	PUNCT
ejpam-6409	161	1	,	,	PUNCT
ejpam-6409	161	2	µ	µ	X
ejpam-6409	161	3	)	)	PUNCT
ejpam-6409	161	4	=	=	PUNCT
ejpam-6409	161	5	λϕs(µ2	λϕs(µ2	ADJ
ejpam-6409	161	6	s−1	s−1	PROPN
ejpam-6409	161	7	)	)	PUNCT
ejpam-6409	161	8	=	=	PUNCT
ejpam-6409	162	1	λµϕs(2	λµϕs(2	NOUN
ejpam-6409	162	2	s−1	s−1	PROPN
ejpam-6409	162	3	)	)	PUNCT
ejpam-6409	162	4	.	.	PUNCT
ejpam-6409	163	1	proposition	proposition	NOUN
ejpam-6409	163	2	3.1	3.1	NUM
ejpam-6409	163	3	:	:	PUNCT
ejpam-6409	163	4	let	let	VERB
ejpam-6409	163	5	u	u	NOUN
ejpam-6409	163	6	,	,	PUNCT
ejpam-6409	163	7	v	v	ADP
ejpam-6409	163	8	∈	∈	NOUN
ejpam-6409	163	9	z2s	z2s	X
ejpam-6409	164	1	[	[	X
ejpam-6409	164	2	ω	ω	X
ejpam-6409	164	3	]	]	X
ejpam-6409	164	4	.	.	PUNCT
ejpam-6409	165	1	then	then	ADV
ejpam-6409	165	2	ϕs(u	ϕs(u	PUNCT
ejpam-6409	165	3	)	)	PUNCT
ejpam-6409	166	1	+	+	NUM
ejpam-6409	166	2	ϕs(v	ϕs(v	NOUN
ejpam-6409	166	3	)	)	PUNCT
ejpam-6409	166	4	=	=	SYM
ejpam-6409	166	5	ϕs(u⊕2	ϕs(u⊕2	NUM
ejpam-6409	166	6	v	v	NOUN
ejpam-6409	166	7	)	)	PUNCT
ejpam-6409	166	8	.	.	PUNCT
ejpam-6409	167	1	proof	proof	NOUN
ejpam-6409	167	2	:	:	PUNCT
ejpam-6409	167	3	let	let	VERB
ejpam-6409	167	4	[	[	X
ejpam-6409	167	5	u0	u0	ADJ
ejpam-6409	167	6	,	,	PUNCT
ejpam-6409	167	7	u1	u1	NOUN
ejpam-6409	167	8	,	,	PUNCT
ejpam-6409	167	9	.	.	PUNCT
ejpam-6409	167	10	.	.	PUNCT
ejpam-6409	168	1	.	.	PUNCT
ejpam-6409	169	1	,	,	PUNCT
ejpam-6409	169	2	us−1]2	us−1]2	PRON
ejpam-6409	169	3	and	and	CCONJ
ejpam-6409	169	4	[	[	X
ejpam-6409	169	5	v0	v0	NOUN
ejpam-6409	169	6	,	,	PUNCT
ejpam-6409	169	7	v1	v1	NOUN
ejpam-6409	169	8	,	,	PUNCT
ejpam-6409	169	9	.	.	PUNCT
ejpam-6409	169	10	.	.	PUNCT
ejpam-6409	170	1	.	.	PUNCT
ejpam-6409	171	1	,	,	PUNCT
ejpam-6409	171	2	vs−1]2	vs−1]2	ADV
ejpam-6409	171	3	be	be	AUX
ejpam-6409	171	4	the	the	DET
ejpam-6409	171	5	2	2	NUM
ejpam-6409	171	6	-	-	PUNCT
ejpam-6409	171	7	ary	ary	NOUN
ejpam-6409	171	8	expansions	expansion	NOUN
ejpam-6409	171	9	of	of	ADP
ejpam-6409	171	10	u	u	NOUN
ejpam-6409	171	11	and	and	CCONJ
ejpam-6409	171	12	v	v	NOUN
ejpam-6409	171	13	,	,	PUNCT
ejpam-6409	171	14	respectively	respectively	ADV
ejpam-6409	171	15	.	.	PUNCT
ejpam-6409	172	1	let	let	VERB
ejpam-6409	172	2	yi	yi	PROPN
ejpam-6409	172	3	be	be	AUX
ejpam-6409	172	4	the	the	DET
ejpam-6409	172	5	(	(	PUNCT
ejpam-6409	172	6	i+	i+	NUM
ejpam-6409	172	7	1)-th	1)-th	NUM
ejpam-6409	172	8	row	row	NOUN
ejpam-6409	172	9	of	of	ADP
ejpam-6409	172	10	y	y	PROPN
ejpam-6409	172	11	,	,	PUNCT
ejpam-6409	172	12	for	for	ADP
ejpam-6409	172	13	0	0	NUM
ejpam-6409	172	14	≤	≤	NUM
ejpam-6409	173	1	i	i	PRON
ejpam-6409	173	2	≤	≤	ADJ
ejpam-6409	173	3	s−	s−	PROPN
ejpam-6409	173	4	2	2	NUM
ejpam-6409	173	5	.	.	PUNCT
ejpam-6409	174	1	then	then	ADV
ejpam-6409	174	2	,	,	PUNCT
ejpam-6409	174	3	ϕs(u	ϕs(u	PUNCT
ejpam-6409	174	4	)	)	PUNCT
ejpam-6409	174	5	=	=	SYM
ejpam-6409	174	6	(	(	PUNCT
ejpam-6409	174	7	us−1	us−1	PROPN
ejpam-6409	174	8	,	,	PUNCT
ejpam-6409	174	9	us−1	us−1	PROPN
ejpam-6409	174	10	,	,	PUNCT
ejpam-6409	174	11	.	.	PUNCT
ejpam-6409	174	12	.	.	PUNCT
ejpam-6409	174	13	.	.	PUNCT
ejpam-6409	175	1	,	,	PUNCT
ejpam-6409	175	2	us−1	us−1	PROPN
ejpam-6409	175	3	)	)	PUNCT
ejpam-6409	176	1	+	+	CCONJ
ejpam-6409	176	2	s−2∑	s−2∑	ADJ
ejpam-6409	176	3	i=0	i=0	ADJ
ejpam-6409	176	4	uiyi	uiyi	NOUN
ejpam-6409	176	5	,	,	PUNCT
ejpam-6409	176	6	muhammad	muhammad	PROPN
ejpam-6409	176	7	sajjad	sajjad	PROPN
ejpam-6409	176	8	et	et	PROPN
ejpam-6409	176	9	al	al	PROPN
ejpam-6409	176	10	.	.	PUNCT
ejpam-6409	176	11	/	/	SYM
ejpam-6409	176	12	eur	eur	PROPN
ejpam-6409	176	13	.	.	PUNCT
ejpam-6409	177	1	j.	j.	PROPN
ejpam-6409	177	2	pure	pure	PROPN
ejpam-6409	177	3	appl	appl	PROPN
ejpam-6409	177	4	.	.	PROPN
ejpam-6409	177	5	math	math	PROPN
ejpam-6409	177	6	,	,	PUNCT
ejpam-6409	177	7	18	18	NUM
ejpam-6409	177	8	(	(	PUNCT
ejpam-6409	177	9	3	3	NUM
ejpam-6409	177	10	)	)	PUNCT
ejpam-6409	177	11	(	(	PUNCT
ejpam-6409	177	12	2025	2025	NUM
ejpam-6409	177	13	)	)	PUNCT
ejpam-6409	177	14	,	,	PUNCT
ejpam-6409	177	15	6409	6409	NUM
ejpam-6409	177	16	7	7	NUM
ejpam-6409	177	17	of	of	ADP
ejpam-6409	177	18	32	32	NUM
ejpam-6409	177	19	ϕs(v	ϕs(v	NOUN
ejpam-6409	177	20	)	)	PUNCT
ejpam-6409	177	21	=	=	SYM
ejpam-6409	178	1	(	(	PUNCT
ejpam-6409	178	2	vs−1	vs−1	ADJ
ejpam-6409	178	3	,	,	PUNCT
ejpam-6409	178	4	vs−1	vs−1	NOUN
ejpam-6409	178	5	,	,	PUNCT
ejpam-6409	178	6	.	.	PUNCT
ejpam-6409	178	7	.	.	PUNCT
ejpam-6409	179	1	.	.	PUNCT
ejpam-6409	180	1	,	,	PUNCT
ejpam-6409	180	2	vs−1	vs−1	ADJ
ejpam-6409	180	3	)	)	PUNCT
ejpam-6409	180	4	+	+	CCONJ
ejpam-6409	180	5	s−2∑	s−2∑	ADJ
ejpam-6409	181	1	i=0	i=0	PROPN
ejpam-6409	181	2	viyi	viyi	ADV
ejpam-6409	181	3	.	.	PUNCT
ejpam-6409	182	1	therefore	therefore	ADV
ejpam-6409	182	2	,	,	PUNCT
ejpam-6409	182	3	ϕs(u	ϕs(u	PUNCT
ejpam-6409	182	4	)	)	PUNCT
ejpam-6409	182	5	+	+	NUM
ejpam-6409	182	6	ϕs(v	ϕs(v	NOUN
ejpam-6409	182	7	)	)	PUNCT
ejpam-6409	182	8	=	=	PRON
ejpam-6409	182	9	(	(	PUNCT
ejpam-6409	182	10	rs−1	rs−1	PROPN
ejpam-6409	182	11	,	,	PUNCT
ejpam-6409	182	12	rs−1	rs−1	NOUN
ejpam-6409	182	13	,	,	PUNCT
ejpam-6409	182	14	.	.	PUNCT
ejpam-6409	182	15	.	.	PUNCT
ejpam-6409	183	1	.	.	PUNCT
ejpam-6409	184	1	,	,	PUNCT
ejpam-6409	184	2	rs−1	rs−1	PROPN
ejpam-6409	184	3	)	)	PUNCT
ejpam-6409	185	1	+	+	CCONJ
ejpam-6409	185	2	s−2∑	s−2∑	PROPN
ejpam-6409	186	1	i=0	i=0	PROPN
ejpam-6409	186	2	riyi	riyi	X
ejpam-6409	186	3	=	=	PUNCT
ejpam-6409	186	4	ϕs(u⊕2	ϕs(u⊕2	NUM
ejpam-6409	186	5	v	v	NOUN
ejpam-6409	186	6	)	)	PUNCT
ejpam-6409	186	7	,	,	PUNCT
ejpam-6409	186	8	where	where	SCONJ
ejpam-6409	186	9	ri	ri	PROPN
ejpam-6409	186	10	=	=	SYM
ejpam-6409	186	11	ui	ui	PROPN
ejpam-6409	186	12	+	+	CCONJ
ejpam-6409	186	13	vi	vi	NOUN
ejpam-6409	186	14	in	in	ADP
ejpam-6409	186	15	z2[ω	z2[ω	NOUN
ejpam-6409	186	16	]	]	PUNCT
ejpam-6409	186	17	for	for	ADP
ejpam-6409	186	18	0	0	NUM
ejpam-6409	186	19	≤	≤	NUM
ejpam-6409	187	1	i	i	PRON
ejpam-6409	187	2	≤	≤	NUM
ejpam-6409	187	3	s−	s−	PROPN
ejpam-6409	187	4	1	1	NUM
ejpam-6409	187	5	.	.	PUNCT
ejpam-6409	188	1	proposition	proposition	NOUN
ejpam-6409	188	2	3.2	3.2	NUM
ejpam-6409	188	3	:	:	PUNCT
ejpam-6409	188	4	let	let	VERB
ejpam-6409	188	5	u	u	NOUN
ejpam-6409	188	6	,	,	PUNCT
ejpam-6409	188	7	v	v	ADP
ejpam-6409	188	8	∈	∈	NOUN
ejpam-6409	188	9	z2s	z2s	X
ejpam-6409	189	1	[	[	X
ejpam-6409	189	2	ω	ω	X
ejpam-6409	189	3	]	]	X
ejpam-6409	189	4	.	.	PUNCT
ejpam-6409	190	1	then	then	ADV
ejpam-6409	190	2	u⊕2	u⊕2	PROPN
ejpam-6409	190	3	v	v	NOUN
ejpam-6409	190	4	=	=	PUNCT
ejpam-6409	190	5	u+	u+	NOUN
ejpam-6409	190	6	v	v	ADP
ejpam-6409	190	7	−	−	PROPN
ejpam-6409	190	8	2(u⊙2	2(u⊙2	NUM
ejpam-6409	190	9	v	v	NOUN
ejpam-6409	190	10	)	)	PUNCT
ejpam-6409	190	11	.	.	PUNCT
ejpam-6409	191	1	proof	proof	NOUN
ejpam-6409	191	2	:	:	PUNCT
ejpam-6409	191	3	let	let	VERB
ejpam-6409	191	4	[	[	X
ejpam-6409	191	5	u0	u0	ADJ
ejpam-6409	191	6	,	,	PUNCT
ejpam-6409	191	7	u1	u1	NOUN
ejpam-6409	191	8	,	,	PUNCT
ejpam-6409	191	9	.	.	PUNCT
ejpam-6409	191	10	.	.	PUNCT
ejpam-6409	192	1	.	.	PUNCT
ejpam-6409	193	1	,	,	PUNCT
ejpam-6409	193	2	us−1]2	us−1]2	NOUN
ejpam-6409	193	3	,	,	PUNCT
ejpam-6409	193	4	[	[	X
ejpam-6409	193	5	v0	v0	NOUN
ejpam-6409	193	6	,	,	PUNCT
ejpam-6409	193	7	v1	v1	NOUN
ejpam-6409	193	8	,	,	PUNCT
ejpam-6409	193	9	.	.	PUNCT
ejpam-6409	193	10	.	.	PUNCT
ejpam-6409	193	11	.	.	PUNCT
ejpam-6409	194	1	,	,	PUNCT
ejpam-6409	194	2	vs−1]2	vs−1]2	ADV
ejpam-6409	194	3	be	be	AUX
ejpam-6409	194	4	the	the	DET
ejpam-6409	194	5	2	2	NUM
ejpam-6409	194	6	-	-	PUNCT
ejpam-6409	194	7	ary	ary	NOUN
ejpam-6409	194	8	expansions	expansion	NOUN
ejpam-6409	194	9	of	of	ADP
ejpam-6409	194	10	u	u	NOUN
ejpam-6409	194	11	and	and	CCONJ
ejpam-6409	194	12	v	v	NOUN
ejpam-6409	194	13	,	,	PUNCT
ejpam-6409	194	14	respectively	respectively	ADV
ejpam-6409	194	15	.	.	PUNCT
ejpam-6409	195	1	note	note	VERB
ejpam-6409	195	2	that	that	SCONJ
ejpam-6409	195	3	0	0	NUM
ejpam-6409	195	4	≤	≤	NUM
ejpam-6409	195	5	ui	ui	NOUN
ejpam-6409	196	1	+	+	CCONJ
ejpam-6409	196	2	vi	vi	ADJ
ejpam-6409	196	3	≤	≤	NUM
ejpam-6409	196	4	2	2	NUM
ejpam-6409	196	5	.	.	PUNCT
ejpam-6409	196	6	by	by	ADP
ejpam-6409	196	7	the	the	DET
ejpam-6409	196	8	division	division	NOUN
ejpam-6409	196	9	algorithm	algorithm	NOUN
ejpam-6409	196	10	,	,	PUNCT
ejpam-6409	196	11	ui	ui	PROPN
ejpam-6409	196	12	+	+	CCONJ
ejpam-6409	196	13	vi	vi	ADJ
ejpam-6409	196	14	=	=	SYM
ejpam-6409	196	15	2ti	2ti	NOUN
ejpam-6409	196	16	+	+	CCONJ
ejpam-6409	196	17	ri	ri	PROPN
ejpam-6409	196	18	,	,	PUNCT
ejpam-6409	196	19	where	where	SCONJ
ejpam-6409	196	20	ti	ti	X
ejpam-6409	196	21	=	=	SYM
ejpam-6409	196	22	1	1	NUM
ejpam-6409	196	23	if	if	SCONJ
ejpam-6409	196	24	ui	ui	PROPN
ejpam-6409	196	25	+	+	NUM
ejpam-6409	196	26	vi	vi	NOUN
ejpam-6409	196	27	≥	≥	NOUN
ejpam-6409	196	28	2	2	NUM
ejpam-6409	196	29	,	,	PUNCT
ejpam-6409	196	30	and	and	CCONJ
ejpam-6409	196	31	ti	ti	X
ejpam-6409	196	32	=	=	SYM
ejpam-6409	196	33	0	0	NUM
ejpam-6409	196	34	otherwise	otherwise	ADV
ejpam-6409	196	35	;	;	PUNCT
ejpam-6409	196	36	also	also	ADV
ejpam-6409	196	37	0	0	X
ejpam-6409	196	38	≤	≤	NUM
ejpam-6409	196	39	ri	ri	X
ejpam-6409	196	40	≤	≤	ADJ
ejpam-6409	196	41	1	1	NUM
ejpam-6409	196	42	.	.	PUNCT
ejpam-6409	197	1	then	then	ADV
ejpam-6409	197	2	we	we	PRON
ejpam-6409	197	3	have	have	VERB
ejpam-6409	197	4	:	:	PUNCT
ejpam-6409	197	5	u+	u+	NOUN
ejpam-6409	197	6	v	v	PART
ejpam-6409	197	7	=	=	SYM
ejpam-6409	197	8	s−1∑	s−1∑	NUM
ejpam-6409	197	9	i=0	i=0	PROPN
ejpam-6409	197	10	(	(	PUNCT
ejpam-6409	197	11	ui	ui	NOUN
ejpam-6409	197	12	+	+	CCONJ
ejpam-6409	197	13	vi)2	vi)2	ADJ
ejpam-6409	197	14	i	i	NOUN
ejpam-6409	197	15	=	=	PROPN
ejpam-6409	197	16	s−1∑	s−1∑	NUM
ejpam-6409	197	17	i=0	i=0	X
ejpam-6409	197	18	(	(	PUNCT
ejpam-6409	197	19	2ti	2ti	NOUN
ejpam-6409	197	20	+	+	CCONJ
ejpam-6409	198	1	ri)2	ri)2	ADJ
ejpam-6409	198	2	i	i	NOUN
ejpam-6409	198	3	=	=	SYM
ejpam-6409	198	4	2	2	NUM
ejpam-6409	198	5	s−1∑	s−1∑	NUM
ejpam-6409	198	6	i=0	i=0	PROPN
ejpam-6409	198	7	ti2	ti2	PROPN
ejpam-6409	198	8	i	i	PRON
ejpam-6409	198	9	+	+	CCONJ
ejpam-6409	198	10	s−1∑	s−1∑	NUM
ejpam-6409	198	11	i=0	i=0	ADJ
ejpam-6409	198	12	ri2	ri2	NOUN
ejpam-6409	199	1	i	i	NOUN
ejpam-6409	199	2	=	=	PUNCT
ejpam-6409	200	1	2(u⊙2	2(u⊙2	NUM
ejpam-6409	200	2	v	v	NOUN
ejpam-6409	200	3	)	)	PUNCT
ejpam-6409	201	1	+	+	CCONJ
ejpam-6409	201	2	u⊕2	u⊕2	PROPN
ejpam-6409	201	3	v.	v.	CCONJ
ejpam-6409	201	4	therefore	therefore	ADV
ejpam-6409	201	5	,	,	PUNCT
ejpam-6409	201	6	u⊕2	u⊕2	PROPN
ejpam-6409	201	7	v	v	NOUN
ejpam-6409	201	8	=	=	PUNCT
ejpam-6409	201	9	u+	u+	NOUN
ejpam-6409	201	10	v	v	ADP
ejpam-6409	201	11	−	−	PROPN
ejpam-6409	201	12	2(u⊙2	2(u⊙2	NUM
ejpam-6409	201	13	v	v	NOUN
ejpam-6409	201	14	)	)	PUNCT
ejpam-6409	201	15	.	.	PUNCT
ejpam-6409	202	1	corollary	corollary	ADJ
ejpam-6409	202	2	3.2	3.2	NUM
ejpam-6409	202	3	:	:	PUNCT
ejpam-6409	202	4	let	let	VERB
ejpam-6409	202	5	u	u	NOUN
ejpam-6409	202	6	,	,	PUNCT
ejpam-6409	202	7	v	v	ADP
ejpam-6409	202	8	∈	∈	NOUN
ejpam-6409	202	9	z2s	z2s	X
ejpam-6409	203	1	[	[	X
ejpam-6409	203	2	ω	ω	X
ejpam-6409	203	3	]	]	X
ejpam-6409	203	4	.	.	PUNCT
ejpam-6409	204	1	then	then	ADV
ejpam-6409	204	2	,	,	PUNCT
ejpam-6409	204	3	ϕs(u	ϕs(u	PUNCT
ejpam-6409	204	4	)	)	PUNCT
ejpam-6409	204	5	+	+	NUM
ejpam-6409	204	6	ϕs(v	ϕs(v	NOUN
ejpam-6409	204	7	)	)	PUNCT
ejpam-6409	205	1	=	=	SYM
ejpam-6409	206	1	ϕs(u+	ϕs(u+	NOUN
ejpam-6409	206	2	v	v	ADP
ejpam-6409	206	3	−	−	PROPN
ejpam-6409	206	4	2(u⊙2	2(u⊙2	NUM
ejpam-6409	206	5	v	v	NOUN
ejpam-6409	206	6	)	)	PUNCT
ejpam-6409	206	7	)	)	PUNCT
ejpam-6409	206	8	.	.	PUNCT
ejpam-6409	207	1	proof	proof	NOUN
ejpam-6409	207	2	:	:	PUNCT
ejpam-6409	207	3	from	from	ADP
ejpam-6409	207	4	proposition	proposition	NOUN
ejpam-6409	207	5	3.1	3.1	NUM
ejpam-6409	207	6	,	,	PUNCT
ejpam-6409	207	7	ϕs(u	ϕs(u	PUNCT
ejpam-6409	207	8	)	)	PUNCT
ejpam-6409	208	1	+	+	NUM
ejpam-6409	208	2	ϕs(v	ϕs(v	NOUN
ejpam-6409	208	3	)	)	PUNCT
ejpam-6409	208	4	=	=	SYM
ejpam-6409	208	5	ϕs(u	ϕs(u	PUNCT
ejpam-6409	208	6	⊕2	⊕2	PROPN
ejpam-6409	208	7	v	v	NOUN
ejpam-6409	208	8	)	)	PUNCT
ejpam-6409	208	9	.	.	PUNCT
ejpam-6409	209	1	from	from	ADP
ejpam-6409	209	2	proposition	proposition	NOUN
ejpam-6409	209	3	3.2	3.2	NUM
ejpam-6409	209	4	,	,	PUNCT
ejpam-6409	209	5	u⊕2	u⊕2	PROPN
ejpam-6409	209	6	v	v	NOUN
ejpam-6409	209	7	=	=	PUNCT
ejpam-6409	209	8	u+	u+	NOUN
ejpam-6409	209	9	v	v	ADP
ejpam-6409	209	10	−	−	PROPN
ejpam-6409	209	11	2(u⊙2	2(u⊙2	NUM
ejpam-6409	209	12	v	v	NOUN
ejpam-6409	209	13	)	)	PUNCT
ejpam-6409	209	14	.	.	PUNCT
ejpam-6409	210	1	so	so	ADV
ejpam-6409	210	2	,	,	PUNCT
ejpam-6409	210	3	ϕs(u	ϕs(u	PUNCT
ejpam-6409	210	4	)	)	PUNCT
ejpam-6409	211	1	+	+	NUM
ejpam-6409	211	2	ϕs(v	ϕs(v	NOUN
ejpam-6409	211	3	)	)	PUNCT
ejpam-6409	211	4	=	=	SYM
ejpam-6409	212	1	ϕs(u⊕2	ϕs(u⊕2	NUM
ejpam-6409	212	2	v	v	NOUN
ejpam-6409	212	3	)	)	PUNCT
ejpam-6409	212	4	=	=	SYM
ejpam-6409	213	1	ϕs(u+	ϕs(u+	NOUN
ejpam-6409	213	2	v	v	ADP
ejpam-6409	213	3	−	−	PROPN
ejpam-6409	213	4	2(u⊙2	2(u⊙2	NUM
ejpam-6409	213	5	v	v	NOUN
ejpam-6409	213	6	)	)	PUNCT
ejpam-6409	213	7	)	)	PUNCT
ejpam-6409	213	8	.	.	PUNCT
ejpam-6409	214	1	corollary	corollary	ADJ
ejpam-6409	214	2	3.3	3.3	NUM
ejpam-6409	214	3	:	:	PUNCT
ejpam-6409	214	4	let	let	VERB
ejpam-6409	214	5	u	u	NOUN
ejpam-6409	214	6	,	,	PUNCT
ejpam-6409	214	7	v	v	ADP
ejpam-6409	214	8	∈	∈	NOUN
ejpam-6409	214	9	z2s	z2s	X
ejpam-6409	215	1	[	[	X
ejpam-6409	215	2	ω	ω	X
ejpam-6409	215	3	]	]	X
ejpam-6409	215	4	.	.	PUNCT
ejpam-6409	216	1	then	then	ADV
ejpam-6409	216	2	,	,	PUNCT
ejpam-6409	216	3	2s−1(u⊕2	2s−1(u⊕2	NUM
ejpam-6409	216	4	v	v	NOUN
ejpam-6409	216	5	)	)	PUNCT
ejpam-6409	216	6	=	=	NOUN
ejpam-6409	216	7	2s−1(u+	2s−1(u+	NUM
ejpam-6409	216	8	v	v	NOUN
ejpam-6409	216	9	)	)	PUNCT
ejpam-6409	216	10	.	.	PUNCT
ejpam-6409	217	1	muhammad	muhammad	PROPN
ejpam-6409	217	2	sajjad	sajjad	PROPN
ejpam-6409	217	3	et	et	PROPN
ejpam-6409	217	4	al	al	PROPN
ejpam-6409	217	5	.	.	PUNCT
ejpam-6409	217	6	/	/	SYM
ejpam-6409	217	7	eur	eur	PROPN
ejpam-6409	217	8	.	.	PUNCT
ejpam-6409	218	1	j.	j.	PROPN
ejpam-6409	218	2	pure	pure	PROPN
ejpam-6409	218	3	appl	appl	PROPN
ejpam-6409	218	4	.	.	PROPN
ejpam-6409	218	5	math	math	PROPN
ejpam-6409	218	6	,	,	PUNCT
ejpam-6409	218	7	18	18	NUM
ejpam-6409	218	8	(	(	PUNCT
ejpam-6409	218	9	3	3	NUM
ejpam-6409	218	10	)	)	PUNCT
ejpam-6409	218	11	(	(	PUNCT
ejpam-6409	218	12	2025	2025	NUM
ejpam-6409	218	13	)	)	PUNCT
ejpam-6409	218	14	,	,	PUNCT
ejpam-6409	218	15	6409	6409	NUM
ejpam-6409	218	16	8	8	NUM
ejpam-6409	218	17	of	of	ADP
ejpam-6409	218	18	32	32	NUM
ejpam-6409	218	19	proof	proof	NOUN
ejpam-6409	218	20	:	:	PUNCT
ejpam-6409	218	21	let	let	VERB
ejpam-6409	218	22	[	[	X
ejpam-6409	218	23	u0	u0	ADJ
ejpam-6409	218	24	,	,	PUNCT
ejpam-6409	218	25	u1	u1	NOUN
ejpam-6409	218	26	,	,	PUNCT
ejpam-6409	218	27	.	.	PUNCT
ejpam-6409	218	28	.	.	PUNCT
ejpam-6409	219	1	.	.	PUNCT
ejpam-6409	220	1	,	,	PUNCT
ejpam-6409	220	2	us−1]2	us−1]2	NOUN
ejpam-6409	220	3	,	,	PUNCT
ejpam-6409	220	4	[	[	X
ejpam-6409	220	5	v0	v0	NOUN
ejpam-6409	220	6	,	,	PUNCT
ejpam-6409	220	7	v1	v1	NOUN
ejpam-6409	220	8	,	,	PUNCT
ejpam-6409	220	9	.	.	PUNCT
ejpam-6409	220	10	.	.	PUNCT
ejpam-6409	220	11	.	.	PUNCT
ejpam-6409	221	1	,	,	PUNCT
ejpam-6409	221	2	vs−1]2	vs−1]2	ADV
ejpam-6409	221	3	be	be	AUX
ejpam-6409	221	4	the	the	DET
ejpam-6409	221	5	binary	binary	ADJ
ejpam-6409	221	6	expansions	expansion	NOUN
ejpam-6409	221	7	of	of	ADP
ejpam-6409	221	8	u	u	NOUN
ejpam-6409	221	9	and	and	CCONJ
ejpam-6409	221	10	v	v	NOUN
ejpam-6409	221	11	,	,	PUNCT
ejpam-6409	221	12	respectively	respectively	ADV
ejpam-6409	221	13	.	.	PUNCT
ejpam-6409	222	1	then	then	ADV
ejpam-6409	222	2	[	[	X
ejpam-6409	222	3	0	0	NUM
ejpam-6409	222	4	,	,	PUNCT
ejpam-6409	222	5	0	0	NUM
ejpam-6409	222	6	,	,	PUNCT
ejpam-6409	222	7	.	.	PUNCT
ejpam-6409	222	8	.	.	PUNCT
ejpam-6409	222	9	.	.	PUNCT
ejpam-6409	223	1	,	,	PUNCT
ejpam-6409	223	2	0	0	NUM
ejpam-6409	223	3	,	,	PUNCT
ejpam-6409	223	4	u0]2	u0]2	ADV
ejpam-6409	223	5	is	be	AUX
ejpam-6409	223	6	the	the	DET
ejpam-6409	223	7	binary	binary	ADJ
ejpam-6409	223	8	expansion	expansion	NOUN
ejpam-6409	223	9	of	of	ADP
ejpam-6409	223	10	2s−1u	2s−1u	NUM
ejpam-6409	223	11	,	,	PUNCT
ejpam-6409	223	12	so	so	ADV
ejpam-6409	223	13	2s−1u⊙2	2s−1u⊙2	ADJ
ejpam-6409	223	14	v	v	NOUN
ejpam-6409	223	15	is	be	AUX
ejpam-6409	223	16	2s−1	2s−1	NUM
ejpam-6409	223	17	if	if	SCONJ
ejpam-6409	223	18	u0	u0	ADJ
ejpam-6409	223	19	+	+	ADJ
ejpam-6409	223	20	vs−1	vs−1	ADJ
ejpam-6409	223	21	≥	≥	NOUN
ejpam-6409	223	22	2	2	NUM
ejpam-6409	223	23	,	,	PUNCT
ejpam-6409	223	24	and	and	CCONJ
ejpam-6409	223	25	0	0	NUM
ejpam-6409	223	26	otherwise	otherwise	ADV
ejpam-6409	223	27	.	.	PUNCT
ejpam-6409	224	1	in	in	ADP
ejpam-6409	224	2	any	any	DET
ejpam-6409	224	3	case	case	NOUN
ejpam-6409	224	4	,	,	PUNCT
ejpam-6409	224	5	2(2s−1u⊙2	2(2s−1u⊙2	NUM
ejpam-6409	224	6	v	v	NOUN
ejpam-6409	224	7	)	)	PUNCT
ejpam-6409	224	8	=	=	SYM
ejpam-6409	225	1	0	0	X
ejpam-6409	225	2	.	.	PUNCT
ejpam-6409	226	1	hence	hence	ADV
ejpam-6409	226	2	,	,	PUNCT
ejpam-6409	226	3	by	by	ADP
ejpam-6409	226	4	proposition	proposition	NOUN
ejpam-6409	226	5	3.2	3.2	NUM
ejpam-6409	226	6	,	,	PUNCT
ejpam-6409	226	7	the	the	DET
ejpam-6409	226	8	result	result	NOUN
ejpam-6409	226	9	follows	follow	VERB
ejpam-6409	226	10	.	.	PUNCT
ejpam-6409	227	1	corollary	corollary	ADJ
ejpam-6409	227	2	3.4	3.4	NUM
ejpam-6409	227	3	:	:	PUNCT
ejpam-6409	227	4	let	let	VERB
ejpam-6409	227	5	u	u	PRON
ejpam-6409	227	6	∈	∈	PROPN
ejpam-6409	227	7	z2s	z2s	X
ejpam-6409	228	1	[	[	X
ejpam-6409	228	2	ω	ω	X
ejpam-6409	228	3	]	]	PUNCT
ejpam-6409	228	4	and	and	CCONJ
ejpam-6409	228	5	[	[	X
ejpam-6409	228	6	u0	u0	ADJ
ejpam-6409	228	7	,	,	PUNCT
ejpam-6409	228	8	u1	u1	NOUN
ejpam-6409	228	9	,	,	PUNCT
ejpam-6409	228	10	.	.	PUNCT
ejpam-6409	228	11	.	.	PUNCT
ejpam-6409	229	1	.	.	PUNCT
ejpam-6409	230	1	,	,	PUNCT
ejpam-6409	230	2	us−1]2	us−1]2	PRON
ejpam-6409	230	3	be	be	AUX
ejpam-6409	230	4	its	its	PRON
ejpam-6409	230	5	binary	binary	ADJ
ejpam-6409	230	6	expansion	expansion	NOUN
ejpam-6409	230	7	.	.	PUNCT
ejpam-6409	231	1	then	then	ADV
ejpam-6409	231	2	,	,	PUNCT
ejpam-6409	231	3	for	for	ADP
ejpam-6409	231	4	any	any	DET
ejpam-6409	231	5	i	i	PROPN
ejpam-6409	231	6	∈	∈	PROPN
ejpam-6409	231	7	{	{	PUNCT
ejpam-6409	231	8	0	0	NUM
ejpam-6409	231	9	,	,	PUNCT
ejpam-6409	231	10	.	.	PUNCT
ejpam-6409	231	11	.	.	PUNCT
ejpam-6409	231	12	.	.	PUNCT
ejpam-6409	232	1	,	,	PUNCT
ejpam-6409	232	2	s−	s−	PROPN
ejpam-6409	232	3	2	2	NUM
ejpam-6409	232	4	}	}	PUNCT
ejpam-6409	232	5	,	,	PUNCT
ejpam-6409	232	6	ϕs(u	ϕs(u	PUNCT
ejpam-6409	232	7	)	)	PUNCT
ejpam-6409	233	1	+	+	CCONJ
ejpam-6409	233	2	ϕs(2	ϕs(2	PROPN
ejpam-6409	233	3	i	i	NOUN
ejpam-6409	233	4	)	)	PUNCT
ejpam-6409	234	1	=	=	SYM
ejpam-6409	234	2	ϕs(u+	ϕs(u+	NOUN
ejpam-6409	234	3	2i	2i	NOUN
ejpam-6409	234	4	−	−	PROPN
ejpam-6409	234	5	2i+1ti	2i+1ti	NUM
ejpam-6409	234	6	)	)	PUNCT
ejpam-6409	234	7	,	,	PUNCT
ejpam-6409	234	8	where	where	SCONJ
ejpam-6409	234	9	ti	ti	NOUN
ejpam-6409	234	10	=	=	SYM
ejpam-6409	234	11	{	{	PUNCT
ejpam-6409	234	12	1	1	NUM
ejpam-6409	234	13	if	if	SCONJ
ejpam-6409	234	14	ui	ui	PROPN
ejpam-6409	234	15	≥	≥	NOUN
ejpam-6409	234	16	1	1	NUM
ejpam-6409	234	17	,	,	PUNCT
ejpam-6409	234	18	0	0	NUM
ejpam-6409	234	19	otherwise	otherwise	ADV
ejpam-6409	234	20	.	.	PUNCT
ejpam-6409	235	1	corollary	corollary	ADJ
ejpam-6409	235	2	3.5	3.5	NUM
ejpam-6409	235	3	:	:	PUNCT
ejpam-6409	235	4	let	let	VERB
ejpam-6409	235	5	v	v	PRON
ejpam-6409	235	6	∈	∈	NOUN
ejpam-6409	235	7	z2s	z2s	X
ejpam-6409	236	1	[	[	X
ejpam-6409	236	2	ω	ω	X
ejpam-6409	236	3	]	]	X
ejpam-6409	236	4	.	.	PUNCT
ejpam-6409	237	1	then	then	ADV
ejpam-6409	237	2	,	,	PUNCT
ejpam-6409	237	3	ϕs(2	ϕs(2	PROPN
ejpam-6409	237	4	s−1	s−1	PROPN
ejpam-6409	237	5	+	+	CCONJ
ejpam-6409	237	6	v	v	NOUN
ejpam-6409	237	7	)	)	PUNCT
ejpam-6409	237	8	=	=	PUNCT
ejpam-6409	238	1	ϕs(2	ϕs(2	PROPN
ejpam-6409	238	2	s−1	s−1	PROPN
ejpam-6409	238	3	)	)	PUNCT
ejpam-6409	238	4	+	+	NUM
ejpam-6409	238	5	ϕs(v	ϕs(v	NUM
ejpam-6409	238	6	)	)	PUNCT
ejpam-6409	238	7	.	.	PUNCT
ejpam-6409	239	1	corollary	corollary	ADJ
ejpam-6409	239	2	3.6	3.6	NUM
ejpam-6409	239	3	:	:	PUNCT
ejpam-6409	239	4	let	let	VERB
ejpam-6409	239	5	u	u	NOUN
ejpam-6409	239	6	,	,	PUNCT
ejpam-6409	239	7	v	v	ADP
ejpam-6409	239	8	∈	∈	NOUN
ejpam-6409	239	9	z2s	z2s	X
ejpam-6409	240	1	[	[	X
ejpam-6409	240	2	ω	ω	X
ejpam-6409	240	3	]	]	X
ejpam-6409	240	4	.	.	PUNCT
ejpam-6409	241	1	then	then	ADV
ejpam-6409	241	2	,	,	PUNCT
ejpam-6409	241	3	ϕs(2	ϕs(2	PROPN
ejpam-6409	241	4	s−1u+	s−1u+	NOUN
ejpam-6409	241	5	v	v	NOUN
ejpam-6409	241	6	)	)	PUNCT
ejpam-6409	241	7	=	=	PUNCT
ejpam-6409	242	1	ϕs(2	ϕs(2	PRON
ejpam-6409	242	2	s−1u	s−1u	NOUN
ejpam-6409	242	3	)	)	PUNCT
ejpam-6409	242	4	+	+	NUM
ejpam-6409	242	5	ϕs(v	ϕs(v	NUM
ejpam-6409	242	6	)	)	PUNCT
ejpam-6409	242	7	.	.	PUNCT
ejpam-6409	243	1	lemma	lemma	PROPN
ejpam-6409	243	2	3.2	3.2	NUM
ejpam-6409	243	3	:	:	PUNCT
ejpam-6409	243	4	let	let	VERB
ejpam-6409	243	5	u	u	PRON
ejpam-6409	243	6	∈	∈	PROPN
ejpam-6409	243	7	{	{	PUNCT
ejpam-6409	243	8	(	(	PUNCT
ejpam-6409	243	9	01)2s−2	01)2s−2	PROPN
ejpam-6409	243	10	,	,	PUNCT
ejpam-6409	243	11	(	(	PUNCT
ejpam-6409	243	12	03)2s−2	03)2s−2	NOUN
ejpam-6409	243	13	,	,	PUNCT
ejpam-6409	243	14	.	.	PUNCT
ejpam-6409	243	15	.	.	PUNCT
ejpam-6409	243	16	.	.	PUNCT
ejpam-6409	244	1	,	,	PUNCT
ejpam-6409	244	2	(	(	PUNCT
ejpam-6409	244	3	31)2s−1	31)2s−1	NUM
ejpam-6409	244	4	,	,	PUNCT
ejpam-6409	244	5	(	(	PUNCT
ejpam-6409	244	6	33)2s−1	33)2s−1	NUM
ejpam-6409	244	7	}	}	PUNCT
ejpam-6409	244	8	⊂	⊂	PROPN
ejpam-6409	244	9	z2s	z2s	X
ejpam-6409	245	1	[	[	X
ejpam-6409	245	2	ω	ω	X
ejpam-6409	245	3	]	]	X
ejpam-6409	245	4	.	.	PUNCT
ejpam-6409	246	1	then	then	ADV
ejpam-6409	246	2	,	,	PUNCT
ejpam-6409	246	3	ϕs(u	ϕs(u	PUNCT
ejpam-6409	246	4	)	)	PUNCT
ejpam-6409	247	1	+	+	CCONJ
ejpam-6409	247	2	ϕs(0	ϕs(0	ADP
ejpam-6409	247	3	·	·	PUNCT
ejpam-6409	247	4	2s−2	2s−2	NUM
ejpam-6409	247	5	)	)	PUNCT
ejpam-6409	248	1	=	=	SYM
ejpam-6409	248	2	ϕs(u+	ϕs(u+	NOUN
ejpam-6409	248	3	0	0	PUNCT
ejpam-6409	248	4	·	·	PUNCT
ejpam-6409	248	5	2s−2	2s−2	NUM
ejpam-6409	248	6	+	+	CCONJ
ejpam-6409	248	7	0	0	NUM
ejpam-6409	248	8	·	·	SYM
ejpam-6409	248	9	2s−1	2s−1	NUM
ejpam-6409	248	10	)	)	PUNCT
ejpam-6409	248	11	.	.	PUNCT
ejpam-6409	249	1	corollary	corollary	NOUN
ejpam-6409	249	2	3.7	3.7	NUM
ejpam-6409	249	3	:	:	PUNCT
ejpam-6409	249	4	let	let	VERB
ejpam-6409	249	5	v	v	X
ejpam-6409	249	6	∈	∈	PROPN
ejpam-6409	249	7	{	{	PUNCT
ejpam-6409	249	8	(	(	PUNCT
ejpam-6409	249	9	01)2s−2	01)2s−2	PROPN
ejpam-6409	249	10	,	,	PUNCT
ejpam-6409	249	11	(	(	PUNCT
ejpam-6409	249	12	03)2s−2	03)2s−2	NOUN
ejpam-6409	249	13	,	,	PUNCT
ejpam-6409	249	14	(	(	PUNCT
ejpam-6409	249	15	21)2s−2	21)2s−2	NUM
ejpam-6409	249	16	,	,	PUNCT
ejpam-6409	249	17	(	(	PUNCT
ejpam-6409	249	18	23)2s−2	23)2s−2	NUM
ejpam-6409	249	19	}	}	PUNCT
ejpam-6409	249	20	and	and	CCONJ
ejpam-6409	249	21	u	u	NOUN
ejpam-6409	249	22	=	=	PUNCT
ejpam-6409	249	23	{	{	PUNCT
ejpam-6409	249	24	(	(	PUNCT
ejpam-6409	249	25	01)2s−2	01)2s−2	PROPN
ejpam-6409	249	26	,	,	PUNCT
ejpam-6409	249	27	(	(	PUNCT
ejpam-6409	249	28	03)2s−2	03)2s−2	NOUN
ejpam-6409	249	29	,	,	PUNCT
ejpam-6409	249	30	.	.	PUNCT
ejpam-6409	249	31	.	.	PUNCT
ejpam-6409	249	32	.	.	PUNCT
ejpam-6409	250	1	,	,	PUNCT
ejpam-6409	250	2	(	(	PUNCT
ejpam-6409	250	3	31)2s−1	31)2s−1	NUM
ejpam-6409	250	4	,	,	PUNCT
ejpam-6409	250	5	(	(	PUNCT
ejpam-6409	250	6	33)2s−1	33)2s−1	NUM
ejpam-6409	250	7	}	}	PUNCT
ejpam-6409	250	8	⊂	⊂	PROPN
ejpam-6409	250	9	z2s	z2s	X
ejpam-6409	251	1	[	[	X
ejpam-6409	251	2	ω	ω	X
ejpam-6409	251	3	]	]	X
ejpam-6409	251	4	.	.	PUNCT
ejpam-6409	252	1	then	then	ADV
ejpam-6409	252	2	,	,	PUNCT
ejpam-6409	252	3	ϕs(u	ϕs(u	PUNCT
ejpam-6409	252	4	)	)	PUNCT
ejpam-6409	252	5	+	+	NUM
ejpam-6409	252	6	ϕs(v	ϕs(v	NOUN
ejpam-6409	252	7	)	)	PUNCT
ejpam-6409	252	8	=	=	PRON
ejpam-6409	252	9	{	{	PUNCT
ejpam-6409	252	10	ϕs(u+	ϕs(u+	PROPN
ejpam-6409	252	11	v	v	PROPN
ejpam-6409	252	12	+	+	NOUN
ejpam-6409	252	13	0	0	NUM
ejpam-6409	252	14	·	·	SYM
ejpam-6409	252	15	2s−1	2s−1	NUM
ejpam-6409	252	16	)	)	PUNCT
ejpam-6409	252	17	if	if	SCONJ
ejpam-6409	252	18	u	u	PROPN
ejpam-6409	252	19	∈	∈	PROPN
ejpam-6409	252	20	u	u	NOUN
ejpam-6409	252	21	,	,	PUNCT
ejpam-6409	252	22	ϕs(u+	ϕs(u+	PROPN
ejpam-6409	252	23	v	v	NOUN
ejpam-6409	252	24	)	)	PUNCT
ejpam-6409	252	25	if	if	SCONJ
ejpam-6409	252	26	u	u	PROPN
ejpam-6409	252	27	∈	∈	X
ejpam-6409	252	28	z2s	z2s	X
ejpam-6409	252	29	[	[	X
ejpam-6409	252	30	ω	ω	X
ejpam-6409	252	31	]	]	PUNCT
ejpam-6409	252	32	\	\	PROPN
ejpam-6409	252	33	u.	u.	PROPN
ejpam-6409	252	34	lemma	lemma	PROPN
ejpam-6409	252	35	3.3	3.3	NUM
ejpam-6409	252	36	:	:	PUNCT
ejpam-6409	252	37	let	let	VERB
ejpam-6409	252	38	u	u	PRON
ejpam-6409	252	39	∈	∈	PROPN
ejpam-6409	252	40	{	{	PUNCT
ejpam-6409	252	41	(	(	PUNCT
ejpam-6409	252	42	10)2s−2	10)2s−2	NUM
ejpam-6409	252	43	,	,	PUNCT
ejpam-6409	252	44	.	.	PUNCT
ejpam-6409	252	45	.	.	PUNCT
ejpam-6409	252	46	.	.	PUNCT
ejpam-6409	253	1	,	,	PUNCT
ejpam-6409	253	2	(	(	PUNCT
ejpam-6409	253	3	13)2s−2	13)2s−2	NUM
ejpam-6409	253	4	,	,	PUNCT
ejpam-6409	253	5	(	(	PUNCT
ejpam-6409	253	6	30)2s−1	30)2s−1	NUM
ejpam-6409	253	7	,	,	PUNCT
ejpam-6409	253	8	.	.	PUNCT
ejpam-6409	253	9	.	.	PUNCT
ejpam-6409	253	10	.	.	PUNCT
ejpam-6409	254	1	,	,	PUNCT
ejpam-6409	254	2	(	(	PUNCT
ejpam-6409	254	3	33)2s−1	33)2s−1	NUM
ejpam-6409	254	4	}	}	PUNCT
ejpam-6409	254	5	⊂	⊂	PROPN
ejpam-6409	254	6	z2s	z2s	X
ejpam-6409	255	1	[	[	X
ejpam-6409	255	2	ω	ω	X
ejpam-6409	255	3	]	]	X
ejpam-6409	255	4	.	.	PUNCT
ejpam-6409	256	1	then	then	ADV
ejpam-6409	256	2	,	,	PUNCT
ejpam-6409	256	3	ϕs(u	ϕs(u	PUNCT
ejpam-6409	256	4	)	)	PUNCT
ejpam-6409	257	1	+	+	CCONJ
ejpam-6409	257	2	ϕs(2	ϕs(2	PROPN
ejpam-6409	257	3	s−2	s−2	PROPN
ejpam-6409	257	4	·	·	PUNCT
ejpam-6409	257	5	0	0	X
ejpam-6409	257	6	)	)	PUNCT
ejpam-6409	258	1	=	=	SYM
ejpam-6409	258	2	ϕs(u+	ϕs(u+	NOUN
ejpam-6409	258	3	2s−2	2s−2	NUM
ejpam-6409	258	4	·	·	PUNCT
ejpam-6409	258	5	0	0	PUNCT
ejpam-6409	259	1	+	+	CCONJ
ejpam-6409	259	2	2s−1	2s−1	NUM
ejpam-6409	259	3	·	·	PUNCT
ejpam-6409	259	4	0	0	NUM
ejpam-6409	259	5	)	)	PUNCT
ejpam-6409	259	6	.	.	PUNCT
ejpam-6409	260	1	corollary	corollary	ADJ
ejpam-6409	260	2	3.8	3.8	NUM
ejpam-6409	260	3	:	:	PUNCT
ejpam-6409	260	4	let	let	VERB
ejpam-6409	260	5	v	v	X
ejpam-6409	260	6	∈	∈	PROPN
ejpam-6409	260	7	{	{	PUNCT
ejpam-6409	260	8	(	(	PUNCT
ejpam-6409	260	9	10)2s−2	10)2s−2	NUM
ejpam-6409	260	10	,	,	PUNCT
ejpam-6409	260	11	(	(	PUNCT
ejpam-6409	260	12	12)2s−2	12)2s−2	NOUN
ejpam-6409	260	13	,	,	PUNCT
ejpam-6409	260	14	(	(	PUNCT
ejpam-6409	260	15	30)2s−2	30)2s−2	NUM
ejpam-6409	260	16	,	,	PUNCT
ejpam-6409	260	17	(	(	PUNCT
ejpam-6409	260	18	32)2s−2	32)2s−2	NOUN
ejpam-6409	260	19	}	}	PUNCT
ejpam-6409	260	20	and	and	CCONJ
ejpam-6409	260	21	let	let	VERB
ejpam-6409	260	22	u	u	PRON
ejpam-6409	260	23	′	′	NOUN
ejpam-6409	260	24	=	=	SYM
ejpam-6409	260	25	{	{	PUNCT
ejpam-6409	260	26	(	(	PUNCT
ejpam-6409	260	27	10)2s−2	10)2s−2	NUM
ejpam-6409	260	28	,	,	PUNCT
ejpam-6409	260	29	.	.	PUNCT
ejpam-6409	260	30	.	.	PUNCT
ejpam-6409	261	1	.	.	PUNCT
ejpam-6409	262	1	,	,	PUNCT
ejpam-6409	262	2	(	(	PUNCT
ejpam-6409	262	3	13)2s−2	13)2s−2	NUM
ejpam-6409	262	4	,	,	PUNCT
ejpam-6409	262	5	(	(	PUNCT
ejpam-6409	262	6	30)2s−1	30)2s−1	NUM
ejpam-6409	262	7	,	,	PUNCT
ejpam-6409	262	8	.	.	PUNCT
ejpam-6409	262	9	.	.	PUNCT
ejpam-6409	262	10	.	.	PUNCT
ejpam-6409	263	1	,	,	PUNCT
ejpam-6409	263	2	(	(	PUNCT
ejpam-6409	263	3	33)2s−1	33)2s−1	NUM
ejpam-6409	263	4	}	}	PUNCT
ejpam-6409	263	5	⊂	⊂	PROPN
ejpam-6409	263	6	z2s	z2s	X
ejpam-6409	264	1	[	[	X
ejpam-6409	264	2	ω	ω	X
ejpam-6409	264	3	]	]	X
ejpam-6409	264	4	.	.	PUNCT
ejpam-6409	265	1	then	then	ADV
ejpam-6409	265	2	,	,	PUNCT
ejpam-6409	265	3	ϕs(u	ϕs(u	PUNCT
ejpam-6409	265	4	)	)	PUNCT
ejpam-6409	265	5	+	+	NUM
ejpam-6409	265	6	ϕs(v	ϕs(v	NOUN
ejpam-6409	265	7	)	)	PUNCT
ejpam-6409	265	8	=	=	PRON
ejpam-6409	265	9	{	{	PUNCT
ejpam-6409	265	10	ϕs(u+	ϕs(u+	PROPN
ejpam-6409	265	11	v	v	PROPN
ejpam-6409	265	12	+	+	CCONJ
ejpam-6409	265	13	2s−1	2s−1	NUM
ejpam-6409	265	14	·	·	PUNCT
ejpam-6409	265	15	0	0	X
ejpam-6409	265	16	)	)	PUNCT
ejpam-6409	265	17	if	if	SCONJ
ejpam-6409	265	18	u	u	PROPN
ejpam-6409	265	19	∈	∈	PROPN
ejpam-6409	265	20	u	u	NOUN
ejpam-6409	265	21	′	′	NOUN
ejpam-6409	265	22	,	,	PUNCT
ejpam-6409	265	23	ϕs(u+	ϕs(u+	PROPN
ejpam-6409	265	24	v	v	NOUN
ejpam-6409	265	25	)	)	PUNCT
ejpam-6409	265	26	if	if	SCONJ
ejpam-6409	265	27	u	u	PROPN
ejpam-6409	265	28	∈	∈	X
ejpam-6409	265	29	z2s	z2s	X
ejpam-6409	266	1	[	[	X
ejpam-6409	266	2	ω	ω	X
ejpam-6409	266	3	]	]	PUNCT
ejpam-6409	266	4	\	\	PROPN
ejpam-6409	266	5	u	u	PROPN
ejpam-6409	266	6	′.	′.	NOUN
ejpam-6409	266	7	corollary	corollary	NOUN
ejpam-6409	266	8	3.9	3.9	NUM
ejpam-6409	266	9	:	:	PUNCT
ejpam-6409	267	1	let	let	VERB
ejpam-6409	267	2	v	v	X
ejpam-6409	267	3	∈	∈	PROPN
ejpam-6409	267	4	{	{	PUNCT
ejpam-6409	267	5	(	(	PUNCT
ejpam-6409	267	6	11)2s−2	11)2s−2	NUM
ejpam-6409	267	7	,	,	PUNCT
ejpam-6409	267	8	(	(	PUNCT
ejpam-6409	267	9	13)2s−2	13)2s−2	NUM
ejpam-6409	267	10	,	,	PUNCT
ejpam-6409	267	11	(	(	PUNCT
ejpam-6409	267	12	31)2s−2	31)2s−2	NUM
ejpam-6409	267	13	,	,	PUNCT
ejpam-6409	267	14	(	(	PUNCT
ejpam-6409	267	15	33)2s−2	33)2s−2	ADJ
ejpam-6409	267	16	}	}	PUNCT
ejpam-6409	267	17	,	,	PUNCT
ejpam-6409	267	18	and	and	CCONJ
ejpam-6409	267	19	define	define	VERB
ejpam-6409	267	20	u1	u1	NOUN
ejpam-6409	267	21	=	=	SYM
ejpam-6409	267	22	{	{	PUNCT
ejpam-6409	267	23	(	(	PUNCT
ejpam-6409	267	24	01)2s−2	01)2s−2	PROPN
ejpam-6409	267	25	,	,	PUNCT
ejpam-6409	267	26	(	(	PUNCT
ejpam-6409	267	27	03)2s−2	03)2s−2	NOUN
ejpam-6409	267	28	,	,	PUNCT
ejpam-6409	267	29	(	(	PUNCT
ejpam-6409	267	30	21)2s−1	21)2s−1	NUM
ejpam-6409	267	31	,	,	PUNCT
ejpam-6409	267	32	(	(	PUNCT
ejpam-6409	267	33	23)2s−1	23)2s−1	NUM
ejpam-6409	267	34	}	}	PUNCT
ejpam-6409	267	35	,	,	PUNCT
ejpam-6409	267	36	u2	u2	NOUN
ejpam-6409	267	37	=	=	SYM
ejpam-6409	267	38	{	{	PUNCT
ejpam-6409	267	39	(	(	PUNCT
ejpam-6409	267	40	10)2s−2	10)2s−2	NUM
ejpam-6409	267	41	,	,	PUNCT
ejpam-6409	267	42	(	(	PUNCT
ejpam-6409	267	43	12)2s−2	12)2s−2	NOUN
ejpam-6409	267	44	,	,	PUNCT
ejpam-6409	267	45	(	(	PUNCT
ejpam-6409	267	46	30)2s−1	30)2s−1	NUM
ejpam-6409	267	47	,	,	PUNCT
ejpam-6409	267	48	(	(	PUNCT
ejpam-6409	267	49	32)2s−1	32)2s−1	NOUN
ejpam-6409	267	50	}	}	PUNCT
ejpam-6409	267	51	,	,	PUNCT
ejpam-6409	267	52	u3	u3	NOUN
ejpam-6409	267	53	=	=	SYM
ejpam-6409	267	54	{	{	PUNCT
ejpam-6409	267	55	(	(	PUNCT
ejpam-6409	267	56	11)2s−2	11)2s−2	NUM
ejpam-6409	267	57	,	,	PUNCT
ejpam-6409	267	58	(	(	PUNCT
ejpam-6409	267	59	13)2s−2	13)2s−2	NUM
ejpam-6409	267	60	,	,	PUNCT
ejpam-6409	267	61	(	(	PUNCT
ejpam-6409	267	62	31)2s−1	31)2s−1	NUM
ejpam-6409	267	63	,	,	PUNCT
ejpam-6409	267	64	(	(	PUNCT
ejpam-6409	267	65	33)2s−1	33)2s−1	NUM
ejpam-6409	267	66	}	}	PUNCT
ejpam-6409	267	67	.	.	PUNCT
ejpam-6409	268	1	muhammad	muhammad	PROPN
ejpam-6409	268	2	sajjad	sajjad	PROPN
ejpam-6409	268	3	et	et	PROPN
ejpam-6409	268	4	al	al	PROPN
ejpam-6409	268	5	.	.	PUNCT
ejpam-6409	268	6	/	/	SYM
ejpam-6409	268	7	eur	eur	PROPN
ejpam-6409	268	8	.	.	PUNCT
ejpam-6409	269	1	j.	j.	PROPN
ejpam-6409	269	2	pure	pure	PROPN
ejpam-6409	269	3	appl	appl	PROPN
ejpam-6409	269	4	.	.	PROPN
ejpam-6409	269	5	math	math	PROPN
ejpam-6409	269	6	,	,	PUNCT
ejpam-6409	269	7	18	18	NUM
ejpam-6409	269	8	(	(	PUNCT
ejpam-6409	269	9	3	3	NUM
ejpam-6409	269	10	)	)	PUNCT
ejpam-6409	269	11	(	(	PUNCT
ejpam-6409	269	12	2025	2025	NUM
ejpam-6409	269	13	)	)	PUNCT
ejpam-6409	269	14	,	,	PUNCT
ejpam-6409	269	15	6409	6409	NUM
ejpam-6409	269	16	9	9	NUM
ejpam-6409	269	17	of	of	ADP
ejpam-6409	269	18	32	32	NUM
ejpam-6409	269	19	then	then	ADV
ejpam-6409	269	20	,	,	PUNCT
ejpam-6409	269	21	ϕs(u	ϕs(u	PUNCT
ejpam-6409	269	22	)	)	PUNCT
ejpam-6409	270	1	+	+	NUM
ejpam-6409	270	2	ϕs(v	ϕs(v	NOUN
ejpam-6409	270	3	)	)	PUNCT
ejpam-6409	270	4	=	=	PUNCT
ejpam-6409	270	5			NOUN
ejpam-6409	270	6	ϕs(u+	ϕs(u+	NOUN
ejpam-6409	270	7	v	v	NOUN
ejpam-6409	270	8	+	+	CCONJ
ejpam-6409	270	9	02s−1	02s−1	NOUN
ejpam-6409	270	10	)	)	PUNCT
ejpam-6409	271	1	if	if	SCONJ
ejpam-6409	271	2	u	u	PROPN
ejpam-6409	271	3	∈	∈	PROPN
ejpam-6409	271	4	u1	u1	NOUN
ejpam-6409	271	5	,	,	PUNCT
ejpam-6409	271	6	ϕs(u+	ϕs(u+	PROPN
ejpam-6409	271	7	v	v	NOUN
ejpam-6409	271	8	+	+	CCONJ
ejpam-6409	271	9	2s−1	2s−1	NUM
ejpam-6409	271	10	·	·	PUNCT
ejpam-6409	271	11	0	0	X
ejpam-6409	271	12	)	)	PUNCT
ejpam-6409	271	13	if	if	SCONJ
ejpam-6409	271	14	u	u	PROPN
ejpam-6409	271	15	∈	∈	PROPN
ejpam-6409	271	16	u2	u2	PROPN
ejpam-6409	271	17	,	,	PUNCT
ejpam-6409	271	18	ϕs(u+	ϕs(u+	NOUN
ejpam-6409	271	19	v	v	NOUN
ejpam-6409	271	20	+	+	CCONJ
ejpam-6409	271	21	2s−1	2s−1	NUM
ejpam-6409	271	22	·	·	SYM
ejpam-6409	271	23	2s−1	2s−1	NUM
ejpam-6409	271	24	)	)	PUNCT
ejpam-6409	271	25	if	if	SCONJ
ejpam-6409	271	26	u	u	PROPN
ejpam-6409	271	27	∈	∈	PROPN
ejpam-6409	271	28	u3	u3	PROPN
ejpam-6409	271	29	,	,	PUNCT
ejpam-6409	271	30	ϕs(u+	ϕs(u+	PROPN
ejpam-6409	271	31	v	v	NOUN
ejpam-6409	271	32	)	)	PUNCT
ejpam-6409	271	33	if	if	SCONJ
ejpam-6409	271	34	u	u	PROPN
ejpam-6409	271	35	∈	∈	X
ejpam-6409	271	36	z2s	z2s	X
ejpam-6409	271	37	[	[	X
ejpam-6409	271	38	ω	ω	X
ejpam-6409	271	39	]	]	X
ejpam-6409	271	40	\	\	PUNCT
ejpam-6409	271	41	(	(	PUNCT
ejpam-6409	271	42	u1	u1	PROPN
ejpam-6409	271	43	∪	∪	VERB
ejpam-6409	271	44	u2	u2	PROPN
ejpam-6409	271	45	∪	∪	NOUN
ejpam-6409	271	46	u3	u3	NOUN
ejpam-6409	271	47	)	)	PUNCT
ejpam-6409	271	48	.	.	PUNCT
ejpam-6409	272	1	lemma	lemma	PROPN
ejpam-6409	272	2	3.4	3.4	NUM
ejpam-6409	272	3	:	:	PUNCT
ejpam-6409	272	4	let	let	VERB
ejpam-6409	272	5	µk	µk	PRON
ejpam-6409	272	6	∈	∈	PROPN
ejpam-6409	272	7	z2[ω	z2[ω	NOUN
ejpam-6409	272	8	]	]	X
ejpam-6409	272	9	,	,	PUNCT
ejpam-6409	272	10	k	k	PROPN
ejpam-6409	272	11	∈	∈	PROPN
ejpam-6409	272	12	{	{	PUNCT
ejpam-6409	272	13	0	0	NUM
ejpam-6409	272	14	,	,	PUNCT
ejpam-6409	272	15	.	.	PUNCT
ejpam-6409	272	16	.	.	PUNCT
ejpam-6409	273	1	.	.	PUNCT
ejpam-6409	274	1	,	,	PUNCT
ejpam-6409	274	2	s−	s−	PROPN
ejpam-6409	274	3	2	2	NUM
ejpam-6409	274	4	}	}	PUNCT
ejpam-6409	274	5	.	.	PUNCT
ejpam-6409	275	1	then	then	ADV
ejpam-6409	275	2	,	,	PUNCT
ejpam-6409	275	3	s−2∑	s−2∑	PROPN
ejpam-6409	275	4	k=0	k=0	PROPN
ejpam-6409	275	5	µkϕs(2	µkϕs(2	NOUN
ejpam-6409	275	6	k	k	NOUN
ejpam-6409	275	7	)	)	PUNCT
ejpam-6409	275	8	=	=	SYM
ejpam-6409	275	9	ϕs	ϕs	INTJ
ejpam-6409	275	10	(	(	PUNCT
ejpam-6409	275	11	s−2∑	s−2∑	PROPN
ejpam-6409	275	12	k=0	k=0	PROPN
ejpam-6409	275	13	µk2	µk2	NOUN
ejpam-6409	275	14	k	k	PROPN
ejpam-6409	275	15	)	)	PUNCT
ejpam-6409	275	16	,	,	PUNCT
ejpam-6409	275	17	where	where	SCONJ
ejpam-6409	275	18	2k	2k	PROPN
ejpam-6409	275	19	∈	∈	PROPN
ejpam-6409	275	20	z2s	z2s	X
ejpam-6409	276	1	[	[	X
ejpam-6409	276	2	ω	ω	X
ejpam-6409	276	3	]	]	X
ejpam-6409	276	4	.	.	PUNCT
ejpam-6409	277	1	proof	proof	NOUN
ejpam-6409	277	2	:	:	PUNCT
ejpam-6409	277	3	let	let	VERB
ejpam-6409	277	4	yk	yk	PROPN
ejpam-6409	277	5	be	be	AUX
ejpam-6409	277	6	the	the	DET
ejpam-6409	277	7	k	k	PROPN
ejpam-6409	277	8	-	-	PUNCT
ejpam-6409	277	9	th	th	VERB
ejpam-6409	277	10	row	row	NOUN
ejpam-6409	277	11	of	of	ADP
ejpam-6409	277	12	matrix	matrix	NOUN
ejpam-6409	277	13	y	y	PROPN
ejpam-6409	277	14	.	.	PUNCT
ejpam-6409	278	1	by	by	ADP
ejpam-6409	278	2	definition	definition	NOUN
ejpam-6409	278	3	,	,	PUNCT
ejpam-6409	278	4	we	we	PRON
ejpam-6409	278	5	have	have	VERB
ejpam-6409	278	6	s−2∑	s−2∑	NUM
ejpam-6409	278	7	k=0	k=0	PROPN
ejpam-6409	278	8	µkϕs(2	µkϕs(2	NOUN
ejpam-6409	278	9	k	k	NOUN
ejpam-6409	278	10	)	)	PUNCT
ejpam-6409	279	1	=	=	SYM
ejpam-6409	279	2	s−2∑	s−2∑	PROPN
ejpam-6409	279	3	k=0	k=0	X
ejpam-6409	279	4	µkek+1y	µkek+1y	VERB
ejpam-6409	279	5	=	=	SYM
ejpam-6409	279	6	s−2∑	s−2∑	NUM
ejpam-6409	279	7	k=0	k=0	PROPN
ejpam-6409	279	8	µkyk+1	µkyk+1	PROPN
ejpam-6409	279	9	=	=	SYM
ejpam-6409	279	10	µy	µy	PROPN
ejpam-6409	279	11	,	,	PUNCT
ejpam-6409	279	12	where	where	SCONJ
ejpam-6409	279	13	µ	µ	X
ejpam-6409	279	14	=	=	SYM
ejpam-6409	279	15	(	(	PUNCT
ejpam-6409	279	16	µ0	µ0	PROPN
ejpam-6409	279	17	,	,	PUNCT
ejpam-6409	279	18	.	.	PUNCT
ejpam-6409	279	19	.	.	PUNCT
ejpam-6409	280	1	.	.	PUNCT
ejpam-6409	281	1	,	,	PUNCT
ejpam-6409	281	2	µs−2	µs−2	PROPN
ejpam-6409	281	3	)	)	PUNCT
ejpam-6409	281	4	.	.	PUNCT
ejpam-6409	282	1	since	since	SCONJ
ejpam-6409	282	2	[	[	X
ejpam-6409	282	3	µ0	µ0	NOUN
ejpam-6409	282	4	,	,	PUNCT
ejpam-6409	282	5	.	.	PUNCT
ejpam-6409	282	6	.	.	PUNCT
ejpam-6409	282	7	.	.	PUNCT
ejpam-6409	283	1	,	,	PUNCT
ejpam-6409	283	2	µs−2	µs−2	PROPN
ejpam-6409	283	3	,	,	PUNCT
ejpam-6409	283	4	0	0	NUM
ejpam-6409	283	5	]	]	PUNCT
ejpam-6409	283	6	is	be	AUX
ejpam-6409	283	7	the	the	DET
ejpam-6409	283	8	binary	binary	ADJ
ejpam-6409	283	9	expansion	expansion	NOUN
ejpam-6409	283	10	of	of	ADP
ejpam-6409	283	11	∑s−2	∑s−2	ADJ
ejpam-6409	283	12	k=0	k=0	PROPN
ejpam-6409	283	13	µk2	µk2	NOUN
ejpam-6409	283	14	k	k	NOUN
ejpam-6409	283	15	,	,	PUNCT
ejpam-6409	283	16	we	we	PRON
ejpam-6409	283	17	conclude	conclude	VERB
ejpam-6409	283	18	that	that	PRON
ejpam-6409	283	19	µy	µy	ADV
ejpam-6409	283	20	=	=	PUNCT
ejpam-6409	284	1	ϕs	ϕs	PROPN
ejpam-6409	284	2	(	(	PUNCT
ejpam-6409	284	3	s−2∑	s−2∑	PROPN
ejpam-6409	284	4	k=0	k=0	PROPN
ejpam-6409	284	5	µk2	µk2	NOUN
ejpam-6409	284	6	k	k	PROPN
ejpam-6409	284	7	)	)	PUNCT
ejpam-6409	284	8	.	.	PUNCT
ejpam-6409	285	1	proposition	proposition	NOUN
ejpam-6409	285	2	3.3	3.3	NUM
ejpam-6409	285	3	.	.	PUNCT
ejpam-6409	286	1	let	let	VERB
ejpam-6409	286	2	u	u	NOUN
ejpam-6409	286	3	,	,	PUNCT
ejpam-6409	286	4	v	v	ADP
ejpam-6409	286	5	∈	∈	NOUN
ejpam-6409	286	6	z2s	z2s	X
ejpam-6409	287	1	[	[	X
ejpam-6409	287	2	ω	ω	X
ejpam-6409	287	3	]	]	X
ejpam-6409	287	4	.	.	PUNCT
ejpam-6409	288	1	then	then	ADV
ejpam-6409	288	2	,	,	PUNCT
ejpam-6409	288	3	ϕs(u	ϕs(u	PUNCT
ejpam-6409	288	4	)	)	PUNCT
ejpam-6409	288	5	+	+	NUM
ejpam-6409	288	6	ϕs(v	ϕs(v	NOUN
ejpam-6409	288	7	)	)	PUNCT
ejpam-6409	288	8	=	=	PRON
ejpam-6409	288	9	ϕs(u−	ϕs(u−	PROPN
ejpam-6409	288	10	v	v	NOUN
ejpam-6409	288	11	)	)	PUNCT
ejpam-6409	288	12	=	=	PUNCT
ejpam-6409	288	13	(	(	PUNCT
ejpam-6409	288	14	µ	µ	NOUN
ejpam-6409	288	15	,	,	PUNCT
ejpam-6409	288	16	.	.	PUNCT
ejpam-6409	288	17	.	.	PUNCT
ejpam-6409	289	1	.	.	PUNCT
ejpam-6409	290	1	,	,	PUNCT
ejpam-6409	290	2	µ	µ	X
ejpam-6409	290	3	)	)	PUNCT
ejpam-6409	290	4	if	if	SCONJ
ejpam-6409	290	5	u−	u−	PROPN
ejpam-6409	290	6	v	v	NOUN
ejpam-6409	290	7	=	=	SYM
ejpam-6409	290	8	µ2s−1	µ2s−1	NOUN
ejpam-6409	290	9	∈	∈	NOUN
ejpam-6409	290	10	2s−1z2s	2s−1z2s	NUM
ejpam-6409	291	1	[	[	X
ejpam-6409	291	2	ω	ω	X
ejpam-6409	291	3	]	]	X
ejpam-6409	291	4	\	\	PUNCT
ejpam-6409	291	5	{	{	PUNCT
ejpam-6409	291	6	00	00	NUM
ejpam-6409	291	7	}	}	PUNCT
ejpam-6409	291	8	,	,	PUNCT
ejpam-6409	291	9	and	and	CCONJ
ejpam-6409	291	10	ϕs(u)−ϕs(v	ϕs(u)−ϕs(v	PROPN
ejpam-6409	291	11	)	)	PUNCT
ejpam-6409	291	12	contains	contain	VERB
ejpam-6409	291	13	each	each	DET
ejpam-6409	291	14	element	element	NOUN
ejpam-6409	291	15	of	of	ADP
ejpam-6409	291	16	z2[ω	z2[ω	NOUN
ejpam-6409	291	17	]	]	PUNCT
ejpam-6409	291	18	exactly	exactly	ADV
ejpam-6409	291	19	22(s−2	22(s−2	NUM
ejpam-6409	291	20	)	)	PUNCT
ejpam-6409	291	21	times	time	NOUN
ejpam-6409	291	22	if	if	SCONJ
ejpam-6409	291	23	u−v	u−v	ADP
ejpam-6409	291	24	∈	∈	PROPN
ejpam-6409	291	25	z2s	z2s	X
ejpam-6409	292	1	[	[	X
ejpam-6409	292	2	ω]\2s−1z2s	ω]\2s−1z2s	PROPN
ejpam-6409	292	3	[	[	X
ejpam-6409	292	4	ω	ω	X
ejpam-6409	292	5	]	]	X
ejpam-6409	292	6	.	.	PUNCT
ejpam-6409	293	1	proof	proof	NOUN
ejpam-6409	293	2	:	:	PUNCT
ejpam-6409	293	3	if	if	SCONJ
ejpam-6409	293	4	u−v	u−v	ADP
ejpam-6409	293	5	=	=	SYM
ejpam-6409	293	6	λ2s−1	λ2s−1	PROPN
ejpam-6409	293	7	∈	∈	NOUN
ejpam-6409	293	8	2s−1z2s	2s−1z2s	NUM
ejpam-6409	294	1	[	[	X
ejpam-6409	294	2	ω]\{0	ω]\{0	ADV
ejpam-6409	294	3	}	}	PUNCT
ejpam-6409	294	4	,	,	PUNCT
ejpam-6409	294	5	then	then	ADV
ejpam-6409	294	6	by	by	ADP
ejpam-6409	294	7	lemma	lemma	PROPN
ejpam-6409	294	8	3.1	3.1	NUM
ejpam-6409	294	9	,	,	PUNCT
ejpam-6409	294	10	ϕs(u	ϕs(u	PUNCT
ejpam-6409	294	11	)	)	PUNCT
ejpam-6409	294	12	=	=	SYM
ejpam-6409	294	13	ϕs(v)+(λ	ϕs(v)+(λ	PROPN
ejpam-6409	294	14	,	,	PUNCT
ejpam-6409	294	15	.	.	PUNCT
ejpam-6409	294	16	.	.	PUNCT
ejpam-6409	294	17	.	.	PUNCT
ejpam-6409	294	18	,	,	PUNCT
ejpam-6409	294	19	λ	λ	X
ejpam-6409	294	20	)	)	PUNCT
ejpam-6409	294	21	,	,	PUNCT
ejpam-6409	294	22	so	so	SCONJ
ejpam-6409	294	23	ϕs(u)−	ϕs(u)−	PROPN
ejpam-6409	294	24	ϕs(v	ϕs(v	NOUN
ejpam-6409	294	25	)	)	PUNCT
ejpam-6409	294	26	=	=	PRON
ejpam-6409	294	27	(	(	PUNCT
ejpam-6409	294	28	λ	λ	PROPN
ejpam-6409	294	29	,	,	PUNCT
ejpam-6409	294	30	.	.	PUNCT
ejpam-6409	294	31	.	.	PUNCT
ejpam-6409	294	32	.	.	PUNCT
ejpam-6409	295	1	,	,	PUNCT
ejpam-6409	295	2	λ	λ	X
ejpam-6409	295	3	)	)	PUNCT
ejpam-6409	295	4	=	=	SYM
ejpam-6409	295	5	ϕs(λ2	ϕs(λ2	X
ejpam-6409	295	6	s−1	s−1	PROPN
ejpam-6409	295	7	)	)	PUNCT
ejpam-6409	295	8	=	=	PRON
ejpam-6409	295	9	ϕs(u−	ϕs(u−	PROPN
ejpam-6409	295	10	v	v	NOUN
ejpam-6409	295	11	)	)	PUNCT
ejpam-6409	295	12	.	.	PUNCT
ejpam-6409	296	1	now	now	ADV
ejpam-6409	296	2	assume	assume	VERB
ejpam-6409	296	3	that	that	SCONJ
ejpam-6409	296	4	u	u	PRON
ejpam-6409	296	5	−	−	PROPN
ejpam-6409	296	6	v	v	ADP
ejpam-6409	296	7	∈	∈	PROPN
ejpam-6409	296	8	z2s	z2s	X
ejpam-6409	297	1	[	[	X
ejpam-6409	297	2	ω	ω	X
ejpam-6409	297	3	]	]	PUNCT
ejpam-6409	297	4	\	\	X
ejpam-6409	297	5	2s−1z2s	2s−1z2s	NUM
ejpam-6409	298	1	[	[	X
ejpam-6409	298	2	ω	ω	X
ejpam-6409	298	3	]	]	X
ejpam-6409	298	4	.	.	PUNCT
ejpam-6409	299	1	without	without	ADP
ejpam-6409	299	2	loss	loss	NOUN
ejpam-6409	299	3	of	of	ADP
ejpam-6409	299	4	generality	generality	NOUN
ejpam-6409	299	5	,	,	PUNCT
ejpam-6409	299	6	either	either	CCONJ
ejpam-6409	299	7	u	u	PROPN
ejpam-6409	299	8	∈	∈	PROPN
ejpam-6409	299	9	2s−1z2s	2s−1z2s	NUM
ejpam-6409	300	1	[	[	X
ejpam-6409	300	2	ω	ω	X
ejpam-6409	300	3	]	]	X
ejpam-6409	300	4	,	,	PUNCT
ejpam-6409	300	5	v	v	PROPN
ejpam-6409	300	6	∈	∈	NOUN
ejpam-6409	300	7	z2s	z2s	X
ejpam-6409	301	1	[	[	X
ejpam-6409	301	2	ω	ω	X
ejpam-6409	301	3	]	]	PUNCT
ejpam-6409	301	4	\	\	X
ejpam-6409	301	5	2s−1z2s	2s−1z2s	NUM
ejpam-6409	302	1	[	[	X
ejpam-6409	302	2	ω	ω	X
ejpam-6409	302	3	]	]	X
ejpam-6409	302	4	or	or	CCONJ
ejpam-6409	302	5	u	u	NOUN
ejpam-6409	302	6	,	,	PUNCT
ejpam-6409	302	7	v	v	ADP
ejpam-6409	302	8	∈	∈	NOUN
ejpam-6409	302	9	z2s	z2s	X
ejpam-6409	302	10	[	[	X
ejpam-6409	302	11	ω	ω	X
ejpam-6409	302	12	]	]	PUNCT
ejpam-6409	302	13	\	\	X
ejpam-6409	302	14	2s−1z2s	2s−1z2s	NUM
ejpam-6409	303	1	[	[	X
ejpam-6409	303	2	ω	ω	X
ejpam-6409	303	3	]	]	X
ejpam-6409	303	4	.	.	PUNCT
ejpam-6409	304	1	for	for	ADP
ejpam-6409	304	2	the	the	DET
ejpam-6409	304	3	first	first	ADJ
ejpam-6409	304	4	case	case	NOUN
ejpam-6409	304	5	,	,	PUNCT
ejpam-6409	304	6	ϕs(u	ϕs(u	PUNCT
ejpam-6409	304	7	)	)	PUNCT
ejpam-6409	304	8	=	=	SYM
ejpam-6409	304	9	(	(	PUNCT
ejpam-6409	304	10	λ1	λ1	ADJ
ejpam-6409	304	11	,	,	PUNCT
ejpam-6409	304	12	.	.	PUNCT
ejpam-6409	304	13	.	.	PUNCT
ejpam-6409	305	1	.	.	PUNCT
ejpam-6409	306	1	,	,	PUNCT
ejpam-6409	306	2	λ1	λ1	ADJ
ejpam-6409	306	3	)	)	PUNCT
ejpam-6409	306	4	and	and	CCONJ
ejpam-6409	306	5	ϕs(v	ϕs(v	NUM
ejpam-6409	306	6	)	)	PUNCT
ejpam-6409	307	1	=	=	SYM
ejpam-6409	307	2	ϕs(v1	ϕs(v1	NOUN
ejpam-6409	307	3	)	)	PUNCT
ejpam-6409	307	4	+	+	CCONJ
ejpam-6409	307	5	(	(	PUNCT
ejpam-6409	307	6	λ2	λ2	NOUN
ejpam-6409	307	7	,	,	PUNCT
ejpam-6409	307	8	.	.	PUNCT
ejpam-6409	307	9	.	.	PUNCT
ejpam-6409	307	10	.	.	PUNCT
ejpam-6409	308	1	,	,	PUNCT
ejpam-6409	308	2	λ2	λ2	NOUN
ejpam-6409	308	3	)	)	PUNCT
ejpam-6409	308	4	,	,	PUNCT
ejpam-6409	308	5	where	where	SCONJ
ejpam-6409	308	6	v1	v1	PROPN
ejpam-6409	308	7	∈	∈	PROPN
ejpam-6409	308	8	{	{	PUNCT
ejpam-6409	308	9	01	01	NUM
ejpam-6409	308	10	,	,	PUNCT
ejpam-6409	308	11	.	.	PUNCT
ejpam-6409	308	12	.	.	PUNCT
ejpam-6409	309	1	.	.	PUNCT
ejpam-6409	310	1	,	,	PUNCT
ejpam-6409	310	2	2s−1	2s−1	NUM
ejpam-6409	310	3	−	−	NOUN
ejpam-6409	310	4	1	1	NUM
ejpam-6409	310	5	,	,	PUNCT
ejpam-6409	310	6	.	.	PUNCT
ejpam-6409	310	7	.	.	PUNCT
ejpam-6409	310	8	.	.	PUNCT
ejpam-6409	311	1	,	,	PUNCT
ejpam-6409	311	2	(	(	PUNCT
ejpam-6409	311	3	2s−1	2s−1	NUM
ejpam-6409	311	4	−	−	NUM
ejpam-6409	311	5	1)(2s−1	1)(2s−1	NUM
ejpam-6409	311	6	−	−	NOUN
ejpam-6409	311	7	1	1	NUM
ejpam-6409	311	8	)	)	PUNCT
ejpam-6409	311	9	}	}	PUNCT
ejpam-6409	311	10	,	,	PUNCT
ejpam-6409	311	11	λ1	λ1	ADJ
ejpam-6409	311	12	,	,	PUNCT
ejpam-6409	311	13	λ2	λ2	PROPN
ejpam-6409	311	14	∈	∈	PROPN
ejpam-6409	311	15	z2[ω	z2[ω	NOUN
ejpam-6409	311	16	]	]	PUNCT
ejpam-6409	311	17	.	.	PUNCT
ejpam-6409	312	1	note	note	VERB
ejpam-6409	312	2	that	that	SCONJ
ejpam-6409	312	3	ϕs(v1	ϕs(v1	NOUN
ejpam-6409	312	4	)	)	PUNCT
ejpam-6409	312	5	is	be	AUX
ejpam-6409	312	6	a	a	DET
ejpam-6409	312	7	nonzero	nonzero	NOUN
ejpam-6409	312	8	row	row	NOUN
ejpam-6409	312	9	of	of	ADP
ejpam-6409	312	10	the	the	DET
ejpam-6409	312	11	gh	gh	PROPN
ejpam-6409	312	12	matrix	matrix	NOUN
ejpam-6409	312	13	h(22	h(22	PROPN
ejpam-6409	312	14	,	,	PUNCT
ejpam-6409	312	15	22(s−2	22(s−2	NUM
ejpam-6409	312	16	)	)	PUNCT
ejpam-6409	312	17	)	)	PUNCT
ejpam-6409	312	18	corresponding	correspond	VERB
ejpam-6409	312	19	to	to	ADP
ejpam-6409	312	20	the	the	DET
ejpam-6409	312	21	gh	gh	PROPN
ejpam-6409	312	22	code	code	NOUN
ejpam-6409	312	23	ϕs(z2s	ϕs(z2s	PROPN
ejpam-6409	313	1	[	[	X
ejpam-6409	313	2	ω	ω	NOUN
ejpam-6409	313	3	]	]	X
ejpam-6409	313	4	)	)	PUNCT
ejpam-6409	313	5	.	.	PUNCT
ejpam-6409	314	1	therefore	therefore	ADV
ejpam-6409	314	2	,	,	PUNCT
ejpam-6409	314	3	ϕs(v1	ϕs(v1	NOUN
ejpam-6409	314	4	)	)	PUNCT
ejpam-6409	314	5	contains	contain	VERB
ejpam-6409	314	6	each	each	DET
ejpam-6409	314	7	element	element	NOUN
ejpam-6409	314	8	of	of	ADP
ejpam-6409	314	9	z2[ω	z2[ω	NOUN
ejpam-6409	314	10	]	]	PUNCT
ejpam-6409	314	11	exactly	exactly	ADV
ejpam-6409	314	12	22(s−2	22(s−2	NUM
ejpam-6409	314	13	)	)	PUNCT
ejpam-6409	314	14	times	time	NOUN
ejpam-6409	314	15	and	and	CCONJ
ejpam-6409	314	16	hence	hence	ADV
ejpam-6409	314	17	ϕs(u)−ϕs(v	ϕs(u)−ϕs(v	PROPN
ejpam-6409	314	18	)	)	PUNCT
ejpam-6409	314	19	contains	contain	VERB
ejpam-6409	314	20	each	each	DET
ejpam-6409	314	21	element	element	NOUN
ejpam-6409	314	22	of	of	ADP
ejpam-6409	314	23	z2[ω	z2[ω	NOUN
ejpam-6409	314	24	]	]	PUNCT
ejpam-6409	314	25	exactly	exactly	ADV
ejpam-6409	314	26	22(s−2	22(s−2	NUM
ejpam-6409	314	27	)	)	PUNCT
ejpam-6409	314	28	times	time	NOUN
ejpam-6409	314	29	.	.	PUNCT
ejpam-6409	315	1	for	for	ADP
ejpam-6409	315	2	the	the	DET
ejpam-6409	315	3	second	second	ADJ
ejpam-6409	315	4	case	case	NOUN
ejpam-6409	315	5	,	,	PUNCT
ejpam-6409	315	6	ϕs(u	ϕs(u	PUNCT
ejpam-6409	315	7	)	)	PUNCT
ejpam-6409	315	8	=	=	SYM
ejpam-6409	315	9	ϕs(u1	ϕs(u1	X
ejpam-6409	315	10	)	)	PUNCT
ejpam-6409	315	11	+	+	CCONJ
ejpam-6409	315	12	(	(	PUNCT
ejpam-6409	315	13	λ1	λ1	ADJ
ejpam-6409	315	14	,	,	PUNCT
ejpam-6409	315	15	.	.	PUNCT
ejpam-6409	315	16	.	.	PUNCT
ejpam-6409	315	17	.	.	PUNCT
ejpam-6409	316	1	,	,	PUNCT
ejpam-6409	316	2	λ1	λ1	ADJ
ejpam-6409	316	3	)	)	PUNCT
ejpam-6409	316	4	and	and	CCONJ
ejpam-6409	316	5	ϕs(v	ϕs(v	NUM
ejpam-6409	316	6	)	)	PUNCT
ejpam-6409	317	1	=	=	SYM
ejpam-6409	317	2	ϕs(v1	ϕs(v1	NOUN
ejpam-6409	317	3	)	)	PUNCT
ejpam-6409	317	4	+	+	CCONJ
ejpam-6409	317	5	(	(	PUNCT
ejpam-6409	317	6	λ2	λ2	NOUN
ejpam-6409	317	7	,	,	PUNCT
ejpam-6409	317	8	.	.	PUNCT
ejpam-6409	317	9	.	.	PUNCT
ejpam-6409	317	10	.	.	PUNCT
ejpam-6409	318	1	,	,	PUNCT
ejpam-6409	318	2	λ2	λ2	NOUN
ejpam-6409	318	3	)	)	PUNCT
ejpam-6409	318	4	,	,	PUNCT
ejpam-6409	318	5	where	where	SCONJ
ejpam-6409	318	6	u1	u1	NOUN
ejpam-6409	318	7	,	,	PUNCT
ejpam-6409	318	8	v1	v1	PROPN
ejpam-6409	318	9	∈	∈	PROPN
ejpam-6409	318	10	{	{	PUNCT
ejpam-6409	318	11	01	01	NUM
ejpam-6409	318	12	,	,	PUNCT
ejpam-6409	318	13	.	.	PUNCT
ejpam-6409	318	14	.	.	PUNCT
ejpam-6409	319	1	.	.	PUNCT
ejpam-6409	320	1	,	,	PUNCT
ejpam-6409	320	2	2s−1	2s−1	NUM
ejpam-6409	320	3	−	−	NOUN
ejpam-6409	320	4	1	1	NUM
ejpam-6409	320	5	,	,	PUNCT
ejpam-6409	320	6	.	.	PUNCT
ejpam-6409	320	7	.	.	PUNCT
ejpam-6409	320	8	.	.	PUNCT
ejpam-6409	321	1	,	,	PUNCT
ejpam-6409	321	2	(	(	PUNCT
ejpam-6409	321	3	2s−1	2s−1	NUM
ejpam-6409	321	4	−	−	NUM
ejpam-6409	321	5	1)(2s−1	1)(2s−1	NUM
ejpam-6409	321	6	−	−	NOUN
ejpam-6409	321	7	1	1	NUM
ejpam-6409	321	8	)	)	PUNCT
ejpam-6409	321	9	}	}	PUNCT
ejpam-6409	321	10	and	and	CCONJ
ejpam-6409	321	11	λ1	λ1	ADJ
ejpam-6409	321	12	,	,	PUNCT
ejpam-6409	321	13	λ2	λ2	PROPN
ejpam-6409	321	14	∈	∈	PROPN
ejpam-6409	321	15	z2[ω	z2[ω	NOUN
ejpam-6409	321	16	]	]	PUNCT
ejpam-6409	321	17	.	.	PUNCT
ejpam-6409	322	1	note	note	VERB
ejpam-6409	322	2	that	that	SCONJ
ejpam-6409	322	3	both	both	DET
ejpam-6409	322	4	ϕs(u1	ϕs(u1	NOUN
ejpam-6409	322	5	)	)	PUNCT
ejpam-6409	322	6	and	and	CCONJ
ejpam-6409	322	7	ϕs(v1	ϕs(v1	NOUN
ejpam-6409	322	8	)	)	PUNCT
ejpam-6409	322	9	are	be	AUX
ejpam-6409	322	10	nonzero	nonzero	NOUN
ejpam-6409	322	11	rows	row	NOUN
ejpam-6409	322	12	of	of	ADP
ejpam-6409	322	13	h(22	h(22	PROPN
ejpam-6409	322	14	,	,	PUNCT
ejpam-6409	322	15	22(s−2	22(s−2	NUM
ejpam-6409	322	16	)	)	PUNCT
ejpam-6409	322	17	)	)	PUNCT
ejpam-6409	322	18	,	,	PUNCT
ejpam-6409	322	19	so	so	ADV
ejpam-6409	322	20	they	they	PRON
ejpam-6409	322	21	contain	contain	VERB
ejpam-6409	322	22	each	each	DET
ejpam-6409	322	23	element	element	NOUN
ejpam-6409	322	24	of	of	ADP
ejpam-6409	322	25	muhammad	muhammad	PROPN
ejpam-6409	322	26	sajjad	sajjad	PROPN
ejpam-6409	322	27	et	et	PROPN
ejpam-6409	322	28	al	al	PROPN
ejpam-6409	322	29	.	.	PUNCT
ejpam-6409	322	30	/	/	SYM
ejpam-6409	322	31	eur	eur	PROPN
ejpam-6409	322	32	.	.	PUNCT
ejpam-6409	323	1	j.	j.	PROPN
ejpam-6409	323	2	pure	pure	PROPN
ejpam-6409	323	3	appl	appl	PROPN
ejpam-6409	323	4	.	.	PROPN
ejpam-6409	323	5	math	math	PROPN
ejpam-6409	323	6	,	,	PUNCT
ejpam-6409	323	7	18	18	NUM
ejpam-6409	323	8	(	(	PUNCT
ejpam-6409	323	9	3	3	NUM
ejpam-6409	323	10	)	)	PUNCT
ejpam-6409	323	11	(	(	PUNCT
ejpam-6409	323	12	2025	2025	NUM
ejpam-6409	323	13	)	)	PUNCT
ejpam-6409	323	14	,	,	PUNCT
ejpam-6409	323	15	6409	6409	NUM
ejpam-6409	323	16	10	10	NUM
ejpam-6409	323	17	of	of	ADP
ejpam-6409	323	18	32	32	NUM
ejpam-6409	323	19	z2[ω	z2[ω	NOUN
ejpam-6409	323	20	]	]	PUNCT
ejpam-6409	323	21	exactly	exactly	ADV
ejpam-6409	323	22	22(s−2	22(s−2	NUM
ejpam-6409	323	23	)	)	PUNCT
ejpam-6409	323	24	times	time	NOUN
ejpam-6409	323	25	,	,	PUNCT
ejpam-6409	323	26	and	and	CCONJ
ejpam-6409	323	27	hence	hence	ADV
ejpam-6409	323	28	ϕs(u)−ϕs(v	ϕs(u)−ϕs(v	PROPN
ejpam-6409	323	29	)	)	PUNCT
ejpam-6409	323	30	contains	contain	VERB
ejpam-6409	323	31	each	each	DET
ejpam-6409	323	32	element	element	NOUN
ejpam-6409	323	33	of	of	ADP
ejpam-6409	323	34	z2[ω	z2[ω	NOUN
ejpam-6409	323	35	]	]	PUNCT
ejpam-6409	323	36	exactly	exactly	ADV
ejpam-6409	323	37	22(s−2	22(s−2	NUM
ejpam-6409	323	38	)	)	PUNCT
ejpam-6409	323	39	times	time	NOUN
ejpam-6409	323	40	.	.	PUNCT
ejpam-6409	324	1	proposition	proposition	NOUN
ejpam-6409	324	2	3.4	3.4	NUM
ejpam-6409	324	3	.	.	PUNCT
ejpam-6409	325	1	let	let	VERB
ejpam-6409	325	2	u	u	NOUN
ejpam-6409	325	3	,	,	PUNCT
ejpam-6409	325	4	v	v	ADP
ejpam-6409	325	5	∈	∈	NOUN
ejpam-6409	325	6	z2s	z2s	X
ejpam-6409	326	1	[	[	X
ejpam-6409	326	2	ω	ω	X
ejpam-6409	326	3	]	]	X
ejpam-6409	326	4	.	.	PUNCT
ejpam-6409	327	1	then	then	ADV
ejpam-6409	327	2	,	,	PUNCT
ejpam-6409	327	3	dh(ϕs(u	dh(ϕs(u	NOUN
ejpam-6409	327	4	)	)	PUNCT
ejpam-6409	327	5	,	,	PUNCT
ejpam-6409	327	6	ϕs(v	ϕs(v	NUM
ejpam-6409	327	7	)	)	PUNCT
ejpam-6409	327	8	)	)	PUNCT
ejpam-6409	328	1	=	=	PUNCT
ejpam-6409	328	2	wth(ϕs(u−	wth(ϕs(u−	PRON
ejpam-6409	328	3	v	v	NOUN
ejpam-6409	328	4	)	)	PUNCT
ejpam-6409	328	5	)	)	PUNCT
ejpam-6409	328	6	.	.	PUNCT
ejpam-6409	329	1	proof	proof	NOUN
ejpam-6409	329	2	:	:	PUNCT
ejpam-6409	329	3	if	if	SCONJ
ejpam-6409	329	4	u	u	PROPN
ejpam-6409	329	5	=	=	NOUN
ejpam-6409	329	6	0	0	NUM
ejpam-6409	329	7	or	or	CCONJ
ejpam-6409	329	8	v	v	NOUN
ejpam-6409	329	9	=	=	SYM
ejpam-6409	329	10	0	0	NUM
ejpam-6409	329	11	,	,	PUNCT
ejpam-6409	329	12	the	the	DET
ejpam-6409	329	13	result	result	NOUN
ejpam-6409	329	14	is	be	AUX
ejpam-6409	329	15	trivially	trivially	ADV
ejpam-6409	329	16	true	true	ADJ
ejpam-6409	329	17	.	.	PUNCT
ejpam-6409	330	1	assume	assume	VERB
ejpam-6409	330	2	u	u	PRON
ejpam-6409	330	3	̸=	̸=	PROPN
ejpam-6409	330	4	0	0	NUM
ejpam-6409	330	5	and	and	CCONJ
ejpam-6409	330	6	v	v	ADP
ejpam-6409	330	7	̸=	̸=	PROPN
ejpam-6409	330	8	0	0	NUM
ejpam-6409	330	9	,	,	PUNCT
ejpam-6409	330	10	and	and	CCONJ
ejpam-6409	330	11	consider	consider	VERB
ejpam-6409	330	12	three	three	NUM
ejpam-6409	330	13	cases	case	NOUN
ejpam-6409	330	14	.	.	PUNCT
ejpam-6409	331	1	first	first	ADV
ejpam-6409	331	2	,	,	PUNCT
ejpam-6409	331	3	if	if	SCONJ
ejpam-6409	331	4	u	u	PROPN
ejpam-6409	331	5	=	=	PROPN
ejpam-6409	331	6	v	v	PROPN
ejpam-6409	331	7	,	,	PUNCT
ejpam-6409	331	8	the	the	DET
ejpam-6409	331	9	result	result	NOUN
ejpam-6409	331	10	is	be	AUX
ejpam-6409	331	11	trivially	trivially	ADV
ejpam-6409	331	12	true	true	ADJ
ejpam-6409	331	13	.	.	PUNCT
ejpam-6409	332	1	second	second	ADJ
ejpam-6409	332	2	,	,	PUNCT
ejpam-6409	332	3	if	if	SCONJ
ejpam-6409	332	4	u−	u−	PROPN
ejpam-6409	332	5	v	v	ADP
ejpam-6409	332	6	∈	∈	NOUN
ejpam-6409	332	7	2s−1z2s	2s−1z2s	NUM
ejpam-6409	333	1	[	[	X
ejpam-6409	333	2	ω]\{0	ω]\{0	ADV
ejpam-6409	333	3	}	}	PUNCT
ejpam-6409	333	4	,	,	PUNCT
ejpam-6409	333	5	then	then	ADV
ejpam-6409	333	6	by	by	ADP
ejpam-6409	333	7	proposition	proposition	NOUN
ejpam-6409	333	8	3.3	3.3	NUM
ejpam-6409	333	9	,	,	PUNCT
ejpam-6409	333	10	ϕs(u)−ϕs(v	ϕs(u)−ϕs(v	PROPN
ejpam-6409	333	11	)	)	PUNCT
ejpam-6409	334	1	=	=	PRON
ejpam-6409	334	2	ϕs(u−	ϕs(u−	PROPN
ejpam-6409	334	3	v	v	NOUN
ejpam-6409	334	4	)	)	PUNCT
ejpam-6409	334	5	,	,	PUNCT
ejpam-6409	334	6	and	and	CCONJ
ejpam-6409	334	7	hence	hence	ADV
ejpam-6409	334	8	dh(ϕs(u	dh(ϕs(u	NUM
ejpam-6409	334	9	)	)	PUNCT
ejpam-6409	334	10	,	,	PUNCT
ejpam-6409	334	11	ϕs(v	ϕs(v	NUM
ejpam-6409	334	12	)	)	PUNCT
ejpam-6409	334	13	)	)	PUNCT
ejpam-6409	335	1	=	=	PUNCT
ejpam-6409	335	2	wth(ϕs(u−	wth(ϕs(u−	PRON
ejpam-6409	335	3	v	v	NOUN
ejpam-6409	335	4	)	)	PUNCT
ejpam-6409	335	5	)	)	PUNCT
ejpam-6409	335	6	.	.	PUNCT
ejpam-6409	336	1	finally	finally	ADV
ejpam-6409	336	2	,	,	PUNCT
ejpam-6409	336	3	assume	assume	VERB
ejpam-6409	336	4	that	that	SCONJ
ejpam-6409	336	5	u	u	NOUN
ejpam-6409	336	6	,	,	PUNCT
ejpam-6409	336	7	v	v	ADP
ejpam-6409	336	8	∈	∈	NOUN
ejpam-6409	336	9	z2s	z2s	X
ejpam-6409	337	1	[	[	X
ejpam-6409	337	2	ω	ω	X
ejpam-6409	337	3	]	]	PUNCT
ejpam-6409	337	4	\	\	X
ejpam-6409	337	5	2s−1z2s	2s−1z2s	NUM
ejpam-6409	338	1	[	[	X
ejpam-6409	338	2	ω	ω	X
ejpam-6409	338	3	]	]	X
ejpam-6409	338	4	.	.	PUNCT
ejpam-6409	339	1	by	by	ADP
ejpam-6409	339	2	proposition	proposition	NOUN
ejpam-6409	339	3	3.3	3.3	NUM
ejpam-6409	339	4	,	,	PUNCT
ejpam-6409	339	5	ϕs(u	ϕs(u	PUNCT
ejpam-6409	339	6	)	)	PUNCT
ejpam-6409	339	7	−	−	NOUN
ejpam-6409	339	8	ϕs(v	ϕs(v	NUM
ejpam-6409	339	9	)	)	PUNCT
ejpam-6409	339	10	contains	contain	VERB
ejpam-6409	339	11	each	each	DET
ejpam-6409	339	12	element	element	NOUN
ejpam-6409	339	13	of	of	ADP
ejpam-6409	339	14	z2[ω	z2[ω	NOUN
ejpam-6409	339	15	]	]	PUNCT
ejpam-6409	339	16	exactly	exactly	ADV
ejpam-6409	339	17	22(s−2	22(s−2	NUM
ejpam-6409	339	18	)	)	PUNCT
ejpam-6409	339	19	times	time	NOUN
ejpam-6409	339	20	,	,	PUNCT
ejpam-6409	339	21	and	and	CCONJ
ejpam-6409	339	22	hence	hence	ADV
ejpam-6409	339	23	dh(ϕs(u	dh(ϕs(u	NUM
ejpam-6409	339	24	)	)	PUNCT
ejpam-6409	339	25	,	,	PUNCT
ejpam-6409	339	26	ϕs(v	ϕs(v	NUM
ejpam-6409	339	27	)	)	PUNCT
ejpam-6409	339	28	)	)	PUNCT
ejpam-6409	340	1	=	=	SYM
ejpam-6409	340	2	3	3	NUM
ejpam-6409	340	3	·	·	SYM
ejpam-6409	340	4	22(s−2	22(s−2	NUM
ejpam-6409	340	5	)	)	PUNCT
ejpam-6409	340	6	=	=	PUNCT
ejpam-6409	340	7	wth(ϕs(u−	wth(ϕs(u−	VERB
ejpam-6409	340	8	v	v	NOUN
ejpam-6409	340	9	)	)	PUNCT
ejpam-6409	340	10	)	)	PUNCT
ejpam-6409	340	11	.	.	PUNCT
ejpam-6409	341	1	4	4	X
ejpam-6409	341	2	.	.	X
ejpam-6409	341	3	construction	construction	NOUN
ejpam-6409	341	4	of	of	ADP
ejpam-6409	341	5	gh	gh	PROPN
ejpam-6409	341	6	codes	code	NOUN
ejpam-6409	341	7	over	over	ADP
ejpam-6409	341	8	z2s	z2s	PROPN
ejpam-6409	341	9	[	[	X
ejpam-6409	341	10	ω	ω	X
ejpam-6409	341	11	]	]	X
ejpam-6409	341	12	let	let	VERB
ejpam-6409	341	13	ti	ti	NOUN
ejpam-6409	341	14	=	=	SYM
ejpam-6409	341	15	{	{	PUNCT
ejpam-6409	341	16	jk	jk	PROPN
ejpam-6409	341	17	·	·	PUNCT
ejpam-6409	341	18	2i−1	2i−1	NUM
ejpam-6409	341	19	:	:	PUNCT
ejpam-6409	341	20	j	j	PROPN
ejpam-6409	341	21	,	,	PUNCT
ejpam-6409	341	22	k	k	PROPN
ejpam-6409	341	23	∈	∈	PROPN
ejpam-6409	341	24	{	{	PUNCT
ejpam-6409	341	25	0	0	NUM
ejpam-6409	341	26	,	,	PUNCT
ejpam-6409	341	27	1	1	NUM
ejpam-6409	341	28	,	,	PUNCT
ejpam-6409	341	29	.	.	PUNCT
ejpam-6409	341	30	.	.	PUNCT
ejpam-6409	342	1	.	.	PUNCT
ejpam-6409	343	1	,	,	PUNCT
ejpam-6409	343	2	2s−i+1	2s−i+1	NUM
ejpam-6409	343	3	−	−	NOUN
ejpam-6409	343	4	1	1	NUM
ejpam-6409	343	5	}	}	PUNCT
ejpam-6409	343	6	}	}	PUNCT
ejpam-6409	343	7	for	for	ADP
ejpam-6409	343	8	all	all	PRON
ejpam-6409	343	9	i	i	PRON
ejpam-6409	343	10	∈	∈	PROPN
ejpam-6409	343	11	{	{	PUNCT
ejpam-6409	343	12	1	1	NUM
ejpam-6409	343	13	,	,	PUNCT
ejpam-6409	343	14	.	.	PUNCT
ejpam-6409	343	15	.	.	PUNCT
ejpam-6409	343	16	.	.	PUNCT
ejpam-6409	344	1	,	,	PUNCT
ejpam-6409	344	2	s	s	X
ejpam-6409	344	3	}	}	PUNCT
ejpam-6409	344	4	.	.	PUNCT
ejpam-6409	345	1	note	note	VERB
ejpam-6409	345	2	that	that	SCONJ
ejpam-6409	345	3	t1	t1	NOUN
ejpam-6409	345	4	=	=	PUNCT
ejpam-6409	345	5	{	{	PUNCT
ejpam-6409	345	6	00	00	NUM
ejpam-6409	345	7	,	,	PUNCT
ejpam-6409	345	8	01	01	NUM
ejpam-6409	345	9	,	,	PUNCT
ejpam-6409	345	10	.	.	PUNCT
ejpam-6409	345	11	.	.	PUNCT
ejpam-6409	346	1	.	.	PUNCT
ejpam-6409	347	1	,	,	PUNCT
ejpam-6409	347	2	2s	2s	NOUN
ejpam-6409	347	3	−	−	NUM
ejpam-6409	347	4	1	1	NUM
ejpam-6409	347	5	2s−1	2s−1	NUM
ejpam-6409	347	6	−	−	NOUN
ejpam-6409	347	7	1	1	NUM
ejpam-6409	347	8	}	}	PUNCT
ejpam-6409	347	9	.	.	PUNCT
ejpam-6409	348	1	let	let	VERB
ejpam-6409	348	2	t1	t1	NOUN
ejpam-6409	348	3	,	,	PUNCT
ejpam-6409	348	4	t2	t2	NOUN
ejpam-6409	348	5	,	,	PUNCT
ejpam-6409	348	6	.	.	PUNCT
ejpam-6409	348	7	.	.	PUNCT
ejpam-6409	349	1	.	.	PUNCT
ejpam-6409	350	1	,	,	PUNCT
ejpam-6409	350	2	ts	ts	AUX
ejpam-6409	350	3	be	be	AUX
ejpam-6409	350	4	nonnegative	nonnegative	ADJ
ejpam-6409	350	5	integers	integer	NOUN
ejpam-6409	350	6	with	with	ADP
ejpam-6409	350	7	t1	t1	PROPN
ejpam-6409	350	8	≥	≥	NUM
ejpam-6409	350	9	1	1	NUM
ejpam-6409	350	10	.	.	PUNCT
ejpam-6409	351	1	consider	consider	VERB
ejpam-6409	351	2	the	the	DET
ejpam-6409	351	3	matrix	matrix	NOUN
ejpam-6409	351	4	a	a	DET
ejpam-6409	351	5	(	(	PUNCT
ejpam-6409	351	6	t1,	t1,	NUM
ejpam-6409	351	7	...	...	PUNCT
ejpam-6409	351	8	,ts	,ts	PUNCT
ejpam-6409	351	9	)	)	PUNCT
ejpam-6409	351	10	2	2	NUM
ejpam-6409	352	1	whose	whose	DET
ejpam-6409	352	2	columns	column	NOUN
ejpam-6409	352	3	are	be	AUX
ejpam-6409	352	4	exactly	exactly	ADV
ejpam-6409	352	5	all	all	DET
ejpam-6409	352	6	the	the	DET
ejpam-6409	352	7	vectors	vector	NOUN
ejpam-6409	352	8	of	of	ADP
ejpam-6409	352	9	the	the	DET
ejpam-6409	352	10	form	form	NOUN
ejpam-6409	352	11	zt	zt	PROPN
ejpam-6409	352	12	,	,	PUNCT
ejpam-6409	352	13	where	where	SCONJ
ejpam-6409	352	14	z	z	PROPN
ejpam-6409	352	15	∈	∈	PROPN
ejpam-6409	352	16	{	{	PUNCT
ejpam-6409	352	17	0	0	NUM
ejpam-6409	352	18	}	}	PUNCT
ejpam-6409	352	19	×	×	NOUN
ejpam-6409	352	20	t	t	NOUN
ejpam-6409	352	21	t1−1	t1−1	PROPN
ejpam-6409	352	22	1	1	NUM
ejpam-6409	352	23	×	×	NOUN
ejpam-6409	352	24	t	t	NOUN
ejpam-6409	352	25	t2	t2	NOUN
ejpam-6409	352	26	2	2	NUM
ejpam-6409	352	27	×	×	NOUN
ejpam-6409	352	28	·	·	PUNCT
ejpam-6409	352	29	·	·	PUNCT
ejpam-6409	352	30	·	·	PUNCT
ejpam-6409	353	1	×	×	NOUN
ejpam-6409	353	2	t	t	NOUN
ejpam-6409	353	3	ts	ts	X
ejpam-6409	353	4	s	s	PROPN
ejpam-6409	353	5	.	.	PUNCT
ejpam-6409	354	1	let	let	VERB
ejpam-6409	354	2	00	00	NUM
ejpam-6409	354	3	,	,	PUNCT
ejpam-6409	354	4	01	01	NUM
ejpam-6409	354	5	,	,	PUNCT
ejpam-6409	354	6	.	.	PUNCT
ejpam-6409	354	7	.	.	PUNCT
ejpam-6409	355	1	.	.	PUNCT
ejpam-6409	356	1	,	,	PUNCT
ejpam-6409	356	2	2s	2s	NOUN
ejpam-6409	356	3	−	−	PROPN
ejpam-6409	356	4	1	1	NUM
ejpam-6409	356	5	2s	2s	NOUN
ejpam-6409	356	6	−	−	NUM
ejpam-6409	356	7	1	1	NUM
ejpam-6409	356	8	be	be	AUX
ejpam-6409	356	9	the	the	DET
ejpam-6409	356	10	vectors	vector	NOUN
ejpam-6409	356	11	having	have	VERB
ejpam-6409	356	12	the	the	DET
ejpam-6409	356	13	same	same	ADJ
ejpam-6409	356	14	element	element	NOUN
ejpam-6409	356	15	00	00	NUM
ejpam-6409	356	16	,	,	PUNCT
ejpam-6409	356	17	01	01	NUM
ejpam-6409	356	18	,	,	PUNCT
ejpam-6409	356	19	.	.	PUNCT
ejpam-6409	356	20	.	.	PUNCT
ejpam-6409	357	1	.	.	PUNCT
ejpam-6409	358	1	,	,	PUNCT
ejpam-6409	358	2	2s	2s	NOUN
ejpam-6409	358	3	−	−	NOUN
ejpam-6409	358	4	1	1	NUM
ejpam-6409	358	5	from	from	ADP
ejpam-6409	358	6	z2s	z2s	PROPN
ejpam-6409	358	7	[	[	X
ejpam-6409	358	8	ω	ω	X
ejpam-6409	358	9	]	]	X
ejpam-6409	358	10	in	in	ADP
ejpam-6409	358	11	all	all	DET
ejpam-6409	358	12	coordinates	coordinate	NOUN
ejpam-6409	358	13	,	,	PUNCT
ejpam-6409	358	14	respectively	respectively	ADV
ejpam-6409	358	15	.	.	PUNCT
ejpam-6409	359	1	any	any	DET
ejpam-6409	359	2	matrix	matrix	NOUN
ejpam-6409	359	3	a	a	DET
ejpam-6409	359	4	(	(	PUNCT
ejpam-6409	359	5	t1,	t1,	NUM
ejpam-6409	359	6	...	...	PUNCT
ejpam-6409	359	7	,ts	,ts	PUNCT
ejpam-6409	359	8	)	)	PUNCT
ejpam-6409	359	9	2	2	NUM
ejpam-6409	359	10	can	can	AUX
ejpam-6409	359	11	also	also	ADV
ejpam-6409	359	12	be	be	AUX
ejpam-6409	359	13	obtained	obtain	VERB
ejpam-6409	359	14	by	by	ADP
ejpam-6409	359	15	applying	apply	VERB
ejpam-6409	359	16	the	the	DET
ejpam-6409	359	17	recursive	recursive	ADJ
ejpam-6409	359	18	construction	construction	NOUN
ejpam-6409	359	19	given	give	VERB
ejpam-6409	359	20	below	below	ADV
ejpam-6409	359	21	.	.	PUNCT
ejpam-6409	360	1	start	start	VERB
ejpam-6409	360	2	with	with	ADP
ejpam-6409	360	3	the	the	DET
ejpam-6409	360	4	matrix	matrix	NOUN
ejpam-6409	360	5	a	a	DET
ejpam-6409	360	6	(	(	PUNCT
ejpam-6409	360	7	1,0,	1,0,	NOUN
ejpam-6409	360	8	...	...	PUNCT
ejpam-6409	360	9	,0	,0	PUNCT
ejpam-6409	360	10	)	)	PUNCT
ejpam-6409	360	11	2	2	NUM
ejpam-6409	360	12	=	=	SYM
ejpam-6409	360	13	(	(	PUNCT
ejpam-6409	360	14	1	1	NUM
ejpam-6409	360	15	0	0	NUM
ejpam-6409	360	16	)	)	PUNCT
ejpam-6409	360	17	.	.	PUNCT
ejpam-6409	361	1	if	if	SCONJ
ejpam-6409	361	2	we	we	PRON
ejpam-6409	361	3	have	have	VERB
ejpam-6409	361	4	a	a	DET
ejpam-6409	361	5	matrix	matrix	NOUN
ejpam-6409	361	6	a	a	DET
ejpam-6409	361	7	(	(	PUNCT
ejpam-6409	361	8	t1,	t1,	NUM
ejpam-6409	361	9	...	...	PUNCT
ejpam-6409	361	10	,ts	,ts	PUNCT
ejpam-6409	361	11	)	)	PUNCT
ejpam-6409	361	12	2	2	NUM
ejpam-6409	361	13	,	,	PUNCT
ejpam-6409	361	14	then	then	ADV
ejpam-6409	361	15	for	for	ADP
ejpam-6409	361	16	any	any	DET
ejpam-6409	361	17	i	i	PROPN
ejpam-6409	361	18	∈	∈	PROPN
ejpam-6409	361	19	{	{	PUNCT
ejpam-6409	361	20	1	1	NUM
ejpam-6409	361	21	,	,	PUNCT
ejpam-6409	361	22	.	.	PUNCT
ejpam-6409	361	23	.	.	PUNCT
ejpam-6409	362	1	.	.	PUNCT
ejpam-6409	363	1	,	,	PUNCT
ejpam-6409	363	2	s	s	X
ejpam-6409	363	3	}	}	PUNCT
ejpam-6409	363	4	,	,	PUNCT
ejpam-6409	363	5	we	we	PRON
ejpam-6409	363	6	can	can	AUX
ejpam-6409	363	7	construct	construct	VERB
ejpam-6409	363	8	the	the	DET
ejpam-6409	363	9	matrix	matrix	NOUN
ejpam-6409	363	10	ai	ai	VERB
ejpam-6409	363	11	=	=	PUNCT
ejpam-6409	363	12	[	[	PUNCT
ejpam-6409	363	13	a	a	DET
ejpam-6409	363	14	a	a	PRON
ejpam-6409	363	15	·	·	PUNCT
ejpam-6409	363	16	·	·	PUNCT
ejpam-6409	363	17	·	·	PUNCT
ejpam-6409	364	1	a	a	PRON
ejpam-6409	364	2	·	·	PUNCT
ejpam-6409	364	3	·	·	PUNCT
ejpam-6409	364	4	·	·	PUNCT
ejpam-6409	364	5	a	a	DET
ejpam-6409	364	6	·	·	PUNCT
ejpam-6409	364	7	·	·	PUNCT
ejpam-6409	364	8	·	·	PUNCT
ejpam-6409	364	9	a	a	DET
ejpam-6409	364	10	2i−1	2i−1	NUM
ejpam-6409	364	11	·	·	PUNCT
ejpam-6409	364	12	00	00	PUNCT
ejpam-6409	365	1	2i−1	2i−1	NUM
ejpam-6409	365	2	·	·	PUNCT
ejpam-6409	365	3	01	01	NUM
ejpam-6409	365	4	·	·	PUNCT
ejpam-6409	365	5	·	·	PUNCT
ejpam-6409	365	6	·	·	PUNCT
ejpam-6409	366	1	2i−1	2i−1	NUM
ejpam-6409	366	2	·	·	PUNCT
ejpam-6409	366	3	0(2s−i+1	0(2s−i+1	NOUN
ejpam-6409	367	1	−	−	NOUN
ejpam-6409	367	2	1	1	NUM
ejpam-6409	367	3	)	)	PUNCT
ejpam-6409	367	4	·	·	PUNCT
ejpam-6409	367	5	·	·	PUNCT
ejpam-6409	367	6	·	·	PUNCT
ejpam-6409	368	1	2i−1	2i−1	NUM
ejpam-6409	368	2	·	·	PUNCT
ejpam-6409	368	3	(	(	PUNCT
ejpam-6409	368	4	2s−i+1	2s−i+1	NUM
ejpam-6409	368	5	−	−	NUM
ejpam-6409	368	6	1)0	1)0	NUM
ejpam-6409	368	7	·	·	PUNCT
ejpam-6409	368	8	·	·	PUNCT
ejpam-6409	368	9	·	·	PUNCT
ejpam-6409	369	1	2i−1	2i−1	NUM
ejpam-6409	369	2	·	·	PUNCT
ejpam-6409	369	3	(	(	PUNCT
ejpam-6409	369	4	2s−i+1	2s−i+1	NUM
ejpam-6409	369	5	−	−	NUM
ejpam-6409	369	6	1)(2s−i+1	1)(2s−i+1	NUM
ejpam-6409	369	7	−	−	NOUN
ejpam-6409	369	8	1	1	NUM
ejpam-6409	369	9	)	)	PUNCT
ejpam-6409	369	10	]	]	PUNCT
ejpam-6409	370	1	muhammad	muhammad	PROPN
ejpam-6409	370	2	sajjad	sajjad	PROPN
ejpam-6409	370	3	et	et	PROPN
ejpam-6409	370	4	al	al	PROPN
ejpam-6409	370	5	.	.	PUNCT
ejpam-6409	370	6	/	/	SYM
ejpam-6409	370	7	eur	eur	PROPN
ejpam-6409	370	8	.	.	PUNCT
ejpam-6409	371	1	j.	j.	PROPN
ejpam-6409	371	2	pure	pure	PROPN
ejpam-6409	371	3	appl	appl	PROPN
ejpam-6409	371	4	.	.	PROPN
ejpam-6409	371	5	math	math	PROPN
ejpam-6409	371	6	,	,	PUNCT
ejpam-6409	371	7	18	18	NUM
ejpam-6409	371	8	(	(	PUNCT
ejpam-6409	371	9	3	3	NUM
ejpam-6409	371	10	)	)	PUNCT
ejpam-6409	371	11	(	(	PUNCT
ejpam-6409	371	12	2025	2025	NUM
ejpam-6409	371	13	)	)	PUNCT
ejpam-6409	371	14	,	,	PUNCT
ejpam-6409	371	15	6409	6409	NUM
ejpam-6409	371	16	11	11	NUM
ejpam-6409	371	17	of	of	ADP
ejpam-6409	371	18	32	32	NUM
ejpam-6409	371	19	finally	finally	ADV
ejpam-6409	371	20	,	,	PUNCT
ejpam-6409	371	21	by	by	ADP
ejpam-6409	371	22	permuting	permute	VERB
ejpam-6409	371	23	the	the	DET
ejpam-6409	371	24	rows	row	NOUN
ejpam-6409	371	25	of	of	ADP
ejpam-6409	371	26	ai	ai	NOUN
ejpam-6409	371	27	,	,	PUNCT
ejpam-6409	371	28	we	we	PRON
ejpam-6409	371	29	obtain	obtain	VERB
ejpam-6409	371	30	a	a	DET
ejpam-6409	371	31	matrix	matrix	NOUN
ejpam-6409	371	32	a	a	DET
ejpam-6409	371	33	(	(	PUNCT
ejpam-6409	371	34	t′1,	t′1,	ADV
ejpam-6409	371	35	...	...	PUNCT
ejpam-6409	371	36	,t	,t	PUNCT
ejpam-6409	372	1	′	′	NUM
ejpam-6409	372	2	s	s	X
ejpam-6409	372	3	)	)	PUNCT
ejpam-6409	372	4	2	2	NUM
ejpam-6409	372	5	where	where	SCONJ
ejpam-6409	372	6	t′j	t′j	PROPN
ejpam-6409	372	7	=	=	PROPN
ejpam-6409	372	8	tj	tj	PROPN
ejpam-6409	372	9	for	for	ADP
ejpam-6409	372	10	j	j	PROPN
ejpam-6409	372	11	̸=	̸=	PROPN
ejpam-6409	372	12	i.	i.	NOUN
ejpam-6409	372	13	note	note	VERB
ejpam-6409	372	14	that	that	SCONJ
ejpam-6409	372	15	by	by	ADP
ejpam-6409	372	16	permuting	permute	VERB
ejpam-6409	372	17	the	the	DET
ejpam-6409	372	18	columns	column	NOUN
ejpam-6409	372	19	of	of	ADP
ejpam-6409	372	20	ai	ai	NOUN
ejpam-6409	372	21	,	,	PUNCT
ejpam-6409	372	22	another	another	DET
ejpam-6409	372	23	matrix	matrix	NOUN
ejpam-6409	372	24	a	a	PRON
ejpam-6409	372	25	(	(	PUNCT
ejpam-6409	372	26	t′1,	t′1,	ADV
ejpam-6409	372	27	...	...	PUNCT
ejpam-6409	372	28	,t	,t	PUNCT
ejpam-6409	373	1	′	′	NUM
ejpam-6409	373	2	s	s	X
ejpam-6409	373	3	)	)	PUNCT
ejpam-6409	373	4	2	2	NUM
ejpam-6409	373	5	can	can	AUX
ejpam-6409	373	6	also	also	ADV
ejpam-6409	373	7	be	be	AUX
ejpam-6409	373	8	obtained	obtain	VERB
ejpam-6409	373	9	.	.	PUNCT
ejpam-6409	374	1	to	to	PART
ejpam-6409	374	2	construct	construct	VERB
ejpam-6409	374	3	matrices	matrix	NOUN
ejpam-6409	374	4	recursively	recursively	ADV
ejpam-6409	374	5	,	,	PUNCT
ejpam-6409	374	6	starting	start	VERB
ejpam-6409	374	7	from	from	ADP
ejpam-6409	374	8	the	the	DET
ejpam-6409	374	9	base	base	NOUN
ejpam-6409	374	10	matrix	matrix	NOUN
ejpam-6409	374	11	a	a	DET
ejpam-6409	374	12	(	(	PUNCT
ejpam-6409	374	13	1,0,	1,0,	NOUN
ejpam-6409	374	14	...	...	PUNCT
ejpam-6409	374	15	,0	,0	PUNCT
ejpam-6409	374	16	)	)	PUNCT
ejpam-6409	374	17	2	2	NUM
ejpam-6409	374	18	,	,	PUNCT
ejpam-6409	374	19	proceed	proceed	VERB
ejpam-6409	374	20	in	in	ADP
ejpam-6409	374	21	the	the	DET
ejpam-6409	374	22	following	following	ADJ
ejpam-6409	374	23	way	way	NOUN
ejpam-6409	374	24	.	.	PUNCT
ejpam-6409	375	1	first	first	ADV
ejpam-6409	375	2	,	,	PUNCT
ejpam-6409	375	3	to	to	PART
ejpam-6409	375	4	obtain	obtain	VERB
ejpam-6409	375	5	matrix	matrix	NOUN
ejpam-6409	375	6	a	a	DET
ejpam-6409	375	7	(	(	PUNCT
ejpam-6409	375	8	t1,0,	t1,0,	NOUN
ejpam-6409	375	9	...	...	PUNCT
ejpam-6409	375	10	,0	,0	PUNCT
ejpam-6409	375	11	)	)	PUNCT
ejpam-6409	375	12	2	2	NUM
ejpam-6409	375	13	,	,	PUNCT
ejpam-6409	375	14	we	we	PRON
ejpam-6409	375	15	add	add	VERB
ejpam-6409	375	16	t1	t1	NOUN
ejpam-6409	375	17	−	−	NUM
ejpam-6409	375	18	1	1	NUM
ejpam-6409	375	19	rows	row	NOUN
ejpam-6409	375	20	of	of	ADP
ejpam-6409	375	21	order	order	NOUN
ejpam-6409	375	22	2s	2s	NOUN
ejpam-6409	375	23	,	,	PUNCT
ejpam-6409	375	24	then	then	ADV
ejpam-6409	375	25	t2	t2	NOUN
ejpam-6409	375	26	rows	row	NOUN
ejpam-6409	375	27	of	of	ADP
ejpam-6409	375	28	order	order	NOUN
ejpam-6409	375	29	2s−1	2s−1	NUM
ejpam-6409	375	30	,	,	PUNCT
ejpam-6409	375	31	and	and	CCONJ
ejpam-6409	375	32	so	so	ADV
ejpam-6409	375	33	on	on	ADV
ejpam-6409	375	34	,	,	PUNCT
ejpam-6409	375	35	up	up	ADP
ejpam-6409	375	36	to	to	PART
ejpam-6409	375	37	generate	generate	VERB
ejpam-6409	375	38	a	a	DET
ejpam-6409	375	39	(	(	PUNCT
ejpam-6409	375	40	t1,t2,	t1,t2,	PROPN
ejpam-6409	375	41	...	...	PUNCT
ejpam-6409	375	42	,0	,0	PUNCT
ejpam-6409	375	43	)	)	PUNCT
ejpam-6409	375	44	2	2	NUM
ejpam-6409	375	45	;	;	PUNCT
ejpam-6409	375	46	and	and	CCONJ
ejpam-6409	375	47	finally	finally	ADV
ejpam-6409	375	48	we	we	PRON
ejpam-6409	375	49	add	add	VERB
ejpam-6409	375	50	ts	ts	ADP
ejpam-6409	375	51	rows	row	NOUN
ejpam-6409	375	52	of	of	ADP
ejpam-6409	375	53	order	order	NOUN
ejpam-6409	375	54	2	2	NUM
ejpam-6409	375	55	to	to	PART
ejpam-6409	375	56	achieve	achieve	VERB
ejpam-6409	375	57	a	a	DET
ejpam-6409	375	58	(	(	PUNCT
ejpam-6409	375	59	t1,	t1,	NUM
ejpam-6409	375	60	...	...	PUNCT
ejpam-6409	375	61	,ts	,ts	PUNCT
ejpam-6409	375	62	)	)	PUNCT
ejpam-6409	375	63	2	2	NUM
ejpam-6409	375	64	.	.	PUNCT
ejpam-6409	376	1	let	let	VERB
ejpam-6409	376	2	h̃	h̃	PROPN
ejpam-6409	376	3	(	(	PUNCT
ejpam-6409	376	4	t1,	t1,	NUM
ejpam-6409	376	5	...	...	PUNCT
ejpam-6409	376	6	,ts	,ts	PUNCT
ejpam-6409	376	7	)	)	PUNCT
ejpam-6409	376	8	2	2	NUM
ejpam-6409	376	9	be	be	AUX
ejpam-6409	376	10	the	the	DET
ejpam-6409	376	11	z2s	z2s	PROPN
ejpam-6409	377	1	[	[	X
ejpam-6409	377	2	ω]-additive	ω]-additive	X
ejpam-6409	377	3	code	code	NOUN
ejpam-6409	377	4	of	of	ADP
ejpam-6409	377	5	type	type	NOUN
ejpam-6409	377	6	(	(	PUNCT
ejpam-6409	377	7	n	n	CCONJ
ejpam-6409	377	8	,	,	PUNCT
ejpam-6409	377	9	t1	t1	NOUN
ejpam-6409	377	10	,	,	PUNCT
ejpam-6409	377	11	.	.	PUNCT
ejpam-6409	377	12	.	.	PUNCT
ejpam-6409	378	1	.	.	PUNCT
ejpam-6409	379	1	,	,	PUNCT
ejpam-6409	379	2	ts	ts	AUX
ejpam-6409	379	3	)	)	PUNCT
ejpam-6409	379	4	generated	generate	VERB
ejpam-6409	379	5	by	by	ADP
ejpam-6409	379	6	the	the	DET
ejpam-6409	379	7	matrix	matrix	NOUN
ejpam-6409	379	8	a	a	DET
ejpam-6409	379	9	(	(	PUNCT
ejpam-6409	379	10	t1,	t1,	NUM
ejpam-6409	379	11	...	...	PUNCT
ejpam-6409	379	12	,ts	,ts	PUNCT
ejpam-6409	379	13	)	)	PUNCT
ejpam-6409	379	14	2	2	NUM
ejpam-6409	379	15	,	,	PUNCT
ejpam-6409	379	16	where	where	SCONJ
ejpam-6409	379	17	t1	t1	NOUN
ejpam-6409	379	18	,	,	PUNCT
ejpam-6409	379	19	t2	t2	NOUN
ejpam-6409	379	20	,	,	PUNCT
ejpam-6409	379	21	.	.	PUNCT
ejpam-6409	379	22	.	.	PUNCT
ejpam-6409	380	1	.	.	PUNCT
ejpam-6409	381	1	,	,	PUNCT
ejpam-6409	381	2	ts	ts	PROPN
ejpam-6409	381	3	are	be	AUX
ejpam-6409	381	4	nonnegative	nonnegative	ADJ
ejpam-6409	381	5	integers	integer	NOUN
ejpam-6409	381	6	with	with	ADP
ejpam-6409	381	7	t1	t1	PROPN
ejpam-6409	381	8	≥	≥	NUM
ejpam-6409	381	9	1	1	NUM
ejpam-6409	381	10	.	.	PUNCT
ejpam-6409	382	1	let	let	VERB
ejpam-6409	382	2	n	n	NOUN
ejpam-6409	382	3	=	=	PUNCT
ejpam-6409	382	4	22(t−s+1	22(t−s+1	NUM
ejpam-6409	382	5	)	)	PUNCT
ejpam-6409	382	6	,	,	PUNCT
ejpam-6409	382	7	where	where	SCONJ
ejpam-6409	382	8	t	t	NOUN
ejpam-6409	382	9	=	=	SYM
ejpam-6409	382	10	(	(	PUNCT
ejpam-6409	382	11	s∑	s∑	PROPN
ejpam-6409	382	12	i=1	i=1	PROPN
ejpam-6409	383	1	(	(	PUNCT
ejpam-6409	383	2	s−	s−	PROPN
ejpam-6409	383	3	i+	i+	PUNCT
ejpam-6409	383	4	1)ti	1)ti	PROPN
ejpam-6409	383	5	)	)	PUNCT
ejpam-6409	384	1	−	−	PROPN
ejpam-6409	385	1	1	1	X
ejpam-6409	385	2	.	.	PUNCT
ejpam-6409	386	1	the	the	DET
ejpam-6409	386	2	code	code	NOUN
ejpam-6409	386	3	h̃	h̃	PROPN
ejpam-6409	386	4	(	(	PUNCT
ejpam-6409	386	5	t1,	t1,	NUM
ejpam-6409	386	6	...	...	PUNCT
ejpam-6409	386	7	,ts	,ts	PUNCT
ejpam-6409	386	8	)	)	PUNCT
ejpam-6409	386	9	2	2	NUM
ejpam-6409	386	10	has	have	VERB
ejpam-6409	386	11	length	length	NOUN
ejpam-6409	386	12	n	n	CCONJ
ejpam-6409	386	13	,	,	PUNCT
ejpam-6409	386	14	and	and	CCONJ
ejpam-6409	386	15	the	the	DET
ejpam-6409	386	16	corresponding	correspond	VERB
ejpam-6409	386	17	z2s	z2s	PROPN
ejpam-6409	386	18	[	[	X
ejpam-6409	386	19	ω]-linear	ω]-linear	ADJ
ejpam-6409	386	20	code	code	NOUN
ejpam-6409	386	21	h	h	NOUN
ejpam-6409	386	22	(	(	PUNCT
ejpam-6409	386	23	t1,	t1,	NUM
ejpam-6409	386	24	...	...	PUNCT
ejpam-6409	386	25	,ts	,ts	PUNCT
ejpam-6409	386	26	)	)	PUNCT
ejpam-6409	386	27	2	2	NUM
ejpam-6409	386	28	=	=	SYM
ejpam-6409	386	29	φs	φs	PROPN
ejpam-6409	386	30	(	(	PUNCT
ejpam-6409	386	31	h̃	h̃	PROPN
ejpam-6409	386	32	(	(	PUNCT
ejpam-6409	386	33	t1,	t1,	NUM
ejpam-6409	386	34	...	...	PUNCT
ejpam-6409	386	35	,ts	,ts	PUNCT
ejpam-6409	386	36	)	)	PUNCT
ejpam-6409	386	37	2	2	NUM
ejpam-6409	386	38	)	)	PUNCT
ejpam-6409	386	39	is	be	AUX
ejpam-6409	386	40	a	a	DET
ejpam-6409	386	41	generalized	generalized	ADJ
ejpam-6409	386	42	hadamard	hadamard	ADJ
ejpam-6409	386	43	code	code	NOUN
ejpam-6409	386	44	of	of	ADP
ejpam-6409	386	45	length	length	NOUN
ejpam-6409	386	46	22	22	NUM
ejpam-6409	386	47	t.	t.	NOUN
ejpam-6409	386	48	example	example	NOUN
ejpam-6409	386	49	4.1	4.1	NUM
ejpam-6409	386	50	:	:	PUNCT
ejpam-6409	386	51	for	for	ADP
ejpam-6409	386	52	s	s	NOUN
ejpam-6409	386	53	=	=	SYM
ejpam-6409	386	54	2	2	NUM
ejpam-6409	386	55	,	,	PUNCT
ejpam-6409	386	56	we	we	PRON
ejpam-6409	386	57	have	have	VERB
ejpam-6409	386	58	the	the	DET
ejpam-6409	386	59	following	follow	VERB
ejpam-6409	386	60	matrices	matrix	NOUN
ejpam-6409	386	61	which	which	PRON
ejpam-6409	386	62	generate	generate	VERB
ejpam-6409	386	63	codes	code	NOUN
ejpam-6409	386	64	over	over	ADP
ejpam-6409	386	65	z4[ω	z4[ω	NOUN
ejpam-6409	386	66	]	]	PUNCT
ejpam-6409	386	67	.	.	PUNCT
ejpam-6409	387	1	a1,1	a1,1	NOUN
ejpam-6409	388	1	=	=	PUNCT
ejpam-6409	388	2	[	[	PUNCT
ejpam-6409	388	3	10	10	NUM
ejpam-6409	388	4	10	10	NUM
ejpam-6409	388	5	10	10	NUM
ejpam-6409	388	6	10	10	NUM
ejpam-6409	388	7	00	00	NUM
ejpam-6409	388	8	02	02	NUM
ejpam-6409	388	9	20	20	NUM
ejpam-6409	388	10	22	22	NUM
ejpam-6409	388	11	]	]	PUNCT
ejpam-6409	388	12	a1,2	a1,2	PROPN
ejpam-6409	388	13	=	=	SYM
ejpam-6409	388	14			PROPN
ejpam-6409	388	15	10	10	NUM
ejpam-6409	388	16	10	10	NUM
ejpam-6409	388	17	10	10	NUM
ejpam-6409	388	18	10	10	NUM
ejpam-6409	388	19	10	10	NUM
ejpam-6409	388	20	10	10	NUM
ejpam-6409	388	21	10	10	NUM
ejpam-6409	388	22	10	10	NUM
ejpam-6409	388	23	10	10	NUM
ejpam-6409	388	24	10	10	NUM
ejpam-6409	388	25	10	10	NUM
ejpam-6409	388	26	10	10	NUM
ejpam-6409	388	27	10	10	NUM
ejpam-6409	388	28	10	10	NUM
ejpam-6409	388	29	10	10	NUM
ejpam-6409	388	30	10	10	NUM
ejpam-6409	388	31	00	00	NUM
ejpam-6409	388	32	02	02	NUM
ejpam-6409	388	33	20	20	NUM
ejpam-6409	388	34	22	22	NUM
ejpam-6409	388	35	00	00	NUM
ejpam-6409	388	36	02	02	NUM
ejpam-6409	388	37	20	20	NUM
ejpam-6409	388	38	22	22	NUM
ejpam-6409	388	39	00	00	NUM
ejpam-6409	388	40	02	02	NUM
ejpam-6409	388	41	20	20	NUM
ejpam-6409	388	42	22	22	NUM
ejpam-6409	388	43	00	00	NUM
ejpam-6409	388	44	02	02	NUM
ejpam-6409	388	45	20	20	NUM
ejpam-6409	388	46	22	22	NUM
ejpam-6409	388	47	00	00	NUM
ejpam-6409	388	48	00	00	NUM
ejpam-6409	388	49	00	00	NUM
ejpam-6409	388	50	00	00	NUM
ejpam-6409	388	51	02	02	NUM
ejpam-6409	388	52	02	02	NUM
ejpam-6409	388	53	02	02	NUM
ejpam-6409	388	54	02	02	NUM
ejpam-6409	388	55	20	20	NUM
ejpam-6409	388	56	20	20	NUM
ejpam-6409	388	57	20	20	NUM
ejpam-6409	388	58	20	20	NUM
ejpam-6409	388	59	22	22	NUM
ejpam-6409	388	60	22	22	NUM
ejpam-6409	388	61	22	22	NUM
ejpam-6409	388	62	22	22	NUM
ejpam-6409	388	63			NOUN
ejpam-6409	388	64	a2,0	a2,0	NOUN
ejpam-6409	389	1	=	=	PUNCT
ejpam-6409	390	1	[	[	PUNCT
ejpam-6409	390	2	10	10	NUM
ejpam-6409	390	3	10	10	NUM
ejpam-6409	390	4	10	10	NUM
ejpam-6409	390	5	10	10	NUM
ejpam-6409	390	6	10	10	NUM
ejpam-6409	390	7	10	10	NUM
ejpam-6409	390	8	10	10	NUM
ejpam-6409	390	9	10	10	NUM
ejpam-6409	390	10	10	10	NUM
ejpam-6409	390	11	10	10	NUM
ejpam-6409	390	12	10	10	NUM
ejpam-6409	390	13	10	10	NUM
ejpam-6409	390	14	10	10	NUM
ejpam-6409	390	15	10	10	NUM
ejpam-6409	390	16	10	10	NUM
ejpam-6409	390	17	10	10	NUM
ejpam-6409	390	18	00	00	NUM
ejpam-6409	390	19	01	01	NUM
ejpam-6409	390	20	02	02	NUM
ejpam-6409	390	21	03	03	NUM
ejpam-6409	390	22	10	10	NUM
ejpam-6409	390	23	11	11	NUM
ejpam-6409	390	24	12	12	NUM
ejpam-6409	390	25	13	13	NUM
ejpam-6409	390	26	20	20	NUM
ejpam-6409	390	27	21	21	NUM
ejpam-6409	390	28	22	22	NUM
ejpam-6409	390	29	23	23	NUM
ejpam-6409	390	30	30	30	NUM
ejpam-6409	390	31	31	31	NUM
ejpam-6409	390	32	32	32	NUM
ejpam-6409	390	33	33	33	NUM
ejpam-6409	390	34	]	]	PUNCT
ejpam-6409	390	35	example	example	NOUN
ejpam-6409	390	36	4.2	4.2	NUM
ejpam-6409	390	37	:	:	PUNCT
ejpam-6409	390	38	for	for	ADP
ejpam-6409	390	39	s	s	NOUN
ejpam-6409	390	40	=	=	SYM
ejpam-6409	390	41	3	3	NUM
ejpam-6409	390	42	,	,	PUNCT
ejpam-6409	390	43	the	the	DET
ejpam-6409	390	44	following	follow	VERB
ejpam-6409	390	45	are	be	AUX
ejpam-6409	390	46	generator	generator	NOUN
ejpam-6409	390	47	matrices	matrix	NOUN
ejpam-6409	390	48	.	.	PUNCT
ejpam-6409	391	1	a1,0,1	a1,0,1	NOUN
ejpam-6409	391	2	=	=	SYM
ejpam-6409	392	1	[	[	PUNCT
ejpam-6409	392	2	10	10	NUM
ejpam-6409	392	3	10	10	NUM
ejpam-6409	392	4	10	10	NUM
ejpam-6409	392	5	10	10	NUM
ejpam-6409	392	6	00	00	NUM
ejpam-6409	392	7	04	04	NUM
ejpam-6409	392	8	40	40	NUM
ejpam-6409	392	9	44	44	NUM
ejpam-6409	392	10	]	]	PUNCT
ejpam-6409	392	11	muhammad	muhammad	PROPN
ejpam-6409	392	12	sajjad	sajjad	PROPN
ejpam-6409	392	13	et	et	PROPN
ejpam-6409	392	14	al	al	PROPN
ejpam-6409	392	15	.	.	PUNCT
ejpam-6409	392	16	/	/	SYM
ejpam-6409	392	17	eur	eur	PROPN
ejpam-6409	392	18	.	.	PUNCT
ejpam-6409	393	1	j.	j.	PROPN
ejpam-6409	393	2	pure	pure	PROPN
ejpam-6409	393	3	appl	appl	PROPN
ejpam-6409	393	4	.	.	PROPN
ejpam-6409	393	5	math	math	PROPN
ejpam-6409	393	6	,	,	PUNCT
ejpam-6409	393	7	18	18	NUM
ejpam-6409	393	8	(	(	PUNCT
ejpam-6409	393	9	3	3	NUM
ejpam-6409	393	10	)	)	PUNCT
ejpam-6409	393	11	(	(	PUNCT
ejpam-6409	393	12	2025	2025	NUM
ejpam-6409	393	13	)	)	PUNCT
ejpam-6409	393	14	,	,	PUNCT
ejpam-6409	393	15	6409	6409	NUM
ejpam-6409	393	16	12	12	NUM
ejpam-6409	393	17	of	of	ADP
ejpam-6409	393	18	32	32	NUM
ejpam-6409	393	19	a1,1,0	a1,1,0	NOUN
ejpam-6409	393	20	=	=	PUNCT
ejpam-6409	393	21	[	[	PUNCT
ejpam-6409	393	22	10	10	NUM
ejpam-6409	393	23	10	10	NUM
ejpam-6409	393	24	10	10	NUM
ejpam-6409	393	25	10	10	NUM
ejpam-6409	393	26	10	10	NUM
ejpam-6409	393	27	10	10	NUM
ejpam-6409	393	28	10	10	NUM
ejpam-6409	393	29	10	10	NUM
ejpam-6409	393	30	10	10	NUM
ejpam-6409	393	31	10	10	NUM
ejpam-6409	393	32	10	10	NUM
ejpam-6409	393	33	10	10	NUM
ejpam-6409	393	34	10	10	NUM
ejpam-6409	393	35	10	10	NUM
ejpam-6409	393	36	10	10	NUM
ejpam-6409	393	37	10	10	NUM
ejpam-6409	393	38	00	00	NUM
ejpam-6409	393	39	02	02	NUM
ejpam-6409	393	40	04	04	NUM
ejpam-6409	393	41	06	06	NUM
ejpam-6409	394	1	20	20	NUM
ejpam-6409	394	2	22	22	NUM
ejpam-6409	394	3	24	24	NUM
ejpam-6409	394	4	26	26	NUM
ejpam-6409	394	5	40	40	NUM
ejpam-6409	394	6	42	42	NUM
ejpam-6409	394	7	44	44	NUM
ejpam-6409	394	8	46	46	NUM
ejpam-6409	394	9	60	60	NUM
ejpam-6409	394	10	62	62	NUM
ejpam-6409	394	11	64	64	NUM
ejpam-6409	394	12	66	66	NUM
ejpam-6409	394	13	]	]	PUNCT
ejpam-6409	394	14	example	example	NOUN
ejpam-6409	394	15	4.3	4.3	NUM
ejpam-6409	394	16	:	:	PUNCT
ejpam-6409	394	17	the	the	DET
ejpam-6409	394	18	additive	additive	ADJ
ejpam-6409	394	19	code	code	NOUN
ejpam-6409	394	20	h̄(1,0,	h̄(1,0,	NOUN
ejpam-6409	394	21	...	...	PUNCT
ejpam-6409	394	22	,0	,0	PUNCT
ejpam-6409	394	23	)	)	PUNCT
ejpam-6409	394	24	is	be	AUX
ejpam-6409	394	25	generated	generate	VERB
ejpam-6409	394	26	by	by	ADP
ejpam-6409	394	27	a	a	DET
ejpam-6409	394	28	(	(	PUNCT
ejpam-6409	394	29	1,0,	1,0,	NOUN
ejpam-6409	394	30	...	...	PUNCT
ejpam-6409	394	31	,0	,0	PUNCT
ejpam-6409	394	32	)	)	PUNCT
ejpam-6409	394	33	2	2	NUM
ejpam-6409	394	34	=	=	SYM
ejpam-6409	394	35	(	(	PUNCT
ejpam-6409	394	36	10	10	NUM
ejpam-6409	394	37	)	)	PUNCT
ejpam-6409	394	38	,	,	PUNCT
ejpam-6409	395	1	so	so	ADV
ejpam-6409	395	2	h̄(1,0,	h̄(1,0,	ADJ
ejpam-6409	395	3	...	...	PUNCT
ejpam-6409	395	4	,0	,0	PUNCT
ejpam-6409	395	5	)	)	PUNCT
ejpam-6409	396	1	=	=	PUNCT
ejpam-6409	396	2	z2s	z2s	PROPN
ejpam-6409	397	1	[	[	X
ejpam-6409	397	2	ω	ω	X
ejpam-6409	397	3	]	]	X
ejpam-6409	397	4	.	.	PUNCT
ejpam-6409	398	1	this	this	DET
ejpam-6409	398	2	additive	additive	ADJ
ejpam-6409	398	3	code	code	NOUN
ejpam-6409	398	4	has	have	VERB
ejpam-6409	398	5	length	length	NOUN
ejpam-6409	398	6	n	n	NOUN
ejpam-6409	398	7	=	=	SYM
ejpam-6409	398	8	1	1	NUM
ejpam-6409	398	9	,	,	PUNCT
ejpam-6409	398	10	cardinality	cardinality	NOUN
ejpam-6409	398	11	22s	22s	NOUN
ejpam-6409	398	12	,	,	PUNCT
ejpam-6409	398	13	and	and	CCONJ
ejpam-6409	398	14	minimum	minimum	NOUN
ejpam-6409	398	15	distance	distance	NOUN
ejpam-6409	398	16	1	1	NUM
ejpam-6409	398	17	.	.	PUNCT
ejpam-6409	399	1	thus	thus	ADV
ejpam-6409	399	2	,	,	PUNCT
ejpam-6409	399	3	h(1,0,	h(1,0,	NOUN
ejpam-6409	399	4	...	...	PUNCT
ejpam-6409	399	5	,0	,0	PUNCT
ejpam-6409	399	6	)	)	PUNCT
ejpam-6409	399	7	=	=	SYM
ejpam-6409	399	8	φs(h̄(1,0,	φs(h̄(1,0,	NOUN
ejpam-6409	399	9	...	...	PUNCT
ejpam-6409	399	10	,0	,0	PUNCT
ejpam-6409	399	11	)	)	PUNCT
ejpam-6409	399	12	)	)	PUNCT
ejpam-6409	399	13	has	have	VERB
ejpam-6409	399	14	length	length	NOUN
ejpam-6409	399	15	n	n	NOUN
ejpam-6409	399	16	=	=	SYM
ejpam-6409	399	17	22(s−1	22(s−1	NUM
ejpam-6409	399	18	)	)	PUNCT
ejpam-6409	399	19	,	,	PUNCT
ejpam-6409	399	20	cardinality	cardinality	NOUN
ejpam-6409	399	21	4n	4n	NOUN
ejpam-6409	399	22	=	=	SYM
ejpam-6409	399	23	22s	22s	NOUN
ejpam-6409	399	24	,	,	PUNCT
ejpam-6409	399	25	and	and	CCONJ
ejpam-6409	399	26	minimum	minimum	NOUN
ejpam-6409	399	27	distance	distance	NOUN
ejpam-6409	399	28	3	3	NUM
ejpam-6409	399	29	.	.	PUNCT
ejpam-6409	400	1	since	since	SCONJ
ejpam-6409	400	2	n	n	ADV
ejpam-6409	400	3	4	4	NUM
ejpam-6409	400	4	=	=	SYM
ejpam-6409	400	5	3	3	NUM
ejpam-6409	400	6	·	·	SYM
ejpam-6409	400	7	22(s−2	22(s−2	NUM
ejpam-6409	400	8	)	)	PUNCT
ejpam-6409	400	9	,	,	PUNCT
ejpam-6409	400	10	it	it	PRON
ejpam-6409	400	11	is	be	AUX
ejpam-6409	400	12	clearly	clearly	ADV
ejpam-6409	400	13	a	a	DET
ejpam-6409	400	14	binary	binary	ADJ
ejpam-6409	400	15	generalized	generalize	VERB
ejpam-6409	400	16	hadamard	hadamard	ADJ
ejpam-6409	400	17	code	code	NOUN
ejpam-6409	400	18	and	and	CCONJ
ejpam-6409	400	19	also	also	ADV
ejpam-6409	400	20	a	a	DET
ejpam-6409	400	21	linear	linear	PROPN
ejpam-6409	400	22	code	code	NOUN
ejpam-6409	400	23	.	.	PUNCT
ejpam-6409	401	1	example	example	NOUN
ejpam-6409	401	2	4.4	4.4	NUM
ejpam-6409	401	3	:	:	PUNCT
ejpam-6409	401	4	for	for	ADP
ejpam-6409	401	5	λ	λ	PROPN
ejpam-6409	401	6	=	=	SYM
ejpam-6409	401	7	1	1	NUM
ejpam-6409	401	8	,	,	PUNCT
ejpam-6409	401	9	the	the	DET
ejpam-6409	401	10	normalized	normalize	VERB
ejpam-6409	401	11	gh	gh	PROPN
ejpam-6409	401	12	matrix	matrix	NOUN
ejpam-6409	401	13	is	be	AUX
ejpam-6409	401	14	given	give	VERB
ejpam-6409	401	15	as	as	ADP
ejpam-6409	401	16	:	:	PUNCT
ejpam-6409	401	17	h(4	h(4	PROPN
ejpam-6409	401	18	,	,	PUNCT
ejpam-6409	401	19	1	1	NUM
ejpam-6409	401	20	)	)	PUNCT
ejpam-6409	401	21	=	=	SYM
ejpam-6409	401	22			NOUN
ejpam-6409	401	23	00	00	NUM
ejpam-6409	401	24	00	00	NUM
ejpam-6409	401	25	00	00	NUM
ejpam-6409	401	26	00	00	NUM
ejpam-6409	401	27	00	00	NUM
ejpam-6409	401	28	01	01	NUM
ejpam-6409	402	1	10	10	NUM
ejpam-6409	402	2	11	11	NUM
ejpam-6409	402	3	00	00	NUM
ejpam-6409	402	4	11	11	NUM
ejpam-6409	402	5	01	01	NUM
ejpam-6409	402	6	10	10	NUM
ejpam-6409	402	7	00	00	NUM
ejpam-6409	402	8	10	10	NUM
ejpam-6409	402	9	11	11	NUM
ejpam-6409	402	10	01	01	NUM
ejpam-6409	402	11			NOUN
ejpam-6409	402	12	then	then	ADV
ejpam-6409	402	13	,	,	PUNCT
ejpam-6409	402	14	fh	fh	PROPN
ejpam-6409	402	15	=	=	PUNCT
ejpam-6409	402	16	{	{	PUNCT
ejpam-6409	402	17	(	(	PUNCT
ejpam-6409	402	18	00	00	NUM
ejpam-6409	402	19	,	,	PUNCT
ejpam-6409	402	20	00	00	NUM
ejpam-6409	402	21	,	,	PUNCT
ejpam-6409	402	22	00	00	NUM
ejpam-6409	402	23	,	,	PUNCT
ejpam-6409	402	24	00	00	NUM
ejpam-6409	402	25	)	)	PUNCT
ejpam-6409	402	26	,	,	PUNCT
ejpam-6409	402	27	(	(	PUNCT
ejpam-6409	402	28	00	00	NUM
ejpam-6409	402	29	,	,	PUNCT
ejpam-6409	402	30	01	01	NUM
ejpam-6409	402	31	,	,	PUNCT
ejpam-6409	402	32	10	10	NUM
ejpam-6409	402	33	,	,	PUNCT
ejpam-6409	402	34	11	11	NUM
ejpam-6409	402	35	)	)	PUNCT
ejpam-6409	402	36	,	,	PUNCT
ejpam-6409	402	37	(	(	PUNCT
ejpam-6409	402	38	00	00	NUM
ejpam-6409	402	39	,	,	PUNCT
ejpam-6409	402	40	11	11	NUM
ejpam-6409	402	41	,	,	PUNCT
ejpam-6409	402	42	01	01	NUM
ejpam-6409	402	43	,	,	PUNCT
ejpam-6409	402	44	10	10	NUM
ejpam-6409	402	45	)	)	PUNCT
ejpam-6409	402	46	,	,	PUNCT
ejpam-6409	402	47	(	(	PUNCT
ejpam-6409	402	48	00	00	NUM
ejpam-6409	402	49	,	,	PUNCT
ejpam-6409	402	50	10	10	NUM
ejpam-6409	402	51	,	,	PUNCT
ejpam-6409	402	52	11	11	NUM
ejpam-6409	402	53	,	,	PUNCT
ejpam-6409	402	54	01	01	NUM
ejpam-6409	402	55	)	)	PUNCT
ejpam-6409	402	56	}	}	PUNCT
ejpam-6409	402	57	,	,	PUNCT
ejpam-6409	402	58	and	and	CCONJ
ejpam-6409	402	59	ch	ch	NOUN
ejpam-6409	402	60	=	=	PUNCT
ejpam-6409	402	61	⋃	⋃	PROPN
ejpam-6409	402	62	α∈z2[ω	α∈z2[ω	ADV
ejpam-6409	402	63	]	]	PUNCT
ejpam-6409	402	64	(	(	PUNCT
ejpam-6409	402	65	fh	fh	PROPN
ejpam-6409	402	66	+	+	CCONJ
ejpam-6409	402	67	α	α	PROPN
ejpam-6409	402	68	·	·	PUNCT
ejpam-6409	402	69	10	10	NUM
ejpam-6409	402	70	)	)	PUNCT
ejpam-6409	402	71	.	.	PUNCT
ejpam-6409	403	1	here	here	ADV
ejpam-6409	403	2	,	,	PUNCT
ejpam-6409	403	3	ch	ch	NOUN
ejpam-6409	403	4	is	be	AUX
ejpam-6409	403	5	a	a	DET
ejpam-6409	403	6	linear	linear	PROPN
ejpam-6409	403	7	gh	gh	PROPN
ejpam-6409	403	8	code	code	PROPN
ejpam-6409	403	9	over	over	ADP
ejpam-6409	403	10	z2[ω	z2[ω	NOUN
ejpam-6409	403	11	]	]	PUNCT
ejpam-6409	403	12	of	of	ADP
ejpam-6409	403	13	length	length	NOUN
ejpam-6409	403	14	4	4	NUM
ejpam-6409	403	15	,	,	PUNCT
ejpam-6409	403	16	and	and	CCONJ
ejpam-6409	403	17	ch	ch	NOUN
ejpam-6409	403	18	=	=	SYM
ejpam-6409	403	19	h1,0	h1,0	PROPN
ejpam-6409	403	20	=	=	SYM
ejpam-6409	403	21	φs(h̄1,0	φs(h̄1,0	PROPN
ejpam-6409	403	22	)	)	PUNCT
ejpam-6409	403	23	=	=	PUNCT
ejpam-6409	404	1	φs(z4[ω	φs(z4[ω	ADV
ejpam-6409	404	2	]	]	PUNCT
ejpam-6409	404	3	)	)	PUNCT
ejpam-6409	404	4	,	,	PUNCT
ejpam-6409	404	5	where	where	SCONJ
ejpam-6409	404	6	h̄1,0	h̄1,0	PROPN
ejpam-6409	404	7	is	be	AUX
ejpam-6409	404	8	generated	generate	VERB
ejpam-6409	404	9	by	by	ADP
ejpam-6409	404	10	a1,0	a1,0	PROPN
ejpam-6409	404	11	2	2	NUM
ejpam-6409	404	12	=	=	SYM
ejpam-6409	404	13	(	(	PUNCT
ejpam-6409	404	14	10	10	NUM
ejpam-6409	404	15	)	)	PUNCT
ejpam-6409	404	16	.	.	PUNCT
ejpam-6409	405	1	theorem	theorem	VERB
ejpam-6409	405	2	4.1	4.1	NUM
ejpam-6409	405	3	:	:	PUNCT
ejpam-6409	405	4	let	let	VERB
ejpam-6409	405	5	t1	t1	VERB
ejpam-6409	405	6	,	,	PUNCT
ejpam-6409	405	7	.	.	PUNCT
ejpam-6409	405	8	.	.	PUNCT
ejpam-6409	406	1	.	.	PUNCT
ejpam-6409	407	1	,	,	PUNCT
ejpam-6409	407	2	ts	ts	AUX
ejpam-6409	407	3	be	be	AUX
ejpam-6409	407	4	non	non	ADJ
ejpam-6409	407	5	-	-	ADJ
ejpam-6409	407	6	negative	negative	ADJ
ejpam-6409	407	7	integers	integer	NOUN
ejpam-6409	407	8	with	with	ADP
ejpam-6409	407	9	t1	t1	PROPN
ejpam-6409	407	10	≥	≥	NUM
ejpam-6409	407	11	1	1	NUM
ejpam-6409	407	12	.	.	PUNCT
ejpam-6409	408	1	the	the	DET
ejpam-6409	408	2	z2s	z2s	PROPN
ejpam-6409	408	3	[	[	X
ejpam-6409	408	4	ω]-linear	ω]-linear	ADJ
ejpam-6409	408	5	code	code	NOUN
ejpam-6409	408	6	h(t1,	h(t1,	NOUN
ejpam-6409	408	7	...	...	PUNCT
ejpam-6409	408	8	,ts	,ts	PUNCT
ejpam-6409	408	9	)	)	PUNCT
ejpam-6409	408	10	of	of	ADP
ejpam-6409	408	11	type	type	NOUN
ejpam-6409	408	12	(	(	PUNCT
ejpam-6409	408	13	n	n	CCONJ
ejpam-6409	408	14	,	,	PUNCT
ejpam-6409	408	15	t1	t1	NOUN
ejpam-6409	408	16	,	,	PUNCT
ejpam-6409	408	17	.	.	PUNCT
ejpam-6409	408	18	.	.	PUNCT
ejpam-6409	408	19	.	.	PUNCT
ejpam-6409	409	1	,	,	PUNCT
ejpam-6409	409	2	ts	ts	PROPN
ejpam-6409	409	3	)	)	PUNCT
ejpam-6409	409	4	is	be	AUX
ejpam-6409	409	5	a	a	DET
ejpam-6409	409	6	generalized	generalized	ADJ
ejpam-6409	409	7	hadamard	hadamard	NOUN
ejpam-6409	409	8	(	(	PUNCT
ejpam-6409	409	9	gh	gh	PROPN
ejpam-6409	409	10	)	)	PUNCT
ejpam-6409	409	11	code	code	NOUN
ejpam-6409	409	12	over	over	ADP
ejpam-6409	409	13	z2s	z2s	PROPN
ejpam-6409	410	1	[	[	X
ejpam-6409	410	2	ω	ω	X
ejpam-6409	410	3	]	]	X
ejpam-6409	410	4	of	of	ADP
ejpam-6409	410	5	length	length	NOUN
ejpam-6409	410	6	22	22	NUM
ejpam-6409	410	7	t	t	PROPN
ejpam-6409	410	8	,	,	PUNCT
ejpam-6409	410	9	where	where	SCONJ
ejpam-6409	410	10	t	t	NOUN
ejpam-6409	410	11	=	=	SYM
ejpam-6409	410	12	(	(	PUNCT
ejpam-6409	410	13	s∑	s∑	PROPN
ejpam-6409	410	14	i=1	i=1	PROPN
ejpam-6409	411	1	(	(	PUNCT
ejpam-6409	411	2	s−	s−	PROPN
ejpam-6409	411	3	i+	i+	PUNCT
ejpam-6409	411	4	1	1	NUM
ejpam-6409	411	5	)	)	PUNCT
ejpam-6409	411	6	·	·	PUNCT
ejpam-6409	411	7	ti	ti	X
ejpam-6409	411	8	)	)	PUNCT
ejpam-6409	411	9	−	−	PROPN
ejpam-6409	411	10	1	1	NUM
ejpam-6409	411	11	and	and	CCONJ
ejpam-6409	411	12	n	n	CCONJ
ejpam-6409	411	13	=	=	SYM
ejpam-6409	411	14	22(t−s+1	22(t−s+1	NUM
ejpam-6409	411	15	)	)	PUNCT
ejpam-6409	411	16	.	.	PUNCT
ejpam-6409	412	1	muhammad	muhammad	PROPN
ejpam-6409	412	2	sajjad	sajjad	PROPN
ejpam-6409	412	3	et	et	PROPN
ejpam-6409	412	4	al	al	PROPN
ejpam-6409	412	5	.	.	PUNCT
ejpam-6409	412	6	/	/	SYM
ejpam-6409	412	7	eur	eur	PROPN
ejpam-6409	412	8	.	.	PUNCT
ejpam-6409	413	1	j.	j.	PROPN
ejpam-6409	413	2	pure	pure	PROPN
ejpam-6409	413	3	appl	appl	PROPN
ejpam-6409	413	4	.	.	PROPN
ejpam-6409	413	5	math	math	PROPN
ejpam-6409	413	6	,	,	PUNCT
ejpam-6409	413	7	18	18	NUM
ejpam-6409	413	8	(	(	PUNCT
ejpam-6409	413	9	3	3	NUM
ejpam-6409	413	10	)	)	PUNCT
ejpam-6409	413	11	(	(	PUNCT
ejpam-6409	413	12	2025	2025	NUM
ejpam-6409	413	13	)	)	PUNCT
ejpam-6409	413	14	,	,	PUNCT
ejpam-6409	413	15	6409	6409	NUM
ejpam-6409	413	16	13	13	NUM
ejpam-6409	413	17	of	of	ADP
ejpam-6409	413	18	32	32	NUM
ejpam-6409	413	19	proof	proof	NOUN
ejpam-6409	413	20	.	.	PUNCT
ejpam-6409	414	1	let	let	VERB
ejpam-6409	414	2	h̄	h̄	PRON
ejpam-6409	414	3	=	=	SYM
ejpam-6409	414	4	h̄(t1,	h̄(t1,	PROPN
ejpam-6409	414	5	...	...	PUNCT
ejpam-6409	414	6	,ts	,ts	PUNCT
ejpam-6409	414	7	)	)	PUNCT
ejpam-6409	414	8	be	be	AUX
ejpam-6409	414	9	the	the	DET
ejpam-6409	414	10	z2s	z2s	PROPN
ejpam-6409	414	11	[	[	X
ejpam-6409	414	12	ω]-additive	ω]-additive	X
ejpam-6409	414	13	code	code	NOUN
ejpam-6409	414	14	of	of	ADP
ejpam-6409	414	15	length	length	NOUN
ejpam-6409	414	16	n.	n.	PROPN
ejpam-6409	414	17	consider	consider	VERB
ejpam-6409	414	18	the	the	DET
ejpam-6409	414	19	matrix	matrix	NOUN
ejpam-6409	414	20	a	a	DET
ejpam-6409	414	21	=	=	PUNCT
ejpam-6409	414	22	a	a	X
ejpam-6409	414	23	(	(	PUNCT
ejpam-6409	414	24	t1,t2,	t1,t2,	NOUN
ejpam-6409	414	25	...	...	PUNCT
ejpam-6409	414	26	,ts	,ts	PUNCT
ejpam-6409	414	27	)	)	PUNCT
ejpam-6409	414	28	2	2	NUM
ejpam-6409	414	29	as	as	ADP
ejpam-6409	414	30	its	its	PRON
ejpam-6409	414	31	generator	generator	NOUN
ejpam-6409	414	32	.	.	PUNCT
ejpam-6409	415	1	the	the	DET
ejpam-6409	415	2	additive	additive	ADJ
ejpam-6409	415	3	code	code	NOUN
ejpam-6409	415	4	can	can	AUX
ejpam-6409	415	5	be	be	AUX
ejpam-6409	415	6	written	write	VERB
ejpam-6409	415	7	as	as	ADP
ejpam-6409	415	8	h̄	h̄	NOUN
ejpam-6409	415	9	=	=	PUNCT
ejpam-6409	415	10	⋃	⋃	NOUN
ejpam-6409	415	11	λ∈z2[ω	λ∈z2[ω	NOUN
ejpam-6409	415	12	]	]	PUNCT
ejpam-6409	415	13	(	(	PUNCT
ejpam-6409	415	14	ah̄	ah̄	NOUN
ejpam-6409	415	15	+	+	CCONJ
ejpam-6409	415	16	λ	λ	PROPN
ejpam-6409	415	17	·	·	PUNCT
ejpam-6409	415	18	2s−1	2s−1	NUM
ejpam-6409	415	19	)	)	PUNCT
ejpam-6409	415	20	,	,	PUNCT
ejpam-6409	415	21	where	where	SCONJ
ejpam-6409	415	22	ah̄	ah̄	PROPN
ejpam-6409	415	23	=	=	PUNCT
ejpam-6409	415	24	{	{	PUNCT
ejpam-6409	415	25	h	h	NOUN
ejpam-6409	415	26	mod	mod	PROPN
ejpam-6409	415	27	2s−1	2s−1	NUM
ejpam-6409	415	28	:	:	PUNCT
ejpam-6409	415	29	h	h	PROPN
ejpam-6409	415	30	∈	∈	PROPN
ejpam-6409	415	31	h̄	h̄	NOUN
ejpam-6409	415	32	}	}	PUNCT
ejpam-6409	415	33	and	and	CCONJ
ejpam-6409	415	34	ah̄	ah̄	PROPN
ejpam-6409	415	35	+	+	CCONJ
ejpam-6409	415	36	λ	λ	PROPN
ejpam-6409	415	37	·	·	SYM
ejpam-6409	415	38	2s−1	2s−1	NUM
ejpam-6409	415	39	=	=	SYM
ejpam-6409	415	40	{	{	PUNCT
ejpam-6409	415	41	h+	h+	X
ejpam-6409	415	42	λ	λ	X
ejpam-6409	415	43	·	·	PUNCT
ejpam-6409	415	44	2s−1	2s−1	NUM
ejpam-6409	415	45	:	:	PUNCT
ejpam-6409	415	46	h	h	PROPN
ejpam-6409	415	47	∈	∈	PROPN
ejpam-6409	415	48	h̄	h̄	NOUN
ejpam-6409	415	49	}	}	PUNCT
ejpam-6409	415	50	.	.	PUNCT
ejpam-6409	416	1	then	then	ADV
ejpam-6409	416	2	by	by	ADP
ejpam-6409	416	3	lemma	lemma	PROPN
ejpam-6409	416	4	3.1	3.1	NUM
ejpam-6409	416	5	,	,	PUNCT
ejpam-6409	416	6	h	h	NOUN
ejpam-6409	416	7	=	=	SYM
ejpam-6409	416	8	φs(h̄	φs(h̄	PROPN
ejpam-6409	416	9	)	)	PUNCT
ejpam-6409	416	10	=	=	PUNCT
ejpam-6409	416	11	⋃	⋃	NOUN
ejpam-6409	416	12	λ∈z2[ω	λ∈z2[ω	NOUN
ejpam-6409	416	13	]	]	PUNCT
ejpam-6409	416	14	(	(	PUNCT
ejpam-6409	416	15	φs(ah̄	φs(ah̄	NUM
ejpam-6409	416	16	)	)	PUNCT
ejpam-6409	416	17	+	+	CCONJ
ejpam-6409	416	18	λ	λ	X
ejpam-6409	416	19	·	·	PUNCT
ejpam-6409	416	20	10	10	NUM
ejpam-6409	416	21	)	)	PUNCT
ejpam-6409	416	22	.	.	PUNCT
ejpam-6409	417	1	the	the	DET
ejpam-6409	417	2	code	code	NOUN
ejpam-6409	417	3	h	h	NOUN
ejpam-6409	417	4	has	have	VERB
ejpam-6409	417	5	length	length	NOUN
ejpam-6409	417	6	22	22	NUM
ejpam-6409	417	7	t	t	NOUN
ejpam-6409	417	8	=	=	SYM
ejpam-6409	417	9	n	n	PROPN
ejpam-6409	417	10	·	·	PUNCT
ejpam-6409	417	11	22(s−1	22(s−1	NUM
ejpam-6409	417	12	)	)	PUNCT
ejpam-6409	417	13	and	and	CCONJ
ejpam-6409	417	14	cardinality	cardinality	NOUN
ejpam-6409	417	15	22(t+1	22(t+1	NUM
ejpam-6409	417	16	)	)	PUNCT
ejpam-6409	417	17	=	=	SYM
ejpam-6409	417	18	n	n	PART
ejpam-6409	417	19	·	·	PUNCT
ejpam-6409	417	20	22s	22s	NUM
ejpam-6409	417	21	.	.	PUNCT
ejpam-6409	418	1	it	it	PRON
ejpam-6409	418	2	is	be	AUX
ejpam-6409	418	3	enough	enough	ADJ
ejpam-6409	418	4	to	to	PART
ejpam-6409	418	5	show	show	VERB
ejpam-6409	418	6	that	that	SCONJ
ejpam-6409	418	7	φs(ah̄	φs(ah̄	NUM
ejpam-6409	418	8	)	)	PUNCT
ejpam-6409	418	9	is	be	AUX
ejpam-6409	418	10	the	the	DET
ejpam-6409	418	11	set	set	NOUN
ejpam-6409	418	12	of	of	ADP
ejpam-6409	418	13	rows	row	NOUN
ejpam-6409	418	14	of	of	ADP
ejpam-6409	418	15	a	a	DET
ejpam-6409	418	16	generalized	generalized	ADJ
ejpam-6409	418	17	hadamard	hadamard	ADJ
ejpam-6409	418	18	matrix	matrix	NOUN
ejpam-6409	418	19	h(22	h(22	PROPN
ejpam-6409	418	20	,	,	PUNCT
ejpam-6409	418	21	22(s−2)n	22(s−2)n	NUM
ejpam-6409	418	22	)	)	PUNCT
ejpam-6409	418	23	.	.	PUNCT
ejpam-6409	419	1	let	let	VERB
ejpam-6409	419	2	us	we	PRON
ejpam-6409	419	3	consider	consider	VERB
ejpam-6409	419	4	two	two	NUM
ejpam-6409	419	5	distinct	distinct	ADJ
ejpam-6409	419	6	elements	element	NOUN
ejpam-6409	419	7	u	u	NOUN
ejpam-6409	419	8	,	,	PUNCT
ejpam-6409	419	9	v	v	PROPN
ejpam-6409	419	10	∈	∈	PROPN
ejpam-6409	419	11	ah̄.	ah̄.	NOUN
ejpam-6409	419	12	we	we	PRON
ejpam-6409	419	13	need	need	VERB
ejpam-6409	419	14	to	to	PART
ejpam-6409	419	15	show	show	VERB
ejpam-6409	419	16	that	that	SCONJ
ejpam-6409	419	17	φs(u)−φs(v	φs(u)−φs(v	NOUN
ejpam-6409	419	18	)	)	PUNCT
ejpam-6409	419	19	contains	contain	VERB
ejpam-6409	419	20	each	each	DET
ejpam-6409	419	21	element	element	NOUN
ejpam-6409	419	22	of	of	ADP
ejpam-6409	419	23	z2[ω	z2[ω	NOUN
ejpam-6409	419	24	]	]	PUNCT
ejpam-6409	419	25	exactly	exactly	ADV
ejpam-6409	419	26	22(s−2)n	22(s−2)n	NUM
ejpam-6409	419	27	times	time	NOUN
ejpam-6409	419	28	.	.	PUNCT
ejpam-6409	420	1	we	we	PRON
ejpam-6409	420	2	analyze	analyze	VERB
ejpam-6409	420	3	two	two	NUM
ejpam-6409	420	4	scenarios	scenario	NOUN
ejpam-6409	420	5	based	base	VERB
ejpam-6409	420	6	on	on	ADP
ejpam-6409	420	7	the	the	DET
ejpam-6409	420	8	order	order	NOUN
ejpam-6409	420	9	of	of	ADP
ejpam-6409	420	10	u	u	NOUN
ejpam-6409	420	11	−	−	PROPN
ejpam-6409	420	12	v	v	NOUN
ejpam-6409	420	13	:	:	PUNCT
ejpam-6409	420	14	case	case	NOUN
ejpam-6409	420	15	1	1	NUM
ejpam-6409	420	16	:	:	PUNCT
ejpam-6409	420	17	if	if	SCONJ
ejpam-6409	420	18	ord(u	ord(u	PROPN
ejpam-6409	420	19	−	−	PROPN
ejpam-6409	420	20	v	v	NOUN
ejpam-6409	420	21	)	)	PUNCT
ejpam-6409	420	22	=	=	SYM
ejpam-6409	420	23	2	2	NUM
ejpam-6409	420	24	,	,	PUNCT
ejpam-6409	420	25	then	then	ADV
ejpam-6409	420	26	by	by	ADP
ejpam-6409	420	27	the	the	DET
ejpam-6409	420	28	established	establish	VERB
ejpam-6409	420	29	construction	construction	NOUN
ejpam-6409	420	30	,	,	PUNCT
ejpam-6409	420	31	u	u	NOUN
ejpam-6409	420	32	−	−	PROPN
ejpam-6409	420	33	v	v	NOUN
ejpam-6409	420	34	includes	include	VERB
ejpam-6409	420	35	all	all	DET
ejpam-6409	420	36	elements	element	NOUN
ejpam-6409	420	37	of	of	ADP
ejpam-6409	420	38	2s−1	2s−1	NUM
ejpam-6409	420	39	·	·	PUNCT
ejpam-6409	420	40	z2s	z2s	PUNCT
ejpam-6409	421	1	[	[	X
ejpam-6409	421	2	ω	ω	X
ejpam-6409	421	3	]	]	X
ejpam-6409	421	4	exactly	exactly	ADV
ejpam-6409	421	5	n/4	n/4	NUM
ejpam-6409	421	6	times	time	NOUN
ejpam-6409	421	7	.	.	PUNCT
ejpam-6409	422	1	therefore	therefore	ADV
ejpam-6409	422	2	,	,	PUNCT
ejpam-6409	422	3	φs(u	φs(u	PUNCT
ejpam-6409	422	4	−	−	PROPN
ejpam-6409	422	5	v	v	NOUN
ejpam-6409	422	6	)	)	PUNCT
ejpam-6409	422	7	includes	include	VERB
ejpam-6409	422	8	all	all	DET
ejpam-6409	422	9	elements	element	NOUN
ejpam-6409	422	10	of	of	ADP
ejpam-6409	422	11	z2[ω	z2[ω	NOUN
ejpam-6409	422	12	]	]	PUNCT
ejpam-6409	422	13	exactly	exactly	ADV
ejpam-6409	422	14	22(s−1	22(s−1	NUM
ejpam-6409	422	15	)	)	PUNCT
ejpam-6409	422	16	·	·	PUNCT
ejpam-6409	423	1	n	n	CCONJ
ejpam-6409	423	2	4	4	NUM
ejpam-6409	423	3	=	=	SYM
ejpam-6409	423	4	22(s−2)n	22(s−2)n	NUM
ejpam-6409	423	5	times	time	NOUN
ejpam-6409	423	6	.	.	PUNCT
ejpam-6409	424	1	by	by	ADP
ejpam-6409	424	2	proposition	proposition	NOUN
ejpam-6409	424	3	3.3	3.3	NUM
ejpam-6409	424	4	,	,	PUNCT
ejpam-6409	424	5	φs(u	φs(u	PUNCT
ejpam-6409	424	6	−	−	PROPN
ejpam-6409	424	7	v	v	NOUN
ejpam-6409	424	8	)	)	PUNCT
ejpam-6409	424	9	=	=	PUNCT
ejpam-6409	424	10	φs(u	φs(u	X
ejpam-6409	424	11	)	)	PUNCT
ejpam-6409	424	12	−	−	NOUN
ejpam-6409	424	13	φs(v	φs(v	NUM
ejpam-6409	424	14	)	)	PUNCT
ejpam-6409	424	15	,	,	PUNCT
ejpam-6409	424	16	and	and	CCONJ
ejpam-6409	424	17	hence	hence	ADV
ejpam-6409	424	18	φs(u	φs(u	NUM
ejpam-6409	424	19	)	)	PUNCT
ejpam-6409	424	20	−	−	ADP
ejpam-6409	424	21	φs(v	φs(v	NUM
ejpam-6409	424	22	)	)	PUNCT
ejpam-6409	424	23	comprises	comprise	VERB
ejpam-6409	424	24	all	all	DET
ejpam-6409	424	25	elements	element	NOUN
ejpam-6409	424	26	of	of	ADP
ejpam-6409	424	27	z2[ω	z2[ω	NOUN
ejpam-6409	424	28	]	]	PUNCT
ejpam-6409	424	29	exactly	exactly	ADV
ejpam-6409	424	30	22(s−2)n	22(s−2)n	NUM
ejpam-6409	424	31	times	time	NOUN
ejpam-6409	424	32	.	.	PUNCT
ejpam-6409	425	1	case	case	NOUN
ejpam-6409	425	2	2	2	NUM
ejpam-6409	425	3	:	:	PUNCT
ejpam-6409	425	4	if	if	SCONJ
ejpam-6409	425	5	ord(u	ord(u	PROPN
ejpam-6409	425	6	−	−	PROPN
ejpam-6409	425	7	v	v	NOUN
ejpam-6409	425	8	)	)	PUNCT
ejpam-6409	425	9	≥	≥	NOUN
ejpam-6409	425	10	2	2	NUM
ejpam-6409	425	11	,	,	PUNCT
ejpam-6409	425	12	then	then	ADV
ejpam-6409	425	13	following	follow	VERB
ejpam-6409	425	14	the	the	DET
ejpam-6409	425	15	construction	construction	NOUN
ejpam-6409	425	16	,	,	PUNCT
ejpam-6409	425	17	u	u	NOUN
ejpam-6409	425	18	−	−	PROPN
ejpam-6409	425	19	v	v	NOUN
ejpam-6409	425	20	includes	include	VERB
ejpam-6409	425	21	all	all	DET
ejpam-6409	425	22	elements	element	NOUN
ejpam-6409	425	23	of	of	ADP
ejpam-6409	425	24	2s−1	2s−1	NUM
ejpam-6409	425	25	·	·	PUNCT
ejpam-6409	425	26	z2s	z2s	PUNCT
ejpam-6409	426	1	[	[	X
ejpam-6409	426	2	ω	ω	X
ejpam-6409	426	3	]	]	X
ejpam-6409	426	4	exactly	exactly	ADV
ejpam-6409	426	5	α	α	NUM
ejpam-6409	426	6	times	time	NOUN
ejpam-6409	426	7	(	(	PUNCT
ejpam-6409	426	8	α	α	PRON
ejpam-6409	426	9	≥	≥	NOUN
ejpam-6409	426	10	0	0	NUM
ejpam-6409	426	11	)	)	PUNCT
ejpam-6409	426	12	,	,	PUNCT
ejpam-6409	426	13	and	and	CCONJ
ejpam-6409	426	14	the	the	DET
ejpam-6409	426	15	remaining	remain	VERB
ejpam-6409	426	16	n−	n−	NOUN
ejpam-6409	426	17	4α	4α	NOUN
ejpam-6409	426	18	coordinates	coordinate	NOUN
ejpam-6409	426	19	are	be	AUX
ejpam-6409	426	20	from	from	ADP
ejpam-6409	426	21	z2s	z2s	PROPN
ejpam-6409	426	22	[	[	X
ejpam-6409	426	23	ω	ω	X
ejpam-6409	426	24	]	]	PUNCT
ejpam-6409	426	25	\	\	PROPN
ejpam-6409	426	26	2s−1	2s−1	NUM
ejpam-6409	426	27	·	·	PUNCT
ejpam-6409	426	28	z2s	z2s	PUNCT
ejpam-6409	427	1	[	[	X
ejpam-6409	427	2	ω	ω	X
ejpam-6409	427	3	]	]	X
ejpam-6409	427	4	.	.	PUNCT
ejpam-6409	428	1	then	then	ADV
ejpam-6409	428	2	by	by	ADP
ejpam-6409	428	3	proposition	proposition	NOUN
ejpam-6409	428	4	3.3	3.3	NUM
ejpam-6409	428	5	,	,	PUNCT
ejpam-6409	428	6	we	we	PRON
ejpam-6409	428	7	have	have	AUX
ejpam-6409	428	8	:	:	PUNCT
ejpam-6409	428	9	φs(u)−φs(v	φs(u)−φs(v	NOUN
ejpam-6409	428	10	)	)	PUNCT
ejpam-6409	428	11	includes	include	VERB
ejpam-6409	428	12	all	all	DET
ejpam-6409	428	13	elements	element	NOUN
ejpam-6409	428	14	of	of	ADP
ejpam-6409	428	15	z2[ω	z2[ω	NOUN
ejpam-6409	428	16	]	]	PUNCT
ejpam-6409	428	17	exactly	exactly	ADV
ejpam-6409	428	18	α·22(s−1)+(n−4α)·22(s−2	α·22(s−1)+(n−4α)·22(s−2	X
ejpam-6409	428	19	)	)	PUNCT
ejpam-6409	429	1	=	=	SYM
ejpam-6409	430	1	22(s−2)n	22(s−2)n	NUM
ejpam-6409	430	2	times	time	NOUN
ejpam-6409	430	3	.	.	PUNCT
ejpam-6409	431	1	therefore	therefore	ADV
ejpam-6409	431	2	,	,	PUNCT
ejpam-6409	431	3	h	h	NOUN
ejpam-6409	431	4	is	be	AUX
ejpam-6409	431	5	a	a	DET
ejpam-6409	431	6	generalized	generalized	ADJ
ejpam-6409	431	7	hadamard	hadamard	ADJ
ejpam-6409	431	8	code	code	NOUN
ejpam-6409	431	9	over	over	ADP
ejpam-6409	431	10	z2s	z2s	PROPN
ejpam-6409	431	11	[	[	X
ejpam-6409	431	12	ω	ω	X
ejpam-6409	431	13	]	]	X
ejpam-6409	431	14	.	.	PUNCT
ejpam-6409	432	1	5	5	X
ejpam-6409	432	2	.	.	X
ejpam-6409	432	3	linearity	linearity	NOUN
ejpam-6409	432	4	of	of	ADP
ejpam-6409	432	5	z2s	z2s	PROPN
ejpam-6409	433	1	[	[	X
ejpam-6409	433	2	ω]-linear	ω]-linear	ADJ
ejpam-6409	433	3	gh	gh	PROPN
ejpam-6409	433	4	codes	code	NOUN
ejpam-6409	433	5	this	this	DET
ejpam-6409	433	6	section	section	NOUN
ejpam-6409	433	7	establishes	establish	VERB
ejpam-6409	433	8	several	several	ADJ
ejpam-6409	433	9	results	result	NOUN
ejpam-6409	433	10	concerning	concern	VERB
ejpam-6409	433	11	the	the	DET
ejpam-6409	433	12	linearity	linearity	NOUN
ejpam-6409	433	13	of	of	ADP
ejpam-6409	433	14	z2s	z2s	PROPN
ejpam-6409	434	1	[	[	X
ejpam-6409	434	2	ω]-linear	ω]-linear	ADJ
ejpam-6409	434	3	gh	gh	PROPN
ejpam-6409	434	4	codes	code	NOUN
ejpam-6409	434	5	by	by	ADP
ejpam-6409	434	6	generalizing	generalize	VERB
ejpam-6409	434	7	the	the	DET
ejpam-6409	434	8	results	result	NOUN
ejpam-6409	434	9	given	give	VERB
ejpam-6409	434	10	in	in	ADP
ejpam-6409	434	11	section	section	NOUN
ejpam-6409	434	12	4	4	NUM
ejpam-6409	434	13	.	.	PUNCT
ejpam-6409	434	14	theorem	theorem	VERB
ejpam-6409	434	15	5.1	5.1	NUM
ejpam-6409	434	16	:	:	PUNCT
ejpam-6409	434	17	the	the	DET
ejpam-6409	434	18	z2s	z2s	PROPN
ejpam-6409	434	19	[	[	X
ejpam-6409	434	20	ω]-linear	ω]-linear	ADJ
ejpam-6409	434	21	hadamard	hadamard	ADJ
ejpam-6409	434	22	codes	code	NOUN
ejpam-6409	434	23	h(1,0,	h(1,0,	NOUN
ejpam-6409	434	24	...	...	PUNCT
ejpam-6409	434	25	,0	,0	PUNCT
ejpam-6409	434	26	)	)	PUNCT
ejpam-6409	434	27	and	and	CCONJ
ejpam-6409	434	28	h(1,0,	h(1,0,	NOUN
ejpam-6409	434	29	...	...	PUNCT
ejpam-6409	434	30	,0,1,0	,0,1,0	PUNCT
ejpam-6409	434	31	)	)	PUNCT
ejpam-6409	434	32	,	,	PUNCT
ejpam-6409	434	33	with	with	ADP
ejpam-6409	434	34	s	s	PROPN
ejpam-6409	434	35	>	>	X
ejpam-6409	434	36	2	2	NUM
ejpam-6409	434	37	,	,	PUNCT
ejpam-6409	434	38	are	be	AUX
ejpam-6409	434	39	linear	linear	ADJ
ejpam-6409	434	40	.	.	PUNCT
ejpam-6409	435	1	proof	proof	NOUN
ejpam-6409	435	2	:	:	PUNCT
ejpam-6409	435	3	by	by	ADP
ejpam-6409	435	4	example	example	NOUN
ejpam-6409	435	5	4.4	4.4	NUM
ejpam-6409	435	6	,	,	PUNCT
ejpam-6409	435	7	we	we	PRON
ejpam-6409	435	8	know	know	VERB
ejpam-6409	435	9	that	that	SCONJ
ejpam-6409	435	10	h(1,0,	h(1,0,	NOUN
ejpam-6409	435	11	...	...	PUNCT
ejpam-6409	435	12	,0	,0	PUNCT
ejpam-6409	435	13	)	)	PUNCT
ejpam-6409	435	14	is	be	AUX
ejpam-6409	435	15	linear	linear	ADJ
ejpam-6409	435	16	.	.	PUNCT
ejpam-6409	436	1	now	now	ADV
ejpam-6409	436	2	,	,	PUNCT
ejpam-6409	436	3	we	we	PRON
ejpam-6409	436	4	examine	examine	VERB
ejpam-6409	436	5	h̃	h̃	PROPN
ejpam-6409	436	6	=	=	SYM
ejpam-6409	436	7	h̃(1,0,	h̃(1,0,	ADJ
ejpam-6409	436	8	...	...	PUNCT
ejpam-6409	436	9	,0,1,0	,0,1,0	PUNCT
ejpam-6409	436	10	)	)	PUNCT
ejpam-6409	436	11	and	and	CCONJ
ejpam-6409	436	12	h	h	NOUN
ejpam-6409	436	13	=	=	NOUN
ejpam-6409	436	14	φs(h̃	φs(h̃	PROPN
ejpam-6409	436	15	)	)	PUNCT
ejpam-6409	436	16	.	.	PUNCT
ejpam-6409	437	1	recall	recall	VERB
ejpam-6409	437	2	that	that	SCONJ
ejpam-6409	437	3	the	the	DET
ejpam-6409	437	4	code	code	NOUN
ejpam-6409	437	5	h	h	NOUN
ejpam-6409	437	6	of	of	ADP
ejpam-6409	437	7	length	length	NOUN
ejpam-6409	437	8	16	16	NUM
ejpam-6409	437	9	is	be	AUX
ejpam-6409	437	10	constructed	construct	VERB
ejpam-6409	437	11	from	from	ADP
ejpam-6409	437	12	:	:	PUNCT
ejpam-6409	437	13	muhammad	muhammad	PROPN
ejpam-6409	437	14	sajjad	sajjad	PROPN
ejpam-6409	437	15	et	et	PROPN
ejpam-6409	437	16	al	al	PROPN
ejpam-6409	437	17	.	.	PUNCT
ejpam-6409	437	18	/	/	SYM
ejpam-6409	437	19	eur	eur	PROPN
ejpam-6409	437	20	.	.	PUNCT
ejpam-6409	438	1	j.	j.	PROPN
ejpam-6409	438	2	pure	pure	PROPN
ejpam-6409	438	3	appl	appl	PROPN
ejpam-6409	438	4	.	.	PROPN
ejpam-6409	438	5	math	math	PROPN
ejpam-6409	438	6	,	,	PUNCT
ejpam-6409	438	7	18	18	NUM
ejpam-6409	438	8	(	(	PUNCT
ejpam-6409	438	9	3	3	NUM
ejpam-6409	438	10	)	)	PUNCT
ejpam-6409	438	11	(	(	PUNCT
ejpam-6409	438	12	2025	2025	NUM
ejpam-6409	438	13	)	)	PUNCT
ejpam-6409	438	14	,	,	PUNCT
ejpam-6409	438	15	6409	6409	NUM
ejpam-6409	438	16	14	14	NUM
ejpam-6409	438	17	of	of	ADP
ejpam-6409	438	18	32	32	NUM
ejpam-6409	438	19	a	a	PRON
ejpam-6409	438	20	(	(	PUNCT
ejpam-6409	438	21	1,0,	1,0,	NOUN
ejpam-6409	438	22	...	...	PUNCT
ejpam-6409	438	23	,0,1,0	,0,1,0	PUNCT
ejpam-6409	438	24	)	)	PUNCT
ejpam-6409	438	25	2	2	NUM
ejpam-6409	438	26	=	=	SYM
ejpam-6409	438	27	(	(	PUNCT
ejpam-6409	438	28	10	10	NUM
ejpam-6409	438	29	10	10	NUM
ejpam-6409	438	30	·	·	PUNCT
ejpam-6409	438	31	·	·	PUNCT
ejpam-6409	438	32	·	·	PUNCT
ejpam-6409	439	1	10	10	NUM
ejpam-6409	439	2	10	10	NUM
ejpam-6409	439	3	00	00	NUM
ejpam-6409	439	4	01	01	NUM
ejpam-6409	439	5	·	·	PUNCT
ejpam-6409	439	6	2s−2	2s−2	NUM
ejpam-6409	439	7	·	·	PUNCT
ejpam-6409	439	8	·	·	PUNCT
ejpam-6409	439	9	·	·	PUNCT
ejpam-6409	440	1	32	32	NUM
ejpam-6409	440	2	·	·	PUNCT
ejpam-6409	440	3	2s−2	2s−2	NUM
ejpam-6409	440	4	33	33	NUM
ejpam-6409	440	5	·	·	SYM
ejpam-6409	440	6	2s−2	2s−2	NUM
ejpam-6409	440	7	)	)	PUNCT
ejpam-6409	440	8	let	let	VERB
ejpam-6409	440	9	αi	αi	X
ejpam-6409	440	10	=	=	PUNCT
ejpam-6409	440	11	(	(	PUNCT
ejpam-6409	440	12	2i	2i	NUM
ejpam-6409	440	13	0	0	NUM
ejpam-6409	440	14	,	,	PUNCT
ejpam-6409	440	15	2i	2i	NOUN
ejpam-6409	440	16	0	0	NUM
ejpam-6409	440	17	,	,	PUNCT
ejpam-6409	440	18	.	.	PUNCT
ejpam-6409	440	19	.	.	PUNCT
ejpam-6409	441	1	.	.	PUNCT
ejpam-6409	442	1	,	,	PUNCT
ejpam-6409	442	2	2i	2i	NOUN
ejpam-6409	442	3	0	0	NUM
ejpam-6409	442	4	,	,	PUNCT
ejpam-6409	442	5	2i	2i	NOUN
ejpam-6409	442	6	0	0	NUM
ejpam-6409	442	7	)	)	PUNCT
ejpam-6409	442	8	for	for	ADP
ejpam-6409	442	9	0	0	NUM
ejpam-6409	442	10	≤	≤	NUM
ejpam-6409	442	11	i	i	PRON
ejpam-6409	442	12	≤	≤	NUM
ejpam-6409	442	13	s−	s−	PROPN
ejpam-6409	442	14	1	1	NUM
ejpam-6409	442	15	,	,	PUNCT
ejpam-6409	442	16	αs	αs	INTJ
ejpam-6409	442	17	=(	=(	NOUN
ejpam-6409	442	18	00,0	00,0	NOUN
ejpam-6409	442	19	2s−1,00,0	2s−1,00,0	NUM
ejpam-6409	442	20	2s−1,2s−1	2s−1,2s−1	NUM
ejpam-6409	442	21	0,2s−1	0,2s−1	NUM
ejpam-6409	443	1	2s−1,2s−1	2s−1,2s−1	NUM
ejpam-6409	443	2	0,2s−1	0,2s−1	NUM
ejpam-6409	443	3	2s−1,00,0	2s−1,00,0	NUM
ejpam-6409	443	4	2s−1,00,0	2s−1,00,0	NUM
ejpam-6409	443	5	2s−1,2s−1	2s−1,2s−1	NUM
ejpam-6409	443	6	0,2s−1	0,2s−1	NUM
ejpam-6409	443	7	2s−1,2s−1	2s−1,2s−1	NUM
ejpam-6409	443	8	0,2s−1	0,2s−1	NUM
ejpam-6409	443	9	2s−1	2s−1	NUM
ejpam-6409	443	10	)	)	PUNCT
ejpam-6409	443	11	αs+1	αs+1	NOUN
ejpam-6409	443	12	=	=	SYM
ejpam-6409	443	13	(	(	PUNCT
ejpam-6409	443	14	00	00	NUM
ejpam-6409	443	15	,	,	PUNCT
ejpam-6409	443	16	01	01	NUM
ejpam-6409	443	17	·	·	SYM
ejpam-6409	443	18	2s−2	2s−2	NUM
ejpam-6409	443	19	,	,	PUNCT
ejpam-6409	443	20	.	.	PUNCT
ejpam-6409	443	21	.	.	PUNCT
ejpam-6409	444	1	.	.	PUNCT
ejpam-6409	445	1	,	,	PUNCT
ejpam-6409	445	2	33	33	NUM
ejpam-6409	445	3	·	·	SYM
ejpam-6409	445	4	2s−2	2s−2	NUM
ejpam-6409	445	5	)	)	PUNCT
ejpam-6409	445	6	suppose	suppose	VERB
ejpam-6409	445	7	c	c	NOUN
ejpam-6409	445	8	represents	represent	VERB
ejpam-6409	445	9	the	the	DET
ejpam-6409	445	10	linear	linear	PROPN
ejpam-6409	445	11	code	code	NOUN
ejpam-6409	445	12	constructed	construct	VERB
ejpam-6409	445	13	from	from	ADP
ejpam-6409	445	14	b	b	X
ejpam-6409	445	15	=	=	PUNCT
ejpam-6409	445	16	{	{	PUNCT
ejpam-6409	445	17	φs(αi	φs(αi	PROPN
ejpam-6409	445	18	)	)	PUNCT
ejpam-6409	445	19	:	:	PUNCT
ejpam-6409	445	20	0	0	NUM
ejpam-6409	445	21	≤	≤	NUM
ejpam-6409	445	22	i	i	PRON
ejpam-6409	445	23	≤	≤	NOUN
ejpam-6409	445	24	s+	s+	PUNCT
ejpam-6409	445	25	1	1	NUM
ejpam-6409	445	26	}	}	PUNCT
ejpam-6409	445	27	.	.	PUNCT
ejpam-6409	446	1	we	we	PRON
ejpam-6409	446	2	now	now	ADV
ejpam-6409	446	3	prove	prove	VERB
ejpam-6409	446	4	that	that	SCONJ
ejpam-6409	446	5	c	c	PROPN
ejpam-6409	446	6	⊆	⊆	NUM
ejpam-6409	446	7	h.	h.	NOUN
ejpam-6409	446	8	let	let	VERB
ejpam-6409	446	9	c	c	NOUN
ejpam-6409	446	10	=	=	SYM
ejpam-6409	446	11	∑s+1	∑s+1	PROPN
ejpam-6409	446	12	i=0	i=0	PROPN
ejpam-6409	446	13	λiφs(αi	λiφs(αi	NOUN
ejpam-6409	446	14	)	)	PUNCT
ejpam-6409	446	15	∈	∈	PROPN
ejpam-6409	446	16	c	c	NOUN
ejpam-6409	446	17	,	,	PUNCT
ejpam-6409	446	18	where	where	SCONJ
ejpam-6409	446	19	λi	λi	ADP
ejpam-6409	446	20	∈	∈	PROPN
ejpam-6409	446	21	z2[ω	z2[ω	NOUN
ejpam-6409	446	22	]	]	PUNCT
ejpam-6409	446	23	.	.	PUNCT
ejpam-6409	447	1	by	by	ADP
ejpam-6409	447	2	corollary	corollary	ADJ
ejpam-6409	447	3	3.5	3.5	NUM
ejpam-6409	447	4	,	,	PUNCT
ejpam-6409	447	5	it	it	PRON
ejpam-6409	447	6	is	be	AUX
ejpam-6409	447	7	sufficient	sufficient	ADJ
ejpam-6409	447	8	to	to	PART
ejpam-6409	447	9	observe	observe	VERB
ejpam-6409	447	10	:	:	PUNCT
ejpam-6409	447	11	c′	c′	ADJ
ejpam-6409	447	12	=	=	SYM
ejpam-6409	447	13	λs+1φs(αs+1	λs+1φs(αs+1	NOUN
ejpam-6409	447	14	)	)	PUNCT
ejpam-6409	448	1	+	+	CCONJ
ejpam-6409	448	2	s−2∑	s−2∑	PROPN
ejpam-6409	448	3	i=0	i=0	PROPN
ejpam-6409	448	4	λiφs(αi	λiφs(αi	NOUN
ejpam-6409	448	5	)	)	PUNCT
ejpam-6409	448	6	∈	∈	PROPN
ejpam-6409	448	7	h.	h.	NOUN
ejpam-6409	449	1	if	if	SCONJ
ejpam-6409	449	2	λs+1	λs+1	PROPN
ejpam-6409	449	3	=	=	SYM
ejpam-6409	449	4	0	0	NUM
ejpam-6409	449	5	,	,	PUNCT
ejpam-6409	449	6	then	then	ADV
ejpam-6409	449	7	c′	c′	NOUN
ejpam-6409	449	8	∈	∈	PROPN
ejpam-6409	449	9	h	h	NOUN
ejpam-6409	449	10	since	since	SCONJ
ejpam-6409	449	11	s−2∑	s−2∑	PROPN
ejpam-6409	449	12	k=0	k=0	PROPN
ejpam-6409	449	13	λiφs(αi	λiφs(αi	NOUN
ejpam-6409	449	14	)	)	PUNCT
ejpam-6409	449	15	=	=	SYM
ejpam-6409	449	16	φs	φs	PROPN
ejpam-6409	449	17	(	(	PUNCT
ejpam-6409	449	18	s−2∑	s−2∑	ADV
ejpam-6409	449	19	k=0	k=0	PROPN
ejpam-6409	449	20	λiαi	λiαi	NOUN
ejpam-6409	449	21	)	)	PUNCT
ejpam-6409	449	22	.	.	PUNCT
ejpam-6409	450	1	if	if	SCONJ
ejpam-6409	450	2	λs+1	λs+1	PROPN
ejpam-6409	450	3	=	=	SYM
ejpam-6409	450	4	10	10	NUM
ejpam-6409	450	5	,	,	PUNCT
ejpam-6409	450	6	then	then	ADV
ejpam-6409	450	7	:	:	PUNCT
ejpam-6409	450	8	c′	c′	NOUN
ejpam-6409	451	1	=	=	SYM
ejpam-6409	451	2	φs(00	φs(00	PROPN
ejpam-6409	451	3	,	,	PUNCT
ejpam-6409	451	4	01	01	NUM
ejpam-6409	451	5	·	·	SYM
ejpam-6409	451	6	2s−2	2s−2	NUM
ejpam-6409	451	7	,	,	PUNCT
ejpam-6409	451	8	.	.	PUNCT
ejpam-6409	451	9	.	.	PUNCT
ejpam-6409	451	10	.	.	PUNCT
ejpam-6409	452	1	,	,	PUNCT
ejpam-6409	452	2	33	33	NUM
ejpam-6409	452	3	·	·	SYM
ejpam-6409	452	4	2s−2	2s−2	NUM
ejpam-6409	452	5	)	)	PUNCT
ejpam-6409	453	1	+	+	CCONJ
ejpam-6409	453	2	ϕs(u	ϕs(u	ADJ
ejpam-6409	453	3	,	,	PUNCT
ejpam-6409	453	4	u	u	NOUN
ejpam-6409	453	5	,	,	PUNCT
ejpam-6409	453	6	.	.	PUNCT
ejpam-6409	453	7	.	.	PUNCT
ejpam-6409	454	1	.	.	PUNCT
ejpam-6409	455	1	,	,	PUNCT
ejpam-6409	455	2	u	u	NOUN
ejpam-6409	455	3	)	)	PUNCT
ejpam-6409	455	4	,	,	PUNCT
ejpam-6409	456	1	where	where	SCONJ
ejpam-6409	456	2	u	u	NOUN
ejpam-6409	456	3	=	=	NOUN
ejpam-6409	456	4	∑s−2	∑s−2	ADJ
ejpam-6409	456	5	k=0	k=0	PROPN
ejpam-6409	456	6	λi2	λi2	VERB
ejpam-6409	456	7	i	i	PRON
ejpam-6409	456	8	0	0	NUM
ejpam-6409	456	9	.	.	PUNCT
ejpam-6409	457	1	let	let	VERB
ejpam-6409	457	2	us	we	PRON
ejpam-6409	457	3	define	define	VERB
ejpam-6409	457	4	:	:	PUNCT
ejpam-6409	457	5	u	u	NOUN
ejpam-6409	457	6	=	=	PRON
ejpam-6409	457	7	{	{	PUNCT
ejpam-6409	457	8	(	(	PUNCT
ejpam-6409	457	9	01	01	NUM
ejpam-6409	457	10	)	)	PUNCT
ejpam-6409	457	11	·	·	PUNCT
ejpam-6409	458	1	2s−2	2s−2	NUM
ejpam-6409	458	2	,	,	PUNCT
ejpam-6409	458	3	(	(	PUNCT
ejpam-6409	458	4	03	03	NUM
ejpam-6409	458	5	)	)	PUNCT
ejpam-6409	458	6	·	·	PUNCT
ejpam-6409	459	1	2s−2	2s−2	NUM
ejpam-6409	459	2	,	,	PUNCT
ejpam-6409	459	3	.	.	PUNCT
ejpam-6409	459	4	.	.	PUNCT
ejpam-6409	459	5	.	.	PUNCT
ejpam-6409	460	1	,	,	PUNCT
ejpam-6409	460	2	(	(	PUNCT
ejpam-6409	460	3	31	31	NUM
ejpam-6409	460	4	)	)	PUNCT
ejpam-6409	460	5	·	·	PUNCT
ejpam-6409	461	1	2s−1	2s−1	NUM
ejpam-6409	461	2	,	,	PUNCT
ejpam-6409	461	3	(	(	PUNCT
ejpam-6409	461	4	33	33	NUM
ejpam-6409	461	5	)	)	PUNCT
ejpam-6409	461	6	·	·	PUNCT
ejpam-6409	461	7	2s−1	2s−1	NUM
ejpam-6409	461	8	}	}	PUNCT
ejpam-6409	461	9	u	u	NOUN
ejpam-6409	461	10	′	′	NOUN
ejpam-6409	461	11	=	=	SYM
ejpam-6409	461	12	{	{	PUNCT
ejpam-6409	461	13	(	(	PUNCT
ejpam-6409	461	14	10	10	NUM
ejpam-6409	461	15	)	)	PUNCT
ejpam-6409	461	16	·	·	PUNCT
ejpam-6409	461	17	2s−2	2s−2	NUM
ejpam-6409	461	18	,	,	PUNCT
ejpam-6409	461	19	.	.	PUNCT
ejpam-6409	461	20	.	.	PUNCT
ejpam-6409	461	21	.	.	PUNCT
ejpam-6409	462	1	,	,	PUNCT
ejpam-6409	462	2	(	(	PUNCT
ejpam-6409	462	3	13	13	NUM
ejpam-6409	462	4	)	)	PUNCT
ejpam-6409	462	5	·	·	PUNCT
ejpam-6409	463	1	2s−2	2s−2	NUM
ejpam-6409	463	2	,	,	PUNCT
ejpam-6409	463	3	(	(	PUNCT
ejpam-6409	463	4	30	30	NUM
ejpam-6409	463	5	)	)	PUNCT
ejpam-6409	463	6	·	·	PUNCT
ejpam-6409	464	1	2s−1	2s−1	NUM
ejpam-6409	464	2	,	,	PUNCT
ejpam-6409	464	3	.	.	PUNCT
ejpam-6409	464	4	.	.	PUNCT
ejpam-6409	464	5	.	.	PUNCT
ejpam-6409	465	1	,	,	PUNCT
ejpam-6409	465	2	(	(	PUNCT
ejpam-6409	465	3	33	33	NUM
ejpam-6409	465	4	)	)	PUNCT
ejpam-6409	465	5	·	·	PUNCT
ejpam-6409	466	1	2s−1	2s−1	NUM
ejpam-6409	466	2	}	}	PUNCT
ejpam-6409	466	3	u1	u1	NOUN
ejpam-6409	466	4	=	=	SYM
ejpam-6409	466	5	{	{	PUNCT
ejpam-6409	466	6	(	(	PUNCT
ejpam-6409	466	7	01	01	NUM
ejpam-6409	466	8	)	)	PUNCT
ejpam-6409	466	9	·	·	PUNCT
ejpam-6409	466	10	2s−2	2s−2	NUM
ejpam-6409	466	11	,	,	PUNCT
ejpam-6409	466	12	(	(	PUNCT
ejpam-6409	466	13	03	03	NUM
ejpam-6409	466	14	)	)	PUNCT
ejpam-6409	466	15	·	·	PUNCT
ejpam-6409	467	1	2s−2	2s−2	NUM
ejpam-6409	467	2	,	,	PUNCT
ejpam-6409	467	3	(	(	PUNCT
ejpam-6409	467	4	21	21	NUM
ejpam-6409	467	5	)	)	PUNCT
ejpam-6409	467	6	·	·	PUNCT
ejpam-6409	468	1	2s−1	2s−1	NUM
ejpam-6409	468	2	,	,	PUNCT
ejpam-6409	468	3	(	(	PUNCT
ejpam-6409	468	4	23	23	NUM
ejpam-6409	468	5	)	)	PUNCT
ejpam-6409	468	6	·	·	PUNCT
ejpam-6409	468	7	2s−1	2s−1	NUM
ejpam-6409	468	8	}	}	PUNCT
ejpam-6409	468	9	u2	u2	NOUN
ejpam-6409	468	10	=	=	SYM
ejpam-6409	468	11	{	{	PUNCT
ejpam-6409	468	12	(	(	PUNCT
ejpam-6409	468	13	10	10	NUM
ejpam-6409	468	14	)	)	PUNCT
ejpam-6409	468	15	·	·	PUNCT
ejpam-6409	468	16	2s−2	2s−2	NUM
ejpam-6409	468	17	,	,	PUNCT
ejpam-6409	468	18	(	(	PUNCT
ejpam-6409	468	19	12	12	NUM
ejpam-6409	468	20	)	)	PUNCT
ejpam-6409	468	21	·	·	PUNCT
ejpam-6409	468	22	2s−2	2s−2	NUM
ejpam-6409	468	23	,	,	PUNCT
ejpam-6409	468	24	(	(	PUNCT
ejpam-6409	468	25	30	30	NUM
ejpam-6409	468	26	)	)	PUNCT
ejpam-6409	468	27	·	·	PUNCT
ejpam-6409	469	1	2s−1	2s−1	NUM
ejpam-6409	469	2	,	,	PUNCT
ejpam-6409	469	3	(	(	PUNCT
ejpam-6409	469	4	32	32	NUM
ejpam-6409	469	5	)	)	PUNCT
ejpam-6409	469	6	·	·	PUNCT
ejpam-6409	469	7	2s−1	2s−1	X
ejpam-6409	469	8	}	}	PUNCT
ejpam-6409	469	9	u3	u3	NOUN
ejpam-6409	469	10	=	=	SYM
ejpam-6409	469	11	{	{	PUNCT
ejpam-6409	469	12	(	(	PUNCT
ejpam-6409	469	13	11	11	NUM
ejpam-6409	469	14	)	)	PUNCT
ejpam-6409	469	15	·	·	PUNCT
ejpam-6409	469	16	2s−2	2s−2	NUM
ejpam-6409	469	17	,	,	PUNCT
ejpam-6409	469	18	(	(	PUNCT
ejpam-6409	469	19	13	13	NUM
ejpam-6409	469	20	)	)	PUNCT
ejpam-6409	469	21	·	·	PUNCT
ejpam-6409	469	22	2s−2	2s−2	NUM
ejpam-6409	469	23	,	,	PUNCT
ejpam-6409	469	24	(	(	PUNCT
ejpam-6409	469	25	31	31	NUM
ejpam-6409	469	26	)	)	PUNCT
ejpam-6409	469	27	·	·	PUNCT
ejpam-6409	470	1	2s−1	2s−1	NUM
ejpam-6409	470	2	,	,	PUNCT
ejpam-6409	470	3	(	(	PUNCT
ejpam-6409	470	4	33	33	NUM
ejpam-6409	470	5	)	)	PUNCT
ejpam-6409	470	6	·	·	PUNCT
ejpam-6409	471	1	2s−1	2s−1	NUM
ejpam-6409	471	2	}	}	PUNCT
ejpam-6409	471	3	then	then	ADV
ejpam-6409	471	4	,	,	PUNCT
ejpam-6409	471	5	by	by	ADP
ejpam-6409	471	6	corollaries	corollary	NOUN
ejpam-6409	471	7	3.7	3.7	NUM
ejpam-6409	471	8	,	,	PUNCT
ejpam-6409	471	9	3.8	3.8	NUM
ejpam-6409	471	10	,	,	PUNCT
ejpam-6409	471	11	and	and	CCONJ
ejpam-6409	471	12	3.9	3.9	NUM
ejpam-6409	471	13	:	:	PUNCT
ejpam-6409	471	14	c′	c′	NOUN
ejpam-6409	471	15	=	=	SYM
ejpam-6409	471	16	φs(00	φs(00	PROPN
ejpam-6409	471	17	,	,	PUNCT
ejpam-6409	471	18	01	01	NUM
ejpam-6409	471	19	·	·	SYM
ejpam-6409	471	20	2s−2	2s−2	NUM
ejpam-6409	471	21	,	,	PUNCT
ejpam-6409	471	22	.	.	PUNCT
ejpam-6409	471	23	.	.	PUNCT
ejpam-6409	471	24	.	.	PUNCT
ejpam-6409	472	1	,	,	PUNCT
ejpam-6409	472	2	33	33	NUM
ejpam-6409	472	3	·	·	SYM
ejpam-6409	472	4	2s−2	2s−2	NUM
ejpam-6409	472	5	)	)	PUNCT
ejpam-6409	473	1	+	+	NUM
ejpam-6409	473	2	φs(u	φs(u	NUM
ejpam-6409	473	3	,	,	PUNCT
ejpam-6409	473	4	u	u	NOUN
ejpam-6409	473	5	,	,	PUNCT
ejpam-6409	473	6	.	.	PUNCT
ejpam-6409	473	7	.	.	PUNCT
ejpam-6409	474	1	.	.	PUNCT
ejpam-6409	475	1	,	,	PUNCT
ejpam-6409	475	2	u	u	NOUN
ejpam-6409	475	3	)	)	PUNCT
ejpam-6409	476	1	+	+	CCONJ
ejpam-6409	477	1	αs	αs	INTJ
ejpam-6409	477	2	.	.	PUNCT
ejpam-6409	478	1	if	if	SCONJ
ejpam-6409	478	2	u	u	PROPN
ejpam-6409	478	3	∈	∈	PROPN
ejpam-6409	478	4	u	u	NOUN
ejpam-6409	478	5	∪	∪	VERB
ejpam-6409	478	6	u	u	NOUN
ejpam-6409	478	7	′	′	NOUN
ejpam-6409	478	8	,	,	PUNCT
ejpam-6409	478	9	then	then	ADV
ejpam-6409	478	10	c′	c′	PROPN
ejpam-6409	478	11	=	=	SYM
ejpam-6409	478	12	φs(00	φs(00	PROPN
ejpam-6409	478	13	,	,	PUNCT
ejpam-6409	478	14	01	01	NUM
ejpam-6409	478	15	·	·	SYM
ejpam-6409	478	16	2s−2	2s−2	NUM
ejpam-6409	478	17	,	,	PUNCT
ejpam-6409	478	18	.	.	PUNCT
ejpam-6409	478	19	.	.	PUNCT
ejpam-6409	478	20	.	.	PUNCT
ejpam-6409	479	1	,	,	PUNCT
ejpam-6409	479	2	33	33	NUM
ejpam-6409	479	3	·	·	SYM
ejpam-6409	479	4	2s−2	2s−2	NUM
ejpam-6409	479	5	)	)	PUNCT
ejpam-6409	480	1	+	+	CCONJ
ejpam-6409	480	2	ϕs(u	ϕs(u	ADJ
ejpam-6409	480	3	,	,	PUNCT
ejpam-6409	480	4	u	u	NOUN
ejpam-6409	480	5	,	,	PUNCT
ejpam-6409	480	6	.	.	PUNCT
ejpam-6409	480	7	.	.	PUNCT
ejpam-6409	481	1	.	.	PUNCT
ejpam-6409	482	1	,	,	PUNCT
ejpam-6409	482	2	u	u	NOUN
ejpam-6409	482	3	)	)	PUNCT
ejpam-6409	482	4	,	,	PUNCT
ejpam-6409	482	5	muhammad	muhammad	PROPN
ejpam-6409	482	6	sajjad	sajjad	PROPN
ejpam-6409	482	7	et	et	PROPN
ejpam-6409	482	8	al	al	PROPN
ejpam-6409	482	9	.	.	PUNCT
ejpam-6409	482	10	/	/	SYM
ejpam-6409	482	11	eur	eur	PROPN
ejpam-6409	482	12	.	.	PUNCT
ejpam-6409	483	1	j.	j.	PROPN
ejpam-6409	483	2	pure	pure	PROPN
ejpam-6409	483	3	appl	appl	PROPN
ejpam-6409	483	4	.	.	PROPN
ejpam-6409	483	5	math	math	PROPN
ejpam-6409	483	6	,	,	PUNCT
ejpam-6409	483	7	18	18	NUM
ejpam-6409	483	8	(	(	PUNCT
ejpam-6409	483	9	3	3	NUM
ejpam-6409	483	10	)	)	PUNCT
ejpam-6409	483	11	(	(	PUNCT
ejpam-6409	483	12	2025	2025	NUM
ejpam-6409	483	13	)	)	PUNCT
ejpam-6409	483	14	,	,	PUNCT
ejpam-6409	483	15	6409	6409	NUM
ejpam-6409	483	16	15	15	NUM
ejpam-6409	483	17	of	of	ADP
ejpam-6409	483	18	32	32	NUM
ejpam-6409	483	19	otherwise	otherwise	ADV
ejpam-6409	483	20	.	.	PUNCT
ejpam-6409	484	1	in	in	ADP
ejpam-6409	484	2	both	both	DET
ejpam-6409	484	3	cases	case	NOUN
ejpam-6409	484	4	,	,	PUNCT
ejpam-6409	484	5	c′	c′	PROPN
ejpam-6409	484	6	∈	∈	PROPN
ejpam-6409	484	7	h.	h.	PROPN
ejpam-6409	484	8	for	for	ADP
ejpam-6409	484	9	λs+1	λs+1	PROPN
ejpam-6409	484	10	=	=	SYM
ejpam-6409	484	11	01	01	NUM
ejpam-6409	484	12	and	and	CCONJ
ejpam-6409	484	13	λs+1	λs+1	PROPN
ejpam-6409	484	14	=	=	SYM
ejpam-6409	484	15	11	11	NUM
ejpam-6409	484	16	,	,	PUNCT
ejpam-6409	484	17	it	it	PRON
ejpam-6409	484	18	can	can	AUX
ejpam-6409	484	19	be	be	AUX
ejpam-6409	484	20	proven	prove	VERB
ejpam-6409	484	21	similarly	similarly	ADV
ejpam-6409	484	22	using	use	VERB
ejpam-6409	484	23	corollaries	corollary	NOUN
ejpam-6409	484	24	3.7	3.7	NUM
ejpam-6409	484	25	,	,	PUNCT
ejpam-6409	484	26	3.8	3.8	NUM
ejpam-6409	484	27	,	,	PUNCT
ejpam-6409	484	28	and	and	CCONJ
ejpam-6409	484	29	3.9	3.9	NUM
ejpam-6409	484	30	.	.	PUNCT
ejpam-6409	485	1	since	since	SCONJ
ejpam-6409	485	2	|c|	|c|	PROPN
ejpam-6409	485	3	=	=	SYM
ejpam-6409	485	4	|h|	|h|	PROPN
ejpam-6409	485	5	=	=	SYM
ejpam-6409	485	6	22(s+2	22(s+2	NUM
ejpam-6409	485	7	)	)	PUNCT
ejpam-6409	485	8	,	,	PUNCT
ejpam-6409	485	9	it	it	PRON
ejpam-6409	485	10	follows	follow	VERB
ejpam-6409	485	11	that	that	SCONJ
ejpam-6409	485	12	c	c	NOUN
ejpam-6409	485	13	=	=	SYM
ejpam-6409	485	14	h	h	NOUN
ejpam-6409	485	15	,	,	PUNCT
ejpam-6409	485	16	and	and	CCONJ
ejpam-6409	485	17	therefore	therefore	ADV
ejpam-6409	485	18	,	,	PUNCT
ejpam-6409	485	19	h	h	NOUN
ejpam-6409	485	20	exhibits	exhibit	VERB
ejpam-6409	485	21	linearity	linearity	NOUN
ejpam-6409	485	22	.	.	PUNCT
ejpam-6409	486	1	theorem	theorem	VERB
ejpam-6409	486	2	5.2	5.2	NUM
ejpam-6409	486	3	:	:	PUNCT
ejpam-6409	486	4	the	the	DET
ejpam-6409	486	5	codes	code	NOUN
ejpam-6409	486	6	h(1,0,	h(1,0,	NOUN
ejpam-6409	486	7	...	...	PUNCT
ejpam-6409	486	8	,0,1,ts	,0,1,ts	PUNCT
ejpam-6409	486	9	)	)	PUNCT
ejpam-6409	486	10	and	and	CCONJ
ejpam-6409	486	11	h(1,0,	h(1,0,	NOUN
ejpam-6409	486	12	...	...	PUNCT
ejpam-6409	486	13	,0,ts	,0,ts	PROPN
ejpam-6409	486	14	)	)	PUNCT
ejpam-6409	486	15	,	,	PUNCT
ejpam-6409	486	16	with	with	ADP
ejpam-6409	486	17	s	s	PROPN
ejpam-6409	486	18	>	>	X
ejpam-6409	486	19	2	2	NUM
ejpam-6409	486	20	and	and	CCONJ
ejpam-6409	486	21	ts	ts	ADP
ejpam-6409	486	22	≥	≥	NOUN
ejpam-6409	486	23	0	0	NUM
ejpam-6409	486	24	,	,	PUNCT
ejpam-6409	486	25	are	be	AUX
ejpam-6409	486	26	the	the	DET
ejpam-6409	486	27	only	only	ADJ
ejpam-6409	486	28	z2s	z2	NOUN
ejpam-6409	487	1	[	[	X
ejpam-6409	487	2	ω]-linear	ω]-linear	ADJ
ejpam-6409	487	3	hadamard	hadamard	ADJ
ejpam-6409	487	4	codes	code	NOUN
ejpam-6409	487	5	that	that	PRON
ejpam-6409	487	6	exhibit	exhibit	VERB
ejpam-6409	487	7	linearity	linearity	NOUN
ejpam-6409	487	8	.	.	PUNCT
ejpam-6409	488	1	proof	proof	NOUN
ejpam-6409	488	2	:	:	PUNCT
ejpam-6409	488	3	initially	initially	ADV
ejpam-6409	488	4	,	,	PUNCT
ejpam-6409	488	5	we	we	PRON
ejpam-6409	488	6	prove	prove	VERB
ejpam-6409	488	7	the	the	DET
ejpam-6409	488	8	linearity	linearity	NOUN
ejpam-6409	488	9	of	of	ADP
ejpam-6409	488	10	these	these	DET
ejpam-6409	488	11	codes	code	NOUN
ejpam-6409	488	12	using	use	VERB
ejpam-6409	488	13	induction	induction	NOUN
ejpam-6409	488	14	on	on	ADP
ejpam-6409	488	15	ts	ts	PROPN
ejpam-6409	488	16	.	.	PUNCT
ejpam-6409	489	1	by	by	ADP
ejpam-6409	489	2	theorem	theorem	NOUN
ejpam-6409	489	3	5.1	5.1	NUM
ejpam-6409	489	4	,	,	PUNCT
ejpam-6409	489	5	the	the	DET
ejpam-6409	489	6	codes	code	NOUN
ejpam-6409	489	7	h(1,0,	h(1,0,	NOUN
ejpam-6409	489	8	...	...	PUNCT
ejpam-6409	489	9	,0	,0	PUNCT
ejpam-6409	489	10	)	)	PUNCT
ejpam-6409	489	11	and	and	CCONJ
ejpam-6409	489	12	h(1,0,	h(1,0,	NOUN
ejpam-6409	489	13	...	...	PUNCT
ejpam-6409	489	14	,0,1,0	,0,1,0	PUNCT
ejpam-6409	489	15	)	)	PUNCT
ejpam-6409	489	16	exhibit	exhibit	VERB
ejpam-6409	489	17	linearity	linearity	NOUN
ejpam-6409	489	18	.	.	PUNCT
ejpam-6409	490	1	we	we	PRON
ejpam-6409	490	2	hypothesize	hypothesize	VERB
ejpam-6409	490	3	that	that	SCONJ
ejpam-6409	490	4	h	h	NOUN
ejpam-6409	490	5	=	=	SYM
ejpam-6409	490	6	φs(h̄	φs(h̄	PROPN
ejpam-6409	490	7	)	)	PUNCT
ejpam-6409	490	8	,	,	PUNCT
ejpam-6409	490	9	where	where	SCONJ
ejpam-6409	490	10	h̄	h̄	NOUN
ejpam-6409	490	11	=	=	SYM
ejpam-6409	490	12	h(1,0,	h(1,0,	NOUN
ejpam-6409	490	13	...	...	PUNCT
ejpam-6409	490	14	,0,ts−1,ts	,0,ts−1,ts	PUNCT
ejpam-6409	490	15	)	)	PUNCT
ejpam-6409	490	16	,	,	PUNCT
ejpam-6409	490	17	ts−1	ts−1	NOUN
ejpam-6409	490	18	∈	∈	PROPN
ejpam-6409	490	19	{	{	PUNCT
ejpam-6409	490	20	0	0	NUM
ejpam-6409	490	21	,	,	PUNCT
ejpam-6409	490	22	1	1	NUM
ejpam-6409	490	23	}	}	PUNCT
ejpam-6409	490	24	and	and	CCONJ
ejpam-6409	490	25	ts	ts	ADP
ejpam-6409	490	26	≥	≥	NOUN
ejpam-6409	490	27	0	0	NUM
ejpam-6409	490	28	,	,	PUNCT
ejpam-6409	490	29	is	be	AUX
ejpam-6409	490	30	linear	linear	ADJ
ejpam-6409	490	31	.	.	PUNCT
ejpam-6409	491	1	now	now	ADV
ejpam-6409	491	2	,	,	PUNCT
ejpam-6409	491	3	we	we	PRON
ejpam-6409	491	4	prove	prove	VERB
ejpam-6409	491	5	that	that	SCONJ
ejpam-6409	491	6	hs	hs	PROPN
ejpam-6409	491	7	=	=	ADJ
ejpam-6409	491	8	h(1,0,	h(1,0,	PROPN
ejpam-6409	491	9	...	...	PUNCT
ejpam-6409	491	10	,0,ts−1,ts+1	,0,ts−1,ts+1	PUNCT
ejpam-6409	491	11	)	)	PUNCT
ejpam-6409	491	12	is	be	AUX
ejpam-6409	491	13	linear	linear	ADJ
ejpam-6409	491	14	.	.	PUNCT
ejpam-6409	492	1	through	through	ADP
ejpam-6409	492	2	iterative	iterative	NOUN
ejpam-6409	492	3	construction	construction	NOUN
ejpam-6409	492	4	,	,	PUNCT
ejpam-6409	492	5	hs	hs	PROPN
ejpam-6409	492	6	=	=	X
ejpam-6409	492	7	{	{	PUNCT
ejpam-6409	492	8	φs	φs	X
ejpam-6409	492	9	(	(	PUNCT
ejpam-6409	492	10	(	(	PUNCT
ejpam-6409	492	11	h	h	NOUN
ejpam-6409	492	12	,	,	PUNCT
ejpam-6409	492	13	h	h	NOUN
ejpam-6409	492	14	,	,	PUNCT
ejpam-6409	492	15	h	h	NOUN
ejpam-6409	492	16	,	,	PUNCT
ejpam-6409	492	17	h	h	NOUN
ejpam-6409	492	18	)	)	PUNCT
ejpam-6409	492	19	+	+	CCONJ
ejpam-6409	493	1	λ(00	λ(00	VERB
ejpam-6409	493	2	,	,	PUNCT
ejpam-6409	493	3	(	(	PUNCT
ejpam-6409	493	4	01)2s−1	01)2s−1	NOUN
ejpam-6409	493	5	,	,	PUNCT
ejpam-6409	493	6	(	(	PUNCT
ejpam-6409	493	7	10)2s−1	10)2s−1	NOUN
ejpam-6409	493	8	,	,	PUNCT
ejpam-6409	493	9	(	(	PUNCT
ejpam-6409	493	10	11)2s−1	11)2s−1	NOUN
ejpam-6409	493	11	)	)	PUNCT
ejpam-6409	493	12	)	)	PUNCT
ejpam-6409	493	13	:	:	PUNCT
ejpam-6409	493	14	h	h	PROPN
ejpam-6409	493	15	∈	∈	PROPN
ejpam-6409	493	16	h̄	h̄	PROPN
ejpam-6409	493	17	,	,	PUNCT
ejpam-6409	493	18	λ	λ	PROPN
ejpam-6409	493	19	∈	∈	PROPN
ejpam-6409	493	20	z2[ω	z2[ω	NOUN
ejpam-6409	493	21	]	]	PUNCT
ejpam-6409	493	22	}	}	PUNCT
ejpam-6409	493	23	,	,	PUNCT
ejpam-6409	493	24	which	which	PRON
ejpam-6409	493	25	simplifies	simplify	VERB
ejpam-6409	493	26	to	to	PART
ejpam-6409	493	27	:	:	PUNCT
ejpam-6409	493	28	{	{	PUNCT
ejpam-6409	493	29	(	(	PUNCT
ejpam-6409	493	30	φs(h	φs(h	PROPN
ejpam-6409	493	31	)	)	PUNCT
ejpam-6409	493	32	,	,	PUNCT
ejpam-6409	493	33	φs(h+	φs(h+	PROPN
ejpam-6409	493	34	λ(01)2s−1	λ(01)2s−1	NUM
ejpam-6409	493	35	)	)	PUNCT
ejpam-6409	493	36	,	,	PUNCT
ejpam-6409	493	37	φs(h+	φs(h+	PROPN
ejpam-6409	493	38	λ(10)2s−1	λ(10)2s−1	NUM
ejpam-6409	493	39	)	)	PUNCT
ejpam-6409	493	40	,	,	PUNCT
ejpam-6409	493	41	φs(h+	φs(h+	PROPN
ejpam-6409	493	42	λ(11)2s−1	λ(11)2s−1	NUM
ejpam-6409	493	43	)	)	PUNCT
ejpam-6409	493	44	)	)	PUNCT
ejpam-6409	493	45	:	:	PUNCT
ejpam-6409	493	46	h	h	PROPN
ejpam-6409	493	47	∈	∈	PROPN
ejpam-6409	493	48	h̄	h̄	PROPN
ejpam-6409	493	49	,	,	PUNCT
ejpam-6409	493	50	λ	λ	PROPN
ejpam-6409	493	51	∈	∈	PROPN
ejpam-6409	493	52	z2[ω	z2[ω	NOUN
ejpam-6409	493	53	]	]	PUNCT
ejpam-6409	493	54	}	}	PUNCT
ejpam-6409	493	55	.	.	PUNCT
ejpam-6409	494	1	by	by	ADP
ejpam-6409	494	2	corollaries	corollary	NOUN
ejpam-6409	494	3	3.1	3.1	NUM
ejpam-6409	494	4	and	and	CCONJ
ejpam-6409	494	5	3.6	3.6	NUM
ejpam-6409	494	6	,	,	PUNCT
ejpam-6409	494	7	this	this	PRON
ejpam-6409	494	8	equals	equal	VERB
ejpam-6409	494	9	:	:	PUNCT
ejpam-6409	494	10	{	{	PUNCT
ejpam-6409	494	11	(	(	PUNCT
ejpam-6409	494	12	h′,h′	h′,h′	PROPN
ejpam-6409	494	13	+	+	PROPN
ejpam-6409	494	14	λ	λ	PROPN
ejpam-6409	494	15	·	·	PUNCT
ejpam-6409	494	16	01,h′	01,h′	PROPN
ejpam-6409	495	1	+	+	CCONJ
ejpam-6409	495	2	λ	λ	PROPN
ejpam-6409	495	3	·	·	PUNCT
ejpam-6409	495	4	10,h′	10,h′	PROPN
ejpam-6409	496	1	+	+	CCONJ
ejpam-6409	496	2	λ	λ	PROPN
ejpam-6409	496	3	·	·	PUNCT
ejpam-6409	496	4	11	11	NUM
ejpam-6409	496	5	)	)	PUNCT
ejpam-6409	496	6	:	:	PUNCT
ejpam-6409	496	7	h′	h′	PROPN
ejpam-6409	496	8	∈	∈	PROPN
ejpam-6409	496	9	h	h	NOUN
ejpam-6409	496	10	,	,	PUNCT
ejpam-6409	496	11	λ	λ	PROPN
ejpam-6409	496	12	∈	∈	PROPN
ejpam-6409	496	13	z2[ω	z2[ω	NOUN
ejpam-6409	496	14	]	]	PUNCT
ejpam-6409	496	15	}	}	PUNCT
ejpam-6409	496	16	.	.	PUNCT
ejpam-6409	497	1	thus	thus	ADV
ejpam-6409	497	2	,	,	PUNCT
ejpam-6409	497	3	it	it	PRON
ejpam-6409	497	4	is	be	AUX
ejpam-6409	497	5	possible	possible	ADJ
ejpam-6409	497	6	to	to	PART
ejpam-6409	497	7	partition	partition	VERB
ejpam-6409	497	8	hs	hs	PROPN
ejpam-6409	497	9	into	into	ADP
ejpam-6409	497	10	4	4	NUM
ejpam-6409	497	11	-	-	PUNCT
ejpam-6409	497	12	blocks	block	NOUN
ejpam-6409	497	13	as	as	SCONJ
ejpam-6409	497	14	follows	follow	VERB
ejpam-6409	497	15	:	:	PUNCT
ejpam-6409	497	16	hs00	hs00	PROPN
ejpam-6409	497	17	=	=	SYM
ejpam-6409	497	18	{	{	PUNCT
ejpam-6409	497	19	(	(	PUNCT
ejpam-6409	497	20	h′,h′,h′,h′	h′,h′,h′,h′	PROPN
ejpam-6409	497	21	)	)	PUNCT
ejpam-6409	497	22	:	:	PUNCT
ejpam-6409	497	23	h′	h′	PROPN
ejpam-6409	497	24	∈	∈	PROPN
ejpam-6409	497	25	h	h	NOUN
ejpam-6409	497	26	}	}	PUNCT
ejpam-6409	497	27	,	,	PUNCT
ejpam-6409	497	28	hs01	hs01	PROPN
ejpam-6409	497	29	=	=	SYM
ejpam-6409	497	30	{	{	PUNCT
ejpam-6409	497	31	(	(	PUNCT
ejpam-6409	497	32	h′,h′	h′,h′	PROPN
ejpam-6409	497	33	+	+	PROPN
ejpam-6409	497	34	01	01	NUM
ejpam-6409	497	35	·	·	SYM
ejpam-6409	497	36	01	01	NUM
ejpam-6409	497	37	,	,	PUNCT
ejpam-6409	497	38	h′	h′	X
ejpam-6409	497	39	+	+	CCONJ
ejpam-6409	497	40	01	01	NUM
ejpam-6409	497	41	·	·	SYM
ejpam-6409	497	42	10,h′	10,h′	PROPN
ejpam-6409	498	1	+	+	CCONJ
ejpam-6409	498	2	01	01	NUM
ejpam-6409	498	3	·	·	SYM
ejpam-6409	498	4	11	11	NUM
ejpam-6409	498	5	)	)	PUNCT
ejpam-6409	498	6	:	:	PUNCT
ejpam-6409	498	7	h′	h′	PROPN
ejpam-6409	498	8	∈	∈	PROPN
ejpam-6409	498	9	h	h	NOUN
ejpam-6409	498	10	}	}	PUNCT
ejpam-6409	498	11	,	,	PUNCT
ejpam-6409	498	12	hs10	hs10	PROPN
ejpam-6409	498	13	=	=	PRON
ejpam-6409	498	14	{	{	PUNCT
ejpam-6409	498	15	(	(	PUNCT
ejpam-6409	498	16	h′,h′	h′,h′	NOUN
ejpam-6409	498	17	+	+	NUM
ejpam-6409	498	18	10	10	NUM
ejpam-6409	498	19	·	·	SYM
ejpam-6409	498	20	01	01	NUM
ejpam-6409	498	21	,	,	PUNCT
ejpam-6409	498	22	h′	h′	X
ejpam-6409	498	23	+	+	CCONJ
ejpam-6409	498	24	10	10	NUM
ejpam-6409	498	25	·	·	SYM
ejpam-6409	498	26	10,h′	10,h′	NUM
ejpam-6409	499	1	+	+	CCONJ
ejpam-6409	499	2	10	10	NUM
ejpam-6409	499	3	·	·	SYM
ejpam-6409	499	4	11	11	NUM
ejpam-6409	499	5	)	)	PUNCT
ejpam-6409	499	6	:	:	PUNCT
ejpam-6409	499	7	h′	h′	PROPN
ejpam-6409	499	8	∈	∈	PROPN
ejpam-6409	499	9	h	h	NOUN
ejpam-6409	499	10	}	}	PUNCT
ejpam-6409	499	11	,	,	PUNCT
ejpam-6409	499	12	hs11	hs11	PROPN
ejpam-6409	499	13	=	=	PRON
ejpam-6409	499	14	{	{	PUNCT
ejpam-6409	499	15	(	(	PUNCT
ejpam-6409	499	16	h′,h′	h′,h′	NOUN
ejpam-6409	499	17	+	+	NUM
ejpam-6409	499	18	11	11	NUM
ejpam-6409	499	19	·	·	SYM
ejpam-6409	499	20	01	01	NUM
ejpam-6409	499	21	,	,	PUNCT
ejpam-6409	499	22	h′	h′	X
ejpam-6409	499	23	+	+	CCONJ
ejpam-6409	500	1	11	11	NUM
ejpam-6409	500	2	·	·	SYM
ejpam-6409	500	3	10,h′	10,h′	NUM
ejpam-6409	500	4	+	+	CCONJ
ejpam-6409	500	5	11	11	NUM
ejpam-6409	500	6	·	·	SYM
ejpam-6409	500	7	11	11	NUM
ejpam-6409	500	8	)	)	PUNCT
ejpam-6409	500	9	:	:	PUNCT
ejpam-6409	500	10	h′	h′	PROPN
ejpam-6409	500	11	∈	∈	PROPN
ejpam-6409	500	12	h	h	NOUN
ejpam-6409	500	13	}	}	PUNCT
ejpam-6409	500	14	.	.	PUNCT
ejpam-6409	501	1	given	give	VERB
ejpam-6409	501	2	that	that	PRON
ejpam-6409	501	3	h	h	NOUN
ejpam-6409	501	4	is	be	AUX
ejpam-6409	501	5	linear	linear	ADJ
ejpam-6409	501	6	,	,	PUNCT
ejpam-6409	501	7	it	it	PRON
ejpam-6409	501	8	is	be	AUX
ejpam-6409	501	9	clear	clear	ADJ
ejpam-6409	501	10	that	that	SCONJ
ejpam-6409	501	11	the	the	DET
ejpam-6409	501	12	sum	sum	NOUN
ejpam-6409	501	13	of	of	ADP
ejpam-6409	501	14	any	any	DET
ejpam-6409	501	15	two	two	NUM
ejpam-6409	501	16	vectors	vector	NOUN
ejpam-6409	501	17	from	from	ADP
ejpam-6409	501	18	hs	hs	PROPN
ejpam-6409	501	19	will	will	AUX
ejpam-6409	501	20	lie	lie	VERB
ejpam-6409	501	21	in	in	ADP
ejpam-6409	501	22	one	one	NUM
ejpam-6409	501	23	of	of	ADP
ejpam-6409	501	24	the	the	DET
ejpam-6409	501	25	blocks	block	NOUN
ejpam-6409	501	26	hs00	hs00	PROPN
ejpam-6409	501	27	,	,	PUNCT
ejpam-6409	501	28	hs01	hs01	PROPN
ejpam-6409	501	29	,	,	PUNCT
ejpam-6409	501	30	hs10	hs10	PROPN
ejpam-6409	501	31	,	,	PUNCT
ejpam-6409	501	32	hs11	hs11	PROPN
ejpam-6409	501	33	.	.	PUNCT
ejpam-6409	502	1	therefore	therefore	ADV
ejpam-6409	502	2	,	,	PUNCT
ejpam-6409	502	3	hs	hs	PROPN
ejpam-6409	502	4	is	be	AUX
ejpam-6409	502	5	linear	linear	ADJ
ejpam-6409	502	6	.	.	PUNCT
ejpam-6409	503	1	we	we	PRON
ejpam-6409	503	2	now	now	ADV
ejpam-6409	503	3	demonstrate	demonstrate	VERB
ejpam-6409	503	4	the	the	DET
ejpam-6409	503	5	nonlinearity	nonlinearity	NOUN
ejpam-6409	503	6	of	of	ADP
ejpam-6409	503	7	h	h	NOUN
ejpam-6409	503	8	=	=	SYM
ejpam-6409	503	9	φs(h̄	φs(h̄	PROPN
ejpam-6409	503	10	)	)	PUNCT
ejpam-6409	503	11	,	,	PUNCT
ejpam-6409	503	12	where	where	SCONJ
ejpam-6409	503	13	h̄	h̄	NOUN
ejpam-6409	503	14	=	=	SYM
ejpam-6409	503	15	h(1,0,	h(1,0,	NOUN
ejpam-6409	503	16	...	...	PUNCT
ejpam-6409	503	17	,0,2,0	,0,2,0	AUX
ejpam-6409	503	18	)	)	PUNCT
ejpam-6409	503	19	.	.	PUNCT
ejpam-6409	504	1	let	let	VERB
ejpam-6409	504	2	r	r	NOUN
ejpam-6409	504	3	=	=	SYM
ejpam-6409	504	4	(	(	PUNCT
ejpam-6409	504	5	00,012s−2	00,012s−2	PROPN
ejpam-6409	504	6	,	,	PUNCT
ejpam-6409	504	7	.	.	PUNCT
ejpam-6409	504	8	.	.	PUNCT
ejpam-6409	505	1	.	.	PUNCT
ejpam-6409	506	1	,	,	PUNCT
ejpam-6409	506	2	332s−2	332s−2	NUM
ejpam-6409	506	3	)	)	PUNCT
ejpam-6409	506	4	.	.	PUNCT
ejpam-6409	507	1	h	h	PROPN
ejpam-6409	507	2	has	have	VERB
ejpam-6409	507	3	length	length	NOUN
ejpam-6409	507	4	256	256	NUM
ejpam-6409	507	5	and	and	CCONJ
ejpam-6409	507	6	is	be	AUX
ejpam-6409	507	7	constructed	construct	VERB
ejpam-6409	507	8	from	from	ADP
ejpam-6409	507	9	a	a	DET
ejpam-6409	507	10	(	(	PUNCT
ejpam-6409	507	11	1,0,	1,0,	NOUN
ejpam-6409	507	12	...	...	PUNCT
ejpam-6409	507	13	,0,2,0	,0,2,0	PUNCT
ejpam-6409	507	14	)	)	PUNCT
ejpam-6409	508	1	2	2	NUM
ejpam-6409	509	1	=	=	SYM
ejpam-6409	509	2	10	10	NUM
ejpam-6409	509	3	10	10	NUM
ejpam-6409	509	4	10	10	NUM
ejpam-6409	509	5	·	·	PUNCT
ejpam-6409	509	6	·	·	PUNCT
ejpam-6409	509	7	·	·	PUNCT
ejpam-6409	510	1	10	10	NUM
ejpam-6409	510	2	10	10	NUM
ejpam-6409	510	3	r	r	NOUN
ejpam-6409	510	4	r	r	NOUN
ejpam-6409	510	5	r	r	NOUN
ejpam-6409	510	6	·	·	PUNCT
ejpam-6409	510	7	·	·	PUNCT
ejpam-6409	510	8	·	·	PUNCT
ejpam-6409	510	9	r	r	NOUN
ejpam-6409	510	10	r	r	NOUN
ejpam-6409	510	11	00	00	PUNCT
ejpam-6409	510	12	(	(	PUNCT
ejpam-6409	510	13	01)2s−2	01)2s−2	PROPN
ejpam-6409	510	14	(	(	PUNCT
ejpam-6409	510	15	02)2s−2	02)2s−2	PROPN
ejpam-6409	510	16	·	·	PUNCT
ejpam-6409	510	17	·	·	PUNCT
ejpam-6409	510	18	·	·	PUNCT
ejpam-6409	510	19	(	(	PUNCT
ejpam-6409	510	20	32	32	NUM
ejpam-6409	510	21	)	)	PUNCT
ejpam-6409	510	22	(	(	PUNCT
ejpam-6409	510	23	33)2s−2	33)2s−2	NUM
ejpam-6409	510	24			PROPN
ejpam-6409	510	25	.	.	PUNCT
ejpam-6409	511	1	by	by	ADP
ejpam-6409	511	2	corollaries	corollary	NOUN
ejpam-6409	511	3	3.5	3.5	NUM
ejpam-6409	511	4	,	,	PUNCT
ejpam-6409	511	5	3.7	3.7	NUM
ejpam-6409	511	6	,	,	PUNCT
ejpam-6409	511	7	3.8	3.8	NUM
ejpam-6409	511	8	,	,	PUNCT
ejpam-6409	511	9	and	and	CCONJ
ejpam-6409	511	10	3.9	3.9	NUM
ejpam-6409	511	11	,	,	PUNCT
ejpam-6409	511	12	we	we	PRON
ejpam-6409	511	13	have	have	VERB
ejpam-6409	511	14	:	:	PUNCT
ejpam-6409	511	15	φs(r	φs(r	NUM
ejpam-6409	511	16	,	,	PUNCT
ejpam-6409	511	17	r	r	NOUN
ejpam-6409	511	18	,	,	PUNCT
ejpam-6409	511	19	.	.	PUNCT
ejpam-6409	511	20	.	.	PUNCT
ejpam-6409	512	1	.	.	PUNCT
ejpam-6409	513	1	,	,	PUNCT
ejpam-6409	513	2	r	r	NOUN
ejpam-6409	513	3	,	,	PUNCT
ejpam-6409	513	4	r	r	NOUN
ejpam-6409	513	5	)	)	PUNCT
ejpam-6409	514	1	+	+	NOUN
ejpam-6409	514	2	φs(00	φs(00	PROPN
ejpam-6409	514	3	,	,	PUNCT
ejpam-6409	514	4	(	(	PUNCT
ejpam-6409	514	5	01)2	01)2	NUM
ejpam-6409	514	6	s−2	s−2	PROPN
ejpam-6409	514	7	,	,	PUNCT
ejpam-6409	514	8	.	.	PUNCT
ejpam-6409	514	9	.	.	PUNCT
ejpam-6409	514	10	.	.	PUNCT
ejpam-6409	515	1	,	,	PUNCT
ejpam-6409	515	2	(	(	PUNCT
ejpam-6409	515	3	33)2s−2	33)2s−2	ADJ
ejpam-6409	515	4	)	)	PUNCT
ejpam-6409	515	5	=	=	PUNCT
ejpam-6409	515	6	φs(z	φs(z	X
ejpam-6409	515	7	)	)	PUNCT
ejpam-6409	515	8	,	,	PUNCT
ejpam-6409	515	9	where	where	SCONJ
ejpam-6409	515	10	z	z	NOUN
ejpam-6409	515	11	=	=	SYM
ejpam-6409	515	12	(	(	PUNCT
ejpam-6409	515	13	r	r	NOUN
ejpam-6409	515	14	,	,	PUNCT
ejpam-6409	515	15	r	r	NOUN
ejpam-6409	515	16	,	,	PUNCT
ejpam-6409	515	17	.	.	PUNCT
ejpam-6409	515	18	.	.	PUNCT
ejpam-6409	515	19	.	.	PUNCT
ejpam-6409	516	1	,	,	PUNCT
ejpam-6409	516	2	r	r	X
ejpam-6409	516	3	)	)	PUNCT
ejpam-6409	516	4	+	+	CCONJ
ejpam-6409	516	5	(	(	PUNCT
ejpam-6409	516	6	00	00	NUM
ejpam-6409	516	7	,	,	PUNCT
ejpam-6409	516	8	(	(	PUNCT
ejpam-6409	516	9	01)2s−2	01)2s−2	NOUN
ejpam-6409	516	10	,	,	PUNCT
ejpam-6409	516	11	.	.	PUNCT
ejpam-6409	516	12	.	.	PUNCT
ejpam-6409	517	1	.	.	PUNCT
ejpam-6409	518	1	,	,	PUNCT
ejpam-6409	518	2	(	(	PUNCT
ejpam-6409	518	3	33)2s−2	33)2s−2	ADJ
ejpam-6409	518	4	)	)	PUNCT
ejpam-6409	519	1	+	+	CCONJ
ejpam-6409	519	2	p	p	X
ejpam-6409	519	3	,	,	PUNCT
ejpam-6409	519	4	muhammad	muhammad	PROPN
ejpam-6409	519	5	sajjad	sajjad	PROPN
ejpam-6409	519	6	et	et	PROPN
ejpam-6409	519	7	al	al	PROPN
ejpam-6409	519	8	.	.	PUNCT
ejpam-6409	519	9	/	/	SYM
ejpam-6409	519	10	eur	eur	PROPN
ejpam-6409	519	11	.	.	PUNCT
ejpam-6409	520	1	j.	j.	PROPN
ejpam-6409	520	2	pure	pure	PROPN
ejpam-6409	520	3	appl	appl	PROPN
ejpam-6409	520	4	.	.	PROPN
ejpam-6409	520	5	math	math	PROPN
ejpam-6409	520	6	,	,	PUNCT
ejpam-6409	520	7	18	18	NUM
ejpam-6409	520	8	(	(	PUNCT
ejpam-6409	520	9	3	3	NUM
ejpam-6409	520	10	)	)	PUNCT
ejpam-6409	520	11	(	(	PUNCT
ejpam-6409	520	12	2025	2025	NUM
ejpam-6409	520	13	)	)	PUNCT
ejpam-6409	520	14	,	,	PUNCT
ejpam-6409	520	15	6409	6409	NUM
ejpam-6409	520	16	16	16	NUM
ejpam-6409	520	17	of	of	ADP
ejpam-6409	520	18	32	32	NUM
ejpam-6409	520	19	with	with	ADP
ejpam-6409	520	20	p	p	NOUN
ejpam-6409	520	21	=	=	SYM
ejpam-6409	520	22	(	(	PUNCT
ejpam-6409	520	23	0	0	NUM
ejpam-6409	520	24	,	,	PUNCT
ejpam-6409	520	25	u1	u1	NOUN
ejpam-6409	520	26	,	,	PUNCT
ejpam-6409	520	27	0	0	NUM
ejpam-6409	520	28	,	,	PUNCT
ejpam-6409	520	29	u1	u1	NOUN
ejpam-6409	520	30	,	,	PUNCT
ejpam-6409	520	31	u2	u2	NOUN
ejpam-6409	520	32	,	,	PUNCT
ejpam-6409	520	33	u3	u3	NOUN
ejpam-6409	520	34	,	,	PUNCT
ejpam-6409	520	35	u2	u2	NOUN
ejpam-6409	520	36	,	,	PUNCT
ejpam-6409	520	37	u3	u3	NOUN
ejpam-6409	520	38	,	,	PUNCT
ejpam-6409	520	39	0	0	NUM
ejpam-6409	520	40	,	,	PUNCT
ejpam-6409	520	41	u1	u1	NOUN
ejpam-6409	520	42	,	,	PUNCT
ejpam-6409	520	43	0	0	NUM
ejpam-6409	520	44	,	,	PUNCT
ejpam-6409	520	45	u1	u1	NOUN
ejpam-6409	520	46	,	,	PUNCT
ejpam-6409	520	47	u2	u2	NOUN
ejpam-6409	520	48	,	,	PUNCT
ejpam-6409	520	49	u3	u3	NOUN
ejpam-6409	520	50	,	,	PUNCT
ejpam-6409	520	51	u2	u2	NOUN
ejpam-6409	520	52	,	,	PUNCT
ejpam-6409	520	53	u3	u3	NOUN
ejpam-6409	520	54	)	)	PUNCT
ejpam-6409	520	55	,	,	PUNCT
ejpam-6409	520	56	and	and	CCONJ
ejpam-6409	520	57	u1	u1	NOUN
ejpam-6409	520	58	=	=	SYM
ejpam-6409	520	59	(	(	PUNCT
ejpam-6409	520	60	00	00	NUM
ejpam-6409	520	61	,	,	PUNCT
ejpam-6409	520	62	2s−1	2s−1	NUM
ejpam-6409	520	63	,	,	PUNCT
ejpam-6409	520	64	00	00	NUM
ejpam-6409	520	65	,	,	PUNCT
ejpam-6409	520	66	2s−1	2s−1	NUM
ejpam-6409	520	67	,	,	PUNCT
ejpam-6409	520	68	.	.	PUNCT
ejpam-6409	520	69	.	.	PUNCT
ejpam-6409	521	1	.	.	PUNCT
ejpam-6409	522	1	,	,	PUNCT
ejpam-6409	522	2	00	00	NUM
ejpam-6409	522	3	,	,	PUNCT
ejpam-6409	522	4	2s−1	2s−1	NUM
ejpam-6409	522	5	)	)	PUNCT
ejpam-6409	522	6	,	,	PUNCT
ejpam-6409	522	7	u2	u2	NOUN
ejpam-6409	522	8	=	=	PUNCT
ejpam-6409	522	9	(	(	PUNCT
ejpam-6409	522	10	00	00	NUM
ejpam-6409	522	11	,	,	PUNCT
ejpam-6409	522	12	00	00	NUM
ejpam-6409	522	13	,	,	PUNCT
ejpam-6409	522	14	00	00	NUM
ejpam-6409	522	15	,	,	PUNCT
ejpam-6409	522	16	00	00	NUM
ejpam-6409	522	17	,	,	PUNCT
ejpam-6409	522	18	2s−10	2s−10	NUM
ejpam-6409	522	19	,	,	PUNCT
ejpam-6409	522	20	.	.	PUNCT
ejpam-6409	522	21	.	.	PUNCT
ejpam-6409	523	1	.	.	PUNCT
ejpam-6409	524	1	,	,	PUNCT
ejpam-6409	524	2	2s−10	2s−10	NUM
ejpam-6409	524	3	)	)	PUNCT
ejpam-6409	524	4	,	,	PUNCT
ejpam-6409	524	5	u3	u3	NOUN
ejpam-6409	524	6	=	=	SYM
ejpam-6409	524	7	(	(	PUNCT
ejpam-6409	524	8	00	00	NUM
ejpam-6409	524	9	,	,	PUNCT
ejpam-6409	524	10	2s−1	2s−1	NUM
ejpam-6409	524	11	,	,	PUNCT
ejpam-6409	524	12	00	00	NUM
ejpam-6409	524	13	,	,	PUNCT
ejpam-6409	524	14	2s−1	2s−1	NUM
ejpam-6409	524	15	,	,	PUNCT
ejpam-6409	524	16	2s−10	2s−10	NUM
ejpam-6409	524	17	,	,	PUNCT
ejpam-6409	524	18	2s−12s−1	2s−12s−1	NUM
ejpam-6409	524	19	,	,	PUNCT
ejpam-6409	524	20	.	.	PUNCT
ejpam-6409	524	21	.	.	PUNCT
ejpam-6409	524	22	.	.	PUNCT
ejpam-6409	525	1	,	,	PUNCT
ejpam-6409	525	2	2s−12s−1	2s−12s−1	NUM
ejpam-6409	525	3	)	)	PUNCT
ejpam-6409	525	4	.	.	PUNCT
ejpam-6409	526	1	since	since	SCONJ
ejpam-6409	526	2	φs(p	φs(p	NOUN
ejpam-6409	526	3	)	)	PUNCT
ejpam-6409	526	4	=	=	SYM
ejpam-6409	526	5	28	28	NUM
ejpam-6409	526	6	·	·	SYM
ejpam-6409	526	7	22(s−1	22(s−1	NUM
ejpam-6409	526	8	)	)	PUNCT
ejpam-6409	526	9	<	<	X
ejpam-6409	526	10	3n	3n	NUM
ejpam-6409	526	11	4	4	NUM
ejpam-6409	526	12	,	,	PUNCT
ejpam-6409	526	13	where	where	SCONJ
ejpam-6409	526	14	n	n	PRON
ejpam-6409	526	15	is	be	AUX
ejpam-6409	526	16	the	the	DET
ejpam-6409	526	17	length	length	NOUN
ejpam-6409	526	18	of	of	ADP
ejpam-6409	526	19	h	h	NOUN
ejpam-6409	526	20	,	,	PUNCT
ejpam-6409	526	21	we	we	PRON
ejpam-6409	526	22	have	have	VERB
ejpam-6409	526	23	φs(p	φs(p	NOUN
ejpam-6409	526	24	)	)	PUNCT
ejpam-6409	526	25	/∈	/∈	PUNCT
ejpam-6409	527	1	h	h	NOUN
ejpam-6409	527	2	,	,	PUNCT
ejpam-6409	527	3	and	and	CCONJ
ejpam-6409	527	4	thus	thus	ADV
ejpam-6409	527	5	φs(z	φs(z	NUM
ejpam-6409	527	6	)	)	PUNCT
ejpam-6409	527	7	/∈	/∈	PUNCT
ejpam-6409	528	1	h.	h.	PROPN
ejpam-6409	528	2	therefore	therefore	ADV
ejpam-6409	528	3	,	,	PUNCT
ejpam-6409	528	4	h	h	NOUN
ejpam-6409	528	5	=	=	PUNCT
ejpam-6409	528	6	h(1,0,	h(1,0,	NOUN
ejpam-6409	528	7	...	...	PUNCT
ejpam-6409	528	8	,0,2,0	,0,2,0	PUNCT
ejpam-6409	528	9	)	)	PUNCT
ejpam-6409	528	10	is	be	AUX
ejpam-6409	528	11	nonlinear	nonlinear	ADJ
ejpam-6409	528	12	.	.	PUNCT
ejpam-6409	529	1	let	let	VERB
ejpam-6409	529	2	h	h	NOUN
ejpam-6409	529	3	=	=	PUNCT
ejpam-6409	529	4	φs(h̄	φs(h̄	PROPN
ejpam-6409	529	5	)	)	PUNCT
ejpam-6409	529	6	,	,	PUNCT
ejpam-6409	529	7	where	where	SCONJ
ejpam-6409	529	8	h̄	h̄	NOUN
ejpam-6409	529	9	=	=	ADJ
ejpam-6409	529	10	h(t1,	h(t1,	NOUN
ejpam-6409	529	11	...	...	PUNCT
ejpam-6409	529	12	,ts	,ts	PUNCT
ejpam-6409	529	13	)	)	PUNCT
ejpam-6409	529	14	.	.	PUNCT
ejpam-6409	530	1	for	for	ADP
ejpam-6409	530	2	any	any	DET
ejpam-6409	530	3	i	i	PROPN
ejpam-6409	530	4	∈	∈	PROPN
ejpam-6409	530	5	{	{	PUNCT
ejpam-6409	530	6	1	1	NUM
ejpam-6409	530	7	,	,	PUNCT
ejpam-6409	530	8	.	.	PUNCT
ejpam-6409	530	9	.	.	PUNCT
ejpam-6409	530	10	.	.	PUNCT
ejpam-6409	531	1	,	,	PUNCT
ejpam-6409	531	2	s	s	X
ejpam-6409	531	3	}	}	PUNCT
ejpam-6409	531	4	,	,	PUNCT
ejpam-6409	531	5	define	define	VERB
ejpam-6409	531	6	hi	hi	INTJ
ejpam-6409	531	7	=	=	SYM
ejpam-6409	531	8	φs(h̄i	φs(h̄i	PROPN
ejpam-6409	531	9	)	)	PUNCT
ejpam-6409	531	10	,	,	PUNCT
ejpam-6409	531	11	where	where	SCONJ
ejpam-6409	531	12	h̄i	h̄i	PROPN
ejpam-6409	531	13	=	=	PROPN
ejpam-6409	531	14	h(t′1,	h(t′1,	PROPN
ejpam-6409	531	15	...	...	PUNCT
ejpam-6409	531	16	,t	,t	PUNCT
ejpam-6409	531	17	′	′	NUM
ejpam-6409	531	18	s	s	NOUN
ejpam-6409	531	19	)	)	PUNCT
ejpam-6409	531	20	,	,	PUNCT
ejpam-6409	531	21	t′i	t′i	ADV
ejpam-6409	531	22	=	=	SYM
ejpam-6409	531	23	ti	ti	NOUN
ejpam-6409	531	24	+	+	CCONJ
ejpam-6409	531	25	1	1	NUM
ejpam-6409	531	26	and	and	CCONJ
ejpam-6409	531	27	t′j	t′j	NOUN
ejpam-6409	531	28	=	=	NOUN
ejpam-6409	531	29	tj	tj	PROPN
ejpam-6409	531	30	for	for	ADP
ejpam-6409	531	31	j	j	PROPN
ejpam-6409	531	32	̸=	̸=	PROPN
ejpam-6409	531	33	i.	i.	NOUN
ejpam-6409	531	34	we	we	PRON
ejpam-6409	531	35	consider	consider	VERB
ejpam-6409	531	36	thath	thath	NOUN
ejpam-6409	531	37	=	=	SYM
ejpam-6409	531	38	φs(h̄	φs(h̄	PROPN
ejpam-6409	531	39	)	)	PUNCT
ejpam-6409	531	40	,	,	PUNCT
ejpam-6409	531	41	where	where	SCONJ
ejpam-6409	531	42	h̄	h̄	NOUN
ejpam-6409	531	43	=	=	SYM
ejpam-6409	531	44	h(1,0,	h(1,0,	NOUN
ejpam-6409	531	45	...	...	PUNCT
ejpam-6409	531	46	,0	,0	NUM
ejpam-6409	531	47	)	)	PUNCT
ejpam-6409	531	48	.	.	PUNCT
ejpam-6409	532	1	now	now	ADV
ejpam-6409	532	2	,	,	PUNCT
ejpam-6409	532	3	we	we	PRON
ejpam-6409	532	4	establish	establish	VERB
ejpam-6409	532	5	the	the	DET
ejpam-6409	532	6	nonlinearity	nonlinearity	NOUN
ejpam-6409	532	7	of	of	ADP
ejpam-6409	532	8	hi	hi	INTJ
ejpam-6409	532	9	for	for	ADP
ejpam-6409	532	10	every	every	DET
ejpam-6409	532	11	i	i	PROPN
ejpam-6409	532	12	∈	∈	PROPN
ejpam-6409	532	13	{	{	PUNCT
ejpam-6409	532	14	1	1	NUM
ejpam-6409	532	15	,	,	PUNCT
ejpam-6409	532	16	.	.	PUNCT
ejpam-6409	532	17	.	.	PUNCT
ejpam-6409	533	1	.	.	PUNCT
ejpam-6409	534	1	,	,	PUNCT
ejpam-6409	534	2	s−	s−	PROPN
ejpam-6409	534	3	2	2	NUM
ejpam-6409	534	4	}	}	PUNCT
ejpam-6409	534	5	.	.	PUNCT
ejpam-6409	535	1	the	the	DET
ejpam-6409	535	2	generator	generator	NOUN
ejpam-6409	535	3	matrix	matrix	NOUN
ejpam-6409	535	4	of	of	ADP
ejpam-6409	535	5	h̄i	h̄i	PROPN
ejpam-6409	535	6	contains	contain	VERB
ejpam-6409	535	7	two	two	NUM
ejpam-6409	535	8	nonzero	nonzero	PROPN
ejpam-6409	535	9	rows	row	NOUN
ejpam-6409	535	10	:	:	PUNCT
ejpam-6409	535	11	w1	w1	NOUN
ejpam-6409	535	12	=	=	SYM
ejpam-6409	535	13	10	10	NUM
ejpam-6409	535	14	,	,	PUNCT
ejpam-6409	535	15	w2	w2	NOUN
ejpam-6409	535	16	=	=	PUNCT
ejpam-6409	535	17	2i−1(00	2i−1(00	PROPN
ejpam-6409	535	18	,	,	PUNCT
ejpam-6409	535	19	.	.	PUNCT
ejpam-6409	535	20	.	.	PUNCT
ejpam-6409	536	1	.	.	PUNCT
ejpam-6409	537	1	,	,	PUNCT
ejpam-6409	537	2	0	0	NUM
ejpam-6409	537	3	,	,	PUNCT
ejpam-6409	537	4	10	10	NUM
ejpam-6409	537	5	,	,	PUNCT
ejpam-6409	537	6	.	.	PUNCT
ejpam-6409	537	7	.	.	PUNCT
ejpam-6409	538	1	.	.	PUNCT
ejpam-6409	539	1	,	,	PUNCT
ejpam-6409	539	2	12s+1−i	12s+1−i	NUM
ejpam-6409	539	3	,	,	PUNCT
ejpam-6409	539	4	.	.	PUNCT
ejpam-6409	539	5	.	.	PUNCT
ejpam-6409	539	6	.	.	PUNCT
ejpam-6409	539	7	)	)	PUNCT
ejpam-6409	539	8	.	.	PUNCT
ejpam-6409	540	1	let	let	VERB
ejpam-6409	540	2	w2j	w2j	PRON
ejpam-6409	540	3	be	be	AUX
ejpam-6409	540	4	the	the	DET
ejpam-6409	540	5	j	j	PROPN
ejpam-6409	540	6	-	-	PUNCT
ejpam-6409	540	7	th	th	VERB
ejpam-6409	540	8	coordinate	coordinate	NOUN
ejpam-6409	540	9	of	of	ADP
ejpam-6409	540	10	w2	w2	NOUN
ejpam-6409	540	11	and	and	CCONJ
ejpam-6409	540	12	[	[	X
ejpam-6409	540	13	(	(	PUNCT
ejpam-6409	540	14	w2j)0	w2j)0	PROPN
ejpam-6409	540	15	,	,	PUNCT
ejpam-6409	540	16	(	(	PUNCT
ejpam-6409	540	17	w2j)1	w2j)1	PROPN
ejpam-6409	540	18	,	,	PUNCT
ejpam-6409	540	19	.	.	PUNCT
ejpam-6409	540	20	.	.	PUNCT
ejpam-6409	541	1	.	.	PUNCT
ejpam-6409	542	1	,	,	PUNCT
ejpam-6409	542	2	(	(	PUNCT
ejpam-6409	542	3	w2j)s−1]p	w2j)s−1]p	VERB
ejpam-6409	542	4	its	its	PRON
ejpam-6409	542	5	p	p	ADJ
ejpam-6409	542	6	-	-	PUNCT
ejpam-6409	542	7	ary	ary	NOUN
ejpam-6409	542	8	expansion	expansion	NOUN
ejpam-6409	542	9	.	.	PUNCT
ejpam-6409	543	1	by	by	ADP
ejpam-6409	543	2	corollary	corollary	ADJ
ejpam-6409	543	3	3.4	3.4	NUM
ejpam-6409	543	4	,	,	PUNCT
ejpam-6409	543	5	ϕs(w2j	ϕs(w2j	PROPN
ejpam-6409	543	6	)	)	PUNCT
ejpam-6409	544	1	+	+	CCONJ
ejpam-6409	544	2	ϕs(2	ϕs(2	PROPN
ejpam-6409	544	3	i−1	i−1	PROPN
ejpam-6409	544	4	)	)	PUNCT
ejpam-6409	544	5	=	=	PUNCT
ejpam-6409	545	1	ϕs(w2j	ϕs(w2j	PROPN
ejpam-6409	546	1	+	+	CCONJ
ejpam-6409	546	2	2i−1	2i−1	NUM
ejpam-6409	546	3	−	−	PROPN
ejpam-6409	546	4	zj	zj	PROPN
ejpam-6409	546	5	)	)	PUNCT
ejpam-6409	546	6	,	,	PUNCT
ejpam-6409	546	7	where	where	SCONJ
ejpam-6409	546	8	zj	zj	PROPN
ejpam-6409	546	9	=	=	SYM
ejpam-6409	546	10	2i	2i	PROPN
ejpam-6409	546	11	if	if	SCONJ
ejpam-6409	546	12	(	(	PUNCT
ejpam-6409	546	13	w2j)i−1	w2j)i−1	PROPN
ejpam-6409	546	14	≥	≥	NUM
ejpam-6409	546	15	1	1	NUM
ejpam-6409	546	16	,	,	PUNCT
ejpam-6409	546	17	and	and	CCONJ
ejpam-6409	546	18	0	0	NUM
ejpam-6409	546	19	otherwise	otherwise	ADV
ejpam-6409	546	20	.	.	PUNCT
ejpam-6409	547	1	then	then	ADV
ejpam-6409	547	2	,	,	PUNCT
ejpam-6409	547	3	φs(w2	φs(w2	NOUN
ejpam-6409	547	4	)	)	PUNCT
ejpam-6409	547	5	+	+	NUM
ejpam-6409	547	6	φs(2	φs(2	PROPN
ejpam-6409	547	7	i−1	i−1	PROPN
ejpam-6409	547	8	)	)	PUNCT
ejpam-6409	547	9	=	=	NOUN
ejpam-6409	548	1	φs(w2	φs(w2	NUM
ejpam-6409	548	2	+	+	NUM
ejpam-6409	548	3	2i−1	2i−1	NUM
ejpam-6409	548	4	−	−	PROPN
ejpam-6409	548	5	z	z	NOUN
ejpam-6409	548	6	)	)	PUNCT
ejpam-6409	548	7	,	,	PUNCT
ejpam-6409	548	8	where	where	SCONJ
ejpam-6409	548	9	z	z	NOUN
ejpam-6409	548	10	=	=	SYM
ejpam-6409	548	11	(	(	PUNCT
ejpam-6409	548	12	z1	z1	PROPN
ejpam-6409	548	13	,	,	PUNCT
ejpam-6409	548	14	z2	z2	PROPN
ejpam-6409	548	15	,	,	PUNCT
ejpam-6409	548	16	.	.	PUNCT
ejpam-6409	548	17	.	.	PUNCT
ejpam-6409	548	18	.	.	PUNCT
ejpam-6409	549	1	,	,	PUNCT
ejpam-6409	549	2	z22(s+1−i	z22(s+1−i	NOUN
ejpam-6409	549	3	)	)	PUNCT
ejpam-6409	549	4	)	)	PUNCT
ejpam-6409	550	1	∈	∈	PROPN
ejpam-6409	550	2	z22(s+1−i	z22(s+1−i	PROPN
ejpam-6409	550	3	)	)	PUNCT
ejpam-6409	550	4	2s	2s	PROPN
ejpam-6409	550	5	and	and	CCONJ
ejpam-6409	550	6	zj	zj	X
ejpam-6409	550	7	=	=	SYM
ejpam-6409	550	8	2i	2i	PROPN
ejpam-6409	550	9	for	for	ADP
ejpam-6409	550	10	even	even	ADV
ejpam-6409	550	11	k	k	PROPN
ejpam-6409	550	12	∈	∈	PROPN
ejpam-6409	550	13	{	{	PUNCT
ejpam-6409	550	14	2	2	NUM
ejpam-6409	550	15	,	,	PUNCT
ejpam-6409	550	16	4	4	NUM
ejpam-6409	550	17	,	,	PUNCT
ejpam-6409	550	18	.	.	PUNCT
ejpam-6409	550	19	.	.	PUNCT
ejpam-6409	550	20	.	.	PUNCT
ejpam-6409	551	1	,	,	PUNCT
ejpam-6409	551	2	2s+1−i	2s+1−i	NUM
ejpam-6409	551	3	}	}	PUNCT
ejpam-6409	551	4	and	and	CCONJ
ejpam-6409	551	5	zj	zj	X
ejpam-6409	551	6	=	=	SYM
ejpam-6409	551	7	0	0	PUNCT
ejpam-6409	552	1	otherwise	otherwise	ADV
ejpam-6409	552	2	.	.	PUNCT
ejpam-6409	553	1	we	we	PRON
ejpam-6409	553	2	just	just	ADV
ejpam-6409	553	3	need	need	VERB
ejpam-6409	553	4	to	to	PART
ejpam-6409	553	5	show	show	VERB
ejpam-6409	553	6	z	z	PROPN
ejpam-6409	553	7	/∈	/∈	PUNCT
ejpam-6409	554	1	h̄i	h̄i	PROPN
ejpam-6409	554	2	.	.	PROPN
ejpam-6409	554	3	note	note	VERB
ejpam-6409	554	4	that	that	PRON
ejpam-6409	554	5	wth(φs(z	wth(φs(z	NOUN
ejpam-6409	554	6	)	)	PUNCT
ejpam-6409	554	7	)	)	PUNCT
ejpam-6409	555	1	=	=	SYM
ejpam-6409	555	2	22(s−i	22(s−i	NUM
ejpam-6409	555	3	)	)	PUNCT
ejpam-6409	555	4	·	·	PUNCT
ejpam-6409	556	1	wth(φs(2	wth(φs(2	NOUN
ejpam-6409	556	2	i	i	PROPN
ejpam-6409	556	3	)	)	PUNCT
ejpam-6409	556	4	)	)	PUNCT
ejpam-6409	556	5	.	.	PUNCT
ejpam-6409	557	1	if	if	SCONJ
ejpam-6409	557	2	i	i	PRON
ejpam-6409	557	3	∈	∈	PROPN
ejpam-6409	557	4	{	{	PUNCT
ejpam-6409	557	5	1	1	NUM
ejpam-6409	557	6	,	,	PUNCT
ejpam-6409	557	7	.	.	PUNCT
ejpam-6409	557	8	.	.	PUNCT
ejpam-6409	558	1	.	.	PUNCT
ejpam-6409	559	1	,	,	PUNCT
ejpam-6409	559	2	s−	s−	PROPN
ejpam-6409	559	3	2	2	NUM
ejpam-6409	559	4	}	}	PUNCT
ejpam-6409	559	5	,	,	PUNCT
ejpam-6409	559	6	then	then	ADV
ejpam-6409	559	7	wth(φs(z	wth(φs(z	PROPN
ejpam-6409	559	8	)	)	PUNCT
ejpam-6409	559	9	)	)	PUNCT
ejpam-6409	560	1	=	=	PUNCT
ejpam-6409	560	2	6	6	NUM
ejpam-6409	560	3	·	·	SYM
ejpam-6409	560	4	22(2s−i−2	22(2s−i−2	NUM
ejpam-6409	560	5	)	)	PUNCT
ejpam-6409	560	6	.	.	PUNCT
ejpam-6409	561	1	but	but	CCONJ
ejpam-6409	561	2	the	the	DET
ejpam-6409	561	3	code	code	NOUN
ejpam-6409	561	4	hi	hi	INTJ
ejpam-6409	561	5	has	have	VERB
ejpam-6409	561	6	minimum	minimum	ADJ
ejpam-6409	561	7	distance	distance	NOUN
ejpam-6409	561	8	3	3	NUM
ejpam-6409	561	9	·	·	SYM
ejpam-6409	561	10	22(2s−i−1	22(2s−i−1	NUM
ejpam-6409	561	11	)	)	PUNCT
ejpam-6409	561	12	>	>	X
ejpam-6409	561	13	wth(φs(z	wth(φs(z	PROPN
ejpam-6409	561	14	)	)	PUNCT
ejpam-6409	561	15	)	)	PUNCT
ejpam-6409	561	16	,	,	PUNCT
ejpam-6409	561	17	therefore	therefore	ADV
ejpam-6409	561	18	,	,	PUNCT
ejpam-6409	561	19	φs(z	φs(z	ADV
ejpam-6409	561	20	)	)	PUNCT
ejpam-6409	561	21	/∈	/∈	PUNCT
ejpam-6409	562	1	hi	hi	INTJ
ejpam-6409	562	2	,	,	PUNCT
ejpam-6409	562	3	for	for	ADP
ejpam-6409	562	4	i	i	PRON
ejpam-6409	562	5	∈	∈	PROPN
ejpam-6409	562	6	{	{	PUNCT
ejpam-6409	562	7	1	1	NUM
ejpam-6409	562	8	,	,	PUNCT
ejpam-6409	562	9	.	.	PUNCT
ejpam-6409	562	10	.	.	PUNCT
ejpam-6409	562	11	.	.	PUNCT
ejpam-6409	563	1	,	,	PUNCT
ejpam-6409	563	2	s−	s−	PROPN
ejpam-6409	563	3	2	2	NUM
ejpam-6409	563	4	}	}	PUNCT
ejpam-6409	563	5	.	.	PUNCT
ejpam-6409	564	1	finally	finally	ADV
ejpam-6409	564	2	,	,	PUNCT
ejpam-6409	564	3	in	in	ADP
ejpam-6409	564	4	general	general	ADJ
ejpam-6409	564	5	,	,	PUNCT
ejpam-6409	564	6	for	for	ADP
ejpam-6409	564	7	h	h	NOUN
ejpam-6409	564	8	=	=	SYM
ejpam-6409	564	9	φs(ĥ	φs(ĥ	NOUN
ejpam-6409	564	10	)	)	PUNCT
ejpam-6409	564	11	,	,	PUNCT
ejpam-6409	564	12	where	where	SCONJ
ejpam-6409	564	13	ĥ	ĥ	X
ejpam-6409	564	14	=	=	SYM
ejpam-6409	564	15	ĥ(t1,	ĥ(t1,	PROPN
ejpam-6409	564	16	...	...	PUNCT
ejpam-6409	564	17	,ts	,ts	PUNCT
ejpam-6409	564	18	)	)	PUNCT
ejpam-6409	564	19	,	,	PUNCT
ejpam-6409	564	20	we	we	PRON
ejpam-6409	564	21	demonstrate	demonstrate	VERB
ejpam-6409	564	22	that	that	SCONJ
ejpam-6409	564	23	whenever	whenever	SCONJ
ejpam-6409	564	24	h	h	NOUN
ejpam-6409	564	25	is	be	AUX
ejpam-6409	564	26	nonlinear	nonlinear	ADJ
ejpam-6409	564	27	,	,	PUNCT
ejpam-6409	564	28	hi	hi	INTJ
ejpam-6409	564	29	remains	remain	VERB
ejpam-6409	564	30	nonlinear	nonlinear	ADJ
ejpam-6409	564	31	for	for	ADP
ejpam-6409	564	32	all	all	PRON
ejpam-6409	564	33	i	i	PRON
ejpam-6409	564	34	∈	∈	PROPN
ejpam-6409	564	35	{	{	PUNCT
ejpam-6409	564	36	1	1	NUM
ejpam-6409	564	37	,	,	PUNCT
ejpam-6409	564	38	.	.	PUNCT
ejpam-6409	564	39	.	.	PUNCT
ejpam-6409	565	1	.	.	PUNCT
ejpam-6409	566	1	,	,	PUNCT
ejpam-6409	566	2	s	s	X
ejpam-6409	566	3	}	}	PUNCT
ejpam-6409	566	4	.	.	PUNCT
ejpam-6409	567	1	let	let	VERB
ejpam-6409	567	2	us	we	PRON
ejpam-6409	567	3	assume	assume	VERB
ejpam-6409	567	4	that	that	SCONJ
ejpam-6409	567	5	hi	hi	INTJ
ejpam-6409	567	6	does	do	AUX
ejpam-6409	567	7	not	not	PART
ejpam-6409	567	8	exhibit	exhibit	VERB
ejpam-6409	567	9	linearity	linearity	NOUN
ejpam-6409	567	10	.	.	PUNCT
ejpam-6409	568	1	then	then	ADV
ejpam-6409	568	2	,	,	PUNCT
ejpam-6409	568	3	by	by	ADP
ejpam-6409	568	4	considering	consider	VERB
ejpam-6409	568	5	iterative	iterative	NOUN
ejpam-6409	568	6	construction	construction	NOUN
ejpam-6409	568	7	,	,	PUNCT
ejpam-6409	568	8	for	for	ADP
ejpam-6409	568	9	any	any	DET
ejpam-6409	568	10	u	u	NOUN
ejpam-6409	568	11	,	,	PUNCT
ejpam-6409	568	12	v	v	NOUN
ejpam-6409	568	13	∈	∈	PROPN
ejpam-6409	568	14	ĥ	ĥ	NOUN
ejpam-6409	568	15	,	,	PUNCT
ejpam-6409	568	16	we	we	PRON
ejpam-6409	568	17	have	have	VERB
ejpam-6409	568	18	that	that	DET
ejpam-6409	568	19	(	(	PUNCT
ejpam-6409	568	20	u	u	NOUN
ejpam-6409	568	21	,	,	PUNCT
ejpam-6409	568	22	.	.	PUNCT
ejpam-6409	568	23	.	.	PUNCT
ejpam-6409	568	24	.	.	PUNCT
ejpam-6409	569	1	,	,	PUNCT
ejpam-6409	569	2	u	u	NOUN
ejpam-6409	569	3	)	)	PUNCT
ejpam-6409	569	4	,	,	PUNCT
ejpam-6409	569	5	(	(	PUNCT
ejpam-6409	569	6	v	v	NOUN
ejpam-6409	569	7	,	,	PUNCT
ejpam-6409	569	8	.	.	PUNCT
ejpam-6409	569	9	.	.	PUNCT
ejpam-6409	570	1	.	.	PUNCT
ejpam-6409	571	1	,	,	PUNCT
ejpam-6409	571	2	v	v	X
ejpam-6409	571	3	)	)	PUNCT
ejpam-6409	571	4	∈	∈	PROPN
ejpam-6409	571	5	ĥi	ĥi	NOUN
ejpam-6409	571	6	.	.	PUNCT
ejpam-6409	572	1	moreover	moreover	ADV
ejpam-6409	572	2	,	,	PUNCT
ejpam-6409	572	3	since	since	SCONJ
ejpam-6409	572	4	hi	hi	INTJ
ejpam-6409	572	5	is	be	AUX
ejpam-6409	572	6	linear	linear	ADJ
ejpam-6409	572	7	,	,	PUNCT
ejpam-6409	572	8	muhammad	muhammad	PROPN
ejpam-6409	572	9	sajjad	sajjad	PROPN
ejpam-6409	572	10	et	et	PROPN
ejpam-6409	572	11	al	al	PROPN
ejpam-6409	572	12	.	.	PUNCT
ejpam-6409	572	13	/	/	SYM
ejpam-6409	572	14	eur	eur	PROPN
ejpam-6409	572	15	.	.	PUNCT
ejpam-6409	573	1	j.	j.	PROPN
ejpam-6409	573	2	pure	pure	PROPN
ejpam-6409	573	3	appl	appl	PROPN
ejpam-6409	573	4	.	.	PROPN
ejpam-6409	573	5	math	math	PROPN
ejpam-6409	573	6	,	,	PUNCT
ejpam-6409	573	7	18	18	NUM
ejpam-6409	573	8	(	(	PUNCT
ejpam-6409	573	9	3	3	NUM
ejpam-6409	573	10	)	)	PUNCT
ejpam-6409	573	11	(	(	PUNCT
ejpam-6409	573	12	2025	2025	NUM
ejpam-6409	573	13	)	)	PUNCT
ejpam-6409	573	14	,	,	PUNCT
ejpam-6409	573	15	6409	6409	NUM
ejpam-6409	573	16	17	17	NUM
ejpam-6409	573	17	of	of	ADP
ejpam-6409	573	18	32	32	NUM
ejpam-6409	573	19	φs(u	φs(u	NOUN
ejpam-6409	573	20	,	,	PUNCT
ejpam-6409	573	21	.	.	PUNCT
ejpam-6409	573	22	.	.	PUNCT
ejpam-6409	573	23	.	.	PUNCT
ejpam-6409	574	1	,	,	PUNCT
ejpam-6409	574	2	u	u	NOUN
ejpam-6409	574	3	)	)	PUNCT
ejpam-6409	575	1	+	+	NUM
ejpam-6409	575	2	φs(v	φs(v	NOUN
ejpam-6409	575	3	,	,	PUNCT
ejpam-6409	575	4	.	.	PUNCT
ejpam-6409	575	5	.	.	PUNCT
ejpam-6409	576	1	.	.	PUNCT
ejpam-6409	577	1	,	,	PUNCT
ejpam-6409	577	2	v	v	NOUN
ejpam-6409	577	3	)	)	PUNCT
ejpam-6409	577	4	=	=	NOUN
ejpam-6409	577	5	φs(a	φs(a	NOUN
ejpam-6409	577	6	,	,	PUNCT
ejpam-6409	577	7	.	.	PUNCT
ejpam-6409	577	8	.	.	PUNCT
ejpam-6409	578	1	.	.	PUNCT
ejpam-6409	579	1	,	,	PUNCT
ejpam-6409	579	2	a)+λ·2i−1	a)+λ·2i−1	PROPN
ejpam-6409	579	3	(	(	PUNCT
ejpam-6409	579	4	0	0	NUM
ejpam-6409	579	5	,	,	PUNCT
ejpam-6409	579	6	.	.	PUNCT
ejpam-6409	579	7	.	.	PUNCT
ejpam-6409	579	8	.	.	PUNCT
ejpam-6409	580	1	,	,	PUNCT
ejpam-6409	580	2	0	0	NUM
ejpam-6409	580	3	,	,	PUNCT
ejpam-6409	580	4	2s−i+1−1	2s−i+1−1	NUM
ejpam-6409	580	5	,	,	PUNCT
ejpam-6409	580	6	.	.	PUNCT
ejpam-6409	580	7	.	.	PUNCT
ejpam-6409	580	8	.	.	PUNCT
ejpam-6409	581	1	,	,	PUNCT
ejpam-6409	581	2	2s−i+1−1	2s−i+1−1	NUM
ejpam-6409	581	3	,	,	PUNCT
ejpam-6409	581	4	0	0	NUM
ejpam-6409	581	5	,	,	PUNCT
ejpam-6409	581	6	.	.	PUNCT
ejpam-6409	581	7	.	.	PUNCT
ejpam-6409	582	1	.	.	PUNCT
ejpam-6409	583	1	,	,	PUNCT
ejpam-6409	583	2	2s−i+1−1	2s−i+1−1	NUM
ejpam-6409	583	3	,	,	PUNCT
ejpam-6409	583	4	2s−i+1−1	2s−i+1−1	NUM
ejpam-6409	583	5	)	)	PUNCT
ejpam-6409	583	6	∈	∈	PROPN
ejpam-6409	584	1	hi	hi	INTJ
ejpam-6409	584	2	where	where	SCONJ
ejpam-6409	584	3	a	a	DET
ejpam-6409	584	4	∈	∈	NOUN
ejpam-6409	584	5	ĥ	ĥ	X
ejpam-6409	584	6	and	and	CCONJ
ejpam-6409	584	7	λ	λ	PART
ejpam-6409	584	8	∈	∈	PROPN
ejpam-6409	584	9	z2s	z2s	X
ejpam-6409	585	1	[	[	X
ejpam-6409	585	2	ω	ω	X
ejpam-6409	585	3	]	]	X
ejpam-6409	585	4	.	.	PUNCT
ejpam-6409	586	1	therefore	therefore	ADV
ejpam-6409	586	2	,	,	PUNCT
ejpam-6409	586	3	φs(u	φs(u	PUNCT
ejpam-6409	586	4	)	)	PUNCT
ejpam-6409	586	5	+	+	SYM
ejpam-6409	586	6	φs(v	φs(v	NOUN
ejpam-6409	586	7	)	)	PUNCT
ejpam-6409	586	8	=	=	SYM
ejpam-6409	586	9	φs(a	φs(a	X
ejpam-6409	586	10	)	)	PUNCT
ejpam-6409	586	11	∈	∈	PROPN
ejpam-6409	586	12	h.	h.	PROPN
ejpam-6409	587	1	so	so	ADV
ejpam-6409	587	2	,	,	PUNCT
ejpam-6409	587	3	h	h	NOUN
ejpam-6409	587	4	is	be	AUX
ejpam-6409	587	5	linear	linear	ADJ
ejpam-6409	587	6	.	.	PUNCT
ejpam-6409	588	1	6	6	X
ejpam-6409	588	2	.	.	X
ejpam-6409	588	3	kernel	kernel	PROPN
ejpam-6409	588	4	of	of	ADP
ejpam-6409	588	5	z2s	z2s	PROPN
ejpam-6409	589	1	[	[	X
ejpam-6409	589	2	ω]-linear	ω]-linear	ADJ
ejpam-6409	589	3	gh	gh	PROPN
ejpam-6409	589	4	codes	code	VERB
ejpam-6409	589	5	the	the	DET
ejpam-6409	589	6	method	method	NOUN
ejpam-6409	589	7	to	to	PART
ejpam-6409	589	8	find	find	VERB
ejpam-6409	589	9	the	the	DET
ejpam-6409	589	10	kernel	kernel	NOUN
ejpam-6409	589	11	of	of	ADP
ejpam-6409	589	12	codes	code	NOUN
ejpam-6409	589	13	over	over	ADP
ejpam-6409	589	14	z2s	z2s	PROPN
ejpam-6409	589	15	is	be	AUX
ejpam-6409	589	16	given	give	VERB
ejpam-6409	589	17	in	in	ADP
ejpam-6409	589	18	section	section	NOUN
ejpam-6409	589	19	4	4	NUM
ejpam-6409	589	20	of	of	ADP
ejpam-6409	589	21	[	[	X
ejpam-6409	589	22	6	6	NUM
ejpam-6409	589	23	,	,	PUNCT
ejpam-6409	589	24	17	17	NUM
ejpam-6409	589	25	]	]	PUNCT
ejpam-6409	589	26	.	.	PUNCT
ejpam-6409	590	1	this	this	DET
ejpam-6409	590	2	section	section	NOUN
ejpam-6409	590	3	is	be	AUX
ejpam-6409	590	4	devoted	devote	VERB
ejpam-6409	590	5	to	to	ADP
ejpam-6409	590	6	establishing	establish	VERB
ejpam-6409	590	7	several	several	ADJ
ejpam-6409	590	8	results	result	NOUN
ejpam-6409	590	9	related	relate	VERB
ejpam-6409	590	10	to	to	ADP
ejpam-6409	590	11	the	the	DET
ejpam-6409	590	12	kernel	kernel	NOUN
ejpam-6409	590	13	of	of	ADP
ejpam-6409	590	14	z2s	z2s	PROPN
ejpam-6409	591	1	[	[	X
ejpam-6409	591	2	ω]-linear	ω]-linear	ADJ
ejpam-6409	591	3	codes	code	NOUN
ejpam-6409	591	4	.	.	PUNCT
ejpam-6409	592	1	assume	assume	VERB
ejpam-6409	592	2	a(t1,	a(t1,	NUM
ejpam-6409	592	3	...	...	PUNCT
ejpam-6409	592	4	,ts	,ts	PUNCT
ejpam-6409	592	5	)	)	PUNCT
ejpam-6409	592	6	represents	represent	VERB
ejpam-6409	592	7	the	the	DET
ejpam-6409	592	8	generator	generator	NOUN
ejpam-6409	592	9	matrix	matrix	NOUN
ejpam-6409	592	10	of	of	ADP
ejpam-6409	592	11	ĥ(t1,	ĥ(t1,	NOUN
ejpam-6409	592	12	...	...	PUNCT
ejpam-6409	592	13	,ts	,ts	PUNCT
ejpam-6409	592	14	)	)	PUNCT
ejpam-6409	592	15	and	and	CCONJ
ejpam-6409	592	16	denote	denote	VERB
ejpam-6409	592	17	wi	wi	PROPN
ejpam-6409	592	18	as	as	ADP
ejpam-6409	592	19	the	the	DET
ejpam-6409	592	20	i	i	PROPN
ejpam-6409	592	21	-	-	PUNCT
ejpam-6409	592	22	th	th	X
ejpam-6409	592	23	row	row	NOUN
ejpam-6409	592	24	vector	vector	NOUN
ejpam-6409	592	25	of	of	ADP
ejpam-6409	592	26	a(t1,	a(t1,	PROPN
ejpam-6409	592	27	...	...	PUNCT
ejpam-6409	592	28	,ts	,ts	PUNCT
ejpam-6409	592	29	)	)	PUNCT
ejpam-6409	592	30	.	.	PUNCT
ejpam-6409	593	1	by	by	ADP
ejpam-6409	593	2	established	establish	VERB
ejpam-6409	593	3	construction	construction	NOUN
ejpam-6409	593	4	,	,	PUNCT
ejpam-6409	593	5	w1	w1	NOUN
ejpam-6409	593	6	=	=	SYM
ejpam-6409	593	7	1	1	NUM
ejpam-6409	593	8	and	and	CCONJ
ejpam-6409	593	9	ord(wi	ord(wi	NOUN
ejpam-6409	593	10	)	)	PUNCT
ejpam-6409	593	11	≤	≤	NOUN
ejpam-6409	593	12	ord(wj	ord(wj	ADV
ejpam-6409	593	13	)	)	PUNCT
ejpam-6409	593	14	if	if	SCONJ
ejpam-6409	593	15	i	i	PRON
ejpam-6409	593	16	>	>	X
ejpam-6409	593	17	j.	j.	PROPN
ejpam-6409	593	18	we	we	PRON
ejpam-6409	593	19	introduce	introduce	VERB
ejpam-6409	593	20	σ	σ	PRON
ejpam-6409	593	21	∈	∈	PROPN
ejpam-6409	593	22	{	{	PUNCT
ejpam-6409	593	23	1	1	NUM
ejpam-6409	593	24	,	,	PUNCT
ejpam-6409	593	25	.	.	PUNCT
ejpam-6409	593	26	.	.	PUNCT
ejpam-6409	593	27	.	.	PUNCT
ejpam-6409	594	1	,	,	PUNCT
ejpam-6409	594	2	s	s	X
ejpam-6409	594	3	}	}	PUNCT
ejpam-6409	594	4	as	as	SCONJ
ejpam-6409	594	5	the	the	DET
ejpam-6409	594	6	integer	integer	NOUN
ejpam-6409	594	7	satisfying	satisfy	VERB
ejpam-6409	594	8	the	the	DET
ejpam-6409	594	9	condition	condition	NOUN
ejpam-6409	594	10	that	that	SCONJ
ejpam-6409	594	11	ord(w2	ord(w2	ADP
ejpam-6409	594	12	)	)	PUNCT
ejpam-6409	594	13	=	=	SYM
ejpam-6409	594	14	2s+1−σ	2s+1−σ	X
ejpam-6409	594	15	.	.	PUNCT
ejpam-6409	594	16	note	note	NOUN
ejpam-6409	594	17	σ	σ	NOUN
ejpam-6409	594	18	=	=	SYM
ejpam-6409	594	19	1	1	NUM
ejpam-6409	594	20	if	if	SCONJ
ejpam-6409	594	21	t1	t1	PROPN
ejpam-6409	594	22	>	>	X
ejpam-6409	594	23	1	1	NUM
ejpam-6409	594	24	,	,	PUNCT
ejpam-6409	594	25	and	and	CCONJ
ejpam-6409	594	26	σ	σ	NOUN
ejpam-6409	594	27	=	=	PROPN
ejpam-6409	594	28	min{i	min{i	PROPN
ejpam-6409	594	29	:	:	PUNCT
ejpam-6409	594	30	ti	ti	X
ejpam-6409	594	31	>	>	X
ejpam-6409	594	32	0	0	PROPN
ejpam-6409	594	33	,	,	PUNCT
ejpam-6409	594	34	i	i	PRON
ejpam-6409	594	35	∈	∈	PROPN
ejpam-6409	594	36	{	{	PUNCT
ejpam-6409	594	37	2	2	NUM
ejpam-6409	594	38	,	,	PUNCT
ejpam-6409	594	39	.	.	PUNCT
ejpam-6409	594	40	.	.	PUNCT
ejpam-6409	595	1	.	.	PUNCT
ejpam-6409	596	1	,	,	PUNCT
ejpam-6409	596	2	s	s	X
ejpam-6409	596	3	}	}	PUNCT
ejpam-6409	596	4	}	}	PUNCT
ejpam-6409	596	5	if	if	SCONJ
ejpam-6409	596	6	t1	t1	NOUN
ejpam-6409	596	7	=	=	SYM
ejpam-6409	596	8	1	1	X
ejpam-6409	596	9	.	.	PUNCT
ejpam-6409	597	1	in	in	ADP
ejpam-6409	597	2	this	this	DET
ejpam-6409	597	3	case	case	NOUN
ejpam-6409	597	4	,	,	PUNCT
ejpam-6409	597	5	if	if	SCONJ
ejpam-6409	597	6	σ	σ	PROPN
ejpam-6409	597	7	=	=	SYM
ejpam-6409	597	8	s	s	PROPN
ejpam-6409	597	9	,	,	PUNCT
ejpam-6409	597	10	the	the	DET
ejpam-6409	597	11	code	code	NOUN
ejpam-6409	597	12	is	be	AUX
ejpam-6409	597	13	ĥ(1,0,	ĥ(1,0,	NOUN
ejpam-6409	597	14	...	...	PUNCT
ejpam-6409	597	15	,0,ts	,0,ts	PROPN
ejpam-6409	597	16	)	)	PUNCT
ejpam-6409	597	17	,	,	PUNCT
ejpam-6409	597	18	which	which	PRON
ejpam-6409	597	19	is	be	AUX
ejpam-6409	597	20	linear	linear	ADJ
ejpam-6409	597	21	.	.	PUNCT
ejpam-6409	598	1	let	let	VERB
ejpam-6409	598	2	u	u	PRON
ejpam-6409	598	3	=	=	PUNCT
ejpam-6409	598	4	(	(	PUNCT
ejpam-6409	598	5	u1	u1	PROPN
ejpam-6409	598	6	,	,	PUNCT
ejpam-6409	598	7	.	.	PUNCT
ejpam-6409	598	8	.	.	PUNCT
ejpam-6409	599	1	.	.	PUNCT
ejpam-6409	600	1	,	,	PUNCT
ejpam-6409	600	2	un	un	PROPN
ejpam-6409	600	3	)	)	PUNCT
ejpam-6409	600	4	∈	∈	PROPN
ejpam-6409	601	1	zn	zn	NOUN
ejpam-6409	601	2	2s	2s	X
ejpam-6409	602	1	[	[	X
ejpam-6409	602	2	ω	ω	X
ejpam-6409	602	3	]	]	PUNCT
ejpam-6409	602	4	and	and	CCONJ
ejpam-6409	602	5	[	[	X
ejpam-6409	602	6	uj,0	uj,0	PROPN
ejpam-6409	602	7	,	,	PUNCT
ejpam-6409	602	8	uj,1	uj,1	NOUN
ejpam-6409	602	9	,	,	PUNCT
ejpam-6409	602	10	.	.	PUNCT
ejpam-6409	602	11	.	.	PUNCT
ejpam-6409	602	12	.	.	PUNCT
ejpam-6409	603	1	,	,	PUNCT
ejpam-6409	603	2	uj	uj	PROPN
ejpam-6409	603	3	,	,	PUNCT
ejpam-6409	603	4	s−1]2	s−1]2	ADV
ejpam-6409	603	5	be	be	AUX
ejpam-6409	603	6	the	the	DET
ejpam-6409	603	7	2	2	NUM
ejpam-6409	603	8	-	-	PUNCT
ejpam-6409	603	9	ary	ary	NOUN
ejpam-6409	603	10	expansion	expansion	NOUN
ejpam-6409	603	11	of	of	ADP
ejpam-6409	603	12	uj	uj	PROPN
ejpam-6409	603	13	,	,	PUNCT
ejpam-6409	603	14	where	where	SCONJ
ejpam-6409	603	15	j	j	PROPN
ejpam-6409	603	16	∈	∈	PROPN
ejpam-6409	603	17	{	{	PUNCT
ejpam-6409	603	18	1	1	NUM
ejpam-6409	603	19	,	,	PUNCT
ejpam-6409	603	20	.	.	PUNCT
ejpam-6409	603	21	.	.	PUNCT
ejpam-6409	604	1	.	.	PUNCT
ejpam-6409	604	2	,	,	PUNCT
ejpam-6409	604	3	n	n	CCONJ
ejpam-6409	604	4	}	}	PUNCT
ejpam-6409	604	5	.	.	PUNCT
ejpam-6409	605	1	assume	assume	VERB
ejpam-6409	605	2	i	i	PRON
ejpam-6409	605	3	is	be	AUX
ejpam-6409	605	4	an	an	DET
ejpam-6409	605	5	integer	integer	NOUN
ejpam-6409	605	6	such	such	ADJ
ejpam-6409	605	7	that	that	SCONJ
ejpam-6409	605	8	i	i	PRON
ejpam-6409	605	9	∈	∈	PROPN
ejpam-6409	605	10	{	{	PUNCT
ejpam-6409	605	11	1	1	NUM
ejpam-6409	605	12	,	,	PUNCT
ejpam-6409	605	13	.	.	PUNCT
ejpam-6409	605	14	.	.	PUNCT
ejpam-6409	606	1	.	.	PUNCT
ejpam-6409	607	1	,	,	PUNCT
ejpam-6409	607	2	s	s	VERB
ejpam-6409	607	3	−	−	NOUN
ejpam-6409	607	4	1	1	NUM
ejpam-6409	607	5	}	}	PUNCT
ejpam-6409	607	6	.	.	PUNCT
ejpam-6409	608	1	then	then	ADV
ejpam-6409	608	2	ui	ui	PROPN
ejpam-6409	608	3	denotes	denote	VERB
ejpam-6409	608	4	the	the	DET
ejpam-6409	608	5	vector	vector	NOUN
ejpam-6409	608	6	in	in	ADP
ejpam-6409	608	7	which	which	PRON
ejpam-6409	608	8	the	the	DET
ejpam-6409	608	9	j	j	PROPN
ejpam-6409	608	10	-	-	PUNCT
ejpam-6409	608	11	th	th	VERB
ejpam-6409	608	12	coordinate	coordinate	NOUN
ejpam-6409	608	13	corresponds	correspond	VERB
ejpam-6409	608	14	to	to	ADP
ejpam-6409	608	15	the	the	DET
ejpam-6409	608	16	i	i	PROPN
ejpam-6409	608	17	-	-	PUNCT
ejpam-6409	608	18	th	th	X
ejpam-6409	608	19	element	element	NOUN
ejpam-6409	608	20	of	of	ADP
ejpam-6409	608	21	the	the	DET
ejpam-6409	608	22	2	2	NUM
ejpam-6409	608	23	-	-	PUNCT
ejpam-6409	608	24	ary	ary	NOUN
ejpam-6409	608	25	expansion	expansion	NOUN
ejpam-6409	608	26	of	of	ADP
ejpam-6409	608	27	uj	uj	PROPN
ejpam-6409	608	28	,	,	PUNCT
ejpam-6409	608	29	that	that	ADV
ejpam-6409	608	30	is	is	ADV
ejpam-6409	608	31	,	,	PUNCT
ejpam-6409	608	32	ui	ui	PROPN
ejpam-6409	608	33	=	=	PUNCT
ejpam-6409	608	34	(	(	PUNCT
ejpam-6409	608	35	u1,i	u1,i	PROPN
ejpam-6409	608	36	,	,	PUNCT
ejpam-6409	608	37	.	.	PUNCT
ejpam-6409	608	38	.	.	PUNCT
ejpam-6409	609	1	.	.	PUNCT
ejpam-6409	610	1	,	,	PUNCT
ejpam-6409	610	2	un	un	PROPN
ejpam-6409	610	3	,	,	PUNCT
ejpam-6409	610	4	i	i	NOUN
ejpam-6409	610	5	)	)	PUNCT
ejpam-6409	610	6	∈	∈	PROPN
ejpam-6409	610	7	zn	zn	NUM
ejpam-6409	610	8	2	2	NUM
ejpam-6409	611	1	[	[	X
ejpam-6409	611	2	ω	ω	X
ejpam-6409	611	3	]	]	X
ejpam-6409	611	4	.	.	PUNCT
ejpam-6409	612	1	proposition	proposition	NOUN
ejpam-6409	612	2	6.1	6.1	NUM
ejpam-6409	612	3	let	let	VERB
ejpam-6409	612	4	ĥ	ĥ	X
ejpam-6409	612	5	=	=	SYM
ejpam-6409	612	6	ĥ(t1,	ĥ(t1,	PROPN
ejpam-6409	612	7	...	...	PUNCT
ejpam-6409	612	8	,ts	,ts	PUNCT
ejpam-6409	612	9	)	)	PUNCT
ejpam-6409	612	10	be	be	AUX
ejpam-6409	612	11	the	the	DET
ejpam-6409	612	12	z2s	z2s	PROPN
ejpam-6409	612	13	[	[	X
ejpam-6409	612	14	ω]-additive	ω]-additive	ADJ
ejpam-6409	612	15	hadamard	hadamard	ADJ
ejpam-6409	612	16	code	code	NOUN
ejpam-6409	612	17	of	of	ADP
ejpam-6409	612	18	type	type	NOUN
ejpam-6409	612	19	(	(	PUNCT
ejpam-6409	612	20	n	n	CCONJ
ejpam-6409	612	21	;	;	PUNCT
ejpam-6409	612	22	t1	t1	NOUN
ejpam-6409	612	23	,	,	PUNCT
ejpam-6409	612	24	.	.	PUNCT
ejpam-6409	612	25	.	.	PUNCT
ejpam-6409	613	1	.	.	PUNCT
ejpam-6409	614	1	,	,	PUNCT
ejpam-6409	614	2	ts	ts	NOUN
ejpam-6409	614	3	)	)	PUNCT
ejpam-6409	614	4	such	such	ADJ
ejpam-6409	614	5	that	that	SCONJ
ejpam-6409	614	6	φs(ĥ	φs(ĥ	NOUN
ejpam-6409	614	7	)	)	PUNCT
ejpam-6409	614	8	is	be	AUX
ejpam-6409	614	9	nonlinear	nonlinear	ADJ
ejpam-6409	614	10	.	.	PUNCT
ejpam-6409	615	1	define	define	VERB
ejpam-6409	615	2	ĥb	ĥb	NOUN
ejpam-6409	615	3	as	as	ADP
ejpam-6409	615	4	the	the	DET
ejpam-6409	615	5	subcode	subcode	NOUN
ejpam-6409	615	6	of	of	ADP
ejpam-6409	615	7	ĥ	ĥ	PUNCT
ejpam-6409	615	8	consisting	consist	VERB
ejpam-6409	615	9	of	of	ADP
ejpam-6409	615	10	all	all	DET
ejpam-6409	615	11	codewords	codeword	NOUN
ejpam-6409	615	12	of	of	ADP
ejpam-6409	615	13	order	order	NOUN
ejpam-6409	615	14	two	two	NUM
ejpam-6409	615	15	.	.	PUNCT
ejpam-6409	616	1	let	let	VERB
ejpam-6409	616	2	b	b	NOUN
ejpam-6409	616	3	=	=	PRON
ejpam-6409	616	4	{	{	PUNCT
ejpam-6409	616	5	{	{	PUNCT
ejpam-6409	616	6	2p}σ−2	2p}σ−2	NUM
ejpam-6409	616	7	p=0	p=0	PROPN
ejpam-6409	616	8	if	if	SCONJ
ejpam-6409	616	9	σ	σ	PROPN
ejpam-6409	616	10	≥	≥	NOUN
ejpam-6409	616	11	2	2	NUM
ejpam-6409	616	12	,	,	PUNCT
ejpam-6409	616	13	∅	∅	NOUN
ejpam-6409	616	14	if	if	SCONJ
ejpam-6409	616	15	σ	σ	PROPN
ejpam-6409	616	16	=	=	SYM
ejpam-6409	616	17	1	1	X
ejpam-6409	616	18	.	.	PUNCT
ejpam-6409	617	1	then	then	ADV
ejpam-6409	617	2	,	,	PUNCT
ejpam-6409	617	3	⟨φs(ĥb	⟨φs(ĥb	PROPN
ejpam-6409	617	4	)	)	PUNCT
ejpam-6409	617	5	,	,	PUNCT
ejpam-6409	617	6	φs(b	φs(b	NUM
ejpam-6409	617	7	)	)	PUNCT
ejpam-6409	617	8	,	,	PUNCT
ejpam-6409	617	9	φs	φs	PROPN
ejpam-6409	617	10	(	(	PUNCT
ejpam-6409	617	11	s−2∑	s−2∑	PROPN
ejpam-6409	617	12	i=0	i=0	PROPN
ejpam-6409	617	13	2i	2i	NUM
ejpam-6409	617	14	)	)	PUNCT
ejpam-6409	617	15	⟩	⟩	PROPN
ejpam-6409	617	16	⊆	⊆	NUM
ejpam-6409	617	17	k(φs(ĥ	k(φs(ĥ	NOUN
ejpam-6409	617	18	)	)	PUNCT
ejpam-6409	617	19	)	)	PUNCT
ejpam-6409	618	1	muhammad	muhammad	PROPN
ejpam-6409	618	2	sajjad	sajjad	PROPN
ejpam-6409	618	3	et	et	PROPN
ejpam-6409	618	4	al	al	PROPN
ejpam-6409	618	5	.	.	PUNCT
ejpam-6409	618	6	/	/	SYM
ejpam-6409	618	7	eur	eur	PROPN
ejpam-6409	618	8	.	.	PUNCT
ejpam-6409	619	1	j.	j.	PROPN
ejpam-6409	619	2	pure	pure	PROPN
ejpam-6409	619	3	appl	appl	PROPN
ejpam-6409	619	4	.	.	PROPN
ejpam-6409	619	5	math	math	PROPN
ejpam-6409	619	6	,	,	PUNCT
ejpam-6409	619	7	18	18	NUM
ejpam-6409	619	8	(	(	PUNCT
ejpam-6409	619	9	3	3	NUM
ejpam-6409	619	10	)	)	PUNCT
ejpam-6409	619	11	(	(	PUNCT
ejpam-6409	619	12	2025	2025	NUM
ejpam-6409	619	13	)	)	PUNCT
ejpam-6409	619	14	,	,	PUNCT
ejpam-6409	619	15	6409	6409	NUM
ejpam-6409	619	16	18	18	NUM
ejpam-6409	619	17	of	of	ADP
ejpam-6409	619	18	32	32	NUM
ejpam-6409	619	19	and	and	CCONJ
ejpam-6409	619	20	ker(φs(ĥ	ker(φs(ĥ	PROPN
ejpam-6409	619	21	)	)	PUNCT
ejpam-6409	619	22	)	)	PUNCT
ejpam-6409	620	1	≥	≥	PROPN
ejpam-6409	621	1	σ	σ	NOUN
ejpam-6409	622	1	+	+	CCONJ
ejpam-6409	622	2	s∑	s∑	PROPN
ejpam-6409	622	3	i=1	i=1	PROPN
ejpam-6409	622	4	ti	ti	PROPN
ejpam-6409	622	5	.	.	PUNCT
ejpam-6409	622	6	proof	proof	NOUN
ejpam-6409	622	7	:	:	PUNCT
ejpam-6409	622	8	let	let	VERB
ejpam-6409	622	9	h	h	NOUN
ejpam-6409	622	10	=	=	SYM
ejpam-6409	622	11	φs(ĥ	φs(ĥ	NOUN
ejpam-6409	622	12	)	)	PUNCT
ejpam-6409	622	13	and	and	CCONJ
ejpam-6409	622	14	r	r	NOUN
ejpam-6409	622	15	=	=	PUNCT
ejpam-6409	622	16	∑s	∑s	PROPN
ejpam-6409	622	17	i=1	i=1	X
ejpam-6409	622	18	ti	ti	VERB
ejpam-6409	622	19	.	.	PUNCT
ejpam-6409	622	20	let	let	VERB
ejpam-6409	622	21	q	q	NOUN
ejpam-6409	623	1	=	=	PUNCT
ejpam-6409	623	2	{	{	PUNCT
ejpam-6409	623	3	ord(wj/2	ord(wj/2	NUM
ejpam-6409	623	4	)	)	PUNCT
ejpam-6409	623	5	·	·	PUNCT
ejpam-6409	623	6	wj}rj=0	wj}rj=0	PROPN
ejpam-6409	623	7	.	.	PUNCT
ejpam-6409	624	1	since	since	SCONJ
ejpam-6409	624	2	ĥb	ĥb	NOUN
ejpam-6409	624	3	includes	include	VERB
ejpam-6409	624	4	all	all	DET
ejpam-6409	624	5	elements	element	NOUN
ejpam-6409	624	6	of	of	ADP
ejpam-6409	624	7	ĥ	ĥ	PUNCT
ejpam-6409	624	8	with	with	ADP
ejpam-6409	624	9	order	order	NOUN
ejpam-6409	624	10	two	two	NUM
ejpam-6409	624	11	,	,	PUNCT
ejpam-6409	624	12	the	the	DET
ejpam-6409	624	13	set	set	NOUN
ejpam-6409	624	14	φs(q	φs(q	NOUN
ejpam-6409	624	15	)	)	PUNCT
ejpam-6409	624	16	serves	serve	VERB
ejpam-6409	624	17	as	as	ADP
ejpam-6409	624	18	a	a	DET
ejpam-6409	624	19	basis	basis	NOUN
ejpam-6409	624	20	for	for	ADP
ejpam-6409	624	21	the	the	DET
ejpam-6409	624	22	binary	binary	PROPN
ejpam-6409	624	23	linear	linear	PROPN
ejpam-6409	624	24	subcode	subcode	NOUN
ejpam-6409	624	25	hb	hb	X
ejpam-6409	624	26	=	=	SYM
ejpam-6409	624	27	φs(ĥb	φs(ĥb	PROPN
ejpam-6409	624	28	)	)	PUNCT
ejpam-6409	624	29	of	of	ADP
ejpam-6409	624	30	h.	h.	PROPN
ejpam-6409	624	31	by	by	ADP
ejpam-6409	624	32	corollary	corollary	ADJ
ejpam-6409	624	33	3.6	3.6	NUM
ejpam-6409	624	34	,	,	PUNCT
ejpam-6409	624	35	for	for	ADP
ejpam-6409	624	36	all	all	DET
ejpam-6409	624	37	b	b	PROPN
ejpam-6409	624	38	∈	∈	NOUN
ejpam-6409	624	39	ĥb	ĥb	NOUN
ejpam-6409	624	40	and	and	CCONJ
ejpam-6409	624	41	u	u	NOUN
ejpam-6409	624	42	∈	∈	PROPN
ejpam-6409	624	43	ĥ	ĥ	PROPN
ejpam-6409	624	44	,	,	PUNCT
ejpam-6409	624	45	we	we	PRON
ejpam-6409	624	46	have	have	VERB
ejpam-6409	624	47	φs(b	φs(b	NOUN
ejpam-6409	624	48	)	)	PUNCT
ejpam-6409	625	1	+	+	CCONJ
ejpam-6409	625	2	φs(u	φs(u	NOUN
ejpam-6409	625	3	)	)	PUNCT
ejpam-6409	625	4	=	=	SYM
ejpam-6409	626	1	φs(b	φs(b	PUNCT
ejpam-6409	626	2	+	+	CCONJ
ejpam-6409	626	3	u	u	NOUN
ejpam-6409	626	4	)	)	PUNCT
ejpam-6409	626	5	∈	∈	PROPN
ejpam-6409	626	6	h	h	NOUN
ejpam-6409	626	7	,	,	PUNCT
ejpam-6409	626	8	and	and	CCONJ
ejpam-6409	626	9	therefore	therefore	ADV
ejpam-6409	626	10	,	,	PUNCT
ejpam-6409	626	11	hb	hb	PROPN
ejpam-6409	626	12	⊆	⊆	NUM
ejpam-6409	626	13	k(h	k(h	PROPN
ejpam-6409	626	14	)	)	PUNCT
ejpam-6409	626	15	.	.	PUNCT
ejpam-6409	627	1	assume	assume	VERB
ejpam-6409	627	2	σ	σ	NOUN
ejpam-6409	627	3	≥	≥	NUM
ejpam-6409	627	4	2	2	NUM
ejpam-6409	627	5	.	.	PUNCT
ejpam-6409	628	1	now	now	ADV
ejpam-6409	628	2	,	,	PUNCT
ejpam-6409	628	3	we	we	PRON
ejpam-6409	628	4	prove	prove	VERB
ejpam-6409	628	5	that	that	SCONJ
ejpam-6409	628	6	φs(2	φs(2	VERB
ejpam-6409	628	7	p	p	X
ejpam-6409	628	8	)	)	PUNCT
ejpam-6409	628	9	∈	∈	PROPN
ejpam-6409	628	10	k(h	k(h	PROPN
ejpam-6409	628	11	)	)	PUNCT
ejpam-6409	628	12	for	for	ADP
ejpam-6409	628	13	all	all	DET
ejpam-6409	628	14	p	p	PROPN
ejpam-6409	628	15	∈	∈	PROPN
ejpam-6409	628	16	{	{	PUNCT
ejpam-6409	628	17	0	0	NUM
ejpam-6409	628	18	,	,	PUNCT
ejpam-6409	628	19	.	.	PUNCT
ejpam-6409	628	20	.	.	PUNCT
ejpam-6409	629	1	.	.	PUNCT
ejpam-6409	630	1	,	,	PUNCT
ejpam-6409	631	1	σ	σ	PROPN
ejpam-6409	631	2	−	−	NOUN
ejpam-6409	631	3	2	2	NUM
ejpam-6409	631	4	}	}	PUNCT
ejpam-6409	631	5	.	.	PUNCT
ejpam-6409	632	1	equivalently	equivalently	ADV
ejpam-6409	632	2	,	,	PUNCT
ejpam-6409	632	3	we	we	PRON
ejpam-6409	632	4	show	show	VERB
ejpam-6409	632	5	that	that	SCONJ
ejpam-6409	632	6	φs(2	φs(2	VERB
ejpam-6409	632	7	p	p	X
ejpam-6409	632	8	)	)	PUNCT
ejpam-6409	632	9	+	+	NUM
ejpam-6409	632	10	φs(u	φs(u	X
ejpam-6409	632	11	)	)	PUNCT
ejpam-6409	632	12	∈	∈	PROPN
ejpam-6409	632	13	h	h	NOUN
ejpam-6409	632	14	for	for	ADP
ejpam-6409	632	15	all	all	DET
ejpam-6409	632	16	u	u	PROPN
ejpam-6409	632	17	∈	∈	PROPN
ejpam-6409	632	18	ĥ.	ĥ.	NOUN
ejpam-6409	632	19	if	if	SCONJ
ejpam-6409	632	20	u	u	PROPN
ejpam-6409	632	21	∈	∈	PROPN
ejpam-6409	632	22	ĥ	ĥ	PROPN
ejpam-6409	632	23	,	,	PUNCT
ejpam-6409	632	24	then	then	ADV
ejpam-6409	632	25	u	u	X
ejpam-6409	632	26	=	=	PROPN
ejpam-6409	632	27	µ	µ	X
ejpam-6409	632	28	·	·	PUNCT
ejpam-6409	632	29	1	1	NUM
ejpam-6409	632	30	+	+	NUM
ejpam-6409	632	31	u′	u′	PROPN
ejpam-6409	632	32	,	,	PUNCT
ejpam-6409	632	33	where	where	SCONJ
ejpam-6409	632	34	µ	µ	X
ejpam-6409	632	35	∈	∈	X
ejpam-6409	632	36	z2s	z2s	X
ejpam-6409	633	1	[	[	X
ejpam-6409	633	2	ω	ω	X
ejpam-6409	633	3	]	]	X
ejpam-6409	633	4	and	and	CCONJ
ejpam-6409	633	5	ord(u′	ord(u′	NOUN
ejpam-6409	633	6	)	)	PUNCT
ejpam-6409	633	7	≤	≤	NOUN
ejpam-6409	633	8	ord(w2	ord(w2	ADP
ejpam-6409	633	9	)	)	PUNCT
ejpam-6409	634	1	=	=	SYM
ejpam-6409	634	2	2s+1−σ	2s+1−σ	X
ejpam-6409	634	3	.	.	PUNCT
ejpam-6409	634	4	let	let	VERB
ejpam-6409	634	5	u	u	PRON
ejpam-6409	634	6	=	=	PUNCT
ejpam-6409	634	7	(	(	PUNCT
ejpam-6409	634	8	u1	u1	PROPN
ejpam-6409	634	9	,	,	PUNCT
ejpam-6409	634	10	.	.	PUNCT
ejpam-6409	634	11	.	.	PUNCT
ejpam-6409	635	1	.	.	PUNCT
ejpam-6409	636	1	,	,	PUNCT
ejpam-6409	636	2	un	un	PROPN
ejpam-6409	636	3	)	)	PUNCT
ejpam-6409	636	4	∈	∈	PROPN
ejpam-6409	637	1	zn	zn	NOUN
ejpam-6409	637	2	2s	2s	X
ejpam-6409	638	1	[	[	X
ejpam-6409	638	2	ω	ω	X
ejpam-6409	638	3	]	]	PUNCT
ejpam-6409	638	4	and	and	CCONJ
ejpam-6409	638	5	[	[	X
ejpam-6409	638	6	ui,0	ui,0	PROPN
ejpam-6409	638	7	,	,	PUNCT
ejpam-6409	638	8	ui,1	ui,1	PROPN
ejpam-6409	638	9	,	,	PUNCT
ejpam-6409	638	10	.	.	PUNCT
ejpam-6409	638	11	.	.	PUNCT
ejpam-6409	638	12	.	.	PUNCT
ejpam-6409	639	1	,	,	PUNCT
ejpam-6409	639	2	ui	ui	PROPN
ejpam-6409	639	3	,	,	PUNCT
ejpam-6409	639	4	s−1]2	s−1]2	ADV
ejpam-6409	639	5	be	be	AUX
ejpam-6409	639	6	the	the	DET
ejpam-6409	639	7	binary	binary	ADJ
ejpam-6409	639	8	expansion	expansion	NOUN
ejpam-6409	639	9	of	of	ADP
ejpam-6409	639	10	ui	ui	PROPN
ejpam-6409	639	11	,	,	PUNCT
ejpam-6409	639	12	i	i	PRON
ejpam-6409	639	13	∈	∈	PROPN
ejpam-6409	639	14	{	{	PUNCT
ejpam-6409	639	15	1	1	NUM
ejpam-6409	639	16	,	,	PUNCT
ejpam-6409	639	17	.	.	PUNCT
ejpam-6409	639	18	.	.	PUNCT
ejpam-6409	640	1	.	.	PUNCT
ejpam-6409	640	2	,	,	PUNCT
ejpam-6409	640	3	n	n	CCONJ
ejpam-6409	640	4	}	}	PUNCT
ejpam-6409	640	5	.	.	PUNCT
ejpam-6409	641	1	let	let	VERB
ejpam-6409	641	2	[	[	X
ejpam-6409	641	3	µ0	µ0	NOUN
ejpam-6409	641	4	,	,	PUNCT
ejpam-6409	641	5	µ1	µ1	PROPN
ejpam-6409	641	6	,	,	PUNCT
ejpam-6409	641	7	.	.	PUNCT
ejpam-6409	641	8	.	.	PUNCT
ejpam-6409	642	1	.	.	PUNCT
ejpam-6409	643	1	,	,	PUNCT
ejpam-6409	643	2	µs−1]2	µs−1]2	ADV
ejpam-6409	643	3	be	be	AUX
ejpam-6409	643	4	the	the	DET
ejpam-6409	643	5	binary	binary	ADJ
ejpam-6409	643	6	expansion	expansion	NOUN
ejpam-6409	643	7	of	of	ADP
ejpam-6409	643	8	µ	µ	X
ejpam-6409	643	9	∈	∈	NOUN
ejpam-6409	643	10	z2s	z2s	X
ejpam-6409	644	1	[	[	X
ejpam-6409	644	2	ω	ω	X
ejpam-6409	644	3	]	]	X
ejpam-6409	644	4	.	.	PUNCT
ejpam-6409	645	1	note	note	VERB
ejpam-6409	645	2	that	that	SCONJ
ejpam-6409	645	3	if	if	SCONJ
ejpam-6409	645	4	v	v	NUM
ejpam-6409	645	5	∈	∈	X
ejpam-6409	645	6	z2s	z2s	X
ejpam-6409	646	1	[	[	X
ejpam-6409	646	2	ω	ω	X
ejpam-6409	646	3	]	]	X
ejpam-6409	646	4	is	be	AUX
ejpam-6409	646	5	of	of	ADP
ejpam-6409	646	6	order	order	NOUN
ejpam-6409	646	7	2	2	NUM
ejpam-6409	646	8	i	i	NOUN
ejpam-6409	646	9	,	,	PUNCT
ejpam-6409	646	10	then	then	ADV
ejpam-6409	646	11	its	its	PRON
ejpam-6409	646	12	binary	binary	ADJ
ejpam-6409	646	13	expansion	expansion	NOUN
ejpam-6409	646	14	is	be	AUX
ejpam-6409	646	15	of	of	ADP
ejpam-6409	646	16	the	the	DET
ejpam-6409	646	17	form	form	NOUN
ejpam-6409	646	18	[	[	X
ejpam-6409	646	19	0	0	NUM
ejpam-6409	646	20	,	,	PUNCT
ejpam-6409	646	21	.	.	PUNCT
ejpam-6409	646	22	.	.	PUNCT
ejpam-6409	647	1	.	.	PUNCT
ejpam-6409	648	1	,	,	PUNCT
ejpam-6409	648	2	0	0	NUM
ejpam-6409	648	3	,	,	PUNCT
ejpam-6409	648	4	vs−i	vs−i	NOUN
ejpam-6409	648	5	,	,	PUNCT
ejpam-6409	648	6	vs−i+1	vs−i+1	PROPN
ejpam-6409	648	7	,	,	PUNCT
ejpam-6409	648	8	.	.	PUNCT
ejpam-6409	648	9	.	.	PUNCT
ejpam-6409	649	1	.	.	PUNCT
ejpam-6409	650	1	,	,	PUNCT
ejpam-6409	650	2	vs−1]2	vs−1]2	X
ejpam-6409	650	3	.	.	PUNCT
ejpam-6409	651	1	since	since	SCONJ
ejpam-6409	651	2	p	p	PROPN
ejpam-6409	651	3	∈	∈	PROPN
ejpam-6409	651	4	{	{	PUNCT
ejpam-6409	651	5	0	0	NUM
ejpam-6409	651	6	,	,	PUNCT
ejpam-6409	651	7	.	.	PUNCT
ejpam-6409	651	8	.	.	PUNCT
ejpam-6409	651	9	.	.	PUNCT
ejpam-6409	652	1	,	,	PUNCT
ejpam-6409	653	1	σ	σ	PROPN
ejpam-6409	653	2	−	−	NOUN
ejpam-6409	653	3	2	2	NUM
ejpam-6409	653	4	}	}	PUNCT
ejpam-6409	653	5	and	and	CCONJ
ejpam-6409	653	6	ord(u′	ord(u′	NOUN
ejpam-6409	653	7	)	)	PUNCT
ejpam-6409	653	8	≤	≤	NOUN
ejpam-6409	653	9	2s+1−σ	2s+1−σ	NUM
ejpam-6409	653	10	,	,	PUNCT
ejpam-6409	653	11	we	we	PRON
ejpam-6409	653	12	have	have	VERB
ejpam-6409	653	13	u(p	u(p	NOUN
ejpam-6409	653	14	)	)	PUNCT
ejpam-6409	653	15	=	=	SYM
ejpam-6409	653	16	(	(	PUNCT
ejpam-6409	653	17	u1,p	u1,p	PROPN
ejpam-6409	653	18	,	,	PUNCT
ejpam-6409	653	19	.	.	PUNCT
ejpam-6409	653	20	.	.	PUNCT
ejpam-6409	653	21	.	.	PUNCT
ejpam-6409	654	1	,	,	PUNCT
ejpam-6409	654	2	un	un	PROPN
ejpam-6409	654	3	,	,	PUNCT
ejpam-6409	654	4	p	p	NOUN
ejpam-6409	654	5	)	)	PUNCT
ejpam-6409	654	6	=	=	SYM
ejpam-6409	654	7	(	(	PUNCT
ejpam-6409	654	8	µp	µp	NOUN
ejpam-6409	654	9	,	,	PUNCT
ejpam-6409	654	10	.	.	PUNCT
ejpam-6409	654	11	.	.	PUNCT
ejpam-6409	655	1	.	.	PUNCT
ejpam-6409	656	1	,	,	PUNCT
ejpam-6409	656	2	µp	µp	NOUN
ejpam-6409	656	3	)	)	PUNCT
ejpam-6409	656	4	.	.	PUNCT
ejpam-6409	657	1	by	by	ADP
ejpam-6409	657	2	corollary	corollary	ADJ
ejpam-6409	657	3	3.4	3.4	NUM
ejpam-6409	657	4	,	,	PUNCT
ejpam-6409	657	5	we	we	PRON
ejpam-6409	657	6	have	have	VERB
ejpam-6409	657	7	φs(2	φs(2	NOUN
ejpam-6409	657	8	p	p	X
ejpam-6409	657	9	)	)	PUNCT
ejpam-6409	657	10	+	+	NUM
ejpam-6409	657	11	φs(u	φs(u	NOUN
ejpam-6409	657	12	)	)	PUNCT
ejpam-6409	658	1	=	=	PUNCT
ejpam-6409	658	2	φs(2	φs(2	PROPN
ejpam-6409	658	3	p	p	NOUN
ejpam-6409	658	4	+	+	CCONJ
ejpam-6409	658	5	u−	u−	PROPN
ejpam-6409	658	6	2p+1tp	2p+1tp	NUM
ejpam-6409	658	7	)	)	PUNCT
ejpam-6409	658	8	,	,	PUNCT
ejpam-6409	658	9	where	where	SCONJ
ejpam-6409	658	10	tp	tp	NOUN
ejpam-6409	658	11	=	=	SYM
ejpam-6409	658	12	{	{	PUNCT
ejpam-6409	658	13	1	1	NUM
ejpam-6409	658	14	if	if	SCONJ
ejpam-6409	658	15	µp	µp	PROPN
ejpam-6409	658	16	≥	≥	NUM
ejpam-6409	658	17	10	10	NUM
ejpam-6409	658	18	,	,	PUNCT
ejpam-6409	658	19	0	0	NUM
ejpam-6409	658	20	otherwise	otherwise	ADV
ejpam-6409	658	21	.	.	PUNCT
ejpam-6409	659	1	therefore	therefore	ADV
ejpam-6409	659	2	,	,	PUNCT
ejpam-6409	659	3	2p+1tp	2p+1tp	NUM
ejpam-6409	659	4	is	be	AUX
ejpam-6409	659	5	either	either	CCONJ
ejpam-6409	659	6	0	0	NUM
ejpam-6409	659	7	or	or	CCONJ
ejpam-6409	659	8	2p+1	2p+1	PROPN
ejpam-6409	659	9	.	.	PUNCT
ejpam-6409	660	1	in	in	ADP
ejpam-6409	660	2	both	both	DET
ejpam-6409	660	3	cases	case	NOUN
ejpam-6409	660	4	,	,	PUNCT
ejpam-6409	660	5	2p+1tp	2p+1tp	NUM
ejpam-6409	660	6	∈	∈	NOUN
ejpam-6409	660	7	ĥ	ĥ	NOUN
ejpam-6409	660	8	,	,	PUNCT
ejpam-6409	660	9	so	so	ADV
ejpam-6409	660	10	φs(2	φs(2	PROPN
ejpam-6409	660	11	p	p	X
ejpam-6409	660	12	)	)	PUNCT
ejpam-6409	660	13	+	+	NUM
ejpam-6409	660	14	φs(u	φs(u	NOUN
ejpam-6409	660	15	)	)	PUNCT
ejpam-6409	661	1	=	=	PUNCT
ejpam-6409	661	2	φs(2	φs(2	PROPN
ejpam-6409	661	3	p	p	NOUN
ejpam-6409	661	4	+	+	CCONJ
ejpam-6409	661	5	u−	u−	PROPN
ejpam-6409	661	6	2p+1tp	2p+1tp	NUM
ejpam-6409	661	7	)	)	PUNCT
ejpam-6409	662	1	∈	∈	PROPN
ejpam-6409	662	2	h.	h.	PROPN
ejpam-6409	662	3	muhammad	muhammad	PROPN
ejpam-6409	662	4	sajjad	sajjad	PROPN
ejpam-6409	662	5	et	et	PROPN
ejpam-6409	662	6	al	al	PROPN
ejpam-6409	662	7	.	.	PUNCT
ejpam-6409	662	8	/	/	SYM
ejpam-6409	662	9	eur	eur	PROPN
ejpam-6409	662	10	.	.	PUNCT
ejpam-6409	663	1	j.	j.	PROPN
ejpam-6409	663	2	pure	pure	PROPN
ejpam-6409	663	3	appl	appl	PROPN
ejpam-6409	663	4	.	.	PROPN
ejpam-6409	663	5	math	math	PROPN
ejpam-6409	663	6	,	,	PUNCT
ejpam-6409	663	7	18	18	NUM
ejpam-6409	663	8	(	(	PUNCT
ejpam-6409	663	9	3	3	NUM
ejpam-6409	663	10	)	)	PUNCT
ejpam-6409	663	11	(	(	PUNCT
ejpam-6409	663	12	2025	2025	NUM
ejpam-6409	663	13	)	)	PUNCT
ejpam-6409	663	14	,	,	PUNCT
ejpam-6409	663	15	6409	6409	NUM
ejpam-6409	663	16	19	19	NUM
ejpam-6409	663	17	of	of	ADP
ejpam-6409	663	18	32	32	NUM
ejpam-6409	663	19	next	next	ADV
ejpam-6409	663	20	,	,	PUNCT
ejpam-6409	663	21	it	it	PRON
ejpam-6409	663	22	can	can	AUX
ejpam-6409	663	23	be	be	AUX
ejpam-6409	663	24	easily	easily	ADV
ejpam-6409	663	25	verified	verify	VERB
ejpam-6409	663	26	that	that	SCONJ
ejpam-6409	663	27	φs	φs	PROPN
ejpam-6409	663	28	(	(	PUNCT
ejpam-6409	663	29	s−2∑	s−2∑	PROPN
ejpam-6409	663	30	i=0	i=0	PROPN
ejpam-6409	663	31	2i	2i	NUM
ejpam-6409	663	32	)	)	PUNCT
ejpam-6409	663	33	∈	∈	PROPN
ejpam-6409	663	34	k(h	k(h	PROPN
ejpam-6409	663	35	)	)	PUNCT
ejpam-6409	663	36	.	.	PUNCT
ejpam-6409	664	1	finally	finally	ADV
ejpam-6409	664	2	,	,	PUNCT
ejpam-6409	664	3	it	it	PRON
ejpam-6409	664	4	remains	remain	VERB
ejpam-6409	664	5	to	to	PART
ejpam-6409	664	6	verify	verify	VERB
ejpam-6409	664	7	that	that	SCONJ
ejpam-6409	664	8	the	the	DET
ejpam-6409	664	9	elements	element	NOUN
ejpam-6409	664	10	of	of	ADP
ejpam-6409	664	11	the	the	DET
ejpam-6409	664	12	set	set	NOUN
ejpam-6409	664	13	{	{	PUNCT
ejpam-6409	664	14	φs(ĥb	φs(ĥb	ADP
ejpam-6409	664	15	)	)	PUNCT
ejpam-6409	664	16	,	,	PUNCT
ejpam-6409	664	17	φs(b	φs(b	NUM
ejpam-6409	664	18	)	)	PUNCT
ejpam-6409	664	19	,	,	PUNCT
ejpam-6409	664	20	φs	φs	PROPN
ejpam-6409	664	21	(	(	PUNCT
ejpam-6409	664	22	s−2∑	s−2∑	PROPN
ejpam-6409	664	23	i=0	i=0	PROPN
ejpam-6409	664	24	2i	2i	NUM
ejpam-6409	664	25	)	)	PUNCT
ejpam-6409	664	26	}	}	PUNCT
ejpam-6409	664	27	are	be	AUX
ejpam-6409	664	28	linearly	linearly	ADV
ejpam-6409	664	29	independent	independent	ADJ
ejpam-6409	664	30	.	.	PUNCT
ejpam-6409	665	1	due	due	ADP
ejpam-6409	665	2	to	to	ADP
ejpam-6409	665	3	the	the	DET
ejpam-6409	665	4	block	block	NOUN
ejpam-6409	665	5	upper	upper	ADJ
ejpam-6409	665	6	-	-	PUNCT
ejpam-6409	665	7	triangular	triangular	NOUN
ejpam-6409	665	8	structure	structure	NOUN
ejpam-6409	665	9	of	of	ADP
ejpam-6409	665	10	the	the	DET
ejpam-6409	665	11	generator	generator	NOUN
ejpam-6409	665	12	matrix	matrix	NOUN
ejpam-6409	665	13	,	,	PUNCT
ejpam-6409	665	14	it	it	PRON
ejpam-6409	665	15	is	be	AUX
ejpam-6409	665	16	straightforward	straightforward	ADJ
ejpam-6409	665	17	to	to	PART
ejpam-6409	665	18	verify	verify	VERB
ejpam-6409	665	19	that	that	SCONJ
ejpam-6409	665	20	the	the	DET
ejpam-6409	665	21	codewords	codeword	NOUN
ejpam-6409	665	22	in	in	ADP
ejpam-6409	665	23	φs(q	φs(q	NOUN
ejpam-6409	665	24	)	)	PUNCT
ejpam-6409	665	25	are	be	AUX
ejpam-6409	665	26	linearly	linearly	ADV
ejpam-6409	665	27	independent	independent	ADJ
ejpam-6409	665	28	from	from	ADP
ejpam-6409	665	29	the	the	DET
ejpam-6409	665	30	codewords	codeword	NOUN
ejpam-6409	665	31	in	in	ADP
ejpam-6409	665	32	{	{	PUNCT
ejpam-6409	665	33	φs(b	φs(b	NOUN
ejpam-6409	665	34	)	)	PUNCT
ejpam-6409	665	35	,	,	PUNCT
ejpam-6409	665	36	φs	φs	PROPN
ejpam-6409	665	37	(	(	PUNCT
ejpam-6409	665	38	s−2∑	s−2∑	PROPN
ejpam-6409	665	39	i=0	i=0	PROPN
ejpam-6409	665	40	2i	2i	NUM
ejpam-6409	665	41	)	)	PUNCT
ejpam-6409	665	42	}	}	PUNCT
ejpam-6409	665	43	.	.	PUNCT
ejpam-6409	666	1	note	note	VERB
ejpam-6409	666	2	σ	σ	NOUN
ejpam-6409	666	3	≤	≤	PROPN
ejpam-6409	666	4	s	s	ADJ
ejpam-6409	666	5	since	since	SCONJ
ejpam-6409	666	6	h	h	NOUN
ejpam-6409	666	7	is	be	AUX
ejpam-6409	666	8	nonlinear	nonlinear	ADJ
ejpam-6409	666	9	.	.	PUNCT
ejpam-6409	667	1	thus	thus	ADV
ejpam-6409	667	2	,	,	PUNCT
ejpam-6409	667	3	by	by	ADP
ejpam-6409	667	4	applying	apply	VERB
ejpam-6409	667	5	lemma	lemma	PROPN
ejpam-6409	667	6	3.4	3.4	NUM
ejpam-6409	667	7	,	,	PUNCT
ejpam-6409	667	8	we	we	PRON
ejpam-6409	667	9	readily	readily	ADV
ejpam-6409	667	10	conclude	conclude	VERB
ejpam-6409	667	11	that	that	SCONJ
ejpam-6409	667	12	the	the	DET
ejpam-6409	667	13	codewords	codeword	NOUN
ejpam-6409	667	14	in	in	ADP
ejpam-6409	667	15	{	{	PUNCT
ejpam-6409	667	16	φs(b	φs(b	NOUN
ejpam-6409	667	17	)	)	PUNCT
ejpam-6409	667	18	,	,	PUNCT
ejpam-6409	667	19	φs	φs	PROPN
ejpam-6409	667	20	(	(	PUNCT
ejpam-6409	667	21	s−2∑	s−2∑	PROPN
ejpam-6409	667	22	i=0	i=0	PROPN
ejpam-6409	667	23	2i	2i	NUM
ejpam-6409	667	24	)	)	PUNCT
ejpam-6409	667	25	}	}	PUNCT
ejpam-6409	667	26	are	be	AUX
ejpam-6409	667	27	linearly	linearly	ADV
ejpam-6409	667	28	independent	independent	ADJ
ejpam-6409	667	29	,	,	PUNCT
ejpam-6409	667	30	which	which	PRON
ejpam-6409	667	31	implies	imply	VERB
ejpam-6409	667	32	that	that	SCONJ
ejpam-6409	667	33	the	the	DET
ejpam-6409	667	34	dimension	dimension	NOUN
ejpam-6409	667	35	of	of	ADP
ejpam-6409	667	36	their	their	PRON
ejpam-6409	667	37	linear	linear	ADJ
ejpam-6409	667	38	span	span	NOUN
ejpam-6409	667	39	is	be	AUX
ejpam-6409	667	40	σ+	σ+	ADJ
ejpam-6409	667	41	r	r	NOUN
ejpam-6409	667	42	,	,	PUNCT
ejpam-6409	667	43	so	so	ADV
ejpam-6409	667	44	ker(h	ker(h	PROPN
ejpam-6409	667	45	)	)	PUNCT
ejpam-6409	667	46	≥	≥	NOUN
ejpam-6409	668	1	σ	σ	PROPN
ejpam-6409	668	2	+	+	PROPN
ejpam-6409	668	3	r.	r.	PROPN
ejpam-6409	668	4	lemma	lemma	PROPN
ejpam-6409	668	5	6.1	6.1	NUM
ejpam-6409	668	6	:	:	PUNCT
ejpam-6409	668	7	let	let	VERB
ejpam-6409	668	8	v	v	NOUN
ejpam-6409	668	9	,	,	PUNCT
ejpam-6409	668	10	µ	µ	X
ejpam-6409	668	11	∈	∈	NOUN
ejpam-6409	668	12	z2s	z2s	X
ejpam-6409	669	1	[	[	X
ejpam-6409	669	2	ω	ω	X
ejpam-6409	669	3	]	]	X
ejpam-6409	669	4	.	.	PUNCT
ejpam-6409	670	1	then	then	ADV
ejpam-6409	670	2	,	,	PUNCT
ejpam-6409	670	3	v	v	X
ejpam-6409	670	4	⊙2	⊙2	NOUN
ejpam-6409	670	5	µ	µ	X
ejpam-6409	670	6	=	=	SYM
ejpam-6409	670	7	s−1∑	s−1∑	NUM
ejpam-6409	670	8	i=0	i=0	X
ejpam-6409	670	9	(	(	PUNCT
ejpam-6409	670	10	v	v	NUM
ejpam-6409	670	11	⊙2	⊙2	VERB
ejpam-6409	670	12	µi2	µi2	ADV
ejpam-6409	670	13	i	i	PROPN
ejpam-6409	670	14	)	)	PUNCT
ejpam-6409	670	15	,	,	PUNCT
ejpam-6409	670	16	where	where	SCONJ
ejpam-6409	670	17	[	[	PUNCT
ejpam-6409	670	18	µ0	µ0	NOUN
ejpam-6409	670	19	,	,	PUNCT
ejpam-6409	670	20	.	.	PUNCT
ejpam-6409	670	21	.	.	PUNCT
ejpam-6409	670	22	.	.	PUNCT
ejpam-6409	671	1	,	,	PUNCT
ejpam-6409	671	2	µs−1	µs−1	NOUN
ejpam-6409	671	3	]	]	PUNCT
ejpam-6409	671	4	2	2	NUM
ejpam-6409	671	5	is	be	AUX
ejpam-6409	671	6	the	the	DET
ejpam-6409	671	7	2	2	NUM
ejpam-6409	671	8	-	-	PUNCT
ejpam-6409	671	9	ary	ary	NOUN
ejpam-6409	671	10	expansion	expansion	NOUN
ejpam-6409	671	11	of	of	ADP
ejpam-6409	671	12	µ.	µ.	PROPN
ejpam-6409	671	13	proof	proof	NOUN
ejpam-6409	671	14	:	:	PUNCT
ejpam-6409	671	15	let	let	VERB
ejpam-6409	671	16	v	v	X
ejpam-6409	671	17	∈	∈	NOUN
ejpam-6409	671	18	z2s	z2s	X
ejpam-6409	672	1	[	[	X
ejpam-6409	672	2	ω	ω	X
ejpam-6409	672	3	]	]	X
ejpam-6409	672	4	and	and	CCONJ
ejpam-6409	672	5	[	[	PUNCT
ejpam-6409	672	6	v0	v0	NOUN
ejpam-6409	672	7	,	,	PUNCT
ejpam-6409	672	8	.	.	PUNCT
ejpam-6409	672	9	.	.	PUNCT
ejpam-6409	673	1	.	.	PUNCT
ejpam-6409	674	1	,	,	PUNCT
ejpam-6409	674	2	vs−1	vs−1	ADJ
ejpam-6409	674	3	]	]	PUNCT
ejpam-6409	674	4	2	2	NUM
ejpam-6409	674	5	be	be	AUX
ejpam-6409	674	6	its	its	PRON
ejpam-6409	674	7	2	2	NUM
ejpam-6409	674	8	-	-	PUNCT
ejpam-6409	674	9	ary	ary	NOUN
ejpam-6409	674	10	expansion	expansion	NOUN
ejpam-6409	674	11	.	.	PUNCT
ejpam-6409	675	1	from	from	ADP
ejpam-6409	675	2	the	the	DET
ejpam-6409	675	3	definition	definition	NOUN
ejpam-6409	675	4	,	,	PUNCT
ejpam-6409	675	5	we	we	PRON
ejpam-6409	675	6	have	have	VERB
ejpam-6409	675	7	v	v	NUM
ejpam-6409	675	8	⊙2	⊙2	NOUN
ejpam-6409	675	9	µ	µ	PROPN
ejpam-6409	675	10	=	=	SYM
ejpam-6409	675	11	v	v	NUM
ejpam-6409	675	12	⊙2	⊙2	NOUN
ejpam-6409	675	13	s−1∑	s−1∑	PROPN
ejpam-6409	675	14	i=0	i=0	PROPN
ejpam-6409	675	15	µi2	µi2	ADJ
ejpam-6409	675	16	i	i	NOUN
ejpam-6409	675	17	=	=	SYM
ejpam-6409	675	18	s−1∑	s−1∑	NUM
ejpam-6409	675	19	i=0	i=0	PROPN
ejpam-6409	675	20	ti2	ti2	PROPN
ejpam-6409	675	21	i	i	PRON
ejpam-6409	675	22	,	,	PUNCT
ejpam-6409	675	23	where	where	SCONJ
ejpam-6409	675	24	ti	ti	NOUN
ejpam-6409	675	25	=	=	SYM
ejpam-6409	675	26	{	{	PUNCT
ejpam-6409	675	27	1	1	NUM
ejpam-6409	675	28	if	if	SCONJ
ejpam-6409	675	29	vi	vi	PROPN
ejpam-6409	676	1	+	+	CCONJ
ejpam-6409	676	2	µi	µi	PROPN
ejpam-6409	676	3	≥	≥	NUM
ejpam-6409	676	4	2	2	NUM
ejpam-6409	676	5	,	,	PUNCT
ejpam-6409	676	6	0	0	NUM
ejpam-6409	676	7	otherwise	otherwise	ADV
ejpam-6409	676	8	.	.	PUNCT
ejpam-6409	677	1	note	note	VERB
ejpam-6409	677	2	that	that	SCONJ
ejpam-6409	677	3	ti2	ti2	PROPN
ejpam-6409	677	4	i	i	PRON
ejpam-6409	678	1	=	=	PUNCT
ejpam-6409	678	2	v	v	NUM
ejpam-6409	678	3	⊙2	⊙2	NOUN
ejpam-6409	678	4	µi2	µi2	PROPN
ejpam-6409	679	1	i	i	PROPN
ejpam-6409	679	2	,	,	PUNCT
ejpam-6409	679	3	muhammad	muhammad	PROPN
ejpam-6409	679	4	sajjad	sajjad	PROPN
ejpam-6409	679	5	et	et	PROPN
ejpam-6409	679	6	al	al	PROPN
ejpam-6409	679	7	.	.	PUNCT
ejpam-6409	679	8	/	/	SYM
ejpam-6409	679	9	eur	eur	PROPN
ejpam-6409	679	10	.	.	PUNCT
ejpam-6409	680	1	j.	j.	PROPN
ejpam-6409	680	2	pure	pure	PROPN
ejpam-6409	680	3	appl	appl	PROPN
ejpam-6409	680	4	.	.	PROPN
ejpam-6409	680	5	math	math	PROPN
ejpam-6409	680	6	,	,	PUNCT
ejpam-6409	680	7	18	18	NUM
ejpam-6409	680	8	(	(	PUNCT
ejpam-6409	680	9	3	3	NUM
ejpam-6409	680	10	)	)	PUNCT
ejpam-6409	680	11	(	(	PUNCT
ejpam-6409	680	12	2025	2025	NUM
ejpam-6409	680	13	)	)	PUNCT
ejpam-6409	680	14	,	,	PUNCT
ejpam-6409	680	15	6409	6409	NUM
ejpam-6409	680	16	20	20	NUM
ejpam-6409	680	17	of	of	ADP
ejpam-6409	680	18	32	32	NUM
ejpam-6409	680	19	so	so	ADV
ejpam-6409	680	20	v	v	ADP
ejpam-6409	680	21	⊙2	⊙2	NOUN
ejpam-6409	680	22	s−1∑	s−1∑	PROPN
ejpam-6409	680	23	i=0	i=0	PROPN
ejpam-6409	680	24	µi2	µi2	ADJ
ejpam-6409	680	25	i	i	NOUN
ejpam-6409	680	26	=	=	SYM
ejpam-6409	680	27	s−1∑	s−1∑	NUM
ejpam-6409	680	28	i=0	i=0	X
ejpam-6409	680	29	(	(	PUNCT
ejpam-6409	680	30	v	v	NUM
ejpam-6409	680	31	⊙2	⊙2	VERB
ejpam-6409	680	32	µi2	µi2	ADV
ejpam-6409	680	33	i	i	PROPN
ejpam-6409	680	34	)	)	PUNCT
ejpam-6409	680	35	.	.	PUNCT
ejpam-6409	681	1	lemma	lemma	PROPN
ejpam-6409	681	2	6.2	6.2	NUM
ejpam-6409	681	3	:	:	PUNCT
ejpam-6409	681	4	let	let	VERB
ejpam-6409	681	5	h	h	NOUN
ejpam-6409	681	6	=	=	PUNCT
ejpam-6409	681	7	h(t1,	h(t1,	NOUN
ejpam-6409	681	8	...	...	PUNCT
ejpam-6409	681	9	,ts	,ts	PUNCT
ejpam-6409	681	10	)	)	PUNCT
ejpam-6409	681	11	be	be	AUX
ejpam-6409	681	12	the	the	DET
ejpam-6409	681	13	z2s	z2s	PROPN
ejpam-6409	682	1	[	[	X
ejpam-6409	682	2	ω]-additive	ω]-additive	ADJ
ejpam-6409	682	3	hadamard	hadamard	ADJ
ejpam-6409	682	4	code	code	NOUN
ejpam-6409	682	5	of	of	ADP
ejpam-6409	682	6	type	type	NOUN
ejpam-6409	682	7	(	(	PUNCT
ejpam-6409	682	8	n	n	CCONJ
ejpam-6409	682	9	;	;	PUNCT
ejpam-6409	682	10	t1	t1	NOUN
ejpam-6409	682	11	,	,	PUNCT
ejpam-6409	682	12	.	.	PUNCT
ejpam-6409	682	13	.	.	PUNCT
ejpam-6409	683	1	.	.	PUNCT
ejpam-6409	684	1	,	,	PUNCT
ejpam-6409	684	2	ts	ts	PROPN
ejpam-6409	684	3	)	)	PUNCT
ejpam-6409	684	4	.	.	PUNCT
ejpam-6409	685	1	define	define	VERB
ejpam-6409	685	2	n	n	NOUN
ejpam-6409	685	3	=	=	PRON
ejpam-6409	685	4	{	{	PUNCT
ejpam-6409	685	5	s−1∑	s−1∑	NUM
ejpam-6409	685	6	i=0	i=0	PROPN
ejpam-6409	685	7	µi2	µi2	NOUN
ejpam-6409	686	1	i	i	PRON
ejpam-6409	686	2	:	:	PUNCT
ejpam-6409	686	3	µi	µi	PROPN
ejpam-6409	686	4	∈	∈	PROPN
ejpam-6409	686	5	z2[ω	z2[ω	NOUN
ejpam-6409	686	6	]	]	PUNCT
ejpam-6409	686	7	}	}	PUNCT
ejpam-6409	686	8	\	\	NOUN
ejpam-6409	686	9	{	{	PUNCT
ejpam-6409	686	10	s−1∑	s−1∑	NUM
ejpam-6409	686	11	i=0	i=0	PROPN
ejpam-6409	686	12	2i	2i	NUM
ejpam-6409	686	13	}	}	PUNCT
ejpam-6409	686	14	if	if	SCONJ
ejpam-6409	686	15	σ	σ	PROPN
ejpam-6409	686	16	≤	≤	PROPN
ejpam-6409	686	17	s−	s−	PROPN
ejpam-6409	686	18	1	1	NUM
ejpam-6409	686	19	.	.	PUNCT
ejpam-6409	687	1	then	then	ADV
ejpam-6409	687	2	,	,	PUNCT
ejpam-6409	687	3	φs(n	φs(n	X
ejpam-6409	687	4	)	)	PUNCT
ejpam-6409	687	5	∩k	∩k	NOUN
ejpam-6409	687	6	(	(	PUNCT
ejpam-6409	687	7	φs(h	φs(h	PROPN
ejpam-6409	687	8	)	)	PUNCT
ejpam-6409	687	9	)	)	PUNCT
ejpam-6409	688	1	=	=	PUNCT
ejpam-6409	688	2	{	{	PUNCT
ejpam-6409	688	3	0	0	NUM
ejpam-6409	688	4	}	}	PUNCT
ejpam-6409	688	5	.	.	PUNCT
ejpam-6409	689	1	proof	proof	NOUN
ejpam-6409	689	2	:	:	PUNCT
ejpam-6409	689	3	let	let	VERB
ejpam-6409	689	4	h	h	NOUN
ejpam-6409	689	5	=	=	VERB
ejpam-6409	689	6	φs(h	φs(h	X
ejpam-6409	689	7	)	)	PUNCT
ejpam-6409	689	8	.	.	PUNCT
ejpam-6409	690	1	suppose	suppose	VERB
ejpam-6409	690	2	u	u	PRON
ejpam-6409	690	3	=	=	PROPN
ejpam-6409	690	4	s−1∑	s−1∑	NUM
ejpam-6409	690	5	i=0	i=0	PROPN
ejpam-6409	690	6	µi2	µi2	NOUN
ejpam-6409	690	7	i	i	PRON
ejpam-6409	690	8	∈	∈	PROPN
ejpam-6409	690	9	n	n	PRON
ejpam-6409	690	10	such	such	ADJ
ejpam-6409	690	11	that	that	PRON
ejpam-6409	690	12	φs(u	φs(u	PUNCT
ejpam-6409	690	13	)	)	PUNCT
ejpam-6409	690	14	∈	∈	PROPN
ejpam-6409	690	15	k(h	k(h	PROPN
ejpam-6409	690	16	)	)	PUNCT
ejpam-6409	690	17	.	.	PUNCT
ejpam-6409	691	1	we	we	PRON
ejpam-6409	691	2	want	want	VERB
ejpam-6409	691	3	to	to	PART
ejpam-6409	691	4	show	show	VERB
ejpam-6409	691	5	that	that	SCONJ
ejpam-6409	691	6	u	u	NOUN
ejpam-6409	691	7	=	=	NOUN
ejpam-6409	691	8	0	0	PROPN
ejpam-6409	691	9	.	.	PUNCT
ejpam-6409	692	1	based	base	VERB
ejpam-6409	692	2	on	on	ADP
ejpam-6409	692	3	the	the	DET
ejpam-6409	692	4	construction	construction	NOUN
ejpam-6409	692	5	,	,	PUNCT
ejpam-6409	692	6	the	the	DET
ejpam-6409	692	7	second	second	ADJ
ejpam-6409	692	8	row	row	NOUN
ejpam-6409	692	9	w2	w2	NOUN
ejpam-6409	692	10	of	of	ADP
ejpam-6409	692	11	a(t1,	a(t1,	PROPN
ejpam-6409	692	12	...	...	PUNCT
ejpam-6409	692	13	,ts	,ts	PUNCT
ejpam-6409	692	14	)	)	PUNCT
ejpam-6409	692	15	is	be	AUX
ejpam-6409	692	16	a	a	DET
ejpam-6409	692	17	2t−2s+σ	2t−2s+σ	ADJ
ejpam-6409	692	18	-	-	ADJ
ejpam-6409	692	19	fold	fold	ADJ
ejpam-6409	692	20	replication	replication	NOUN
ejpam-6409	692	21	of	of	ADP
ejpam-6409	692	22	v	v	NOUN
ejpam-6409	692	23	=	=	SYM
ejpam-6409	692	24	2σ−1	2σ−1	NUM
ejpam-6409	692	25	(	(	PUNCT
ejpam-6409	692	26	00	00	NUM
ejpam-6409	692	27	,	,	PUNCT
ejpam-6409	692	28	.	.	PUNCT
ejpam-6409	692	29	.	.	PUNCT
ejpam-6409	693	1	.	.	PUNCT
ejpam-6409	694	1	,	,	PUNCT
ejpam-6409	694	2	0	0	NUM
ejpam-6409	694	3	,	,	PUNCT
ejpam-6409	694	4	2s+1−σ	2s+1−σ	NUM
ejpam-6409	694	5	−	−	NOUN
ejpam-6409	694	6	1	1	NUM
ejpam-6409	694	7	,	,	PUNCT
ejpam-6409	694	8	10	10	NUM
ejpam-6409	694	9	,	,	PUNCT
ejpam-6409	694	10	.	.	PUNCT
ejpam-6409	694	11	.	.	PUNCT
ejpam-6409	694	12	.	.	PUNCT
ejpam-6409	695	1	,	,	PUNCT
ejpam-6409	695	2	2s+1−σ	2s+1−σ	NUM
ejpam-6409	695	3	−	−	NOUN
ejpam-6409	695	4	1	1	NUM
ejpam-6409	695	5	,	,	PUNCT
ejpam-6409	695	6	2s+1−σ	2s+1−σ	NUM
ejpam-6409	695	7	−	−	NOUN
ejpam-6409	695	8	1	1	NUM
ejpam-6409	695	9	)	)	PUNCT
ejpam-6409	695	10	,	,	PUNCT
ejpam-6409	695	11	and	and	CCONJ
ejpam-6409	695	12	ord(w2	ord(w2	ADP
ejpam-6409	695	13	)	)	PUNCT
ejpam-6409	695	14	=	=	SYM
ejpam-6409	695	15	2s+1−σ	2s+1−σ	X
ejpam-6409	695	16	.	.	PUNCT
ejpam-6409	695	17	by	by	ADP
ejpam-6409	695	18	corollary	corollary	ADJ
ejpam-6409	695	19	3.2	3.2	NUM
ejpam-6409	695	20	,	,	PUNCT
ejpam-6409	695	21	we	we	PRON
ejpam-6409	695	22	have	have	VERB
ejpam-6409	695	23	φs(w2	φs(w2	NUM
ejpam-6409	695	24	)	)	PUNCT
ejpam-6409	695	25	+	+	NOUN
ejpam-6409	695	26	φs(u	φs(u	NOUN
ejpam-6409	695	27	)	)	PUNCT
ejpam-6409	695	28	=	=	SYM
ejpam-6409	695	29	φs	φs	PROPN
ejpam-6409	695	30	(	(	PUNCT
ejpam-6409	695	31	w2	w2	NOUN
ejpam-6409	695	32	+	+	CCONJ
ejpam-6409	695	33	u−	u−	PROPN
ejpam-6409	695	34	2(w2	2(w2	NUM
ejpam-6409	695	35	⊙2	⊙2	NOUN
ejpam-6409	695	36	u	u	NOUN
ejpam-6409	695	37	)	)	PUNCT
ejpam-6409	695	38	)	)	PUNCT
ejpam-6409	695	39	.	.	PUNCT
ejpam-6409	696	1	since	since	SCONJ
ejpam-6409	696	2	φs(u	φs(u	NUM
ejpam-6409	696	3	)	)	PUNCT
ejpam-6409	696	4	∈	∈	PROPN
ejpam-6409	696	5	k(h	k(h	PROPN
ejpam-6409	696	6	)	)	PUNCT
ejpam-6409	696	7	,	,	PUNCT
ejpam-6409	696	8	it	it	PRON
ejpam-6409	696	9	follows	follow	VERB
ejpam-6409	696	10	that	that	SCONJ
ejpam-6409	696	11	2(w2	2(w2	NUM
ejpam-6409	696	12	⊙2	⊙2	NOUN
ejpam-6409	696	13	u	u	NOUN
ejpam-6409	696	14	)	)	PUNCT
ejpam-6409	696	15	∈	∈	PROPN
ejpam-6409	696	16	h.	h.	PROPN
ejpam-6409	696	17	write	write	PROPN
ejpam-6409	696	18	w2	w2	PROPN
ejpam-6409	696	19	=	=	SYM
ejpam-6409	696	20	(	(	PUNCT
ejpam-6409	696	21	w1	w1	NOUN
ejpam-6409	696	22	,	,	PUNCT
ejpam-6409	696	23	w2	w2	NOUN
ejpam-6409	696	24	,	,	PUNCT
ejpam-6409	696	25	.	.	PUNCT
ejpam-6409	696	26	.	.	PUNCT
ejpam-6409	697	1	.	.	PUNCT
ejpam-6409	698	1	,	,	PUNCT
ejpam-6409	698	2	wn	wn	PROPN
ejpam-6409	698	3	)	)	PUNCT
ejpam-6409	698	4	,	,	PUNCT
ejpam-6409	698	5	and	and	CCONJ
ejpam-6409	698	6	let	let	VERB
ejpam-6409	698	7	[	[	PUNCT
ejpam-6409	698	8	wj,0	wj,0	PROPN
ejpam-6409	698	9	,	,	PUNCT
ejpam-6409	698	10	wj,1	wj,1	NOUN
ejpam-6409	698	11	,	,	PUNCT
ejpam-6409	698	12	.	.	PUNCT
ejpam-6409	698	13	.	.	PUNCT
ejpam-6409	699	1	.	.	PUNCT
ejpam-6409	700	1	,	,	PUNCT
ejpam-6409	700	2	wj	wj	PROPN
ejpam-6409	700	3	,	,	PUNCT
ejpam-6409	700	4	s−1]2	s−1]2	ADV
ejpam-6409	700	5	be	be	AUX
ejpam-6409	700	6	the	the	DET
ejpam-6409	700	7	2	2	NUM
ejpam-6409	700	8	-	-	PUNCT
ejpam-6409	700	9	ary	ary	NOUN
ejpam-6409	700	10	expansion	expansion	NOUN
ejpam-6409	700	11	of	of	ADP
ejpam-6409	700	12	wj	wj	PROPN
ejpam-6409	700	13	,	,	PUNCT
ejpam-6409	700	14	for	for	ADP
ejpam-6409	700	15	j	j	PROPN
ejpam-6409	700	16	∈	∈	PROPN
ejpam-6409	700	17	{	{	PUNCT
ejpam-6409	700	18	1	1	NUM
ejpam-6409	700	19	,	,	PUNCT
ejpam-6409	700	20	.	.	PUNCT
ejpam-6409	700	21	.	.	PUNCT
ejpam-6409	701	1	.	.	PUNCT
ejpam-6409	701	2	,	,	PUNCT
ejpam-6409	701	3	n	n	CCONJ
ejpam-6409	701	4	}	}	PUNCT
ejpam-6409	701	5	.	.	PUNCT
ejpam-6409	702	1	by	by	ADP
ejpam-6409	702	2	lemma	lemma	PROPN
ejpam-6409	702	3	6.1	6.1	NUM
ejpam-6409	702	4	,	,	PUNCT
ejpam-6409	702	5	2(w2	2(w2	NUM
ejpam-6409	702	6	⊙2	⊙2	NOUN
ejpam-6409	702	7	u	u	NOUN
ejpam-6409	702	8	)	)	PUNCT
ejpam-6409	702	9	=	=	SYM
ejpam-6409	702	10	2	2	NUM
ejpam-6409	702	11	s−2∑	s−2∑	NUM
ejpam-6409	702	12	i	i	PRON
ejpam-6409	702	13	=	=	SYM
ejpam-6409	702	14	σ−1	σ−1	PROPN
ejpam-6409	702	15	(	(	PUNCT
ejpam-6409	702	16	w2	w2	NOUN
ejpam-6409	702	17	⊙2	⊙2	VERB
ejpam-6409	702	18	µi2	µi2	PROPN
ejpam-6409	702	19	i	i	NOUN
ejpam-6409	702	20	)	)	PUNCT
ejpam-6409	702	21	=	=	SYM
ejpam-6409	703	1	2	2	NUM
ejpam-6409	703	2	s−2∑	s−2∑	NUM
ejpam-6409	703	3	i	i	PRON
ejpam-6409	703	4	=	=	PUNCT
ejpam-6409	703	5	σ−1	σ−1	PROPN
ejpam-6409	703	6	ri2	ri2	VERB
ejpam-6409	703	7	i	i	PRON
ejpam-6409	703	8	∈	∈	PROPN
ejpam-6409	703	9	h	h	NOUN
ejpam-6409	703	10	,	,	PUNCT
ejpam-6409	703	11	where	where	SCONJ
ejpam-6409	703	12	ri	ri	NOUN
ejpam-6409	703	13	=	=	SYM
ejpam-6409	703	14	(	(	PUNCT
ejpam-6409	703	15	r1,i	r1,i	PROPN
ejpam-6409	703	16	,	,	PUNCT
ejpam-6409	703	17	r2,i	r2,i	PROPN
ejpam-6409	703	18	,	,	PUNCT
ejpam-6409	703	19	.	.	PUNCT
ejpam-6409	703	20	.	.	PUNCT
ejpam-6409	704	1	.	.	PUNCT
ejpam-6409	705	1	,	,	PUNCT
ejpam-6409	705	2	rn	rn	PROPN
ejpam-6409	705	3	,	,	PUNCT
ejpam-6409	705	4	i	i	PROPN
ejpam-6409	705	5	)	)	PUNCT
ejpam-6409	705	6	,	,	PUNCT
ejpam-6409	705	7	muhammad	muhammad	PROPN
ejpam-6409	705	8	sajjad	sajjad	PROPN
ejpam-6409	705	9	et	et	PROPN
ejpam-6409	705	10	al	al	PROPN
ejpam-6409	705	11	.	.	PUNCT
ejpam-6409	705	12	/	/	SYM
ejpam-6409	705	13	eur	eur	PROPN
ejpam-6409	705	14	.	.	PUNCT
ejpam-6409	706	1	j.	j.	PROPN
ejpam-6409	706	2	pure	pure	PROPN
ejpam-6409	706	3	appl	appl	PROPN
ejpam-6409	706	4	.	.	PROPN
ejpam-6409	706	5	math	math	PROPN
ejpam-6409	706	6	,	,	PUNCT
ejpam-6409	706	7	18	18	NUM
ejpam-6409	706	8	(	(	PUNCT
ejpam-6409	706	9	3	3	NUM
ejpam-6409	706	10	)	)	PUNCT
ejpam-6409	706	11	(	(	PUNCT
ejpam-6409	706	12	2025	2025	NUM
ejpam-6409	706	13	)	)	PUNCT
ejpam-6409	706	14	,	,	PUNCT
ejpam-6409	706	15	6409	6409	NUM
ejpam-6409	706	16	21	21	NUM
ejpam-6409	706	17	of	of	ADP
ejpam-6409	706	18	32	32	NUM
ejpam-6409	706	19	and	and	CCONJ
ejpam-6409	706	20	rj	rj	PROPN
ejpam-6409	706	21	,	,	PUNCT
ejpam-6409	706	22	i	i	PRON
ejpam-6409	706	23	=	=	PUNCT
ejpam-6409	706	24	{	{	PUNCT
ejpam-6409	706	25	1	1	NUM
ejpam-6409	706	26	if	if	SCONJ
ejpam-6409	706	27	wj	wj	PROPN
ejpam-6409	706	28	,	,	PUNCT
ejpam-6409	706	29	i	i	PROPN
ejpam-6409	706	30	+	+	X
ejpam-6409	706	31	µi	µi	PROPN
ejpam-6409	706	32	≥	≥	NUM
ejpam-6409	706	33	2	2	NUM
ejpam-6409	706	34	,	,	PUNCT
ejpam-6409	706	35	0	0	NUM
ejpam-6409	707	1	otherwise	otherwise	ADV
ejpam-6409	707	2	.	.	PUNCT
ejpam-6409	708	1	let	let	VERB
ejpam-6409	709	1	τ	τ	PROPN
ejpam-6409	710	1	=	=	PUNCT
ejpam-6409	710	2	∑s	∑s	PROPN
ejpam-6409	710	3	i=1	i=1	PRON
ejpam-6409	710	4	ti	ti	PROPN
ejpam-6409	710	5	.	.	PUNCT
ejpam-6409	711	1	since	since	SCONJ
ejpam-6409	711	2	σ	σ	NOUN
ejpam-6409	711	3	≤	≤	PROPN
ejpam-6409	711	4	s−	s−	PROPN
ejpam-6409	711	5	1	1	NUM
ejpam-6409	711	6	,	,	PUNCT
ejpam-6409	711	7	we	we	PRON
ejpam-6409	711	8	have	have	VERB
ejpam-6409	711	9	τ	τ	PROPN
ejpam-6409	711	10	≥	≥	NUM
ejpam-6409	711	11	2	2	NUM
ejpam-6409	711	12	.	.	PUNCT
ejpam-6409	712	1	if	if	SCONJ
ejpam-6409	712	2	τ	τ	PROPN
ejpam-6409	712	3	=	=	SYM
ejpam-6409	712	4	2	2	NUM
ejpam-6409	712	5	,	,	PUNCT
ejpam-6409	712	6	then	then	ADV
ejpam-6409	712	7	h	h	NOUN
ejpam-6409	712	8	has	have	VERB
ejpam-6409	712	9	length	length	NOUN
ejpam-6409	712	10	2s+1−σ	2s+1−σ	NUM
ejpam-6409	712	11	,	,	PUNCT
ejpam-6409	712	12	and	and	CCONJ
ejpam-6409	712	13	the	the	DET
ejpam-6409	712	14	only	only	ADJ
ejpam-6409	712	15	vectors	vector	NOUN
ejpam-6409	712	16	in	in	ADP
ejpam-6409	712	17	a(t1,	a(t1,	PROPN
ejpam-6409	712	18	...	...	PUNCT
ejpam-6409	712	19	,ts	,ts	PUNCT
ejpam-6409	712	20	)	)	PUNCT
ejpam-6409	712	21	are	be	AUX
ejpam-6409	712	22	1	1	NUM
ejpam-6409	712	23	and	and	CCONJ
ejpam-6409	712	24	w2	w2	NOUN
ejpam-6409	712	25	=	=	PROPN
ejpam-6409	713	1	v.	v.	CCONJ
ejpam-6409	713	2	if	if	SCONJ
ejpam-6409	713	3	τ	τ	PROPN
ejpam-6409	713	4	≥	≥	NOUN
ejpam-6409	713	5	3	3	NUM
ejpam-6409	713	6	,	,	PUNCT
ejpam-6409	713	7	for	for	ADP
ejpam-6409	713	8	i	i	PROPN
ejpam-6409	713	9	∈	∈	PROPN
ejpam-6409	713	10	{	{	PUNCT
ejpam-6409	713	11	3	3	NUM
ejpam-6409	713	12	,	,	PUNCT
ejpam-6409	713	13	.	.	PUNCT
ejpam-6409	713	14	.	.	PUNCT
ejpam-6409	714	1	.	.	PUNCT
ejpam-6409	715	1	,	,	PUNCT
ejpam-6409	715	2	τ	τ	PROPN
ejpam-6409	715	3	}	}	PUNCT
ejpam-6409	715	4	,	,	PUNCT
ejpam-6409	715	5	the	the	DET
ejpam-6409	715	6	i	i	PROPN
ejpam-6409	715	7	-	-	PUNCT
ejpam-6409	715	8	th	th	X
ejpam-6409	715	9	row	row	NOUN
ejpam-6409	715	10	wi	wi	PROPN
ejpam-6409	715	11	of	of	ADP
ejpam-6409	715	12	a	a	DET
ejpam-6409	715	13	(	(	PUNCT
ejpam-6409	715	14	t1,	t1,	NUM
ejpam-6409	715	15	...	...	PUNCT
ejpam-6409	715	16	,ts	,ts	PUNCT
ejpam-6409	715	17	)	)	PUNCT
ejpam-6409	715	18	has	have	VERB
ejpam-6409	715	19	zeros	zero	NOUN
ejpam-6409	715	20	in	in	ADP
ejpam-6409	715	21	its	its	PRON
ejpam-6409	715	22	first	first	ADJ
ejpam-6409	715	23	2s+1−σ	2s+1−σ	NUM
ejpam-6409	715	24	coordinates	coordinate	NOUN
ejpam-6409	715	25	.	.	PUNCT
ejpam-6409	716	1	since	since	SCONJ
ejpam-6409	716	2	σ	σ	PROPN
ejpam-6409	716	3	≤	≤	PROPN
ejpam-6409	716	4	s	s	PART
ejpam-6409	716	5	−	−	PROPN
ejpam-6409	716	6	1	1	NUM
ejpam-6409	716	7	and	and	CCONJ
ejpam-6409	716	8	τ	τ	PROPN
ejpam-6409	716	9	≥	≥	NUM
ejpam-6409	716	10	2	2	NUM
ejpam-6409	716	11	,	,	PUNCT
ejpam-6409	716	12	every	every	DET
ejpam-6409	716	13	element	element	NOUN
ejpam-6409	716	14	of	of	ADP
ejpam-6409	716	15	h	h	NOUN
ejpam-6409	716	16	,	,	PUNCT
ejpam-6409	716	17	when	when	SCONJ
ejpam-6409	716	18	restricted	restrict	VERB
ejpam-6409	716	19	to	to	ADP
ejpam-6409	716	20	the	the	DET
ejpam-6409	716	21	first	first	ADJ
ejpam-6409	716	22	2s+1−σ	2s+1−σ	NUM
ejpam-6409	716	23	coordinates	coordinate	NOUN
ejpam-6409	716	24	,	,	PUNCT
ejpam-6409	716	25	takes	take	VERB
ejpam-6409	716	26	the	the	DET
ejpam-6409	716	27	form	form	NOUN
ejpam-6409	716	28	µ1	µ1	NOUN
ejpam-6409	716	29	·	·	SYM
ejpam-6409	716	30	1	1	NUM
ejpam-6409	716	31	+	+	NUM
ejpam-6409	716	32	µ2	µ2	PROPN
ejpam-6409	716	33	·	·	PUNCT
ejpam-6409	716	34	v	v	NOUN
ejpam-6409	716	35	for	for	ADP
ejpam-6409	716	36	some	some	DET
ejpam-6409	716	37	µ1	µ1	PROPN
ejpam-6409	716	38	,	,	PUNCT
ejpam-6409	716	39	µ2	µ2	PROPN
ejpam-6409	716	40	∈	∈	PROPN
ejpam-6409	716	41	z2s	z2s	PROPN
ejpam-6409	717	1	[	[	X
ejpam-6409	717	2	ω	ω	X
ejpam-6409	717	3	]	]	X
ejpam-6409	717	4	.	.	PUNCT
ejpam-6409	718	1	now	now	ADV
ejpam-6409	718	2	,	,	PUNCT
ejpam-6409	718	3	2	2	NUM
ejpam-6409	718	4	s−2∑	s−2∑	NUM
ejpam-6409	718	5	i	i	PRON
ejpam-6409	718	6	=	=	PUNCT
ejpam-6409	719	1	σ−1	σ−1	PROPN
ejpam-6409	719	2	ri2	ri2	NOUN
ejpam-6409	719	3	i	i	PRON
ejpam-6409	719	4	restricted	restrict	VERB
ejpam-6409	719	5	to	to	ADP
ejpam-6409	719	6	the	the	DET
ejpam-6409	719	7	first	first	ADJ
ejpam-6409	719	8	m	m	NOUN
ejpam-6409	719	9	=	=	SYM
ejpam-6409	719	10	2s+1−σ	2s+1−σ	NUM
ejpam-6409	719	11	coordinates	coordinate	NOUN
ejpam-6409	719	12	is	be	AUX
ejpam-6409	719	13	2	2	NUM
ejpam-6409	719	14	s−2∑	s−2∑	NUM
ejpam-6409	719	15	i	i	PRON
ejpam-6409	719	16	=	=	NOUN
ejpam-6409	719	17	σ−1	σ−1	PROPN
ejpam-6409	719	18	r′	r′	PROPN
ejpam-6409	719	19	i2	i2	PROPN
ejpam-6409	719	20	i	i	PRON
ejpam-6409	719	21	,	,	PUNCT
ejpam-6409	719	22	where	where	SCONJ
ejpam-6409	719	23	r′	r′	VERB
ejpam-6409	719	24	i	i	PRON
ejpam-6409	719	25	=	=	SYM
ejpam-6409	719	26	(	(	PUNCT
ejpam-6409	719	27	t1,i	t1,i	PROPN
ejpam-6409	719	28	,	,	PUNCT
ejpam-6409	719	29	t2,i	t2,i	ADV
ejpam-6409	719	30	,	,	PUNCT
ejpam-6409	719	31	.	.	PUNCT
ejpam-6409	719	32	.	.	PUNCT
ejpam-6409	719	33	.	.	PUNCT
ejpam-6409	720	1	,	,	PUNCT
ejpam-6409	720	2	tm	tm	NOUN
ejpam-6409	720	3	,	,	PUNCT
ejpam-6409	720	4	i	i	PROPN
ejpam-6409	720	5	)	)	PUNCT
ejpam-6409	720	6	.	.	PUNCT
ejpam-6409	721	1	therefore	therefore	ADV
ejpam-6409	721	2	,	,	PUNCT
ejpam-6409	721	3	we	we	PRON
ejpam-6409	721	4	want	want	VERB
ejpam-6409	721	5	µ1	µ1	PROPN
ejpam-6409	721	6	,	,	PUNCT
ejpam-6409	721	7	µ2	µ2	PROPN
ejpam-6409	721	8	∈	∈	PROPN
ejpam-6409	721	9	z2s	z2s	PROPN
ejpam-6409	722	1	[	[	X
ejpam-6409	722	2	ω	ω	X
ejpam-6409	722	3	]	]	X
ejpam-6409	722	4	such	such	ADJ
ejpam-6409	722	5	that	that	SCONJ
ejpam-6409	722	6	2	2	NUM
ejpam-6409	722	7	s−2∑	s−2∑	NUM
ejpam-6409	723	1	i	i	PRON
ejpam-6409	723	2	=	=	NOUN
ejpam-6409	723	3	σ−1	σ−1	PROPN
ejpam-6409	723	4	r′	r′	PROPN
ejpam-6409	723	5	i2	i2	PROPN
ejpam-6409	724	1	i	i	PRON
ejpam-6409	724	2	=	=	PUNCT
ejpam-6409	724	3	µ1	µ1	PROPN
ejpam-6409	724	4	·	·	SYM
ejpam-6409	724	5	1	1	NUM
ejpam-6409	724	6	+	+	NUM
ejpam-6409	724	7	µ2	µ2	PROPN
ejpam-6409	724	8	·	·	PUNCT
ejpam-6409	724	9	v.	v.	CCONJ
ejpam-6409	724	10	since	since	SCONJ
ejpam-6409	724	11	the	the	DET
ejpam-6409	724	12	initial	initial	ADJ
ejpam-6409	724	13	entry	entry	NOUN
ejpam-6409	724	14	of	of	ADP
ejpam-6409	724	15	v	v	NOUN
ejpam-6409	724	16	is	be	AUX
ejpam-6409	724	17	zero	zero	NUM
ejpam-6409	724	18	,	,	PUNCT
ejpam-6409	724	19	the	the	DET
ejpam-6409	724	20	first	first	ADJ
ejpam-6409	724	21	coordinate	coordinate	NOUN
ejpam-6409	724	22	of	of	ADP
ejpam-6409	724	23	v(i	v(i	NUM
ejpam-6409	724	24	)	)	PUNCT
ejpam-6409	724	25	is	be	AUX
ejpam-6409	724	26	zero	zero	NUM
ejpam-6409	724	27	for	for	ADP
ejpam-6409	724	28	all	all	PRON
ejpam-6409	724	29	i	i	PRON
ejpam-6409	724	30	∈	∈	PROPN
ejpam-6409	724	31	{	{	PUNCT
ejpam-6409	724	32	0	0	NUM
ejpam-6409	724	33	,	,	PUNCT
ejpam-6409	724	34	.	.	PUNCT
ejpam-6409	724	35	.	.	PUNCT
ejpam-6409	725	1	.	.	PUNCT
ejpam-6409	726	1	,	,	PUNCT
ejpam-6409	726	2	s−	s−	PROPN
ejpam-6409	726	3	1	1	NUM
ejpam-6409	726	4	}	}	PUNCT
ejpam-6409	726	5	.	.	PUNCT
ejpam-6409	727	1	thus	thus	ADV
ejpam-6409	727	2	,	,	PUNCT
ejpam-6409	727	3	µ1	µ1	PROPN
ejpam-6409	727	4	=	=	SYM
ejpam-6409	727	5	0	0	NUM
ejpam-6409	727	6	,	,	PUNCT
ejpam-6409	727	7	and	and	CCONJ
ejpam-6409	727	8	2	2	NUM
ejpam-6409	727	9	s−2∑	s−2∑	NOUN
ejpam-6409	728	1	i	i	PRON
ejpam-6409	728	2	=	=	NOUN
ejpam-6409	728	3	σ−1	σ−1	PROPN
ejpam-6409	728	4	r′	r′	PROPN
ejpam-6409	728	5	i2	i2	PROPN
ejpam-6409	728	6	i	i	NOUN
ejpam-6409	728	7	=	=	PUNCT
ejpam-6409	728	8	µ2v	µ2v	PROPN
ejpam-6409	728	9	.	.	PUNCT
ejpam-6409	728	10	note	note	VERB
ejpam-6409	728	11	that	that	SCONJ
ejpam-6409	729	1	v	v	X
ejpam-6409	729	2	=	=	SYM
ejpam-6409	729	3	s−1∑	s−1∑	NUM
ejpam-6409	729	4	i=0	i=0	PROPN
ejpam-6409	729	5	v(i)2i	v(i)2i	X
ejpam-6409	730	1	=	=	PUNCT
ejpam-6409	730	2	s−1∑	s−1∑	NUM
ejpam-6409	730	3	i	i	NOUN
ejpam-6409	730	4	=	=	PROPN
ejpam-6409	730	5	σ−1	σ−1	PROPN
ejpam-6409	730	6	v(i)2i	v(i)2i	NUM
ejpam-6409	730	7	.	.	PUNCT
ejpam-6409	731	1	let	let	VERB
ejpam-6409	731	2	a	a	DET
ejpam-6409	731	3	=	=	SYM
ejpam-6409	731	4	2	2	NUM
ejpam-6409	731	5	s−2∑	s−2∑	NUM
ejpam-6409	732	1	i	i	PRON
ejpam-6409	732	2	=	=	NOUN
ejpam-6409	732	3	σ−1	σ−1	PROPN
ejpam-6409	732	4	r′	r′	PROPN
ejpam-6409	732	5	i2	i2	PROPN
ejpam-6409	732	6	i	i	PROPN
ejpam-6409	732	7	,	,	PUNCT
ejpam-6409	732	8	b	b	PROPN
ejpam-6409	732	9	=	=	SYM
ejpam-6409	732	10	µ2v	µ2v	PROPN
ejpam-6409	732	11	.	.	PUNCT
ejpam-6409	733	1	muhammad	muhammad	PROPN
ejpam-6409	733	2	sajjad	sajjad	PROPN
ejpam-6409	733	3	et	et	PROPN
ejpam-6409	733	4	al	al	PROPN
ejpam-6409	733	5	.	.	PUNCT
ejpam-6409	733	6	/	/	SYM
ejpam-6409	733	7	eur	eur	PROPN
ejpam-6409	733	8	.	.	PUNCT
ejpam-6409	734	1	j.	j.	PROPN
ejpam-6409	734	2	pure	pure	PROPN
ejpam-6409	734	3	appl	appl	PROPN
ejpam-6409	734	4	.	.	PROPN
ejpam-6409	734	5	math	math	PROPN
ejpam-6409	734	6	,	,	PUNCT
ejpam-6409	734	7	18	18	NUM
ejpam-6409	734	8	(	(	PUNCT
ejpam-6409	734	9	3	3	NUM
ejpam-6409	734	10	)	)	PUNCT
ejpam-6409	734	11	(	(	PUNCT
ejpam-6409	734	12	2025	2025	NUM
ejpam-6409	734	13	)	)	PUNCT
ejpam-6409	734	14	,	,	PUNCT
ejpam-6409	734	15	6409	6409	NUM
ejpam-6409	734	16	22	22	NUM
ejpam-6409	734	17	of	of	ADP
ejpam-6409	734	18	32	32	NUM
ejpam-6409	734	19	suppose	suppose	VERB
ejpam-6409	734	20	µ2	µ2	PROPN
ejpam-6409	734	21	∈	∈	PROPN
ejpam-6409	734	22	a	a	PRON
ejpam-6409	734	23	=	=	X
ejpam-6409	734	24	{	{	PUNCT
ejpam-6409	734	25	0	0	NUM
ejpam-6409	734	26	,	,	PUNCT
ejpam-6409	734	27	2s−σ+1	2s−σ+1	NUM
ejpam-6409	734	28	}	}	PUNCT
ejpam-6409	734	29	.	.	PUNCT
ejpam-6409	735	1	then	then	ADV
ejpam-6409	735	2	b	b	X
ejpam-6409	735	3	=	=	SYM
ejpam-6409	735	4	0	0	PROPN
ejpam-6409	735	5	.	.	PUNCT
ejpam-6409	736	1	given	give	VERB
ejpam-6409	736	2	the	the	DET
ejpam-6409	736	3	existence	existence	NOUN
ejpam-6409	736	4	of	of	ADP
ejpam-6409	736	5	some	some	DET
ejpam-6409	736	6	µi0	µi0	NOUN
ejpam-6409	736	7	̸=	̸=	PROPN
ejpam-6409	736	8	0	0	NUM
ejpam-6409	736	9	,	,	PUNCT
ejpam-6409	736	10	the	the	DET
ejpam-6409	736	11	vector	vector	NOUN
ejpam-6409	736	12	r′	r′	PROPN
ejpam-6409	736	13	i0	i0	PROPN
ejpam-6409	736	14	has	have	VERB
ejpam-6409	736	15	at	at	ADV
ejpam-6409	736	16	least	least	ADV
ejpam-6409	736	17	one	one	NUM
ejpam-6409	736	18	nonzero	nonzero	NOUN
ejpam-6409	736	19	coordinate	coordinate	NOUN
ejpam-6409	736	20	,	,	PUNCT
ejpam-6409	736	21	so	so	SCONJ
ejpam-6409	736	22	a	a	DET
ejpam-6409	736	23	̸=	̸=	PROPN
ejpam-6409	736	24	0	0	NUM
ejpam-6409	736	25	,	,	PUNCT
ejpam-6409	736	26	a	a	DET
ejpam-6409	736	27	contradiction	contradiction	NOUN
ejpam-6409	736	28	.	.	PUNCT
ejpam-6409	737	1	on	on	ADP
ejpam-6409	737	2	the	the	DET
ejpam-6409	737	3	other	other	ADJ
ejpam-6409	737	4	hand	hand	NOUN
ejpam-6409	737	5	,	,	PUNCT
ejpam-6409	737	6	if	if	SCONJ
ejpam-6409	737	7	µ2	µ2	PROPN
ejpam-6409	737	8	∈	∈	PROPN
ejpam-6409	737	9	z2s	z2s	X
ejpam-6409	737	10	[	[	X
ejpam-6409	737	11	ω	ω	X
ejpam-6409	737	12	]	]	X
ejpam-6409	737	13	\a	\a	NUM
ejpam-6409	737	14	,	,	PUNCT
ejpam-6409	737	15	let	let	VERB
ejpam-6409	737	16	a(i	a(i	VERB
ejpam-6409	737	17	)	)	PUNCT
ejpam-6409	738	1	=	=	SYM
ejpam-6409	738	2	(	(	PUNCT
ejpam-6409	738	3	a1,i	a1,i	PROPN
ejpam-6409	738	4	,	,	PUNCT
ejpam-6409	738	5	a2,i	a2,i	PROPN
ejpam-6409	738	6	,	,	PUNCT
ejpam-6409	738	7	.	.	PUNCT
ejpam-6409	738	8	.	.	PUNCT
ejpam-6409	739	1	.	.	PUNCT
ejpam-6409	740	1	,	,	PUNCT
ejpam-6409	740	2	an	an	DET
ejpam-6409	740	3	,	,	PUNCT
ejpam-6409	740	4	i	i	NOUN
ejpam-6409	740	5	)	)	PUNCT
ejpam-6409	740	6	,	,	PUNCT
ejpam-6409	740	7	σ	σ	PROPN
ejpam-6409	740	8	≤	≤	NUM
ejpam-6409	740	9	i	i	PRON
ejpam-6409	740	10	≤	≤	NUM
ejpam-6409	740	11	s−	s−	PROPN
ejpam-6409	740	12	1	1	NUM
ejpam-6409	740	13	,	,	PUNCT
ejpam-6409	740	14	where	where	SCONJ
ejpam-6409	740	15	aj	aj	PROPN
ejpam-6409	740	16	,	,	PUNCT
ejpam-6409	740	17	i	i	PRON
ejpam-6409	740	18	∈	∈	PROPN
ejpam-6409	740	19	{	{	PUNCT
ejpam-6409	740	20	0	0	NUM
ejpam-6409	740	21	,	,	PUNCT
ejpam-6409	740	22	1	1	NUM
ejpam-6409	740	23	}	}	PUNCT
ejpam-6409	740	24	for	for	ADP
ejpam-6409	740	25	all	all	DET
ejpam-6409	740	26	j	j	PROPN
ejpam-6409	740	27	∈	∈	PROPN
ejpam-6409	740	28	{	{	PUNCT
ejpam-6409	740	29	1	1	NUM
ejpam-6409	740	30	,	,	PUNCT
ejpam-6409	740	31	.	.	PUNCT
ejpam-6409	740	32	.	.	PUNCT
ejpam-6409	740	33	.	.	PUNCT
ejpam-6409	741	1	,	,	PUNCT
ejpam-6409	741	2	m	m	VERB
ejpam-6409	741	3	}	}	PUNCT
ejpam-6409	741	4	and	and	CCONJ
ejpam-6409	741	5	i	i	PRON
ejpam-6409	741	6	∈	∈	PROPN
ejpam-6409	741	7	{	{	PUNCT
ejpam-6409	741	8	σ	σ	PROPN
ejpam-6409	741	9	,	,	PUNCT
ejpam-6409	741	10	.	.	PUNCT
ejpam-6409	741	11	.	.	PUNCT
ejpam-6409	741	12	.	.	PUNCT
ejpam-6409	742	1	,	,	PUNCT
ejpam-6409	742	2	s−	s−	PROPN
ejpam-6409	742	3	1	1	NUM
ejpam-6409	742	4	}	}	PUNCT
ejpam-6409	742	5	.	.	PUNCT
ejpam-6409	743	1	since	since	SCONJ
ejpam-6409	743	2	v	v	NOUN
ejpam-6409	743	3	=	=	SYM
ejpam-6409	743	4	2σ−1(00	2σ−1(00	NOUN
ejpam-6409	743	5	,	,	PUNCT
ejpam-6409	743	6	.	.	PUNCT
ejpam-6409	743	7	.	.	PUNCT
ejpam-6409	743	8	.	.	PUNCT
ejpam-6409	744	1	,	,	PUNCT
ejpam-6409	744	2	0	0	NUM
ejpam-6409	744	3	,	,	PUNCT
ejpam-6409	744	4	2s+1−σ	2s+1−σ	NUM
ejpam-6409	744	5	−	−	NOUN
ejpam-6409	744	6	1	1	NUM
ejpam-6409	744	7	,	,	PUNCT
ejpam-6409	744	8	10	10	NUM
ejpam-6409	744	9	,	,	PUNCT
ejpam-6409	744	10	.	.	PUNCT
ejpam-6409	744	11	.	.	PUNCT
ejpam-6409	744	12	.	.	PUNCT
ejpam-6409	745	1	,	,	PUNCT
ejpam-6409	745	2	2s+1−σ	2s+1−σ	NUM
ejpam-6409	745	3	−	−	NOUN
ejpam-6409	745	4	1	1	NUM
ejpam-6409	745	5	,	,	PUNCT
ejpam-6409	745	6	2s+1−σ	2s+1−σ	NUM
ejpam-6409	745	7	−	−	NOUN
ejpam-6409	745	8	1	1	NUM
ejpam-6409	745	9	)	)	PUNCT
ejpam-6409	745	10	,	,	PUNCT
ejpam-6409	745	11	there	there	PRON
ejpam-6409	745	12	exists	exist	VERB
ejpam-6409	745	13	some	some	DET
ejpam-6409	745	14	i1	i1	PROPN
ejpam-6409	745	15	∈	∈	PROPN
ejpam-6409	745	16	{	{	PUNCT
ejpam-6409	745	17	σ	σ	PROPN
ejpam-6409	745	18	,	,	PUNCT
ejpam-6409	745	19	.	.	PUNCT
ejpam-6409	745	20	.	.	PUNCT
ejpam-6409	746	1	.	.	PUNCT
ejpam-6409	747	1	,	,	PUNCT
ejpam-6409	747	2	s	s	VERB
ejpam-6409	747	3	−	−	PROPN
ejpam-6409	747	4	1	1	NUM
ejpam-6409	747	5	}	}	PUNCT
ejpam-6409	747	6	such	such	ADJ
ejpam-6409	747	7	that	that	SCONJ
ejpam-6409	747	8	the	the	DET
ejpam-6409	747	9	coordinates	coordinate	NOUN
ejpam-6409	747	10	of	of	ADP
ejpam-6409	747	11	b(i1	b(i1	NOUN
ejpam-6409	747	12	)	)	PUNCT
ejpam-6409	747	13	do	do	AUX
ejpam-6409	747	14	not	not	PART
ejpam-6409	747	15	belong	belong	VERB
ejpam-6409	747	16	to	to	ADP
ejpam-6409	747	17	{	{	PUNCT
ejpam-6409	747	18	0	0	NUM
ejpam-6409	747	19	,	,	PUNCT
ejpam-6409	747	20	1	1	NUM
ejpam-6409	747	21	}	}	PUNCT
ejpam-6409	747	22	,	,	PUNCT
ejpam-6409	747	23	a	a	DET
ejpam-6409	747	24	contradiction	contradiction	NOUN
ejpam-6409	747	25	.	.	PUNCT
ejpam-6409	748	1	therefore	therefore	ADV
ejpam-6409	748	2	,	,	PUNCT
ejpam-6409	748	3	if	if	SCONJ
ejpam-6409	748	4	u	u	PROPN
ejpam-6409	748	5	̸=	̸=	PROPN
ejpam-6409	748	6	0	0	NUM
ejpam-6409	748	7	,	,	PUNCT
ejpam-6409	748	8	then	then	ADV
ejpam-6409	748	9	2(w2	2(w2	NUM
ejpam-6409	748	10	⊙2	⊙2	NOUN
ejpam-6409	748	11	u	u	NOUN
ejpam-6409	748	12	)	)	PUNCT
ejpam-6409	748	13	=	=	SYM
ejpam-6409	748	14	µ1	µ1	PROPN
ejpam-6409	748	15	·	·	SYM
ejpam-6409	748	16	1	1	NUM
ejpam-6409	748	17	+	+	NUM
ejpam-6409	748	18	µ2	µ2	PROPN
ejpam-6409	748	19	·	·	PUNCT
ejpam-6409	748	20	v	v	NOUN
ejpam-6409	748	21	,	,	PUNCT
ejpam-6409	748	22	and	and	CCONJ
ejpam-6409	748	23	hence	hence	ADV
ejpam-6409	748	24	u	u	X
ejpam-6409	748	25	=	=	NOUN
ejpam-6409	748	26	0	0	PROPN
ejpam-6409	748	27	.	.	PUNCT
ejpam-6409	748	28	lemma	lemma	PROPN
ejpam-6409	748	29	6.3	6.3	NUM
ejpam-6409	748	30	:	:	PUNCT
ejpam-6409	748	31	let	let	VERB
ejpam-6409	748	32	ĥ	ĥ	X
ejpam-6409	748	33	=	=	SYM
ejpam-6409	748	34	ĥ(t1,	ĥ(t1,	PROPN
ejpam-6409	748	35	...	...	PUNCT
ejpam-6409	748	36	,ts	,ts	PUNCT
ejpam-6409	748	37	)	)	PUNCT
ejpam-6409	748	38	be	be	AUX
ejpam-6409	748	39	the	the	DET
ejpam-6409	748	40	z2s	z2s	PROPN
ejpam-6409	748	41	[	[	X
ejpam-6409	748	42	ω]-additive	ω]-additive	X
ejpam-6409	748	43	gh	gh	PROPN
ejpam-6409	748	44	code	code	NOUN
ejpam-6409	748	45	of	of	ADP
ejpam-6409	748	46	type	type	NOUN
ejpam-6409	748	47	(	(	PUNCT
ejpam-6409	748	48	n	n	CCONJ
ejpam-6409	748	49	;	;	PUNCT
ejpam-6409	748	50	t1	t1	NOUN
ejpam-6409	748	51	,	,	PUNCT
ejpam-6409	748	52	.	.	PUNCT
ejpam-6409	748	53	.	.	PUNCT
ejpam-6409	748	54	.	.	PUNCT
ejpam-6409	749	1	,	,	PUNCT
ejpam-6409	749	2	ts	ts	PROPN
ejpam-6409	749	3	)	)	PUNCT
ejpam-6409	749	4	.	.	PUNCT
ejpam-6409	750	1	let	let	VERB
ejpam-6409	750	2	wi	wi	PROPN
ejpam-6409	750	3	be	be	AUX
ejpam-6409	750	4	the	the	DET
ejpam-6409	750	5	ith	ith	NOUN
ejpam-6409	750	6	row	row	NOUN
ejpam-6409	750	7	of	of	ADP
ejpam-6409	750	8	a	a	DET
ejpam-6409	750	9	(	(	PUNCT
ejpam-6409	750	10	t1,t2,	t1,t2,	NOUN
ejpam-6409	750	11	...	...	PUNCT
ejpam-6409	750	12	,ts	,ts	PUNCT
ejpam-6409	750	13	)	)	PUNCT
ejpam-6409	750	14	2	2	NUM
ejpam-6409	750	15	and	and	CCONJ
ejpam-6409	751	1	τ	τ	PROPN
ejpam-6409	752	1	=	=	PUNCT
ejpam-6409	752	2	∑s	∑s	PROPN
ejpam-6409	752	3	i=1	i=1	PRON
ejpam-6409	752	4	ti	ti	PROPN
ejpam-6409	752	5	.	.	PUNCT
ejpam-6409	752	6	define	define	VERB
ejpam-6409	752	7	ξ	ξ	NOUN
ejpam-6409	752	8	=	=	SYM
ejpam-6409	752	9	{	{	PUNCT
ejpam-6409	752	10	v	v	NOUN
ejpam-6409	752	11	=	=	PUNCT
ejpam-6409	752	12	τ−ts∑	τ−ts∑	NUM
ejpam-6409	752	13	i=2	i=2	NOUN
ejpam-6409	752	14	µiwi	µiwi	NOUN
ejpam-6409	752	15	:	:	PUNCT
ejpam-6409	752	16	µi	µi	PROPN
ejpam-6409	752	17	∈	∈	PROPN
ejpam-6409	752	18	z2s	z2s	X
ejpam-6409	753	1	[	[	X
ejpam-6409	753	2	ω	ω	X
ejpam-6409	753	3	]	]	X
ejpam-6409	753	4	,	,	PUNCT
ejpam-6409	753	5	ord(v	ord(v	PROPN
ejpam-6409	753	6	)	)	PUNCT
ejpam-6409	753	7	>	>	X
ejpam-6409	753	8	2	2	X
ejpam-6409	753	9	}	}	PUNCT
ejpam-6409	753	10	,	,	PUNCT
ejpam-6409	754	1	n	n	NOUN
ejpam-6409	754	2	=	=	PUNCT
ejpam-6409	754	3	{	{	PUNCT
ejpam-6409	754	4	s−1∑	s−1∑	NUM
ejpam-6409	754	5	i=0	i=0	PROPN
ejpam-6409	754	6	µi2	µi2	NOUN
ejpam-6409	754	7	i	i	PRON
ejpam-6409	754	8	:	:	PUNCT
ejpam-6409	754	9	µi	µi	PROPN
ejpam-6409	754	10	∈	∈	PROPN
ejpam-6409	754	11	z2[ω	z2[ω	NOUN
ejpam-6409	754	12	]	]	X
ejpam-6409	754	13	\	\	PUNCT
ejpam-6409	754	14	{	{	PUNCT
ejpam-6409	754	15	s−1∑	s−1∑	NUM
ejpam-6409	754	16	i=0	i=0	PROPN
ejpam-6409	754	17	2i	2i	NUM
ejpam-6409	754	18	}	}	PUNCT
ejpam-6409	754	19	}	}	PUNCT
ejpam-6409	754	20	if	if	SCONJ
ejpam-6409	754	21	σ	σ	PROPN
ejpam-6409	754	22	≤	≤	PROPN
ejpam-6409	754	23	s−	s−	PROPN
ejpam-6409	754	24	1	1	NUM
ejpam-6409	754	25	,	,	PUNCT
ejpam-6409	754	26	and	and	CCONJ
ejpam-6409	754	27	ξ	ξ	X
ejpam-6409	754	28	+	+	NOUN
ejpam-6409	754	29	n	n	NOUN
ejpam-6409	754	30	=	=	PRON
ejpam-6409	754	31	{	{	PUNCT
ejpam-6409	754	32	vξ	vξ	PROPN
ejpam-6409	754	33	+	+	CCONJ
ejpam-6409	754	34	vn	vn	X
ejpam-6409	754	35	:	:	PUNCT
ejpam-6409	754	36	vξ	vξ	PROPN
ejpam-6409	754	37	∈	∈	PROPN
ejpam-6409	754	38	ξ	ξ	X
ejpam-6409	754	39	∪	∪	X
ejpam-6409	754	40	{	{	PUNCT
ejpam-6409	754	41	0},vn	0},vn	NUM
ejpam-6409	754	42	∈	∈	PROPN
ejpam-6409	754	43	n	n	CCONJ
ejpam-6409	754	44	}	}	PUNCT
ejpam-6409	754	45	.	.	PUNCT
ejpam-6409	755	1	then	then	ADV
ejpam-6409	755	2	φs(ξ	φs(ξ	PUNCT
ejpam-6409	755	3	+	+	ADJ
ejpam-6409	755	4	n	n	NUM
ejpam-6409	755	5	)	)	PUNCT
ejpam-6409	755	6	∩k(φs(ĥ	∩k(φs(ĥ	PUNCT
ejpam-6409	755	7	)	)	PUNCT
ejpam-6409	755	8	)	)	PUNCT
ejpam-6409	756	1	=	=	PRON
ejpam-6409	756	2	{	{	PUNCT
ejpam-6409	756	3	0	0	NUM
ejpam-6409	756	4	}	}	PUNCT
ejpam-6409	756	5	.	.	PUNCT
ejpam-6409	757	1	proof	proof	NOUN
ejpam-6409	757	2	:	:	PUNCT
ejpam-6409	757	3	let	let	VERB
ejpam-6409	757	4	h	h	NOUN
ejpam-6409	757	5	=	=	SYM
ejpam-6409	757	6	φs(ĥ	φs(ĥ	NOUN
ejpam-6409	757	7	)	)	PUNCT
ejpam-6409	757	8	,	,	PUNCT
ejpam-6409	757	9	which	which	PRON
ejpam-6409	757	10	has	have	VERB
ejpam-6409	757	11	length	length	NOUN
ejpam-6409	757	12	n	n	NOUN
ejpam-6409	757	13	=	=	SYM
ejpam-6409	757	14	22	22	NUM
ejpam-6409	757	15	t	t	NOUN
ejpam-6409	757	16	=	=	SYM
ejpam-6409	757	17	n	n	PROPN
ejpam-6409	757	18	·	·	PUNCT
ejpam-6409	757	19	22(s−1	22(s−1	NUM
ejpam-6409	757	20	)	)	PUNCT
ejpam-6409	757	21	.	.	PUNCT
ejpam-6409	758	1	by	by	ADP
ejpam-6409	758	2	lemma	lemma	PROPN
ejpam-6409	758	3	5	5	NUM
ejpam-6409	758	4	,	,	PUNCT
ejpam-6409	758	5	we	we	PRON
ejpam-6409	758	6	know	know	VERB
ejpam-6409	758	7	that	that	SCONJ
ejpam-6409	758	8	φs(n	φs(n	X
ejpam-6409	758	9	)	)	PUNCT
ejpam-6409	758	10	∩k(h	∩k(h	PROPN
ejpam-6409	758	11	)	)	PUNCT
ejpam-6409	758	12	=	=	PRON
ejpam-6409	758	13	{	{	PUNCT
ejpam-6409	758	14	0	0	NUM
ejpam-6409	758	15	}	}	PUNCT
ejpam-6409	758	16	,	,	PUNCT
ejpam-6409	758	17	now	now	ADV
ejpam-6409	758	18	we	we	PRON
ejpam-6409	758	19	prove	prove	VERB
ejpam-6409	758	20	that	that	SCONJ
ejpam-6409	758	21	φs(ξ	φs(ξ	NOUN
ejpam-6409	758	22	)	)	PUNCT
ejpam-6409	758	23	∩k(h	∩k(h	PROPN
ejpam-6409	758	24	)	)	PUNCT
ejpam-6409	758	25	=	=	PUNCT
ejpam-6409	758	26	∅.	∅.	AUX
ejpam-6409	758	27	let	let	VERB
ejpam-6409	758	28	v	v	NOUN
ejpam-6409	758	29	=	=	SYM
ejpam-6409	758	30	∑τ−ts	∑τ−t	NOUN
ejpam-6409	758	31	i=2	i=2	PROPN
ejpam-6409	758	32	µiwi	µiwi	VERB
ejpam-6409	758	33	∈	∈	PROPN
ejpam-6409	758	34	ξ	ξ	PROPN
ejpam-6409	758	35	.	.	PUNCT
ejpam-6409	759	1	since	since	SCONJ
ejpam-6409	759	2	ord(v	ord(v	PROPN
ejpam-6409	759	3	)	)	PUNCT
ejpam-6409	759	4	>	>	X
ejpam-6409	759	5	2	2	NUM
ejpam-6409	759	6	and	and	CCONJ
ejpam-6409	759	7	ord(wi	ord(wi	NOUN
ejpam-6409	759	8	)	)	PUNCT
ejpam-6409	759	9	≤	≤	NOUN
ejpam-6409	759	10	2s+1−σ	2s+1−σ	NUM
ejpam-6409	759	11	,	,	PUNCT
ejpam-6409	759	12	we	we	PRON
ejpam-6409	759	13	have	have	VERB
ejpam-6409	759	14	ord(v	ord(v	NUM
ejpam-6409	759	15	)	)	PUNCT
ejpam-6409	760	1	=	=	PUNCT
ejpam-6409	760	2	2r	2r	NUM
ejpam-6409	760	3	for	for	ADP
ejpam-6409	760	4	some	some	DET
ejpam-6409	760	5	2	2	NUM
ejpam-6409	760	6	≤	≤	NOUN
ejpam-6409	760	7	r	r	NOUN
ejpam-6409	760	8	≤	≤	NOUN
ejpam-6409	760	9	s+1−σ	s+1−σ	NOUN
ejpam-6409	760	10	.	.	PUNCT
ejpam-6409	761	1	through	through	ADP
ejpam-6409	761	2	the	the	DET
ejpam-6409	761	3	iterative	iterative	NOUN
ejpam-6409	761	4	construction	construction	NOUN
ejpam-6409	761	5	of	of	ADP
ejpam-6409	761	6	a(t1,	a(t1,	PROPN
ejpam-6409	761	7	...	...	PUNCT
ejpam-6409	761	8	,ts	,ts	PUNCT
ejpam-6409	761	9	)	)	PUNCT
ejpam-6409	761	10	,	,	PUNCT
ejpam-6409	761	11	it	it	PRON
ejpam-6409	761	12	is	be	AUX
ejpam-6409	761	13	clear	clear	ADJ
ejpam-6409	761	14	that	that	SCONJ
ejpam-6409	761	15	all	all	DET
ejpam-6409	761	16	elements	element	NOUN
ejpam-6409	761	17	of	of	ADP
ejpam-6409	761	18	z2s	z2s	PROPN
ejpam-6409	761	19	[	[	X
ejpam-6409	761	20	ω	ω	X
ejpam-6409	761	21	]	]	X
ejpam-6409	761	22	whose	whose	DET
ejpam-6409	761	23	order	order	NOUN
ejpam-6409	761	24	is	be	AUX
ejpam-6409	761	25	2r	2r	NUM
ejpam-6409	761	26	or	or	CCONJ
ejpam-6409	761	27	less	less	ADV
ejpam-6409	761	28	appear	appear	VERB
ejpam-6409	761	29	as	as	ADP
ejpam-6409	761	30	a	a	DET
ejpam-6409	761	31	coordinate	coordinate	NOUN
ejpam-6409	761	32	of	of	ADP
ejpam-6409	761	33	v.	v.	INTJ
ejpam-6409	761	34	let	let	VERB
ejpam-6409	762	1	[	[	X
ejpam-6409	762	2	vj,0	vj,0	PROPN
ejpam-6409	762	3	,	,	PUNCT
ejpam-6409	762	4	vj,1	vj,1	NOUN
ejpam-6409	762	5	,	,	PUNCT
ejpam-6409	762	6	.	.	PUNCT
ejpam-6409	762	7	.	.	PUNCT
ejpam-6409	763	1	.	.	PUNCT
ejpam-6409	764	1	,	,	PUNCT
ejpam-6409	764	2	vj	vj	INTJ
ejpam-6409	764	3	,	,	PUNCT
ejpam-6409	764	4	s−1]2	s−1]2	ADJ
ejpam-6409	765	1	muhammad	muhammad	PROPN
ejpam-6409	765	2	sajjad	sajjad	PROPN
ejpam-6409	765	3	et	et	PROPN
ejpam-6409	765	4	al	al	PROPN
ejpam-6409	765	5	.	.	PUNCT
ejpam-6409	765	6	/	/	SYM
ejpam-6409	765	7	eur	eur	PROPN
ejpam-6409	765	8	.	.	PUNCT
ejpam-6409	766	1	j.	j.	PROPN
ejpam-6409	766	2	pure	pure	PROPN
ejpam-6409	766	3	appl	appl	PROPN
ejpam-6409	766	4	.	.	PROPN
ejpam-6409	766	5	math	math	PROPN
ejpam-6409	766	6	,	,	PUNCT
ejpam-6409	766	7	18	18	NUM
ejpam-6409	766	8	(	(	PUNCT
ejpam-6409	766	9	3	3	NUM
ejpam-6409	766	10	)	)	PUNCT
ejpam-6409	766	11	(	(	PUNCT
ejpam-6409	766	12	2025	2025	NUM
ejpam-6409	766	13	)	)	PUNCT
ejpam-6409	766	14	,	,	PUNCT
ejpam-6409	766	15	6409	6409	NUM
ejpam-6409	766	16	23	23	NUM
ejpam-6409	766	17	of	of	ADP
ejpam-6409	766	18	32	32	NUM
ejpam-6409	766	19	be	be	AUX
ejpam-6409	766	20	the	the	DET
ejpam-6409	766	21	2	2	NUM
ejpam-6409	766	22	-	-	PUNCT
ejpam-6409	766	23	ary	ary	NOUN
ejpam-6409	766	24	expansion	expansion	NOUN
ejpam-6409	766	25	of	of	ADP
ejpam-6409	766	26	vj	vj	PROPN
ejpam-6409	766	27	,	,	PUNCT
ejpam-6409	766	28	for	for	ADP
ejpam-6409	766	29	j	j	PROPN
ejpam-6409	766	30	∈	∈	PROPN
ejpam-6409	766	31	{	{	PUNCT
ejpam-6409	766	32	1	1	NUM
ejpam-6409	766	33	,	,	PUNCT
ejpam-6409	766	34	.	.	PUNCT
ejpam-6409	766	35	.	.	PUNCT
ejpam-6409	767	1	.	.	PUNCT
ejpam-6409	767	2	,	,	PUNCT
ejpam-6409	767	3	n	n	CCONJ
ejpam-6409	767	4	}	}	PUNCT
ejpam-6409	767	5	.	.	PUNCT
ejpam-6409	768	1	by	by	ADP
ejpam-6409	768	2	corollary	corollary	ADJ
ejpam-6409	768	3	4	4	NUM
ejpam-6409	768	4	,	,	PUNCT
ejpam-6409	768	5	we	we	PRON
ejpam-6409	768	6	have	have	VERB
ejpam-6409	768	7	φs(v	φs(v	NOUN
ejpam-6409	768	8	)	)	PUNCT
ejpam-6409	769	1	+	+	CCONJ
ejpam-6409	769	2	φs(2	φs(2	PRON
ejpam-6409	769	3	s−r	s−r	NOUN
ejpam-6409	769	4	)	)	PUNCT
ejpam-6409	769	5	=	=	SYM
ejpam-6409	769	6	φs	φs	PROPN
ejpam-6409	769	7	(	(	PUNCT
ejpam-6409	769	8	v	v	NOUN
ejpam-6409	769	9	+	+	NOUN
ejpam-6409	769	10	2s−r	2s−r	NUM
ejpam-6409	769	11	−	−	ADP
ejpam-6409	769	12	2s−r+1rs−r	2s−r+1rs−r	NUM
ejpam-6409	769	13	)	)	PUNCT
ejpam-6409	769	14	,	,	PUNCT
ejpam-6409	769	15	where	where	SCONJ
ejpam-6409	769	16	ts−r	ts−r	NOUN
ejpam-6409	769	17	=	=	SYM
ejpam-6409	769	18	(	(	PUNCT
ejpam-6409	769	19	t1,(s−r	t1,(s−r	NUM
ejpam-6409	769	20	)	)	PUNCT
ejpam-6409	769	21	,	,	PUNCT
ejpam-6409	769	22	t2,(s−r	t2,(s−r	NUM
ejpam-6409	769	23	)	)	PUNCT
ejpam-6409	769	24	,	,	PUNCT
ejpam-6409	769	25	.	.	PUNCT
ejpam-6409	769	26	.	.	PUNCT
ejpam-6409	769	27	.	.	PUNCT
ejpam-6409	770	1	,	,	PUNCT
ejpam-6409	770	2	tn,(s−r	tn,(s−r	NOUN
ejpam-6409	770	3	)	)	PUNCT
ejpam-6409	770	4	)	)	PUNCT
ejpam-6409	770	5	,	,	PUNCT
ejpam-6409	770	6	and	and	CCONJ
ejpam-6409	770	7	for	for	ADP
ejpam-6409	770	8	j	j	PROPN
ejpam-6409	770	9	∈	∈	PROPN
ejpam-6409	770	10	{	{	PUNCT
ejpam-6409	770	11	1	1	NUM
ejpam-6409	770	12	,	,	PUNCT
ejpam-6409	770	13	.	.	PUNCT
ejpam-6409	770	14	.	.	PUNCT
ejpam-6409	771	1	.	.	PUNCT
ejpam-6409	771	2	,	,	PUNCT
ejpam-6409	772	1	n	n	CCONJ
ejpam-6409	772	2	}	}	PUNCT
ejpam-6409	772	3	,	,	PUNCT
ejpam-6409	773	1	tj,(s−r	tj,(s−r	NUM
ejpam-6409	773	2	)	)	PUNCT
ejpam-6409	774	1	=	=	NOUN
ejpam-6409	774	2	{	{	PUNCT
ejpam-6409	774	3	1	1	NUM
ejpam-6409	774	4	,	,	PUNCT
ejpam-6409	774	5	if	if	SCONJ
ejpam-6409	774	6	vj,(s−r	vj,(s−r	NOUN
ejpam-6409	774	7	)	)	PUNCT
ejpam-6409	774	8	≥	≥	NOUN
ejpam-6409	774	9	1	1	NUM
ejpam-6409	774	10	,	,	PUNCT
ejpam-6409	774	11	0	0	NUM
ejpam-6409	774	12	,	,	PUNCT
ejpam-6409	774	13	otherwise	otherwise	ADV
ejpam-6409	774	14	.	.	PUNCT
ejpam-6409	775	1	it	it	PRON
ejpam-6409	775	2	suffices	suffice	VERB
ejpam-6409	775	3	to	to	PART
ejpam-6409	775	4	show	show	VERB
ejpam-6409	775	5	that	that	SCONJ
ejpam-6409	775	6	2s−r+1ts−r	2s−r+1ts−r	NUM
ejpam-6409	775	7	/∈	/∈	PUNCT
ejpam-6409	775	8	ĥ	ĥ	X
ejpam-6409	775	9	to	to	PART
ejpam-6409	775	10	prove	prove	VERB
ejpam-6409	775	11	that	that	DET
ejpam-6409	775	12	φs(v	φs(v	NOUN
ejpam-6409	775	13	)	)	PUNCT
ejpam-6409	775	14	/∈	/∈	PUNCT
ejpam-6409	776	1	k(h	k(h	PROPN
ejpam-6409	776	2	)	)	PUNCT
ejpam-6409	776	3	.	.	PUNCT
ejpam-6409	777	1	since	since	SCONJ
ejpam-6409	777	2	v	v	NOUN
ejpam-6409	777	3	=	=	SYM
ejpam-6409	777	4	τ−ts∑	τ−ts∑	VERB
ejpam-6409	777	5	i=2	i=2	NOUN
ejpam-6409	777	6	µiwi	µiwi	NOUN
ejpam-6409	777	7	=	=	PUNCT
ejpam-6409	777	8	(	(	PUNCT
ejpam-6409	777	9	v1	v1	NOUN
ejpam-6409	777	10	,	,	PUNCT
ejpam-6409	777	11	v2	v2	NOUN
ejpam-6409	777	12	,	,	PUNCT
ejpam-6409	777	13	.	.	PUNCT
ejpam-6409	777	14	.	.	PUNCT
ejpam-6409	777	15	.	.	PUNCT
ejpam-6409	778	1	,	,	PUNCT
ejpam-6409	778	2	vn	vn	PROPN
ejpam-6409	778	3	)	)	PUNCT
ejpam-6409	778	4	and	and	CCONJ
ejpam-6409	778	5	ord(v	ord(v	NUM
ejpam-6409	778	6	)	)	PUNCT
ejpam-6409	779	1	=	=	NOUN
ejpam-6409	779	2	pr	pr	NOUN
ejpam-6409	779	3	for	for	ADP
ejpam-6409	779	4	some	some	DET
ejpam-6409	779	5	2	2	NUM
ejpam-6409	779	6	≤	≤	NOUN
ejpam-6409	779	7	r	r	NOUN
ejpam-6409	779	8	≤	≤	NOUN
ejpam-6409	779	9	s	s	PART
ejpam-6409	779	10	+	+	ADJ
ejpam-6409	779	11	1	1	NUM
ejpam-6409	779	12	−	−	PROPN
ejpam-6409	779	13	σ	σ	NOUN
ejpam-6409	779	14	,	,	PUNCT
ejpam-6409	779	15	according	accord	VERB
ejpam-6409	779	16	to	to	ADP
ejpam-6409	779	17	the	the	DET
ejpam-6409	779	18	construction	construction	NOUN
ejpam-6409	779	19	,	,	PUNCT
ejpam-6409	779	20	v	v	NOUN
ejpam-6409	779	21	contains	contain	VERB
ejpam-6409	779	22	each	each	DET
ejpam-6409	779	23	element	element	NOUN
ejpam-6409	779	24	of	of	ADP
ejpam-6409	779	25	2s−1z2s	2s−1z2s	PROPN
ejpam-6409	779	26	[	[	X
ejpam-6409	779	27	ω	ω	X
ejpam-6409	779	28	]	]	X
ejpam-6409	779	29	exactly	exactly	ADV
ejpam-6409	779	30	α	α	PROPN
ejpam-6409	779	31	times	time	NOUN
ejpam-6409	779	32	,	,	PUNCT
ejpam-6409	779	33	α	α	PRON
ejpam-6409	779	34	≥	≥	NOUN
ejpam-6409	779	35	0	0	NUM
ejpam-6409	779	36	,	,	PUNCT
ejpam-6409	779	37	and	and	CCONJ
ejpam-6409	779	38	the	the	DET
ejpam-6409	779	39	remaining	remain	VERB
ejpam-6409	779	40	n−	n−	NOUN
ejpam-6409	779	41	4α	4α	NOUN
ejpam-6409	779	42	coordinates	coordinate	NOUN
ejpam-6409	779	43	come	come	VERB
ejpam-6409	779	44	from	from	ADP
ejpam-6409	779	45	z2s	z2s	PROPN
ejpam-6409	779	46	[	[	X
ejpam-6409	779	47	ω	ω	X
ejpam-6409	779	48	]	]	PUNCT
ejpam-6409	779	49	\	\	X
ejpam-6409	779	50	2s−1z2s	2s−1z2s	NUM
ejpam-6409	780	1	[	[	X
ejpam-6409	780	2	ω	ω	X
ejpam-6409	780	3	]	]	X
ejpam-6409	780	4	.	.	PUNCT
ejpam-6409	781	1	so	so	ADV
ejpam-6409	781	2	,	,	PUNCT
ejpam-6409	781	3	wth	wth	NOUN
ejpam-6409	781	4	(	(	PUNCT
ejpam-6409	781	5	φs(2	φs(2	NOUN
ejpam-6409	781	6	s−r+1rs−r	s−r+1rs−r	PROPN
ejpam-6409	781	7	)	)	PUNCT
ejpam-6409	781	8	)	)	PUNCT
ejpam-6409	781	9	≤	≤	NOUN
ejpam-6409	781	10	(	(	PUNCT
ejpam-6409	781	11	n−	n−	NOUN
ejpam-6409	781	12	4α	4α	NOUN
ejpam-6409	781	13	)	)	PUNCT
ejpam-6409	781	14	·	·	PUNCT
ejpam-6409	781	15	3	3	NUM
ejpam-6409	781	16	·	·	SYM
ejpam-6409	781	17	22(s−2	22(s−2	NUM
ejpam-6409	781	18	)	)	PUNCT
ejpam-6409	781	19	<	<	X
ejpam-6409	781	20	3n	3n	NUM
ejpam-6409	781	21	·	·	PUNCT
ejpam-6409	781	22	22(s−2	22(s−2	NUM
ejpam-6409	781	23	)	)	PUNCT
ejpam-6409	782	1	=	=	SYM
ejpam-6409	782	2	3n	3n	NUM
ejpam-6409	782	3	4	4	NUM
ejpam-6409	782	4	=	=	SYM
ejpam-6409	782	5	d(h	d(h	PROPN
ejpam-6409	782	6	)	)	PUNCT
ejpam-6409	782	7	.	.	PUNCT
ejpam-6409	783	1	therefore	therefore	ADV
ejpam-6409	783	2	,	,	PUNCT
ejpam-6409	783	3	φs(v	φs(v	X
ejpam-6409	783	4	)	)	PUNCT
ejpam-6409	783	5	/∈	/∈	PUNCT
ejpam-6409	784	1	k(h	k(h	PROPN
ejpam-6409	784	2	)	)	PUNCT
ejpam-6409	784	3	,	,	PUNCT
ejpam-6409	784	4	and	and	CCONJ
ejpam-6409	784	5	φs(ξ	φs(ξ	NUM
ejpam-6409	784	6	)	)	PUNCT
ejpam-6409	784	7	∩k(h	∩k(h	NOUN
ejpam-6409	784	8	)	)	PUNCT
ejpam-6409	784	9	=	=	PUNCT
ejpam-6409	784	10	∅.	∅.	NOUN
ejpam-6409	784	11	we	we	PRON
ejpam-6409	784	12	now	now	ADV
ejpam-6409	784	13	proceed	proceed	VERB
ejpam-6409	784	14	to	to	PART
ejpam-6409	784	15	show	show	VERB
ejpam-6409	784	16	that	that	SCONJ
ejpam-6409	784	17	φs(ξ	φs(ξ	PUNCT
ejpam-6409	784	18	+	+	NOUN
ejpam-6409	784	19	n	n	NUM
ejpam-6409	784	20	)	)	PUNCT
ejpam-6409	784	21	∩k(φs(ĥ	∩k(φs(ĥ	PUNCT
ejpam-6409	784	22	)	)	PUNCT
ejpam-6409	784	23	)	)	PUNCT
ejpam-6409	785	1	=	=	PRON
ejpam-6409	785	2	{	{	PUNCT
ejpam-6409	785	3	0	0	NUM
ejpam-6409	785	4	}	}	PUNCT
ejpam-6409	785	5	.	.	PUNCT
ejpam-6409	786	1	let	let	VERB
ejpam-6409	786	2	v	v	NOUN
ejpam-6409	786	3	=	=	SYM
ejpam-6409	786	4	vξ	vξ	PROPN
ejpam-6409	787	1	+	+	NUM
ejpam-6409	787	2	vn	vn	PROPN
ejpam-6409	787	3	∈	∈	PROPN
ejpam-6409	788	1	ξ	ξ	X
ejpam-6409	788	2	+	+	PROPN
ejpam-6409	788	3	n	n	PRON
ejpam-6409	788	4	\	\	NOUN
ejpam-6409	788	5	{	{	PUNCT
ejpam-6409	788	6	0	0	NUM
ejpam-6409	788	7	}	}	PUNCT
ejpam-6409	788	8	,	,	PUNCT
ejpam-6409	788	9	where	where	SCONJ
ejpam-6409	788	10	vξ	vξ	PROPN
ejpam-6409	788	11	∈	∈	PROPN
ejpam-6409	788	12	ξ	ξ	PROPN
ejpam-6409	788	13	and	and	CCONJ
ejpam-6409	788	14	vn	vn	PROPN
ejpam-6409	788	15	∈	∈	PROPN
ejpam-6409	788	16	n	n	CCONJ
ejpam-6409	788	17	.	.	PUNCT
ejpam-6409	789	1	we	we	PRON
ejpam-6409	789	2	previously	previously	ADV
ejpam-6409	789	3	proved	prove	VERB
ejpam-6409	789	4	that	that	SCONJ
ejpam-6409	789	5	φs(v	φs(v	NUM
ejpam-6409	789	6	)	)	PUNCT
ejpam-6409	789	7	/∈	/∈	PUNCT
ejpam-6409	790	1	k(h	k(h	PROPN
ejpam-6409	790	2	)	)	PUNCT
ejpam-6409	791	1	if	if	SCONJ
ejpam-6409	791	2	vξ	vξ	PROPN
ejpam-6409	791	3	=	=	NOUN
ejpam-6409	791	4	0	0	NUM
ejpam-6409	791	5	or	or	CCONJ
ejpam-6409	791	6	vn	vn	X
ejpam-6409	791	7	=	=	NOUN
ejpam-6409	791	8	0	0	PROPN
ejpam-6409	791	9	.	.	PUNCT
ejpam-6409	792	1	hence	hence	ADV
ejpam-6409	792	2	,	,	PUNCT
ejpam-6409	792	3	assume	assume	VERB
ejpam-6409	792	4	vξ	vξ	NUM
ejpam-6409	792	5	̸=	̸=	PROPN
ejpam-6409	792	6	0	0	NUM
ejpam-6409	792	7	and	and	CCONJ
ejpam-6409	792	8	0n	0n	NOUN
ejpam-6409	792	9	̸=	̸=	PROPN
ejpam-6409	792	10	0	0	NUM
ejpam-6409	792	11	.	.	PUNCT
ejpam-6409	793	1	we	we	PRON
ejpam-6409	793	2	know	know	VERB
ejpam-6409	793	3	vn	vn	PROPN
ejpam-6409	793	4	=	=	SYM
ejpam-6409	793	5	(	(	PUNCT
ejpam-6409	793	6	v	v	NOUN
ejpam-6409	793	7	,	,	PUNCT
ejpam-6409	793	8	.	.	PUNCT
ejpam-6409	793	9	.	.	PUNCT
ejpam-6409	794	1	.	.	PUNCT
ejpam-6409	795	1	,	,	PUNCT
ejpam-6409	795	2	v	v	NOUN
ejpam-6409	795	3	)	)	PUNCT
ejpam-6409	795	4	.	.	PUNCT
ejpam-6409	796	1	muhammad	muhammad	PROPN
ejpam-6409	796	2	sajjad	sajjad	PROPN
ejpam-6409	796	3	et	et	PROPN
ejpam-6409	796	4	al	al	PROPN
ejpam-6409	796	5	.	.	PUNCT
ejpam-6409	796	6	/	/	SYM
ejpam-6409	796	7	eur	eur	PROPN
ejpam-6409	796	8	.	.	PUNCT
ejpam-6409	797	1	j.	j.	PROPN
ejpam-6409	797	2	pure	pure	PROPN
ejpam-6409	797	3	appl	appl	PROPN
ejpam-6409	797	4	.	.	PROPN
ejpam-6409	797	5	math	math	PROPN
ejpam-6409	797	6	,	,	PUNCT
ejpam-6409	797	7	18	18	NUM
ejpam-6409	797	8	(	(	PUNCT
ejpam-6409	797	9	3	3	NUM
ejpam-6409	797	10	)	)	PUNCT
ejpam-6409	797	11	(	(	PUNCT
ejpam-6409	797	12	2025	2025	NUM
ejpam-6409	797	13	)	)	PUNCT
ejpam-6409	797	14	,	,	PUNCT
ejpam-6409	797	15	6409	6409	NUM
ejpam-6409	797	16	24	24	NUM
ejpam-6409	797	17	of	of	ADP
ejpam-6409	797	18	32	32	NUM
ejpam-6409	797	19	let	let	VERB
ejpam-6409	798	1	[	[	X
ejpam-6409	798	2	v0	v0	NOUN
ejpam-6409	798	3	,	,	PUNCT
ejpam-6409	798	4	v1	v1	NOUN
ejpam-6409	798	5	,	,	PUNCT
ejpam-6409	798	6	.	.	PUNCT
ejpam-6409	798	7	.	.	PUNCT
ejpam-6409	799	1	.	.	PUNCT
ejpam-6409	800	1	,	,	PUNCT
ejpam-6409	800	2	vs−1]2	vs−1]2	ADV
ejpam-6409	800	3	be	be	AUX
ejpam-6409	800	4	the	the	DET
ejpam-6409	800	5	2	2	NUM
ejpam-6409	800	6	-	-	PUNCT
ejpam-6409	800	7	ary	ary	NOUN
ejpam-6409	800	8	expansion	expansion	NOUN
ejpam-6409	800	9	of	of	ADP
ejpam-6409	800	10	v.	v.	PROPN
ejpam-6409	800	11	consider	consider	VERB
ejpam-6409	800	12	vn1	vn1	NOUN
ejpam-6409	800	13	and	and	CCONJ
ejpam-6409	800	14	vn2	vn2	NOUN
ejpam-6409	800	15	as	as	ADP
ejpam-6409	800	16	the	the	DET
ejpam-6409	800	17	elements	element	NOUN
ejpam-6409	800	18	of	of	ADP
ejpam-6409	800	19	z2s	z2s	PROPN
ejpam-6409	800	20	[	[	X
ejpam-6409	800	21	ω	ω	X
ejpam-6409	800	22	]	]	X
ejpam-6409	800	23	having	have	VERB
ejpam-6409	800	24	2	2	NUM
ejpam-6409	800	25	-	-	PUNCT
ejpam-6409	800	26	ary	ary	NOUN
ejpam-6409	800	27	expansions	expansion	NOUN
ejpam-6409	800	28	[	[	X
ejpam-6409	800	29	0	0	NUM
ejpam-6409	800	30	,	,	PUNCT
ejpam-6409	800	31	.	.	PUNCT
ejpam-6409	800	32	.	.	PUNCT
ejpam-6409	801	1	.	.	PUNCT
ejpam-6409	802	1	,	,	PUNCT
ejpam-6409	802	2	0	0	NUM
ejpam-6409	802	3	,	,	PUNCT
ejpam-6409	802	4	vs−r	vs−r	ADV
ejpam-6409	802	5	,	,	PUNCT
ejpam-6409	802	6	.	.	PUNCT
ejpam-6409	802	7	.	.	PUNCT
ejpam-6409	803	1	.	.	PUNCT
ejpam-6409	804	1	,	,	PUNCT
ejpam-6409	804	2	vs−1]2	vs−1]2	NOUN
ejpam-6409	804	3	and	and	CCONJ
ejpam-6409	804	4	[	[	X
ejpam-6409	804	5	v0	v0	NOUN
ejpam-6409	804	6	,	,	PUNCT
ejpam-6409	804	7	.	.	PUNCT
ejpam-6409	804	8	.	.	PUNCT
ejpam-6409	805	1	.	.	PUNCT
ejpam-6409	806	1	,	,	PUNCT
ejpam-6409	806	2	vs−r−1	vs−r−1	PROPN
ejpam-6409	806	3	,	,	PUNCT
ejpam-6409	806	4	0	0	NUM
ejpam-6409	806	5	,	,	PUNCT
ejpam-6409	806	6	.	.	PUNCT
ejpam-6409	806	7	.	.	PUNCT
ejpam-6409	807	1	.	.	PUNCT
ejpam-6409	808	1	,	,	PUNCT
ejpam-6409	808	2	0]2	0]2	PROPN
ejpam-6409	808	3	,	,	PUNCT
ejpam-6409	808	4	respectively	respectively	ADV
ejpam-6409	808	5	.	.	PUNCT
ejpam-6409	809	1	then	then	ADV
ejpam-6409	809	2	,	,	PUNCT
ejpam-6409	809	3	vn	vn	PROPN
ejpam-6409	809	4	=	=	PUNCT
ejpam-6409	809	5	vn1	vn1	NOUN
ejpam-6409	809	6	+	+	CCONJ
ejpam-6409	809	7	vn2	vn2	PROPN
ejpam-6409	809	8	,	,	PUNCT
ejpam-6409	809	9	where	where	SCONJ
ejpam-6409	809	10	vni	vni	NOUN
ejpam-6409	809	11	=	=	SYM
ejpam-6409	809	12	(	(	PUNCT
ejpam-6409	809	13	vni	vni	INTJ
ejpam-6409	809	14	,	,	PUNCT
ejpam-6409	809	15	.	.	PUNCT
ejpam-6409	809	16	.	.	PUNCT
ejpam-6409	809	17	.	.	PUNCT
ejpam-6409	810	1	,	,	PUNCT
ejpam-6409	810	2	vni	vni	PROPN
ejpam-6409	810	3	)	)	PUNCT
ejpam-6409	810	4	,	,	PUNCT
ejpam-6409	810	5	i	i	PRON
ejpam-6409	810	6	∈	∈	PROPN
ejpam-6409	810	7	{	{	PUNCT
ejpam-6409	810	8	1	1	NUM
ejpam-6409	810	9	,	,	PUNCT
ejpam-6409	810	10	2	2	NUM
ejpam-6409	810	11	}	}	PUNCT
ejpam-6409	810	12	.	.	PUNCT
ejpam-6409	811	1	since	since	SCONJ
ejpam-6409	811	2	ord(vξ	ord(vξ	NOUN
ejpam-6409	811	3	)	)	PUNCT
ejpam-6409	811	4	=	=	SYM
ejpam-6409	811	5	2r	2r	NUM
ejpam-6409	811	6	with	with	ADP
ejpam-6409	811	7	2	2	NUM
ejpam-6409	811	8	≤	≤	NOUN
ejpam-6409	811	9	r	r	NOUN
ejpam-6409	811	10	≤	≤	NUM
ejpam-6409	811	11	s+	s+	PUNCT
ejpam-6409	811	12	1−	1−	NUM
ejpam-6409	811	13	σ	σ	PROPN
ejpam-6409	811	14	,	,	PUNCT
ejpam-6409	811	15	the	the	DET
ejpam-6409	811	16	2	2	NUM
ejpam-6409	811	17	-	-	PUNCT
ejpam-6409	811	18	ary	ary	NOUN
ejpam-6409	811	19	expansion	expansion	NOUN
ejpam-6409	811	20	of	of	ADP
ejpam-6409	811	21	each	each	DET
ejpam-6409	811	22	coordinate	coordinate	NOUN
ejpam-6409	811	23	of	of	ADP
ejpam-6409	811	24	vξ	vξ	NOUN
ejpam-6409	811	25	takes	take	VERB
ejpam-6409	811	26	the	the	DET
ejpam-6409	811	27	form	form	NOUN
ejpam-6409	811	28	[	[	X
ejpam-6409	811	29	0	0	NUM
ejpam-6409	811	30	,	,	PUNCT
ejpam-6409	811	31	.	.	PUNCT
ejpam-6409	811	32	.	.	PUNCT
ejpam-6409	812	1	.	.	PUNCT
ejpam-6409	813	1	,	,	PUNCT
ejpam-6409	813	2	0	0	NUM
ejpam-6409	813	3	,	,	PUNCT
ejpam-6409	813	4	vξ,(s−r	vξ,(s−r	NOUN
ejpam-6409	813	5	)	)	PUNCT
ejpam-6409	813	6	,	,	PUNCT
ejpam-6409	813	7	.	.	PUNCT
ejpam-6409	813	8	.	.	PUNCT
ejpam-6409	814	1	.	.	PUNCT
ejpam-6409	815	1	,	,	PUNCT
ejpam-6409	815	2	vξ,(s−1)]2	vξ,(s−1)]2	PROPN
ejpam-6409	815	3	.	.	PUNCT
ejpam-6409	815	4	note	note	VERB
ejpam-6409	815	5	that	that	SCONJ
ejpam-6409	815	6	ord(vn1	ord(vn1	VERB
ejpam-6409	815	7	)	)	PUNCT
ejpam-6409	815	8	≤	≤	NUM
ejpam-6409	815	9	ord(vξ	ord(vξ	NOUN
ejpam-6409	815	10	)	)	PUNCT
ejpam-6409	815	11	by	by	ADP
ejpam-6409	815	12	construction	construction	NOUN
ejpam-6409	815	13	.	.	PUNCT
ejpam-6409	816	1	it	it	PRON
ejpam-6409	816	2	follows	follow	VERB
ejpam-6409	816	3	that	that	SCONJ
ejpam-6409	816	4	2	2	X
ejpam-6409	816	5	(	(	PUNCT
ejpam-6409	816	6	vn2	vn2	NOUN
ejpam-6409	816	7	⊙2	⊙2	NOUN
ejpam-6409	816	8	2	2	NUM
ejpam-6409	816	9	s−r	s−r	NOUN
ejpam-6409	816	10	)	)	PUNCT
ejpam-6409	817	1	=	=	SYM
ejpam-6409	817	2	0	0	X
ejpam-6409	817	3	.	.	PUNCT
ejpam-6409	818	1	therefore	therefore	ADV
ejpam-6409	818	2	,	,	PUNCT
ejpam-6409	818	3	wth	wth	NOUN
ejpam-6409	818	4	(	(	PUNCT
ejpam-6409	818	5	φs(2(v	φs(2(v	PROPN
ejpam-6409	818	6	⊙2	⊙2	VERB
ejpam-6409	818	7	2	2	NUM
ejpam-6409	818	8	s−r	s−r	NOUN
ejpam-6409	818	9	)	)	PUNCT
ejpam-6409	818	10	)	)	PUNCT
ejpam-6409	818	11	)	)	PUNCT
ejpam-6409	818	12	=	=	PUNCT
ejpam-6409	818	13	wth	wth	NOUN
ejpam-6409	818	14	(	(	PUNCT
ejpam-6409	818	15	φs(2((vξ	φs(2((vξ	X
ejpam-6409	818	16	+	+	CCONJ
ejpam-6409	818	17	vn1)⊙2	vn1)⊙2	PROPN
ejpam-6409	818	18	2	2	NUM
ejpam-6409	818	19	s−r	s−r	NOUN
ejpam-6409	818	20	)	)	PUNCT
ejpam-6409	818	21	)	)	PUNCT
ejpam-6409	818	22	)	)	PUNCT
ejpam-6409	818	23	.	.	PUNCT
ejpam-6409	819	1	since	since	SCONJ
ejpam-6409	819	2	ord(vn1	ord(vn1	NOUN
ejpam-6409	819	3	)	)	PUNCT
ejpam-6409	819	4	≤	≤	NOUN
ejpam-6409	819	5	ord(vξ	ord(vξ	NOUN
ejpam-6409	819	6	)	)	PUNCT
ejpam-6409	819	7	,	,	PUNCT
ejpam-6409	819	8	there	there	PRON
ejpam-6409	819	9	exists	exist	VERB
ejpam-6409	819	10	a	a	DET
ejpam-6409	819	11	permutation	permutation	NOUN
ejpam-6409	819	12	of	of	ADP
ejpam-6409	819	13	coordinates	coordinate	NOUN
ejpam-6409	819	14	π	π	X
ejpam-6409	819	15	satisfying	satisfy	VERB
ejpam-6409	819	16	π(vξ	π(vξ	X
ejpam-6409	819	17	+	+	X
ejpam-6409	819	18	vn1	vn1	NOUN
ejpam-6409	819	19	)	)	PUNCT
ejpam-6409	820	1	=	=	SYM
ejpam-6409	820	2	vξ	vξ	PROPN
ejpam-6409	820	3	.	.	PUNCT
ejpam-6409	820	4	thus	thus	ADV
ejpam-6409	820	5	,	,	PUNCT
ejpam-6409	820	6	wth	wth	NOUN
ejpam-6409	820	7	(	(	PUNCT
ejpam-6409	820	8	φs(2((vξ	φs(2((vξ	X
ejpam-6409	820	9	+	+	CCONJ
ejpam-6409	820	10	vn1)⊙2	vn1)⊙2	PROPN
ejpam-6409	820	11	2	2	NUM
ejpam-6409	820	12	s−r	s−r	NOUN
ejpam-6409	820	13	)	)	PUNCT
ejpam-6409	820	14	)	)	PUNCT
ejpam-6409	820	15	)	)	PUNCT
ejpam-6409	821	1	=	=	PUNCT
ejpam-6409	821	2	wth	wth	NOUN
ejpam-6409	821	3	(	(	PUNCT
ejpam-6409	821	4	φs(2(vξ	φs(2(vξ	NOUN
ejpam-6409	821	5	⊙2	⊙2	NOUN
ejpam-6409	821	6	2	2	NUM
ejpam-6409	821	7	s−r	s−r	NOUN
ejpam-6409	821	8	)	)	PUNCT
ejpam-6409	821	9	)	)	PUNCT
ejpam-6409	821	10	)	)	PUNCT
ejpam-6409	821	11	.	.	PUNCT
ejpam-6409	822	1	since	since	SCONJ
ejpam-6409	822	2	ord(vξ	ord(vξ	NOUN
ejpam-6409	822	3	)	)	PUNCT
ejpam-6409	822	4	=	=	VERB
ejpam-6409	822	5	pr	pr	NOUN
ejpam-6409	822	6	with	with	ADP
ejpam-6409	822	7	2	2	NUM
ejpam-6409	822	8	≤	≤	NOUN
ejpam-6409	822	9	r	r	NOUN
ejpam-6409	822	10	≤	≤	NOUN
ejpam-6409	822	11	s	s	PART
ejpam-6409	822	12	+	+	ADJ
ejpam-6409	822	13	1	1	NUM
ejpam-6409	822	14	−	−	PROPN
ejpam-6409	822	15	σ	σ	NOUN
ejpam-6409	822	16	,	,	PUNCT
ejpam-6409	822	17	as	as	ADP
ejpam-6409	822	18	in	in	ADP
ejpam-6409	822	19	the	the	DET
ejpam-6409	822	20	previous	previous	ADJ
ejpam-6409	822	21	case	case	NOUN
ejpam-6409	822	22	,	,	PUNCT
ejpam-6409	822	23	this	this	PRON
ejpam-6409	822	24	leads	lead	VERB
ejpam-6409	822	25	to	to	ADP
ejpam-6409	822	26	a	a	DET
ejpam-6409	822	27	contradiction	contradiction	NOUN
ejpam-6409	822	28	.	.	PUNCT
ejpam-6409	823	1	therefore	therefore	ADV
ejpam-6409	823	2	,	,	PUNCT
ejpam-6409	823	3	φs(v	φs(v	X
ejpam-6409	823	4	)	)	PUNCT
ejpam-6409	823	5	/∈	/∈	PUNCT
ejpam-6409	824	1	k(h	k(h	PROPN
ejpam-6409	824	2	)	)	PUNCT
ejpam-6409	824	3	and	and	CCONJ
ejpam-6409	824	4	φs(ξ	φs(ξ	ADJ
ejpam-6409	824	5	+	+	ADJ
ejpam-6409	824	6	n	n	NOUN
ejpam-6409	824	7	)	)	PUNCT
ejpam-6409	824	8	∩k(h	∩k(h	PROPN
ejpam-6409	824	9	)	)	PUNCT
ejpam-6409	824	10	=	=	PRON
ejpam-6409	824	11	{	{	PUNCT
ejpam-6409	824	12	0	0	NUM
ejpam-6409	824	13	}	}	PUNCT
ejpam-6409	824	14	.	.	PUNCT
ejpam-6409	825	1	theorem	theorem	VERB
ejpam-6409	825	2	6.1	6.1	NUM
ejpam-6409	825	3	:	:	PUNCT
ejpam-6409	825	4	let	let	VERB
ejpam-6409	825	5	ĥ	ĥ	X
ejpam-6409	825	6	=	=	SYM
ejpam-6409	825	7	ĥ(t1,	ĥ(t1,	PROPN
ejpam-6409	825	8	...	...	PUNCT
ejpam-6409	825	9	,ts	,ts	PUNCT
ejpam-6409	825	10	)	)	PUNCT
ejpam-6409	825	11	be	be	AUX
ejpam-6409	825	12	the	the	DET
ejpam-6409	825	13	z2s	z2s	PROPN
ejpam-6409	825	14	[	[	X
ejpam-6409	825	15	ω]-additive	ω]-additive	ADJ
ejpam-6409	825	16	hadamard	hadamard	ADJ
ejpam-6409	825	17	code	code	NOUN
ejpam-6409	825	18	of	of	ADP
ejpam-6409	825	19	type	type	NOUN
ejpam-6409	825	20	(	(	PUNCT
ejpam-6409	825	21	n	n	CCONJ
ejpam-6409	825	22	;	;	PUNCT
ejpam-6409	825	23	t1	t1	NOUN
ejpam-6409	825	24	,	,	PUNCT
ejpam-6409	825	25	.	.	PUNCT
ejpam-6409	825	26	.	.	PUNCT
ejpam-6409	826	1	.	.	PUNCT
ejpam-6409	827	1	,	,	PUNCT
ejpam-6409	827	2	ts	ts	NOUN
ejpam-6409	827	3	)	)	PUNCT
ejpam-6409	827	4	such	such	ADJ
ejpam-6409	827	5	that	that	SCONJ
ejpam-6409	827	6	φs(ĥ	φs(ĥ	NOUN
ejpam-6409	827	7	)	)	PUNCT
ejpam-6409	827	8	is	be	AUX
ejpam-6409	827	9	nonlinear	nonlinear	ADJ
ejpam-6409	827	10	.	.	PUNCT
ejpam-6409	828	1	define	define	VERB
ejpam-6409	828	2	ĥb	ĥb	NOUN
ejpam-6409	828	3	as	as	ADP
ejpam-6409	828	4	the	the	DET
ejpam-6409	828	5	subcode	subcode	NOUN
ejpam-6409	828	6	of	of	ADP
ejpam-6409	828	7	ĥ	ĥ	PUNCT
ejpam-6409	828	8	consisting	consist	VERB
ejpam-6409	828	9	of	of	ADP
ejpam-6409	828	10	all	all	DET
ejpam-6409	828	11	the	the	DET
ejpam-6409	828	12	codewords	codeword	NOUN
ejpam-6409	828	13	of	of	ADP
ejpam-6409	828	14	order	order	NOUN
ejpam-6409	828	15	two	two	NUM
ejpam-6409	828	16	.	.	PUNCT
ejpam-6409	829	1	let	let	VERB
ejpam-6409	829	2	b	b	NOUN
ejpam-6409	829	3	=	=	PRON
ejpam-6409	829	4	{	{	PUNCT
ejpam-6409	829	5	{	{	PUNCT
ejpam-6409	829	6	2p}σ−2	2p}σ−2	NUM
ejpam-6409	829	7	p=0	p=0	PROPN
ejpam-6409	829	8	if	if	SCONJ
ejpam-6409	829	9	σ	σ	PROPN
ejpam-6409	829	10	≥	≥	NOUN
ejpam-6409	829	11	2	2	NUM
ejpam-6409	829	12	,	,	PUNCT
ejpam-6409	829	13	∅	∅	NOUN
ejpam-6409	829	14	if	if	SCONJ
ejpam-6409	829	15	σ	σ	PROPN
ejpam-6409	829	16	=	=	SYM
ejpam-6409	829	17	1	1	X
ejpam-6409	829	18	.	.	PUNCT
ejpam-6409	830	1	muhammad	muhammad	PROPN
ejpam-6409	830	2	sajjad	sajjad	PROPN
ejpam-6409	830	3	et	et	PROPN
ejpam-6409	830	4	al	al	PROPN
ejpam-6409	830	5	.	.	PUNCT
ejpam-6409	830	6	/	/	SYM
ejpam-6409	830	7	eur	eur	PROPN
ejpam-6409	830	8	.	.	PUNCT
ejpam-6409	831	1	j.	j.	PROPN
ejpam-6409	831	2	pure	pure	PROPN
ejpam-6409	831	3	appl	appl	PROPN
ejpam-6409	831	4	.	.	PROPN
ejpam-6409	831	5	math	math	PROPN
ejpam-6409	831	6	,	,	PUNCT
ejpam-6409	831	7	18	18	NUM
ejpam-6409	831	8	(	(	PUNCT
ejpam-6409	831	9	3	3	NUM
ejpam-6409	831	10	)	)	PUNCT
ejpam-6409	831	11	(	(	PUNCT
ejpam-6409	831	12	2025	2025	NUM
ejpam-6409	831	13	)	)	PUNCT
ejpam-6409	831	14	,	,	PUNCT
ejpam-6409	831	15	6409	6409	NUM
ejpam-6409	831	16	25	25	NUM
ejpam-6409	831	17	of	of	ADP
ejpam-6409	831	18	32	32	NUM
ejpam-6409	831	19	then	then	ADV
ejpam-6409	831	20	,	,	PUNCT
ejpam-6409	831	21	⟨φs(ĥb	⟨φs(ĥb	PROPN
ejpam-6409	831	22	)	)	PUNCT
ejpam-6409	831	23	,	,	PUNCT
ejpam-6409	831	24	φs(b	φs(b	NUM
ejpam-6409	831	25	)	)	PUNCT
ejpam-6409	831	26	,	,	PUNCT
ejpam-6409	831	27	φs	φs	PROPN
ejpam-6409	831	28	(	(	PUNCT
ejpam-6409	831	29	s−2∑	s−2∑	PROPN
ejpam-6409	831	30	i=0	i=0	PROPN
ejpam-6409	831	31	2i	2i	NUM
ejpam-6409	831	32	)	)	PUNCT
ejpam-6409	831	33	⟩	⟩	PROPN
ejpam-6409	831	34	⊆	⊆	NUM
ejpam-6409	831	35	k(φs(ĥ	k(φs(ĥ	NOUN
ejpam-6409	831	36	)	)	PUNCT
ejpam-6409	831	37	)	)	PUNCT
ejpam-6409	831	38	,	,	PUNCT
ejpam-6409	831	39	and	and	CCONJ
ejpam-6409	831	40	ker(φs(ĥ	ker(φs(ĥ	PROPN
ejpam-6409	831	41	)	)	PUNCT
ejpam-6409	831	42	)	)	PUNCT
ejpam-6409	832	1	=	=	PUNCT
ejpam-6409	833	1	σ	σ	NOUN
ejpam-6409	834	1	+	+	CCONJ
ejpam-6409	834	2	s∑	s∑	PROPN
ejpam-6409	834	3	i=1	i=1	PROPN
ejpam-6409	834	4	ti	ti	PROPN
ejpam-6409	834	5	.	.	PUNCT
ejpam-6409	834	6	proof	proof	NOUN
ejpam-6409	834	7	:	:	PUNCT
ejpam-6409	834	8	the	the	DET
ejpam-6409	834	9	conclusion	conclusion	NOUN
ejpam-6409	834	10	directly	directly	ADV
ejpam-6409	834	11	follows	follow	VERB
ejpam-6409	834	12	from	from	ADP
ejpam-6409	834	13	proposition	proposition	NOUN
ejpam-6409	834	14	6.1	6.1	NUM
ejpam-6409	834	15	and	and	CCONJ
ejpam-6409	834	16	lemma	lemma	PROPN
ejpam-6409	834	17	6.3	6.3	NUM
ejpam-6409	834	18	.	.	PUNCT
ejpam-6409	835	1	corollary	corollary	ADJ
ejpam-6409	835	2	6.1	6.1	NUM
ejpam-6409	835	3	:	:	PUNCT
ejpam-6409	835	4	let	let	VERB
ejpam-6409	835	5	ĥ	ĥ	X
ejpam-6409	835	6	=	=	SYM
ejpam-6409	835	7	ĥ(t1,	ĥ(t1,	PROPN
ejpam-6409	835	8	...	...	PUNCT
ejpam-6409	835	9	,ts	,ts	PUNCT
ejpam-6409	835	10	)	)	PUNCT
ejpam-6409	835	11	be	be	AUX
ejpam-6409	835	12	the	the	DET
ejpam-6409	835	13	z2s	z2s	PROPN
ejpam-6409	835	14	[	[	X
ejpam-6409	835	15	ω]-additive	ω]-additive	ADJ
ejpam-6409	835	16	hadamard	hadamard	ADJ
ejpam-6409	835	17	code	code	NOUN
ejpam-6409	835	18	of	of	ADP
ejpam-6409	835	19	type	type	NOUN
ejpam-6409	835	20	(	(	PUNCT
ejpam-6409	835	21	n	n	CCONJ
ejpam-6409	835	22	;	;	PUNCT
ejpam-6409	835	23	t1	t1	NOUN
ejpam-6409	835	24	,	,	PUNCT
ejpam-6409	835	25	.	.	PUNCT
ejpam-6409	835	26	.	.	PUNCT
ejpam-6409	836	1	.	.	PUNCT
ejpam-6409	837	1	,	,	PUNCT
ejpam-6409	837	2	ts	ts	NOUN
ejpam-6409	837	3	)	)	PUNCT
ejpam-6409	837	4	such	such	ADJ
ejpam-6409	837	5	that	that	SCONJ
ejpam-6409	837	6	φs(ĥ	φs(ĥ	NOUN
ejpam-6409	837	7	)	)	PUNCT
ejpam-6409	837	8	is	be	AUX
ejpam-6409	837	9	nonlinear	nonlinear	ADJ
ejpam-6409	837	10	.	.	PUNCT
ejpam-6409	838	1	let	let	VERB
ejpam-6409	838	2	wi	wi	PROPN
ejpam-6409	838	3	be	be	AUX
ejpam-6409	838	4	the	the	DET
ejpam-6409	838	5	ith	ith	NOUN
ejpam-6409	838	6	row	row	NOUN
ejpam-6409	838	7	of	of	ADP
ejpam-6409	838	8	a	a	DET
ejpam-6409	838	9	(	(	PUNCT
ejpam-6409	838	10	t1,t2,	t1,t2,	NOUN
ejpam-6409	838	11	...	...	PUNCT
ejpam-6409	838	12	,ts	,ts	PUNCT
ejpam-6409	838	13	)	)	PUNCT
ejpam-6409	838	14	2	2	NUM
ejpam-6409	838	15	and	and	CCONJ
ejpam-6409	839	1	τ	τ	PROPN
ejpam-6409	840	1	=	=	PUNCT
ejpam-6409	840	2	∑s	∑s	PROPN
ejpam-6409	840	3	i=1	i=1	PRON
ejpam-6409	840	4	ti	ti	VERB
ejpam-6409	840	5	.	.	PUNCT
ejpam-6409	840	6	let	let	VERB
ejpam-6409	840	7	q	q	NOUN
ejpam-6409	840	8	=	=	NOUN
ejpam-6409	840	9	{	{	PUNCT
ejpam-6409	840	10	ord(wj/2)wj}rj=0	ord(wj/2)wj}rj=0	VERB
ejpam-6409	840	11	,	,	PUNCT
ejpam-6409	840	12	b	b	X
ejpam-6409	840	13	=	=	PRON
ejpam-6409	840	14	{	{	PUNCT
ejpam-6409	840	15	{	{	PUNCT
ejpam-6409	840	16	2p}σ−2	2p}σ−2	NUM
ejpam-6409	840	17	p=0	p=0	PROPN
ejpam-6409	840	18	if	if	SCONJ
ejpam-6409	840	19	σ	σ	PROPN
ejpam-6409	840	20	≥	≥	NOUN
ejpam-6409	840	21	2	2	NUM
ejpam-6409	840	22	,	,	PUNCT
ejpam-6409	840	23	∅	∅	NOUN
ejpam-6409	840	24	if	if	SCONJ
ejpam-6409	840	25	σ	σ	PROPN
ejpam-6409	840	26	=	=	SYM
ejpam-6409	840	27	1	1	X
ejpam-6409	840	28	.	.	PUNCT
ejpam-6409	841	1	then	then	ADV
ejpam-6409	841	2	{	{	PUNCT
ejpam-6409	841	3	φs(q	φs(q	NOUN
ejpam-6409	841	4	)	)	PUNCT
ejpam-6409	841	5	,	,	PUNCT
ejpam-6409	841	6	φs(b	φs(b	NUM
ejpam-6409	841	7	)	)	PUNCT
ejpam-6409	841	8	,	,	PUNCT
ejpam-6409	841	9	φs	φs	PROPN
ejpam-6409	841	10	(	(	PUNCT
ejpam-6409	841	11	∑s−2	∑s−2	ADJ
ejpam-6409	841	12	i=0	i=0	PROPN
ejpam-6409	841	13	2	2	NUM
ejpam-6409	841	14	i	i	NOUN
ejpam-6409	841	15	)	)	PUNCT
ejpam-6409	841	16	}	}	PUNCT
ejpam-6409	841	17	forms	form	VERB
ejpam-6409	841	18	a	a	DET
ejpam-6409	841	19	basis	basis	NOUN
ejpam-6409	841	20	for	for	ADP
ejpam-6409	841	21	itk(φ(ĥ	itk(φ(ĥ	PROPN
ejpam-6409	841	22	)	)	PUNCT
ejpam-6409	841	23	)	)	PUNCT
ejpam-6409	841	24	.	.	PUNCT
ejpam-6409	842	1	example	example	NOUN
ejpam-6409	842	2	6.1	6.1	NUM
ejpam-6409	842	3	:	:	PUNCT
ejpam-6409	842	4	let	let	VERB
ejpam-6409	842	5	it	it	PRON
ejpam-6409	842	6	h(2,0,0	h(2,0,0	NOUN
ejpam-6409	842	7	)	)	PUNCT
ejpam-6409	842	8	be	be	VERB
ejpam-6409	842	9	the	the	DET
ejpam-6409	842	10	z8[ω]-linear	z8[ω]-linear	NUM
ejpam-6409	842	11	hadamard	hadamard	PROPN
ejpam-6409	842	12	code	code	NOUN
ejpam-6409	842	13	discussed	discuss	VERB
ejpam-6409	842	14	in	in	ADP
ejpam-6409	842	15	example	example	NOUN
ejpam-6409	842	16	4	4	NUM
ejpam-6409	842	17	.	.	PUNCT
ejpam-6409	843	1	according	accord	VERB
ejpam-6409	843	2	to	to	ADP
ejpam-6409	843	3	theorem	theorem	ADJ
ejpam-6409	843	4	3.4	3.4	NUM
ejpam-6409	843	5	,	,	PUNCT
ejpam-6409	843	6	ker(h(2,0,0	ker(h(2,0,0	NOUN
ejpam-6409	843	7	)	)	PUNCT
ejpam-6409	843	8	)	)	PUNCT
ejpam-6409	843	9	=	=	SYM
ejpam-6409	844	1	3	3	X
ejpam-6409	844	2	.	.	PUNCT
ejpam-6409	844	3	by	by	ADP
ejpam-6409	844	4	corollary	corollary	ADJ
ejpam-6409	844	5	6.1	6.1	NUM
ejpam-6409	844	6	,	,	PUNCT
ejpam-6409	844	7	k(h(2,0,0	k(h(2,0,0	NOUN
ejpam-6409	844	8	)	)	PUNCT
ejpam-6409	844	9	)	)	PUNCT
ejpam-6409	844	10	can	can	AUX
ejpam-6409	844	11	be	be	AUX
ejpam-6409	844	12	constructed	construct	VERB
ejpam-6409	844	13	from	from	ADP
ejpam-6409	844	14	a	a	DET
ejpam-6409	844	15	basis	basis	NOUN
ejpam-6409	844	16	.	.	PUNCT
ejpam-6409	845	1	to	to	PART
ejpam-6409	845	2	begin	begin	VERB
ejpam-6409	845	3	with	with	ADP
ejpam-6409	845	4	,	,	PUNCT
ejpam-6409	845	5	we	we	PRON
ejpam-6409	845	6	have	have	VERB
ejpam-6409	845	7	that	that	DET
ejpam-6409	845	8	q	q	NOUN
ejpam-6409	845	9	=	=	PUNCT
ejpam-6409	845	10	{	{	PUNCT
ejpam-6409	845	11	40	40	NUM
ejpam-6409	845	12	,	,	PUNCT
ejpam-6409	845	13	(	(	PUNCT
ejpam-6409	845	14	00	00	NUM
ejpam-6409	845	15	,	,	PUNCT
ejpam-6409	845	16	04	04	NUM
ejpam-6409	845	17	,	,	PUNCT
ejpam-6409	845	18	.	.	PUNCT
ejpam-6409	845	19	.	.	PUNCT
ejpam-6409	846	1	.	.	PUNCT
ejpam-6409	847	1	,	,	PUNCT
ejpam-6409	847	2	00	00	NUM
ejpam-6409	847	3	,	,	PUNCT
ejpam-6409	847	4	04	04	NUM
ejpam-6409	847	5	,	,	PUNCT
ejpam-6409	847	6	40	40	NUM
ejpam-6409	847	7	,	,	PUNCT
ejpam-6409	847	8	44	44	NUM
ejpam-6409	847	9	,	,	PUNCT
ejpam-6409	847	10	.	.	PUNCT
ejpam-6409	847	11	.	.	PUNCT
ejpam-6409	848	1	.	.	PUNCT
ejpam-6409	849	1	,	,	PUNCT
ejpam-6409	849	2	40	40	NUM
ejpam-6409	849	3	,	,	PUNCT
ejpam-6409	849	4	44	44	NUM
ejpam-6409	849	5	,	,	PUNCT
ejpam-6409	849	6	00	00	NUM
ejpam-6409	849	7	,	,	PUNCT
ejpam-6409	849	8	04	04	NUM
ejpam-6409	849	9	,	,	PUNCT
ejpam-6409	849	10	.	.	PUNCT
ejpam-6409	849	11	.	.	PUNCT
ejpam-6409	850	1	.	.	PUNCT
ejpam-6409	851	1	,	,	PUNCT
ejpam-6409	851	2	40	40	NUM
ejpam-6409	851	3	,	,	PUNCT
ejpam-6409	851	4	44	44	NUM
ejpam-6409	851	5	)	)	PUNCT
ejpam-6409	851	6	}	}	PUNCT
ejpam-6409	851	7	.	.	PUNCT
ejpam-6409	852	1	since	since	SCONJ
ejpam-6409	852	2	σ	σ	PROPN
ejpam-6409	852	3	=	=	SYM
ejpam-6409	852	4	1	1	NUM
ejpam-6409	852	5	,	,	PUNCT
ejpam-6409	852	6	in	in	ADP
ejpam-6409	852	7	this	this	DET
ejpam-6409	852	8	case	case	NOUN
ejpam-6409	852	9	b	b	X
ejpam-6409	852	10	=	=	X
ejpam-6409	852	11	∅.	∅.	NOUN
ejpam-6409	852	12	thus	thus	ADV
ejpam-6409	852	13	,	,	PUNCT
ejpam-6409	852	14	k(h(2,0,0	k(h(2,0,0	NOUN
ejpam-6409	852	15	)	)	PUNCT
ejpam-6409	852	16	)	)	PUNCT
ejpam-6409	852	17	=	=	SYM
ejpam-6409	852	18	⟨φs(40	⟨φs(40	NOUN
ejpam-6409	852	19	)	)	PUNCT
ejpam-6409	852	20	,	,	PUNCT
ejpam-6409	852	21	φs(00	φs(00	PROPN
ejpam-6409	852	22	,	,	PUNCT
ejpam-6409	852	23	04	04	NUM
ejpam-6409	852	24	,	,	PUNCT
ejpam-6409	852	25	.	.	PUNCT
ejpam-6409	852	26	.	.	PUNCT
ejpam-6409	852	27	.	.	PUNCT
ejpam-6409	853	1	,	,	PUNCT
ejpam-6409	853	2	00	00	NUM
ejpam-6409	853	3	,	,	PUNCT
ejpam-6409	853	4	04	04	NUM
ejpam-6409	853	5	,	,	PUNCT
ejpam-6409	853	6	40	40	NUM
ejpam-6409	853	7	,	,	PUNCT
ejpam-6409	853	8	44	44	NUM
ejpam-6409	853	9	,	,	PUNCT
ejpam-6409	853	10	.	.	PUNCT
ejpam-6409	853	11	.	.	PUNCT
ejpam-6409	854	1	.	.	PUNCT
ejpam-6409	855	1	,	,	PUNCT
ejpam-6409	855	2	40	40	NUM
ejpam-6409	855	3	,	,	PUNCT
ejpam-6409	855	4	44	44	NUM
ejpam-6409	855	5	,	,	PUNCT
ejpam-6409	855	6	00	00	NUM
ejpam-6409	855	7	,	,	PUNCT
ejpam-6409	855	8	04	04	NUM
ejpam-6409	855	9	,	,	PUNCT
ejpam-6409	855	10	.	.	PUNCT
ejpam-6409	855	11	.	.	PUNCT
ejpam-6409	856	1	.	.	PUNCT
ejpam-6409	857	1	,	,	PUNCT
ejpam-6409	857	2	40	40	NUM
ejpam-6409	857	3	,	,	PUNCT
ejpam-6409	857	4	44	44	NUM
ejpam-6409	857	5	)	)	PUNCT
ejpam-6409	857	6	,	,	PUNCT
ejpam-6409	857	7	φs(30)⟩.	φs(30)⟩.	VERB
ejpam-6409	857	8	7	7	NUM
ejpam-6409	857	9	.	.	X
ejpam-6409	857	10	classification	classification	NOUN
ejpam-6409	857	11	of	of	ADP
ejpam-6409	857	12	z2s	z2s	PROPN
ejpam-6409	858	1	[	[	X
ejpam-6409	858	2	ω]-linear	ω]-linear	ADJ
ejpam-6409	858	3	hadamard	hadamard	ADJ
ejpam-6409	858	4	codes	code	VERB
ejpam-6409	858	5	our	our	PRON
ejpam-6409	858	6	discussion	discussion	NOUN
ejpam-6409	858	7	here	here	ADV
ejpam-6409	858	8	is	be	AUX
ejpam-6409	858	9	aimed	aim	VERB
ejpam-6409	858	10	at	at	ADP
ejpam-6409	858	11	some	some	DET
ejpam-6409	858	12	aspects	aspect	NOUN
ejpam-6409	858	13	of	of	ADP
ejpam-6409	858	14	classifying	classify	VERB
ejpam-6409	858	15	z2s	z2s	PROPN
ejpam-6409	858	16	[	[	X
ejpam-6409	858	17	ω]-linear	ω]-linear	ADJ
ejpam-6409	858	18	codes	code	NOUN
ejpam-6409	858	19	for	for	ADP
ejpam-6409	858	20	length	length	NOUN
ejpam-6409	858	21	22	22	NUM
ejpam-6409	858	22	t	t	NOUN
ejpam-6409	858	23	for	for	ADP
ejpam-6409	858	24	t	t	PROPN
ejpam-6409	858	25	≥	≥	NUM
ejpam-6409	858	26	3	3	NUM
ejpam-6409	858	27	and	and	CCONJ
ejpam-6409	858	28	s	s	X
ejpam-6409	858	29	>	>	X
ejpam-6409	858	30	2	2	NUM
ejpam-6409	858	31	,	,	PUNCT
ejpam-6409	858	32	but	but	CCONJ
ejpam-6409	858	33	we	we	PRON
ejpam-6409	858	34	realize	realize	VERB
ejpam-6409	858	35	that	that	SCONJ
ejpam-6409	858	36	dimension	dimension	NOUN
ejpam-6409	858	37	of	of	ADP
ejpam-6409	858	38	the	the	DET
ejpam-6409	858	39	kernel	kernel	NOUN
ejpam-6409	858	40	alone	alone	ADV
ejpam-6409	858	41	is	be	AUX
ejpam-6409	858	42	not	not	PART
ejpam-6409	858	43	sufficient	sufficient	ADJ
ejpam-6409	858	44	for	for	ADP
ejpam-6409	858	45	a	a	DET
ejpam-6409	858	46	full	full	ADJ
ejpam-6409	858	47	classification	classification	NOUN
ejpam-6409	858	48	.	.	PUNCT
ejpam-6409	859	1	theorem	theorem	VERB
ejpam-6409	859	2	3	3	NUM
ejpam-6409	859	3	states	state	NOUN
ejpam-6409	859	4	that	that	SCONJ
ejpam-6409	859	5	for	for	ADP
ejpam-6409	859	6	any	any	DET
ejpam-6409	859	7	t	t	PROPN
ejpam-6409	859	8	≥	≥	NOUN
ejpam-6409	859	9	3	3	NUM
ejpam-6409	859	10	and	and	CCONJ
ejpam-6409	859	11	s	s	X
ejpam-6409	859	12	>	>	X
ejpam-6409	859	13	2	2	NUM
ejpam-6409	859	14	,	,	PUNCT
ejpam-6409	859	15	there	there	PRON
ejpam-6409	859	16	are	be	VERB
ejpam-6409	859	17	at	at	ADP
ejpam-6409	859	18	most	most	ADJ
ejpam-6409	859	19	two	two	NUM
ejpam-6409	859	20	z2s	z2	NOUN
ejpam-6409	860	1	[	[	X
ejpam-6409	860	2	ω]-linear	ω]-linear	ADJ
ejpam-6409	860	3	codes	code	NOUN
ejpam-6409	860	4	of	of	ADP
ejpam-6409	860	5	length	length	NOUN
ejpam-6409	860	6	22	22	NUM
ejpam-6409	860	7	t	t	PROPN
ejpam-6409	860	8	,	,	PUNCT
ejpam-6409	860	9	namely	namely	ADV
ejpam-6409	860	10	;	;	PUNCT
ejpam-6409	860	11	h(1,0,	h(1,0,	NOUN
ejpam-6409	860	12	...	...	PUNCT
ejpam-6409	860	13	,0,1,ts	,0,1,ts	PUNCT
ejpam-6409	860	14	)	)	PUNCT
ejpam-6409	860	15	and	and	CCONJ
ejpam-6409	860	16	h(1,0,	h(1,0,	NOUN
ejpam-6409	860	17	...	...	PUNCT
ejpam-6409	860	18	,0,ts	,0,ts	PROPN
ejpam-6409	860	19	)	)	PUNCT
ejpam-6409	860	20	which	which	PRON
ejpam-6409	860	21	are	be	AUX
ejpam-6409	860	22	linear	linear	ADJ
ejpam-6409	860	23	.	.	PUNCT
ejpam-6409	861	1	consequently	consequently	ADV
ejpam-6409	861	2	,	,	PUNCT
ejpam-6409	861	3	our	our	PRON
ejpam-6409	861	4	focus	focus	NOUN
ejpam-6409	861	5	can	can	AUX
ejpam-6409	861	6	be	be	AUX
ejpam-6409	861	7	then	then	ADV
ejpam-6409	861	8	placed	place	VERB
ejpam-6409	861	9	on	on	ADP
ejpam-6409	861	10	t	t	PROPN
ejpam-6409	861	11	≥	≥	NUM
ejpam-6409	861	12	5	5	NUM
ejpam-6409	861	13	and	and	CCONJ
ejpam-6409	861	14	2	2	NUM
ejpam-6409	861	15	≤	≤	NOUN
ejpam-6409	861	16	s	s	PART
ejpam-6409	861	17	≤	≤	NUM
ejpam-6409	861	18	t−	t−	PROPN
ejpam-6409	861	19	2	2	NUM
ejpam-6409	861	20	in	in	ADP
ejpam-6409	861	21	order	order	NOUN
ejpam-6409	861	22	to	to	PART
ejpam-6409	861	23	classify	classify	VERB
ejpam-6409	861	24	the	the	DET
ejpam-6409	861	25	nonlinear	nonlinear	ADJ
ejpam-6409	861	26	codes	code	NOUN
ejpam-6409	861	27	.	.	PUNCT
ejpam-6409	862	1	theorem	theorem	VERB
ejpam-6409	862	2	7.1	7.1	NUM
ejpam-6409	862	3	let	let	VERB
ejpam-6409	862	4	at	at	ADP
ejpam-6409	862	5	,	,	PUNCT
ejpam-6409	862	6	s	s	PART
ejpam-6409	862	7	denote	denote	NOUN
ejpam-6409	862	8	the	the	DET
ejpam-6409	862	9	number	number	NOUN
ejpam-6409	862	10	of	of	ADP
ejpam-6409	862	11	inequivalent	inequivalent	NOUN
ejpam-6409	862	12	z2s	z2s	PROPN
ejpam-6409	863	1	[	[	X
ejpam-6409	863	2	ω]-linear	ω]-linear	ADJ
ejpam-6409	863	3	hadamard	hadamard	ADJ
ejpam-6409	863	4	codes	code	NOUN
ejpam-6409	863	5	of	of	ADP
ejpam-6409	863	6	length	length	NOUN
ejpam-6409	863	7	22	22	NUM
ejpam-6409	863	8	t.	t.	NOUN
ejpam-6409	863	9	then	then	ADV
ejpam-6409	863	10	,	,	PUNCT
ejpam-6409	863	11	at	at	ADP
ejpam-6409	863	12	,	,	PUNCT
ejpam-6409	863	13	s	s	PART
ejpam-6409	863	14	=	=	PUNCT
ejpam-6409	863	15			PROPN
ejpam-6409	863	16	0	0	PUNCT
ejpam-6409	863	17	if	if	SCONJ
ejpam-6409	863	18	t	t	PROPN
ejpam-6409	863	19	≥	≥	NOUN
ejpam-6409	863	20	3	3	NUM
ejpam-6409	863	21	and	and	CCONJ
ejpam-6409	863	22	s	s	PRON
ejpam-6409	863	23	≥	≥	NOUN
ejpam-6409	863	24	t+	t+	PUNCT
ejpam-6409	863	25	2	2	NUM
ejpam-6409	863	26	,	,	PUNCT
ejpam-6409	863	27	1	1	NUM
ejpam-6409	863	28	if	if	SCONJ
ejpam-6409	863	29	t	t	PROPN
ejpam-6409	863	30	≥	≥	NOUN
ejpam-6409	863	31	3	3	NUM
ejpam-6409	863	32	and	and	CCONJ
ejpam-6409	863	33	s	s	PROPN
ejpam-6409	863	34	∈	∈	PROPN
ejpam-6409	863	35	{	{	PUNCT
ejpam-6409	863	36	t−	t−	PROPN
ejpam-6409	863	37	1	1	NUM
ejpam-6409	863	38	,	,	PUNCT
ejpam-6409	863	39	t	t	PROPN
ejpam-6409	863	40	,	,	PUNCT
ejpam-6409	863	41	t+	t+	VERB
ejpam-6409	863	42	1	1	NUM
ejpam-6409	863	43	}	}	PUNCT
ejpam-6409	863	44	,	,	PUNCT
ejpam-6409	863	45	1	1	NUM
ejpam-6409	863	46	if	if	SCONJ
ejpam-6409	863	47	t	t	NOUN
ejpam-6409	863	48	=	=	SYM
ejpam-6409	863	49	4	4	NUM
ejpam-6409	863	50	and	and	CCONJ
ejpam-6409	863	51	s	s	X
ejpam-6409	863	52	=	=	SYM
ejpam-6409	863	53	2	2	NUM
ejpam-6409	863	54	,	,	PUNCT
ejpam-6409	863	55	muhammad	muhammad	PROPN
ejpam-6409	863	56	sajjad	sajjad	PROPN
ejpam-6409	863	57	et	et	PROPN
ejpam-6409	863	58	al	al	PROPN
ejpam-6409	863	59	.	.	PUNCT
ejpam-6409	863	60	/	/	SYM
ejpam-6409	863	61	eur	eur	PROPN
ejpam-6409	863	62	.	.	PUNCT
ejpam-6409	864	1	j.	j.	PROPN
ejpam-6409	864	2	pure	pure	PROPN
ejpam-6409	864	3	appl	appl	PROPN
ejpam-6409	864	4	.	.	PROPN
ejpam-6409	864	5	math	math	PROPN
ejpam-6409	864	6	,	,	PUNCT
ejpam-6409	864	7	18	18	NUM
ejpam-6409	864	8	(	(	PUNCT
ejpam-6409	864	9	3	3	NUM
ejpam-6409	864	10	)	)	PUNCT
ejpam-6409	864	11	(	(	PUNCT
ejpam-6409	864	12	2025	2025	NUM
ejpam-6409	864	13	)	)	PUNCT
ejpam-6409	864	14	,	,	PUNCT
ejpam-6409	864	15	6409	6409	NUM
ejpam-6409	864	16	26	26	NUM
ejpam-6409	864	17	of	of	ADP
ejpam-6409	864	18	32	32	NUM
ejpam-6409	864	19	and	and	CCONJ
ejpam-6409	864	20	the	the	DET
ejpam-6409	864	21	z2s	z2s	PROPN
ejpam-6409	865	1	[	[	X
ejpam-6409	865	2	ω]-linear	ω]-linear	ADJ
ejpam-6409	865	3	hadamard	hadamard	PROPN
ejpam-6409	865	4	code	code	NOUN
ejpam-6409	865	5	is	be	AUX
ejpam-6409	865	6	linear	linear	ADJ
ejpam-6409	865	7	when	when	SCONJ
ejpam-6409	865	8	at	at	ADP
ejpam-6409	865	9	,	,	PUNCT
ejpam-6409	865	10	s	s	NOUN
ejpam-6409	865	11	=	=	NOUN
ejpam-6409	865	12	1	1	X
ejpam-6409	865	13	.	.	PUNCT
ejpam-6409	866	1	moreover	moreover	ADV
ejpam-6409	866	2	,	,	PUNCT
ejpam-6409	866	3	if	if	SCONJ
ejpam-6409	866	4	t	t	PROPN
ejpam-6409	866	5	≥	≥	NUM
ejpam-6409	866	6	5	5	NUM
ejpam-6409	866	7	and	and	CCONJ
ejpam-6409	866	8	2	2	NUM
ejpam-6409	866	9	≤	≤	NUM
ejpam-6409	866	10	s	s	PART
ejpam-6409	866	11	≤	≤	NOUN
ejpam-6409	866	12	t	t	NOUN
ejpam-6409	866	13	−	−	PROPN
ejpam-6409	866	14	2	2	NUM
ejpam-6409	866	15	,	,	PUNCT
ejpam-6409	866	16	then	then	ADV
ejpam-6409	866	17	at	at	ADP
ejpam-6409	866	18	,	,	PUNCT
ejpam-6409	866	19	s	s	PART
ejpam-6409	866	20	≥	≥	NOUN
ejpam-6409	866	21	2	2	NUM
ejpam-6409	866	22	,	,	PUNCT
ejpam-6409	866	23	and	and	CCONJ
ejpam-6409	866	24	there	there	PRON
ejpam-6409	866	25	exist	exist	VERB
ejpam-6409	866	26	one	one	NUM
ejpam-6409	866	27	linear	linear	ADJ
ejpam-6409	866	28	code	code	NOUN
ejpam-6409	866	29	and	and	CCONJ
ejpam-6409	866	30	at	at	ADV
ejpam-6409	866	31	least	least	ADV
ejpam-6409	866	32	one	one	NUM
ejpam-6409	866	33	nonlinear	nonlinear	ADJ
ejpam-6409	866	34	code	code	NOUN
ejpam-6409	866	35	.	.	PUNCT
ejpam-6409	867	1	proof	proof	NOUN
ejpam-6409	867	2	:	:	PUNCT
ejpam-6409	867	3	if	if	SCONJ
ejpam-6409	867	4	t	t	PROPN
ejpam-6409	867	5	≥	≥	NOUN
ejpam-6409	867	6	3	3	NUM
ejpam-6409	867	7	and	and	CCONJ
ejpam-6409	867	8	s	s	PRON
ejpam-6409	867	9	≥	≥	NOUN
ejpam-6409	867	10	t+	t+	PUNCT
ejpam-6409	867	11	2	2	NUM
ejpam-6409	867	12	,	,	PUNCT
ejpam-6409	867	13	then	then	ADV
ejpam-6409	867	14	for	for	ADP
ejpam-6409	867	15	the	the	DET
ejpam-6409	867	16	equation	equation	NOUN
ejpam-6409	867	17	t	t	NOUN
ejpam-6409	867	18	=	=	PUNCT
ejpam-6409	867	19	(	(	PUNCT
ejpam-6409	867	20	s∑	s∑	PROPN
ejpam-6409	867	21	i=1	i=1	PROPN
ejpam-6409	867	22	(	(	PUNCT
ejpam-6409	867	23	s−	s−	PROPN
ejpam-6409	867	24	i+	i+	PUNCT
ejpam-6409	867	25	1)ti	1)ti	PROPN
ejpam-6409	867	26	)	)	PUNCT
ejpam-6409	868	1	−	−	PROPN
ejpam-6409	868	2	1	1	NUM
ejpam-6409	868	3	,	,	PUNCT
ejpam-6409	868	4	with	with	ADP
ejpam-6409	868	5	t1	t1	PROPN
ejpam-6409	868	6	≥	≥	NUM
ejpam-6409	868	7	1	1	NUM
ejpam-6409	868	8	,	,	PUNCT
ejpam-6409	868	9	there	there	PRON
ejpam-6409	868	10	are	be	VERB
ejpam-6409	868	11	no	no	DET
ejpam-6409	868	12	nonnegative	nonnegative	ADJ
ejpam-6409	868	13	integer	integer	NOUN
ejpam-6409	868	14	solutions	solution	NOUN
ejpam-6409	868	15	,	,	PUNCT
ejpam-6409	868	16	so	so	SCONJ
ejpam-6409	868	17	at	at	ADP
ejpam-6409	868	18	,	,	PUNCT
ejpam-6409	868	19	s	s	NOUN
ejpam-6409	868	20	=	=	NOUN
ejpam-6409	868	21	0	0	PROPN
ejpam-6409	868	22	.	.	PUNCT
ejpam-6409	869	1	if	if	SCONJ
ejpam-6409	869	2	t	t	PROPN
ejpam-6409	869	3	≥	≥	NOUN
ejpam-6409	869	4	3	3	NUM
ejpam-6409	869	5	and	and	CCONJ
ejpam-6409	869	6	s	s	NOUN
ejpam-6409	869	7	=	=	X
ejpam-6409	869	8	t	t	PROPN
ejpam-6409	869	9	+	+	NOUN
ejpam-6409	869	10	1	1	NUM
ejpam-6409	869	11	,	,	PUNCT
ejpam-6409	869	12	there	there	PRON
ejpam-6409	869	13	is	be	VERB
ejpam-6409	869	14	only	only	ADV
ejpam-6409	869	15	one	one	NUM
ejpam-6409	869	16	solution	solution	NOUN
ejpam-6409	869	17	(	(	PUNCT
ejpam-6409	869	18	t1	t1	NOUN
ejpam-6409	869	19	,	,	PUNCT
ejpam-6409	869	20	.	.	PUNCT
ejpam-6409	869	21	.	.	PUNCT
ejpam-6409	870	1	.	.	PUNCT
ejpam-6409	871	1	,	,	PUNCT
ejpam-6409	871	2	ts	ts	NOUN
ejpam-6409	871	3	)	)	PUNCT
ejpam-6409	871	4	=	=	SYM
ejpam-6409	871	5	(	(	PUNCT
ejpam-6409	871	6	1	1	NUM
ejpam-6409	871	7	,	,	PUNCT
ejpam-6409	871	8	0	0	NUM
ejpam-6409	871	9	,	,	PUNCT
ejpam-6409	871	10	.	.	PUNCT
ejpam-6409	871	11	.	.	PUNCT
ejpam-6409	872	1	.	.	PUNCT
ejpam-6409	873	1	,	,	PUNCT
ejpam-6409	873	2	0	0	NUM
ejpam-6409	873	3	)	)	PUNCT
ejpam-6409	873	4	.	.	PUNCT
ejpam-6409	874	1	if	if	SCONJ
ejpam-6409	874	2	t	t	PROPN
ejpam-6409	874	3	≥	≥	NOUN
ejpam-6409	874	4	3	3	NUM
ejpam-6409	874	5	and	and	CCONJ
ejpam-6409	874	6	s	s	PROPN
ejpam-6409	874	7	=	=	SYM
ejpam-6409	874	8	t	t	PROPN
ejpam-6409	874	9	,	,	PUNCT
ejpam-6409	874	10	there	there	PRON
ejpam-6409	874	11	is	be	VERB
ejpam-6409	874	12	exactly	exactly	ADV
ejpam-6409	874	13	one	one	NUM
ejpam-6409	874	14	solution	solution	NOUN
ejpam-6409	874	15	(	(	PUNCT
ejpam-6409	874	16	1	1	NUM
ejpam-6409	874	17	,	,	PUNCT
ejpam-6409	874	18	0	0	NUM
ejpam-6409	874	19	,	,	PUNCT
ejpam-6409	874	20	.	.	PUNCT
ejpam-6409	874	21	.	.	PUNCT
ejpam-6409	875	1	.	.	PUNCT
ejpam-6409	876	1	,	,	PUNCT
ejpam-6409	876	2	0	0	NUM
ejpam-6409	876	3	,	,	PUNCT
ejpam-6409	876	4	1	1	NUM
ejpam-6409	876	5	)	)	PUNCT
ejpam-6409	876	6	.	.	PUNCT
ejpam-6409	877	1	if	if	SCONJ
ejpam-6409	877	2	t	t	PROPN
ejpam-6409	877	3	≥	≥	NOUN
ejpam-6409	877	4	3	3	NUM
ejpam-6409	877	5	and	and	CCONJ
ejpam-6409	877	6	s	s	NOUN
ejpam-6409	877	7	=	=	SYM
ejpam-6409	877	8	t	t	PROPN
ejpam-6409	877	9	−	−	NOUN
ejpam-6409	877	10	1	1	NUM
ejpam-6409	877	11	,	,	PUNCT
ejpam-6409	877	12	there	there	PRON
ejpam-6409	877	13	are	be	VERB
ejpam-6409	877	14	two	two	NUM
ejpam-6409	877	15	solutions	solution	NOUN
ejpam-6409	877	16	(	(	PUNCT
ejpam-6409	877	17	1	1	NUM
ejpam-6409	877	18	,	,	PUNCT
ejpam-6409	877	19	0	0	NUM
ejpam-6409	877	20	,	,	PUNCT
ejpam-6409	877	21	.	.	PUNCT
ejpam-6409	877	22	.	.	PUNCT
ejpam-6409	878	1	.	.	PUNCT
ejpam-6409	879	1	,	,	PUNCT
ejpam-6409	879	2	0	0	NUM
ejpam-6409	879	3	,	,	PUNCT
ejpam-6409	879	4	2	2	NUM
ejpam-6409	879	5	)	)	PUNCT
ejpam-6409	879	6	and	and	CCONJ
ejpam-6409	879	7	(	(	PUNCT
ejpam-6409	879	8	1	1	NUM
ejpam-6409	879	9	,	,	PUNCT
ejpam-6409	879	10	0	0	NUM
ejpam-6409	879	11	,	,	PUNCT
ejpam-6409	879	12	.	.	PUNCT
ejpam-6409	879	13	.	.	PUNCT
ejpam-6409	880	1	.	.	PUNCT
ejpam-6409	881	1	,	,	PUNCT
ejpam-6409	881	2	0	0	NUM
ejpam-6409	881	3	,	,	PUNCT
ejpam-6409	881	4	1	1	NUM
ejpam-6409	881	5	,	,	PUNCT
ejpam-6409	881	6	0	0	NUM
ejpam-6409	881	7	)	)	PUNCT
ejpam-6409	881	8	.	.	PUNCT
ejpam-6409	882	1	notably	notably	ADV
ejpam-6409	882	2	,	,	PUNCT
ejpam-6409	882	3	when	when	SCONJ
ejpam-6409	882	4	t	t	PROPN
ejpam-6409	882	5	=	=	SYM
ejpam-6409	882	6	3	3	NUM
ejpam-6409	882	7	and	and	CCONJ
ejpam-6409	882	8	s	s	NOUN
ejpam-6409	882	9	=	=	SYM
ejpam-6409	882	10	2	2	NUM
ejpam-6409	882	11	,	,	PUNCT
ejpam-6409	882	12	both	both	DET
ejpam-6409	882	13	solutions	solution	NOUN
ejpam-6409	882	14	are	be	AUX
ejpam-6409	882	15	(	(	PUNCT
ejpam-6409	882	16	1	1	NUM
ejpam-6409	882	17	,	,	PUNCT
ejpam-6409	882	18	2	2	NUM
ejpam-6409	882	19	)	)	PUNCT
ejpam-6409	882	20	and	and	CCONJ
ejpam-6409	882	21	(	(	PUNCT
ejpam-6409	882	22	2	2	NUM
ejpam-6409	882	23	,	,	PUNCT
ejpam-6409	882	24	0	0	NUM
ejpam-6409	882	25	)	)	PUNCT
ejpam-6409	882	26	.	.	PUNCT
ejpam-6409	883	1	by	by	ADP
ejpam-6409	883	2	theorem	theorem	NOUN
ejpam-6409	883	3	3.3	3.3	NUM
ejpam-6409	883	4	,	,	PUNCT
ejpam-6409	883	5	for	for	ADP
ejpam-6409	883	6	all	all	DET
ejpam-6409	883	7	the	the	DET
ejpam-6409	883	8	above	above	ADJ
ejpam-6409	883	9	solutions	solution	NOUN
ejpam-6409	883	10	,	,	PUNCT
ejpam-6409	883	11	we	we	PRON
ejpam-6409	883	12	obtain	obtain	VERB
ejpam-6409	883	13	a	a	DET
ejpam-6409	883	14	linear	linear	ADJ
ejpam-6409	883	15	code	code	NOUN
ejpam-6409	883	16	h(t1,	h(t1,	NOUN
ejpam-6409	883	17	...	...	PUNCT
ejpam-6409	883	18	,ts	,ts	NUM
ejpam-6409	883	19	)	)	PUNCT
ejpam-6409	883	20	.	.	PUNCT
ejpam-6409	884	1	finally	finally	ADV
ejpam-6409	884	2	,	,	PUNCT
ejpam-6409	884	3	if	if	SCONJ
ejpam-6409	884	4	t	t	PROPN
ejpam-6409	884	5	≥	≥	NUM
ejpam-6409	884	6	5	5	NUM
ejpam-6409	884	7	and	and	CCONJ
ejpam-6409	884	8	2	2	NUM
ejpam-6409	884	9	≤	≤	NOUN
ejpam-6409	884	10	s	s	PART
ejpam-6409	884	11	≤	≤	PROPN
ejpam-6409	884	12	t−2	t−2	PROPN
ejpam-6409	884	13	,	,	PUNCT
ejpam-6409	884	14	the	the	DET
ejpam-6409	884	15	solutions	solution	NOUN
ejpam-6409	884	16	(	(	PUNCT
ejpam-6409	884	17	1	1	NUM
ejpam-6409	884	18	,	,	PUNCT
ejpam-6409	884	19	0	0	NUM
ejpam-6409	884	20	,	,	PUNCT
ejpam-6409	884	21	.	.	PUNCT
ejpam-6409	884	22	.	.	PUNCT
ejpam-6409	884	23	.	.	PUNCT
ejpam-6409	885	1	,	,	PUNCT
ejpam-6409	885	2	0	0	NUM
ejpam-6409	885	3	,	,	PUNCT
ejpam-6409	885	4	t−s+1	t−s+1	PROPN
ejpam-6409	885	5	)	)	PUNCT
ejpam-6409	885	6	and	and	CCONJ
ejpam-6409	885	7	(	(	PUNCT
ejpam-6409	885	8	1	1	NUM
ejpam-6409	885	9	,	,	PUNCT
ejpam-6409	885	10	0	0	NUM
ejpam-6409	885	11	,	,	PUNCT
ejpam-6409	885	12	.	.	PUNCT
ejpam-6409	885	13	.	.	PUNCT
ejpam-6409	886	1	.	.	PUNCT
ejpam-6409	887	1	,	,	PUNCT
ejpam-6409	887	2	0	0	NUM
ejpam-6409	887	3	,	,	PUNCT
ejpam-6409	887	4	1	1	NUM
ejpam-6409	887	5	,	,	PUNCT
ejpam-6409	887	6	t−	t−	PRON
ejpam-6409	887	7	s−	s−	PROPN
ejpam-6409	887	8	1	1	NUM
ejpam-6409	887	9	)	)	PUNCT
ejpam-6409	887	10	always	always	ADV
ejpam-6409	887	11	exist	exist	VERB
ejpam-6409	887	12	,	,	PUNCT
ejpam-6409	887	13	yielding	yield	VERB
ejpam-6409	887	14	a	a	DET
ejpam-6409	887	15	linear	linear	ADJ
ejpam-6409	887	16	code	code	NOUN
ejpam-6409	887	17	.	.	PUNCT
ejpam-6409	888	1	in	in	ADP
ejpam-6409	888	2	these	these	DET
ejpam-6409	888	3	cases	case	NOUN
ejpam-6409	888	4	,	,	PUNCT
ejpam-6409	888	5	there	there	PRON
ejpam-6409	888	6	is	be	VERB
ejpam-6409	888	7	at	at	ADV
ejpam-6409	888	8	least	least	ADJ
ejpam-6409	888	9	one	one	NUM
ejpam-6409	888	10	additional	additional	ADJ
ejpam-6409	888	11	solution	solution	NOUN
ejpam-6409	888	12	.	.	PUNCT
ejpam-6409	889	1	if	if	SCONJ
ejpam-6409	889	2	s	s	PRON
ejpam-6409	889	3	=	=	NOUN
ejpam-6409	889	4	2	2	NUM
ejpam-6409	889	5	,	,	PUNCT
ejpam-6409	889	6	at	at	ADP
ejpam-6409	889	7	,	,	PUNCT
ejpam-6409	889	8	s	s	PART
ejpam-6409	889	9	=	=	PUNCT
ejpam-6409	889	10	⌊	⌊	VERB
ejpam-6409	889	11	t−	t−	ADP
ejpam-6409	889	12	1	1	NUM
ejpam-6409	889	13	2	2	NUM
ejpam-6409	889	14	⌋	⌋	NOUN
ejpam-6409	889	15	≥	≥	NOUN
ejpam-6409	889	16	2	2	NUM
ejpam-6409	889	17	since	since	SCONJ
ejpam-6409	889	18	t	t	PROPN
ejpam-6409	889	19	≥	≥	NUM
ejpam-6409	889	20	5	5	NUM
ejpam-6409	889	21	.	.	PUNCT
ejpam-6409	890	1	on	on	ADP
ejpam-6409	890	2	the	the	DET
ejpam-6409	890	3	other	other	ADJ
ejpam-6409	890	4	hand	hand	NOUN
ejpam-6409	890	5	,	,	PUNCT
ejpam-6409	890	6	if	if	SCONJ
ejpam-6409	890	7	s	s	PART
ejpam-6409	890	8	=	=	SYM
ejpam-6409	890	9	3	3	NUM
ejpam-6409	890	10	,	,	PUNCT
ejpam-6409	890	11	(	(	PUNCT
ejpam-6409	890	12	2	2	NUM
ejpam-6409	890	13	,	,	PUNCT
ejpam-6409	890	14	0	0	NUM
ejpam-6409	890	15	,	,	PUNCT
ejpam-6409	890	16	.	.	PUNCT
ejpam-6409	890	17	.	.	PUNCT
ejpam-6409	891	1	.	.	PUNCT
ejpam-6409	892	1	,	,	PUNCT
ejpam-6409	892	2	0	0	NUM
ejpam-6409	892	3	,	,	PUNCT
ejpam-6409	892	4	t−	t−	PROPN
ejpam-6409	892	5	2s+	2s+	NUM
ejpam-6409	892	6	1	1	NUM
ejpam-6409	892	7	)	)	PUNCT
ejpam-6409	892	8	is	be	AUX
ejpam-6409	892	9	a	a	DET
ejpam-6409	892	10	solution	solution	NOUN
ejpam-6409	892	11	because	because	SCONJ
ejpam-6409	892	12	t	t	PROPN
ejpam-6409	892	13	≥	≥	NUM
ejpam-6409	892	14	2s−	2s−	NUM
ejpam-6409	892	15	1	1	NUM
ejpam-6409	892	16	when	when	SCONJ
ejpam-6409	892	17	t	t	PROPN
ejpam-6409	892	18	≥	≥	NUM
ejpam-6409	892	19	5	5	NUM
ejpam-6409	892	20	;	;	PUNCT
ejpam-6409	892	21	and	and	CCONJ
ejpam-6409	892	22	if	if	SCONJ
ejpam-6409	892	23	s	s	X
ejpam-6409	892	24	≥	≥	NOUN
ejpam-6409	892	25	4	4	NUM
ejpam-6409	892	26	,	,	PUNCT
ejpam-6409	892	27	(	(	PUNCT
ejpam-6409	892	28	1	1	NUM
ejpam-6409	892	29	,	,	PUNCT
ejpam-6409	892	30	0	0	NUM
ejpam-6409	892	31	,	,	PUNCT
ejpam-6409	892	32	.	.	PUNCT
ejpam-6409	892	33	.	.	PUNCT
ejpam-6409	893	1	.	.	PUNCT
ejpam-6409	894	1	,	,	PUNCT
ejpam-6409	894	2	0	0	NUM
ejpam-6409	894	3	,	,	PUNCT
ejpam-6409	894	4	1	1	NUM
ejpam-6409	894	5	,	,	PUNCT
ejpam-6409	894	6	0	0	NUM
ejpam-6409	894	7	,	,	PUNCT
ejpam-6409	894	8	t	t	NOUN
ejpam-6409	894	9	−	−	PROPN
ejpam-6409	894	10	s	s	PART
ejpam-6409	894	11	−	−	PROPN
ejpam-6409	894	12	2	2	NUM
ejpam-6409	894	13	)	)	PUNCT
ejpam-6409	894	14	is	be	AUX
ejpam-6409	894	15	a	a	DET
ejpam-6409	894	16	solution	solution	NOUN
ejpam-6409	894	17	.	.	PUNCT
ejpam-6409	895	1	therefore	therefore	ADV
ejpam-6409	895	2	,	,	PUNCT
ejpam-6409	895	3	for	for	ADP
ejpam-6409	895	4	all	all	DET
ejpam-6409	895	5	the	the	DET
ejpam-6409	895	6	cases	case	NOUN
ejpam-6409	895	7	,	,	PUNCT
ejpam-6409	895	8	at	at	ADP
ejpam-6409	895	9	,	,	PUNCT
ejpam-6409	895	10	s	s	PART
ejpam-6409	895	11	≥	≥	NOUN
ejpam-6409	895	12	2	2	NUM
ejpam-6409	895	13	by	by	ADP
ejpam-6409	895	14	theorem	theorem	NOUN
ejpam-6409	895	15	5.2	5.2	NUM
ejpam-6409	895	16	.	.	PUNCT
ejpam-6409	895	17	example	example	NOUN
ejpam-6409	895	18	7.1	7.1	NUM
ejpam-6409	895	19	:	:	PUNCT
ejpam-6409	895	20	the	the	DET
ejpam-6409	895	21	z8[ω]-linear	z8[ω]-linear	NUM
ejpam-6409	895	22	hadamard	hadamard	ADJ
ejpam-6409	895	23	codes	code	NOUN
ejpam-6409	895	24	of	of	ADP
ejpam-6409	895	25	length	length	NOUN
ejpam-6409	895	26	22	22	NUM
ejpam-6409	895	27	t	t	NOUN
ejpam-6409	895	28	=	=	SYM
ejpam-6409	895	29	65536	65536	NUM
ejpam-6409	895	30	,	,	PUNCT
ejpam-6409	895	31	listed	list	VERB
ejpam-6409	895	32	below	below	ADV
ejpam-6409	895	33	,	,	PUNCT
ejpam-6409	895	34	are	be	AUX
ejpam-6409	895	35	:	:	PUNCT
ejpam-6409	895	36	h(1,0,6	h(1,0,6	NUM
ejpam-6409	895	37	)	)	PUNCT
ejpam-6409	895	38	,	,	PUNCT
ejpam-6409	895	39	h(1,1,4	h(1,1,4	PROPN
ejpam-6409	895	40	)	)	PUNCT
ejpam-6409	895	41	,	,	PUNCT
ejpam-6409	895	42	h(1,2,2	h(1,2,2	PROPN
ejpam-6409	895	43	)	)	PUNCT
ejpam-6409	895	44	,	,	PUNCT
ejpam-6409	895	45	h(1,3,0	h(1,3,0	PROPN
ejpam-6409	895	46	)	)	PUNCT
ejpam-6409	895	47	,	,	PUNCT
ejpam-6409	895	48	h(2,0,3	h(2,0,3	NUM
ejpam-6409	895	49	)	)	PUNCT
ejpam-6409	895	50	,	,	PUNCT
ejpam-6409	895	51	h(2,1,1	h(2,1,1	NUM
ejpam-6409	895	52	)	)	PUNCT
ejpam-6409	895	53	,	,	PUNCT
ejpam-6409	895	54	and	and	CCONJ
ejpam-6409	895	55	h(2,1,1	h(2,1,1	NUM
ejpam-6409	895	56	)	)	PUNCT
ejpam-6409	895	57	.	.	PUNCT
ejpam-6409	896	1	both	both	PRON
ejpam-6409	896	2	of	of	ADP
ejpam-6409	896	3	the	the	DET
ejpam-6409	896	4	first	first	ADJ
ejpam-6409	896	5	two	two	NUM
ejpam-6409	896	6	are	be	AUX
ejpam-6409	896	7	equivalent	equivalent	ADJ
ejpam-6409	896	8	because	because	SCONJ
ejpam-6409	896	9	they	they	PRON
ejpam-6409	896	10	are	be	AUX
ejpam-6409	896	11	linear	linear	PROPN
ejpam-6409	896	12	codes	code	NOUN
ejpam-6409	896	13	by	by	ADP
ejpam-6409	896	14	theorem	theorem	NOUN
ejpam-6409	896	15	3	3	NUM
ejpam-6409	896	16	.	.	PUNCT
ejpam-6409	896	17	according	accord	VERB
ejpam-6409	896	18	to	to	ADP
ejpam-6409	896	19	theorem	theorem	ADJ
ejpam-6409	896	20	4	4	NUM
ejpam-6409	896	21	,	,	PUNCT
ejpam-6409	896	22	the	the	DET
ejpam-6409	896	23	other	other	ADJ
ejpam-6409	896	24	codes	code	NOUN
ejpam-6409	896	25	have	have	VERB
ejpam-6409	896	26	kernel	kernel	NOUN
ejpam-6409	896	27	dimensions	dimension	NOUN
ejpam-6409	896	28	of	of	ADP
ejpam-6409	896	29	7	7	NUM
ejpam-6409	896	30	,	,	PUNCT
ejpam-6409	896	31	6	6	NUM
ejpam-6409	896	32	,	,	PUNCT
ejpam-6409	896	33	6	6	NUM
ejpam-6409	896	34	,	,	PUNCT
ejpam-6409	896	35	5	5	NUM
ejpam-6409	896	36	,	,	PUNCT
ejpam-6409	896	37	and	and	CCONJ
ejpam-6409	896	38	4	4	X
ejpam-6409	896	39	.	.	X
ejpam-6409	897	1	therefore	therefore	ADV
ejpam-6409	897	2	,	,	PUNCT
ejpam-6409	897	3	from	from	ADP
ejpam-6409	897	4	this	this	DET
ejpam-6409	897	5	invariant	invariant	NOUN
ejpam-6409	897	6	,	,	PUNCT
ejpam-6409	897	7	we	we	PRON
ejpam-6409	897	8	conclude	conclude	VERB
ejpam-6409	897	9	that	that	SCONJ
ejpam-6409	897	10	all	all	DET
ejpam-6409	897	11	these	these	DET
ejpam-6409	897	12	codes	code	NOUN
ejpam-6409	897	13	are	be	AUX
ejpam-6409	897	14	different	different	ADJ
ejpam-6409	897	15	except	except	SCONJ
ejpam-6409	897	16	h(1,3,0	h(1,3,0	PROPN
ejpam-6409	897	17	)	)	PUNCT
ejpam-6409	897	18	and	and	CCONJ
ejpam-6409	897	19	h(2,0,3	h(2,0,3	NUM
ejpam-6409	897	20	)	)	PUNCT
ejpam-6409	897	21	that	that	PRON
ejpam-6409	897	22	share	share	VERB
ejpam-6409	897	23	identical	identical	ADJ
ejpam-6409	897	24	kernel	kernel	NOUN
ejpam-6409	897	25	dimensions	dimension	NOUN
ejpam-6409	897	26	.	.	PUNCT
ejpam-6409	898	1	we	we	PRON
ejpam-6409	898	2	have	have	AUX
ejpam-6409	898	3	observed	observe	VERB
ejpam-6409	898	4	in	in	ADP
ejpam-6409	898	5	some	some	DET
ejpam-6409	898	6	instances	instance	NOUN
ejpam-6409	898	7	that	that	PRON
ejpam-6409	898	8	codes	code	NOUN
ejpam-6409	898	9	defined	define	VERB
ejpam-6409	898	10	over	over	ADP
ejpam-6409	898	11	z2s	z2	NOUN
ejpam-6409	898	12	and	and	CCONJ
ejpam-6409	898	13	codes	code	NOUN
ejpam-6409	898	14	over	over	ADP
ejpam-6409	898	15	z2s	z2s	PROPN
ejpam-6409	898	16	[	[	X
ejpam-6409	898	17	ω	ω	X
ejpam-6409	898	18	]	]	X
ejpam-6409	898	19	have	have	VERB
ejpam-6409	898	20	the	the	DET
ejpam-6409	898	21	same	same	ADJ
ejpam-6409	898	22	rank	rank	NOUN
ejpam-6409	898	23	and	and	CCONJ
ejpam-6409	898	24	kernel	kernel	PROPN
ejpam-6409	898	25	dimension	dimension	NOUN
ejpam-6409	898	26	,	,	PUNCT
ejpam-6409	898	27	meaning	mean	VERB
ejpam-6409	898	28	that	that	SCONJ
ejpam-6409	898	29	rank(h(1,3,0	rank(h(1,3,0	VERB
ejpam-6409	898	30	)	)	PUNCT
ejpam-6409	898	31	)	)	PUNCT
ejpam-6409	899	1	=	=	SYM
ejpam-6409	899	2	12	12	NUM
ejpam-6409	899	3	,	,	PUNCT
ejpam-6409	899	4	and	and	CCONJ
ejpam-6409	899	5	rank(h(2,0,3	rank(h(2,0,3	NOUN
ejpam-6409	899	6	)	)	PUNCT
ejpam-6409	899	7	)	)	PUNCT
ejpam-6409	900	1	=	=	PUNCT
ejpam-6409	900	2	11	11	NUM
ejpam-6409	900	3	,	,	PUNCT
ejpam-6409	900	4	which	which	PRON
ejpam-6409	900	5	gives	give	VERB
ejpam-6409	900	6	their	their	PRON
ejpam-6409	900	7	non	non	ADJ
ejpam-6409	900	8	-	-	NOUN
ejpam-6409	900	9	equivalence	equivalence	NOUN
ejpam-6409	900	10	.	.	PUNCT
ejpam-6409	901	1	as	as	ADP
ejpam-6409	901	2	a	a	DET
ejpam-6409	901	3	result	result	NOUN
ejpam-6409	901	4	,	,	PUNCT
ejpam-6409	901	5	in	in	ADP
ejpam-6409	901	6	the	the	DET
ejpam-6409	901	7	case	case	NOUN
ejpam-6409	901	8	of	of	ADP
ejpam-6409	901	9	the	the	DET
ejpam-6409	901	10	z2s	z2s	PROPN
ejpam-6409	901	11	[	[	X
ejpam-6409	901	12	ω]-linear	ω]-linear	ADJ
ejpam-6409	901	13	hadamard	hadamard	ADJ
ejpam-6409	901	14	codes	code	NOUN
ejpam-6409	901	15	of	of	ADP
ejpam-6409	901	16	length	length	NOUN
ejpam-6409	901	17	22	22	NUM
ejpam-6409	901	18	t	t	NOUN
ejpam-6409	901	19	=	=	SYM
ejpam-6409	901	20	65536	65536	NUM
ejpam-6409	901	21	,	,	PUNCT
ejpam-6409	901	22	the	the	DET
ejpam-6409	901	23	rank	rank	NOUN
ejpam-6409	901	24	gives	give	VERB
ejpam-6409	901	25	a	a	DET
ejpam-6409	901	26	complete	complete	ADJ
ejpam-6409	901	27	classification	classification	NOUN
ejpam-6409	901	28	that	that	PRON
ejpam-6409	901	29	does	do	AUX
ejpam-6409	901	30	not	not	PART
ejpam-6409	901	31	require	require	VERB
ejpam-6409	901	32	consideration	consideration	NOUN
ejpam-6409	901	33	of	of	ADP
ejpam-6409	901	34	the	the	DET
ejpam-6409	901	35	kernel	kernel	NOUN
ejpam-6409	901	36	.	.	PUNCT
ejpam-6409	901	37	example	example	NOUN
ejpam-6409	901	38	7.2	7.2	NUM
ejpam-6409	901	39	:	:	PUNCT
ejpam-6409	901	40	theorem	theorem	VERB
ejpam-6409	901	41	6.1	6.1	NUM
ejpam-6409	901	42	verifies	verifie	NOUN
ejpam-6409	901	43	that	that	PRON
ejpam-6409	901	44	for	for	ADP
ejpam-6409	901	45	all	all	DET
ejpam-6409	901	46	5	5	NUM
ejpam-6409	901	47	≤	≤	NUM
ejpam-6409	901	48	t	t	NOUN
ejpam-6409	901	49	≤	≤	NOUN
ejpam-6409	901	50	7	7	NUM
ejpam-6409	901	51	and	and	CCONJ
ejpam-6409	901	52	2	2	NUM
ejpam-6409	901	53	≤	≤	NOUN
ejpam-6409	901	54	s	s	PART
ejpam-6409	901	55	≤	≤	NOUN
ejpam-6409	901	56	t	t	NOUN
ejpam-6409	901	57	−	−	PROPN
ejpam-6409	901	58	2	2	NUM
ejpam-6409	901	59	,	,	PUNCT
ejpam-6409	901	60	the	the	DET
ejpam-6409	901	61	nonlinear	nonlinear	ADJ
ejpam-6409	901	62	z2s	z2s	PROPN
ejpam-6409	902	1	[	[	X
ejpam-6409	902	2	ω]-linear	ω]-linear	ADJ
ejpam-6409	902	3	hadamard	hadamard	ADJ
ejpam-6409	902	4	codes	code	NOUN
ejpam-6409	902	5	of	of	ADP
ejpam-6409	902	6	length	length	NOUN
ejpam-6409	902	7	22	22	NUM
ejpam-6409	902	8	t	t	NOUN
ejpam-6409	902	9	have	have	VERB
ejpam-6409	902	10	different	different	ADJ
ejpam-6409	902	11	kernel	kernel	NOUN
ejpam-6409	902	12	dimensions	dimension	NOUN
ejpam-6409	902	13	and	and	CCONJ
ejpam-6409	902	14	we	we	PRON
ejpam-6409	902	15	are	be	AUX
ejpam-6409	902	16	able	able	ADJ
ejpam-6409	902	17	to	to	PART
ejpam-6409	902	18	classify	classify	VERB
ejpam-6409	902	19	these	these	DET
ejpam-6409	902	20	codes	code	NOUN
ejpam-6409	902	21	according	accord	VERB
ejpam-6409	902	22	to	to	ADP
ejpam-6409	902	23	this	this	DET
ejpam-6409	902	24	invariant	invariant	ADJ
ejpam-6409	902	25	.	.	PUNCT
ejpam-6409	903	1	similar	similar	ADJ
ejpam-6409	903	2	results	result	NOUN
ejpam-6409	903	3	apply	apply	VERB
ejpam-6409	903	4	for	for	ADP
ejpam-6409	903	5	t	t	NOUN
ejpam-6409	903	6	=	=	SYM
ejpam-6409	903	7	8	8	NUM
ejpam-6409	903	8	,	,	PUNCT
ejpam-6409	903	9	9	9	NUM
ejpam-6409	903	10	,	,	PUNCT
ejpam-6409	903	11	10	10	NUM
ejpam-6409	903	12	,	,	PUNCT
ejpam-6409	903	13	and	and	CCONJ
ejpam-6409	903	14	11	11	NUM
ejpam-6409	903	15	;	;	PUNCT
ejpam-6409	903	16	exceptions	exception	NOUN
ejpam-6409	903	17	occur	occur	VERB
ejpam-6409	903	18	for	for	ADP
ejpam-6409	903	19	certain	certain	ADJ
ejpam-6409	903	20	values	value	NOUN
ejpam-6409	903	21	of	of	ADP
ejpam-6409	903	22	s	s	PRON
ejpam-6409	903	23	in	in	ADP
ejpam-6409	903	24	each	each	DET
ejpam-6409	903	25	case	case	NOUN
ejpam-6409	903	26	.	.	PUNCT
ejpam-6409	904	1	in	in	ADP
ejpam-6409	904	2	the	the	DET
ejpam-6409	904	3	particular	particular	ADJ
ejpam-6409	904	4	cases	case	NOUN
ejpam-6409	904	5	of	of	ADP
ejpam-6409	904	6	t	t	PROPN
ejpam-6409	904	7	and	and	CCONJ
ejpam-6409	904	8	s	s	PROPN
ejpam-6409	904	9	,	,	PUNCT
ejpam-6409	904	10	classification	classification	NOUN
ejpam-6409	904	11	according	accord	VERB
ejpam-6409	904	12	to	to	ADP
ejpam-6409	904	13	kernel	kernel	NOUN
ejpam-6409	904	14	analysis	analysis	NOUN
ejpam-6409	904	15	provides	provide	VERB
ejpam-6409	904	16	only	only	ADV
ejpam-6409	904	17	partial	partial	ADJ
ejpam-6409	904	18	results	result	NOUN
ejpam-6409	904	19	.	.	PUNCT
ejpam-6409	905	1	it	it	PRON
ejpam-6409	905	2	was	be	AUX
ejpam-6409	905	3	found	find	VERB
ejpam-6409	905	4	that	that	SCONJ
ejpam-6409	905	5	for	for	ADP
ejpam-6409	905	6	codes	code	NOUN
ejpam-6409	905	7	over	over	ADP
ejpam-6409	905	8	z2s	z2s	PROPN
ejpam-6409	905	9	and	and	CCONJ
ejpam-6409	905	10	codes	code	NOUN
ejpam-6409	905	11	over	over	ADP
ejpam-6409	905	12	z2s	z2s	PROPN
ejpam-6409	905	13	[	[	X
ejpam-6409	905	14	ω	ω	X
ejpam-6409	905	15	]	]	X
ejpam-6409	905	16	,	,	PUNCT
ejpam-6409	905	17	the	the	DET
ejpam-6409	905	18	rank	rank	NOUN
ejpam-6409	905	19	and	and	CCONJ
ejpam-6409	905	20	dimension	dimension	NOUN
ejpam-6409	905	21	of	of	ADP
ejpam-6409	905	22	kernel	kernel	PROPN
ejpam-6409	905	23	are	be	AUX
ejpam-6409	905	24	equal	equal	ADJ
ejpam-6409	905	25	.	.	PUNCT
ejpam-6409	906	1	the	the	DET
ejpam-6409	906	2	software	software	NOUN
ejpam-6409	906	3	magma	magma	NOUN
ejpam-6409	906	4	can	can	AUX
ejpam-6409	906	5	be	be	AUX
ejpam-6409	906	6	used	use	VERB
ejpam-6409	906	7	to	to	PART
ejpam-6409	906	8	determine	determine	VERB
ejpam-6409	906	9	the	the	DET
ejpam-6409	906	10	rank	rank	NOUN
ejpam-6409	906	11	and	and	CCONJ
ejpam-6409	906	12	kernel	kernel	PROPN
ejpam-6409	906	13	muhammad	muhammad	PROPN
ejpam-6409	906	14	sajjad	sajjad	PROPN
ejpam-6409	906	15	et	et	PROPN
ejpam-6409	906	16	al	al	PROPN
ejpam-6409	906	17	.	.	PUNCT
ejpam-6409	906	18	/	/	SYM
ejpam-6409	906	19	eur	eur	PROPN
ejpam-6409	906	20	.	.	PUNCT
ejpam-6409	907	1	j.	j.	PROPN
ejpam-6409	907	2	pure	pure	PROPN
ejpam-6409	907	3	appl	appl	PROPN
ejpam-6409	907	4	.	.	PROPN
ejpam-6409	907	5	math	math	PROPN
ejpam-6409	907	6	,	,	PUNCT
ejpam-6409	907	7	18	18	NUM
ejpam-6409	907	8	(	(	PUNCT
ejpam-6409	907	9	3	3	NUM
ejpam-6409	907	10	)	)	PUNCT
ejpam-6409	907	11	(	(	PUNCT
ejpam-6409	907	12	2025	2025	NUM
ejpam-6409	907	13	)	)	PUNCT
ejpam-6409	907	14	,	,	PUNCT
ejpam-6409	907	15	6409	6409	NUM
ejpam-6409	907	16	27	27	NUM
ejpam-6409	907	17	of	of	ADP
ejpam-6409	907	18	32	32	NUM
ejpam-6409	907	19	dimension	dimension	NOUN
ejpam-6409	907	20	for	for	ADP
ejpam-6409	907	21	any	any	DET
ejpam-6409	907	22	5	5	NUM
ejpam-6409	907	23	≤	≤	NOUN
ejpam-6409	907	24	t	t	NOUN
ejpam-6409	907	25	≤	≤	NUM
ejpam-6409	907	26	11	11	NUM
ejpam-6409	907	27	and	and	CCONJ
ejpam-6409	907	28	2	2	NUM
ejpam-6409	907	29	≤	≤	NOUN
ejpam-6409	907	30	s	s	PART
ejpam-6409	907	31	≤	≤	NUM
ejpam-6409	907	32	t−	t−	PROPN
ejpam-6409	907	33	2	2	NUM
ejpam-6409	907	34	[	[	SYM
ejpam-6409	907	35	6	6	NUM
ejpam-6409	907	36	,	,	PUNCT
ejpam-6409	907	37	17	17	NUM
ejpam-6409	907	38	]	]	PUNCT
ejpam-6409	907	39	.	.	PUNCT
ejpam-6409	908	1	tables	table	NOUN
ejpam-6409	908	2	2	2	NUM
ejpam-6409	908	3	and	and	CCONJ
ejpam-6409	908	4	5	5	NUM
ejpam-6409	908	5	give	give	VERB
ejpam-6409	908	6	the	the	DET
ejpam-6409	908	7	values	value	NOUN
ejpam-6409	908	8	of	of	ADP
ejpam-6409	908	9	(	(	PUNCT
ejpam-6409	908	10	t1	t1	NOUN
ejpam-6409	908	11	,	,	PUNCT
ejpam-6409	908	12	.	.	PUNCT
ejpam-6409	908	13	.	.	PUNCT
ejpam-6409	909	1	.	.	PUNCT
ejpam-6409	910	1	,	,	PUNCT
ejpam-6409	910	2	ts	ts	NOUN
ejpam-6409	910	3	)	)	PUNCT
ejpam-6409	910	4	together	together	ADV
ejpam-6409	910	5	with	with	ADP
ejpam-6409	910	6	the	the	DET
ejpam-6409	910	7	pair	pair	NOUN
ejpam-6409	910	8	(	(	PUNCT
ejpam-6409	910	9	r	r	NOUN
ejpam-6409	910	10	,	,	PUNCT
ejpam-6409	910	11	k	k	NOUN
ejpam-6409	910	12	)	)	PUNCT
ejpam-6409	910	13	,	,	PUNCT
ejpam-6409	910	14	where	where	SCONJ
ejpam-6409	910	15	r	r	NOUN
ejpam-6409	910	16	is	be	AUX
ejpam-6409	910	17	the	the	DET
ejpam-6409	910	18	rank	rank	NOUN
ejpam-6409	910	19	and	and	CCONJ
ejpam-6409	910	20	k	k	NOUN
ejpam-6409	910	21	the	the	DET
ejpam-6409	910	22	dimension	dimension	NOUN
ejpam-6409	910	23	of	of	ADP
ejpam-6409	910	24	the	the	DET
ejpam-6409	910	25	kernel	kernel	NOUN
ejpam-6409	910	26	for	for	ADP
ejpam-6409	910	27	all	all	DET
ejpam-6409	910	28	nonlinear	nonlinear	ADJ
ejpam-6409	910	29	z2s	z2	NOUN
ejpam-6409	910	30	[	[	X
ejpam-6409	910	31	ω]-linear	ω]-linear	ADJ
ejpam-6409	910	32	hadamard	hadamard	ADJ
ejpam-6409	910	33	codes	code	NOUN
ejpam-6409	910	34	of	of	ADP
ejpam-6409	910	35	length	length	NOUN
ejpam-6409	910	36	22	22	NUM
ejpam-6409	910	37	t	t	NOUN
ejpam-6409	910	38	for	for	ADP
ejpam-6409	910	39	5	5	NUM
ejpam-6409	910	40	≤	≤	NOUN
ejpam-6409	910	41	t	t	NOUN
ejpam-6409	910	42	≤	≤	NUM
ejpam-6409	910	43	10	10	NUM
ejpam-6409	910	44	.	.	PUNCT
ejpam-6409	911	1	such	such	ADJ
ejpam-6409	911	2	tables	table	NOUN
ejpam-6409	911	3	show	show	VERB
ejpam-6409	911	4	that	that	SCONJ
ejpam-6409	911	5	each	each	DET
ejpam-6409	911	6	length	length	NOUN
ejpam-6409	911	7	22	22	NUM
ejpam-6409	911	8	t	t	NOUN
ejpam-6409	911	9	code	code	NOUN
ejpam-6409	911	10	has	have	VERB
ejpam-6409	911	11	exactly	exactly	ADV
ejpam-6409	911	12	one	one	NUM
ejpam-6409	911	13	rank	rank	NOUN
ejpam-6409	911	14	value	value	NOUN
ejpam-6409	911	15	when	when	SCONJ
ejpam-6409	911	16	5	5	NUM
ejpam-6409	911	17	≤	≤	NOUN
ejpam-6409	911	18	t	t	X
ejpam-6409	911	19	≤	≤	NUM
ejpam-6409	911	20	10	10	NUM
ejpam-6409	911	21	and	and	CCONJ
ejpam-6409	911	22	2	2	NUM
ejpam-6409	911	23	≤	≤	NUM
ejpam-6409	912	1	s	s	PART
ejpam-6409	912	2	≤	≤	NOUN
ejpam-6409	912	3	t	t	NOUN
ejpam-6409	912	4	−	−	PROPN
ejpam-6409	912	5	2	2	NUM
ejpam-6409	912	6	is	be	AUX
ejpam-6409	912	7	fixed	fix	VERB
ejpam-6409	912	8	.	.	PUNCT
ejpam-6409	913	1	thus	thus	ADV
ejpam-6409	913	2	,	,	PUNCT
ejpam-6409	913	3	all	all	DET
ejpam-6409	913	4	codes	code	NOUN
ejpam-6409	913	5	in	in	ADP
ejpam-6409	913	6	such	such	ADJ
ejpam-6409	913	7	situations	situation	NOUN
ejpam-6409	913	8	are	be	AUX
ejpam-6409	913	9	different	different	ADJ
ejpam-6409	913	10	,	,	PUNCT
ejpam-6409	913	11	allowing	allow	VERB
ejpam-6409	913	12	only	only	ADV
ejpam-6409	913	13	the	the	DET
ejpam-6409	913	14	rank	rank	NOUN
ejpam-6409	913	15	of	of	ADP
ejpam-6409	913	16	the	the	DET
ejpam-6409	913	17	code	code	NOUN
ejpam-6409	913	18	to	to	PART
ejpam-6409	913	19	be	be	AUX
ejpam-6409	913	20	classified	classify	VERB
ejpam-6409	913	21	,	,	PUNCT
ejpam-6409	913	22	from	from	ADP
ejpam-6409	913	23	which	which	PRON
ejpam-6409	913	24	we	we	PRON
ejpam-6409	913	25	obtain	obtain	VERB
ejpam-6409	913	26	the	the	DET
ejpam-6409	913	27	following	follow	VERB
ejpam-6409	913	28	assertion	assertion	NOUN
ejpam-6409	913	29	.	.	PUNCT
ejpam-6409	914	1	table	table	NOUN
ejpam-6409	914	2	2	2	NUM
ejpam-6409	914	3	:	:	PUNCT
ejpam-6409	914	4	rank	rank	NOUN
ejpam-6409	914	5	and	and	CCONJ
ejpam-6409	914	6	dimension	dimension	NOUN
ejpam-6409	914	7	of	of	ADP
ejpam-6409	914	8	kernel	kernel	NOUN
ejpam-6409	914	9	for	for	ADP
ejpam-6409	914	10	all	all	DET
ejpam-6409	914	11	nonlinear	nonlinear	ADJ
ejpam-6409	914	12	z2s	z2	NOUN
ejpam-6409	915	1	[	[	X
ejpam-6409	915	2	ω]-linear	ω]-linear	ADJ
ejpam-6409	915	3	hadamard	hadamard	ADJ
ejpam-6409	915	4	codes	code	NOUN
ejpam-6409	915	5	of	of	ADP
ejpam-6409	915	6	length	length	NOUN
ejpam-6409	915	7	22	22	NUM
ejpam-6409	915	8	t	t	NOUN
ejpam-6409	915	9	t	t	NOUN
ejpam-6409	915	10	=	=	SYM
ejpam-6409	915	11	5	5	NUM
ejpam-6409	915	12	t	t	NOUN
ejpam-6409	915	13	=	=	SYM
ejpam-6409	915	14	6	6	NUM
ejpam-6409	915	15	t	t	NOUN
ejpam-6409	915	16	=	=	SYM
ejpam-6409	915	17	7	7	NUM
ejpam-6409	915	18	(	(	PUNCT
ejpam-6409	915	19	t1	t1	NOUN
ejpam-6409	915	20	,	,	PUNCT
ejpam-6409	915	21	.	.	PUNCT
ejpam-6409	915	22	.	.	PUNCT
ejpam-6409	915	23	.	.	PUNCT
ejpam-6409	916	1	,	,	PUNCT
ejpam-6409	916	2	ts	ts	NOUN
ejpam-6409	916	3	)	)	PUNCT
ejpam-6409	916	4	(	(	PUNCT
ejpam-6409	916	5	r	r	NOUN
ejpam-6409	916	6	,	,	PUNCT
ejpam-6409	916	7	k	k	NOUN
ejpam-6409	916	8	)	)	PUNCT
ejpam-6409	916	9	(	(	PUNCT
ejpam-6409	916	10	t1	t1	NOUN
ejpam-6409	916	11	,	,	PUNCT
ejpam-6409	916	12	.	.	PUNCT
ejpam-6409	916	13	.	.	PUNCT
ejpam-6409	916	14	.	.	PUNCT
ejpam-6409	917	1	,	,	PUNCT
ejpam-6409	917	2	ts	ts	NOUN
ejpam-6409	917	3	)	)	PUNCT
ejpam-6409	917	4	(	(	PUNCT
ejpam-6409	917	5	r	r	NOUN
ejpam-6409	917	6	,	,	PUNCT
ejpam-6409	917	7	k	k	NOUN
ejpam-6409	917	8	)	)	PUNCT
ejpam-6409	917	9	(	(	PUNCT
ejpam-6409	917	10	t1	t1	NOUN
ejpam-6409	917	11	,	,	PUNCT
ejpam-6409	917	12	.	.	PUNCT
ejpam-6409	917	13	.	.	PUNCT
ejpam-6409	917	14	.	.	PUNCT
ejpam-6409	918	1	,	,	PUNCT
ejpam-6409	918	2	ts	ts	NOUN
ejpam-6409	918	3	)	)	PUNCT
ejpam-6409	918	4	(	(	PUNCT
ejpam-6409	918	5	r	r	NOUN
ejpam-6409	918	6	,	,	PUNCT
ejpam-6409	918	7	k	k	NOUN
ejpam-6409	918	8	)	)	PUNCT
ejpam-6409	918	9	z4[ω	z4[ω	NOUN
ejpam-6409	918	10	]	]	PUNCT
ejpam-6409	918	11	(	(	PUNCT
ejpam-6409	918	12	3,0	3,0	NUM
ejpam-6409	918	13	)	)	PUNCT
ejpam-6409	918	14	(	(	PUNCT
ejpam-6409	918	15	7,4	7,4	NUM
ejpam-6409	918	16	)	)	PUNCT
ejpam-6409	918	17	(	(	PUNCT
ejpam-6409	918	18	3,1	3,1	NUM
ejpam-6409	918	19	)	)	PUNCT
ejpam-6409	918	20	(	(	PUNCT
ejpam-6409	918	21	8,5	8,5	NUM
ejpam-6409	918	22	)	)	PUNCT
ejpam-6409	918	23	(	(	PUNCT
ejpam-6409	918	24	3,2	3,2	NUM
ejpam-6409	918	25	)	)	PUNCT
ejpam-6409	918	26	(	(	PUNCT
ejpam-6409	918	27	9,6	9,6	NUM
ejpam-6409	918	28	)	)	PUNCT
ejpam-6409	918	29	(	(	PUNCT
ejpam-6409	918	30	4,0	4,0	NUM
ejpam-6409	918	31	)	)	PUNCT
ejpam-6409	918	32	(	(	PUNCT
ejpam-6409	918	33	11,5	11,5	NUM
ejpam-6409	918	34	)	)	PUNCT
ejpam-6409	918	35	z8[ω	z8[ω	PROPN
ejpam-6409	918	36	]	]	X
ejpam-6409	918	37	(	(	PUNCT
ejpam-6409	918	38	2,0,0	2,0,0	NUM
ejpam-6409	918	39	)	)	PUNCT
ejpam-6409	918	40	(	(	PUNCT
ejpam-6409	918	41	8,3	8,3	NUM
ejpam-6409	918	42	)	)	PUNCT
ejpam-6409	918	43	(	(	PUNCT
ejpam-6409	918	44	1,2,0	1,2,0	NUM
ejpam-6409	918	45	)	)	PUNCT
ejpam-6409	918	46	(	(	PUNCT
ejpam-6409	918	47	8,5	8,5	NUM
ejpam-6409	918	48	)	)	PUNCT
ejpam-6409	918	49	(	(	PUNCT
ejpam-6409	918	50	1,2,1	1,2,1	NUM
ejpam-6409	918	51	)	)	PUNCT
ejpam-6409	918	52	(	(	PUNCT
ejpam-6409	918	53	9,6	9,6	NUM
ejpam-6409	918	54	)	)	PUNCT
ejpam-6409	918	55	(	(	PUNCT
ejpam-6409	918	56	2,0,1	2,0,1	NUM
ejpam-6409	918	57	)	)	PUNCT
ejpam-6409	918	58	(	(	PUNCT
ejpam-6409	918	59	9,4	9,4	NUM
ejpam-6409	918	60	)	)	PUNCT
ejpam-6409	918	61	(	(	PUNCT
ejpam-6409	918	62	2,0,2	2,0,2	NUM
ejpam-6409	918	63	)	)	PUNCT
ejpam-6409	918	64	(	(	PUNCT
ejpam-6409	918	65	10,5	10,5	NUM
ejpam-6409	918	66	)	)	PUNCT
ejpam-6409	918	67	(	(	PUNCT
ejpam-6409	918	68	2,1,0	2,1,0	NUM
ejpam-6409	918	69	)	)	PUNCT
ejpam-6409	918	70	(	(	PUNCT
ejpam-6409	918	71	12,4	12,4	NOUN
ejpam-6409	918	72	)	)	PUNCT
ejpam-6409	918	73	z16[ω	z16[ω	NOUN
ejpam-6409	918	74	]	]	PUNCT
ejpam-6409	918	75	(	(	PUNCT
ejpam-6409	918	76	1,1,0,0	1,1,0,0	NUM
ejpam-6409	918	77	)	)	PUNCT
ejpam-6409	918	78	(	(	PUNCT
ejpam-6409	918	79	9,4	9,4	NUM
ejpam-6409	918	80	)	)	PUNCT
ejpam-6409	918	81	(	(	PUNCT
ejpam-6409	918	82	1,0,2,0	1,0,2,0	NUM
ejpam-6409	918	83	)	)	PUNCT
ejpam-6409	918	84	(	(	PUNCT
ejpam-6409	918	85	9,6	9,6	NUM
ejpam-6409	918	86	)	)	PUNCT
ejpam-6409	918	87	(	(	PUNCT
ejpam-6409	918	88	1,1,0,1	1,1,0,1	NUM
ejpam-6409	918	89	)	)	PUNCT
ejpam-6409	918	90	(	(	PUNCT
ejpam-6409	918	91	10,5	10,5	NUM
ejpam-6409	918	92	)	)	PUNCT
ejpam-6409	918	93	(	(	PUNCT
ejpam-6409	918	94	2,0,0,0	2,0,0,0	NUM
ejpam-6409	918	95	)	)	PUNCT
ejpam-6409	918	96	(	(	PUNCT
ejpam-6409	918	97	14,3	14,3	X
ejpam-6409	918	98	)	)	PUNCT
ejpam-6409	918	99	z32[ω	z32[ω	NOUN
ejpam-6409	918	100	]	]	PUNCT
ejpam-6409	918	101	(	(	PUNCT
ejpam-6409	918	102	1,0,1,0,0	1,0,1,0,0	NUM
ejpam-6409	918	103	)	)	PUNCT
ejpam-6409	918	104	(	(	PUNCT
ejpam-6409	918	105	10,5	10,5	NUM
ejpam-6409	918	106	)	)	PUNCT
ejpam-6409	918	107	theorem	theorem	VERB
ejpam-6409	918	108	7.2	7.2	NUM
ejpam-6409	918	109	:	:	PUNCT
ejpam-6409	918	110	let	let	VERB
ejpam-6409	918	111	at	at	ADP
ejpam-6409	918	112	,	,	PUNCT
ejpam-6409	918	113	s	s	PRON
ejpam-6409	918	114	represent	represent	VERB
ejpam-6409	918	115	the	the	DET
ejpam-6409	918	116	number	number	NOUN
ejpam-6409	918	117	of	of	ADP
ejpam-6409	918	118	inequivalent	inequivalent	NOUN
ejpam-6409	918	119	z2s	z2s	PROPN
ejpam-6409	919	1	[	[	X
ejpam-6409	919	2	ω]-linear	ω]-linear	ADJ
ejpam-6409	919	3	hadamard	hadamard	ADJ
ejpam-6409	919	4	codes	code	NOUN
ejpam-6409	919	5	of	of	ADP
ejpam-6409	919	6	length	length	NOUN
ejpam-6409	919	7	22	22	NUM
ejpam-6409	919	8	t.	t.	NOUN
ejpam-6409	919	9	then	then	ADV
ejpam-6409	919	10	,	,	PUNCT
ejpam-6409	919	11	for	for	ADP
ejpam-6409	919	12	any	any	DET
ejpam-6409	919	13	t	t	PROPN
ejpam-6409	919	14	≥	≥	NOUN
ejpam-6409	919	15	3	3	NUM
ejpam-6409	919	16	and	and	CCONJ
ejpam-6409	919	17	2	2	NUM
ejpam-6409	919	18	≤	≤	NOUN
ejpam-6409	919	19	s	s	PART
ejpam-6409	919	20	≤	≤	NOUN
ejpam-6409	919	21	t−	t−	PROPN
ejpam-6409	919	22	1	1	NUM
ejpam-6409	919	23	,	,	PUNCT
ejpam-6409	919	24	at	at	ADP
ejpam-6409	919	25	,	,	PUNCT
ejpam-6409	919	26	s	s	VERB
ejpam-6409	919	27	≤	≤	NUM
ejpam-6409	919	28	∣∣∣∣∣	∣∣∣∣∣	ADP
ejpam-6409	919	29	{	{	PUNCT
ejpam-6409	919	30	(	(	PUNCT
ejpam-6409	919	31	t1	t1	NOUN
ejpam-6409	919	32	,	,	PUNCT
ejpam-6409	919	33	.	.	PUNCT
ejpam-6409	919	34	.	.	PUNCT
ejpam-6409	919	35	.	.	PUNCT
ejpam-6409	920	1	,	,	PUNCT
ejpam-6409	920	2	ts	ts	NOUN
ejpam-6409	920	3	)	)	PUNCT
ejpam-6409	920	4	∈	∈	PROPN
ejpam-6409	921	1	ns	ns	NUM
ejpam-6409	921	2	:	:	PUNCT
ejpam-6409	921	3	t	t	NOUN
ejpam-6409	921	4	=	=	PUNCT
ejpam-6409	921	5	(	(	PUNCT
ejpam-6409	921	6	s∑	s∑	PROPN
ejpam-6409	921	7	i=1	i=1	PROPN
ejpam-6409	922	1	(	(	PUNCT
ejpam-6409	922	2	s−	s−	PROPN
ejpam-6409	922	3	i+	i+	PUNCT
ejpam-6409	922	4	1)ti	1)ti	PROPN
ejpam-6409	922	5	)	)	PUNCT
ejpam-6409	923	1	−	−	PROPN
ejpam-6409	923	2	1	1	NUM
ejpam-6409	923	3	,	,	PUNCT
ejpam-6409	923	4	t1	t1	NOUN
ejpam-6409	923	5	≥	≥	NUM
ejpam-6409	923	6	1	1	NUM
ejpam-6409	923	7	}	}	PUNCT
ejpam-6409	923	8	∣∣∣∣∣−	∣∣∣∣∣−	PROPN
ejpam-6409	923	9	1	1	NUM
ejpam-6409	923	10	.	.	PUNCT
ejpam-6409	924	1	moreover	moreover	ADV
ejpam-6409	924	2	,	,	PUNCT
ejpam-6409	924	3	for	for	ADP
ejpam-6409	924	4	any	any	DET
ejpam-6409	924	5	values	value	NOUN
ejpam-6409	924	6	of	of	ADP
ejpam-6409	924	7	t	t	PROPN
ejpam-6409	924	8	within	within	ADP
ejpam-6409	924	9	the	the	DET
ejpam-6409	924	10	range	range	NOUN
ejpam-6409	924	11	[	[	X
ejpam-6409	924	12	3	3	NUM
ejpam-6409	924	13	,	,	PUNCT
ejpam-6409	924	14	11	11	NUM
ejpam-6409	924	15	]	]	PUNCT
ejpam-6409	924	16	and	and	CCONJ
ejpam-6409	924	17	s	s	X
ejpam-6409	924	18	in	in	ADP
ejpam-6409	924	19	the	the	DET
ejpam-6409	924	20	range	range	NOUN
ejpam-6409	924	21	[	[	X
ejpam-6409	924	22	2	2	NUM
ejpam-6409	924	23	,	,	PUNCT
ejpam-6409	924	24	t	t	NOUN
ejpam-6409	924	25	−	−	PROPN
ejpam-6409	924	26	1	1	NUM
ejpam-6409	924	27	]	]	PUNCT
ejpam-6409	924	28	,	,	PUNCT
ejpam-6409	924	29	including	include	VERB
ejpam-6409	924	30	the	the	DET
ejpam-6409	924	31	endpoints	endpoint	NOUN
ejpam-6409	924	32	,	,	PUNCT
ejpam-6409	924	33	this	this	DET
ejpam-6409	924	34	bound	bind	VERB
ejpam-6409	924	35	is	be	AUX
ejpam-6409	924	36	sharp	sharp	ADJ
ejpam-6409	924	37	.	.	PUNCT
ejpam-6409	925	1	using	use	VERB
ejpam-6409	925	2	the	the	DET
ejpam-6409	925	3	outputs	output	NOUN
ejpam-6409	925	4	of	of	ADP
ejpam-6409	925	5	theorems	theorem	NOUN
ejpam-6409	925	6	5.1	5.1	NUM
ejpam-6409	925	7	and	and	CCONJ
ejpam-6409	925	8	5.2	5.2	NUM
ejpam-6409	925	9	,	,	PUNCT
ejpam-6409	925	10	we	we	PRON
ejpam-6409	925	11	give	give	VERB
ejpam-6409	925	12	the	the	DET
ejpam-6409	925	13	following	follow	VERB
ejpam-6409	925	14	table	table	NOUN
ejpam-6409	925	15	3	3	NUM
ejpam-6409	925	16	that	that	PRON
ejpam-6409	925	17	lists	list	VERB
ejpam-6409	925	18	the	the	DET
ejpam-6409	925	19	number	number	NOUN
ejpam-6409	925	20	of	of	ADP
ejpam-6409	925	21	nonequivalent	nonequivalent	ADJ
ejpam-6409	925	22	z2s	z2s	PROPN
ejpam-6409	926	1	[	[	X
ejpam-6409	926	2	ω]-linear	ω]-linear	ADJ
ejpam-6409	926	3	hadamard	hadamard	ADJ
ejpam-6409	926	4	codes	code	NOUN
ejpam-6409	926	5	of	of	ADP
ejpam-6409	926	6	length	length	NOUN
ejpam-6409	926	7	22	22	NUM
ejpam-6409	926	8	t	t	NOUN
ejpam-6409	926	9	where	where	SCONJ
ejpam-6409	926	10	3	3	NUM
ejpam-6409	926	11	≤	≤	NOUN
ejpam-6409	926	12	t	t	X
ejpam-6409	926	13	≤	≤	NUM
ejpam-6409	926	14	11	11	NUM
ejpam-6409	926	15	and	and	CCONJ
ejpam-6409	926	16	2	2	NUM
ejpam-6409	926	17	≤	≤	NOUN
ejpam-6409	926	18	s	s	PART
ejpam-6409	926	19	≤	≤	ADJ
ejpam-6409	926	20	9	9	NUM
ejpam-6409	926	21	.	.	PUNCT
ejpam-6409	927	1	classification	classification	NOUN
ejpam-6409	927	2	is	be	AUX
ejpam-6409	927	3	inadequate	inadequate	ADJ
ejpam-6409	927	4	only	only	ADV
ejpam-6409	927	5	when	when	SCONJ
ejpam-6409	927	6	considering	consider	VERB
ejpam-6409	927	7	the	the	DET
ejpam-6409	927	8	kernel	kernel	PROPN
ejpam-6409	927	9	dimension	dimension	NOUN
ejpam-6409	927	10	in	in	ADP
ejpam-6409	927	11	the	the	DET
ejpam-6409	927	12	highlighted	highlight	VERB
ejpam-6409	927	13	cases	case	NOUN
ejpam-6409	927	14	in	in	ADP
ejpam-6409	927	15	bold	bold	ADJ
ejpam-6409	927	16	.	.	PUNCT
ejpam-6409	928	1	however	however	ADV
ejpam-6409	928	2	,	,	PUNCT
ejpam-6409	928	3	in	in	ADP
ejpam-6409	928	4	all	all	DET
ejpam-6409	928	5	cases	case	NOUN
ejpam-6409	928	6	,	,	PUNCT
ejpam-6409	928	7	the	the	DET
ejpam-6409	928	8	aforesaid	aforesaid	NOUN
ejpam-6409	928	9	rank	rank	NOUN
ejpam-6409	928	10	turns	turn	VERB
ejpam-6409	928	11	out	out	ADP
ejpam-6409	928	12	to	to	PART
ejpam-6409	928	13	be	be	AUX
ejpam-6409	928	14	a	a	DET
ejpam-6409	928	15	good	good	ADJ
ejpam-6409	928	16	method	method	NOUN
ejpam-6409	928	17	of	of	ADP
ejpam-6409	928	18	classification	classification	NOUN
ejpam-6409	928	19	.	.	PUNCT
ejpam-6409	929	1	there	there	PRON
ejpam-6409	929	2	appear	appear	VERB
ejpam-6409	929	3	to	to	PART
ejpam-6409	929	4	be	be	AUX
ejpam-6409	929	5	z4[ω]-linear	z4[ω]-linear	PROPN
ejpam-6409	929	6	hadamard	hadamard	ADJ
ejpam-6409	929	7	codes	code	NOUN
ejpam-6409	929	8	that	that	PRON
ejpam-6409	929	9	do	do	AUX
ejpam-6409	929	10	not	not	PART
ejpam-6409	929	11	exist	exist	VERB
ejpam-6409	929	12	as	as	ADP
ejpam-6409	929	13	equivalent	equivalent	ADJ
ejpam-6409	929	14	z2s	z2	NOUN
ejpam-6409	930	1	[	[	X
ejpam-6409	930	2	ω]linear	ω]linear	ADJ
ejpam-6409	930	3	hadamard	hadamard	ADJ
ejpam-6409	930	4	codes	code	NOUN
ejpam-6409	930	5	,	,	PUNCT
ejpam-6409	930	6	for	for	ADP
ejpam-6409	930	7	s	s	PROPN
ejpam-6409	930	8	>	>	X
ejpam-6409	930	9	2	2	NUM
ejpam-6409	930	10	.	.	PUNCT
ejpam-6409	930	11	example	example	NOUN
ejpam-6409	930	12	7.3	7.3	NUM
ejpam-6409	930	13	:	:	PUNCT
ejpam-6409	930	14	let	let	VERB
ejpam-6409	930	15	us	we	PRON
ejpam-6409	930	16	consider	consider	VERB
ejpam-6409	930	17	h(2,0,0	h(2,0,0	NOUN
ejpam-6409	930	18	)	)	PUNCT
ejpam-6409	930	19	as	as	ADP
ejpam-6409	930	20	the	the	DET
ejpam-6409	930	21	z8[ω]-linear	z8[ω]-linear	NUM
ejpam-6409	930	22	hadamard	hadamard	PROPN
ejpam-6409	930	23	code	code	NOUN
ejpam-6409	930	24	of	of	ADP
ejpam-6409	930	25	length	length	NOUN
ejpam-6409	930	26	1024	1024	NUM
ejpam-6409	930	27	.	.	PUNCT
ejpam-6409	931	1	from	from	ADP
ejpam-6409	931	2	theorem	theorem	ADJ
ejpam-6409	931	3	3	3	NUM
ejpam-6409	931	4	,	,	PUNCT
ejpam-6409	931	5	we	we	PRON
ejpam-6409	931	6	know	know	VERB
ejpam-6409	931	7	that	that	SCONJ
ejpam-6409	931	8	ker(h(2,0,0	ker(h(2,0,0	NOUN
ejpam-6409	931	9	)	)	PUNCT
ejpam-6409	931	10	)	)	PUNCT
ejpam-6409	932	1	=	=	SYM
ejpam-6409	932	2	3	3	NUM
ejpam-6409	932	3	,	,	PUNCT
ejpam-6409	932	4	i.e.	i.e.	X
ejpam-6409	932	5	,	,	PUNCT
ejpam-6409	932	6	h(2,0,0	h(2,0,0	NOUN
ejpam-6409	932	7	)	)	PUNCT
ejpam-6409	932	8	can	can	AUX
ejpam-6409	932	9	not	not	PART
ejpam-6409	932	10	be	be	AUX
ejpam-6409	932	11	linear	linear	ADJ
ejpam-6409	932	12	.	.	PUNCT
ejpam-6409	933	1	there	there	PRON
ejpam-6409	933	2	are	be	VERB
ejpam-6409	933	3	acknowledged	acknowledge	VERB
ejpam-6409	933	4	to	to	PART
ejpam-6409	933	5	be	be	AUX
ejpam-6409	933	6	three	three	NUM
ejpam-6409	933	7	z4[ω]-linear	z4[ω]-linear	NUM
ejpam-6409	933	8	hadamard	hadamard	ADJ
ejpam-6409	933	9	codes	code	NOUN
ejpam-6409	933	10	of	of	ADP
ejpam-6409	933	11	length	length	NOUN
ejpam-6409	933	12	1024	1024	NUM
ejpam-6409	933	13	given	give	VERB
ejpam-6409	933	14	by	by	ADP
ejpam-6409	933	15	h(1,4	h(1,4	PROPN
ejpam-6409	933	16	)	)	PUNCT
ejpam-6409	933	17	,	,	PUNCT
ejpam-6409	933	18	h(2,2	h(2,2	PROPN
ejpam-6409	933	19	)	)	PUNCT
ejpam-6409	933	20	,	,	PUNCT
ejpam-6409	933	21	and	and	CCONJ
ejpam-6409	933	22	h(3,0	h(3,0	NOUN
ejpam-6409	933	23	)	)	PUNCT
ejpam-6409	933	24	.	.	PUNCT
ejpam-6409	934	1	the	the	DET
ejpam-6409	934	2	first	first	ADJ
ejpam-6409	934	3	two	two	NUM
ejpam-6409	934	4	codes	code	NOUN
ejpam-6409	934	5	have	have	VERB
ejpam-6409	934	6	a	a	DET
ejpam-6409	934	7	linear	linear	ADJ
ejpam-6409	934	8	structure	structure	NOUN
ejpam-6409	934	9	,	,	PUNCT
ejpam-6409	934	10	while	while	SCONJ
ejpam-6409	934	11	the	the	DET
ejpam-6409	934	12	last	last	ADJ
ejpam-6409	934	13	one	one	NOUN
ejpam-6409	934	14	is	be	AUX
ejpam-6409	934	15	nonlinear	nonlinear	ADJ
ejpam-6409	934	16	,	,	PUNCT
ejpam-6409	934	17	and	and	CCONJ
ejpam-6409	934	18	by	by	ADP
ejpam-6409	934	19	theorem	theorem	NOUN
ejpam-6409	934	20	5.2	5.2	NUM
ejpam-6409	934	21	it	it	PRON
ejpam-6409	934	22	is	be	AUX
ejpam-6409	934	23	shown	show	VERB
ejpam-6409	934	24	that	that	SCONJ
ejpam-6409	934	25	ker(h(3,0	ker(h(3,0	NOUN
ejpam-6409	934	26	)	)	PUNCT
ejpam-6409	934	27	)	)	PUNCT
ejpam-6409	935	1	=	=	PUNCT
ejpam-6409	935	2	4	4	X
ejpam-6409	935	3	.	.	PUNCT
ejpam-6409	935	4	consequently	consequently	ADV
ejpam-6409	935	5	,	,	PUNCT
ejpam-6409	935	6	there	there	PRON
ejpam-6409	935	7	does	do	AUX
ejpam-6409	935	8	not	not	PART
ejpam-6409	935	9	exist	exist	VERB
ejpam-6409	935	10	a	a	DET
ejpam-6409	935	11	z4[ω]-linear	z4[ω]-linear	NUM
ejpam-6409	935	12	hadamard	hadamard	ADJ
ejpam-6409	935	13	code	code	NOUN
ejpam-6409	935	14	equivalent	equivalent	NOUN
ejpam-6409	935	15	to	to	ADP
ejpam-6409	935	16	the	the	DET
ejpam-6409	935	17	z8[ω]-linear	z8[ω]-linear	NUM
ejpam-6409	935	18	hadamard	hadamard	PROPN
ejpam-6409	935	19	code	code	PROPN
ejpam-6409	935	20	h(2,0,0	h(2,0,0	NOUN
ejpam-6409	935	21	)	)	PUNCT
ejpam-6409	935	22	.	.	PUNCT
ejpam-6409	936	1	example	example	NOUN
ejpam-6409	937	1	7.4	7.4	NUM
ejpam-6409	937	2	:	:	PUNCT
ejpam-6409	937	3	it	it	PRON
ejpam-6409	937	4	is	be	AUX
ejpam-6409	937	5	apparent	apparent	ADJ
ejpam-6409	937	6	from	from	ADP
ejpam-6409	937	7	table	table	NOUN
ejpam-6409	937	8	2	2	NUM
ejpam-6409	937	9	that	that	PRON
ejpam-6409	937	10	for	for	ADP
ejpam-6409	937	11	t	t	NOUN
ejpam-6409	937	12	=	=	SYM
ejpam-6409	937	13	5	5	NUM
ejpam-6409	937	14	,	,	PUNCT
ejpam-6409	937	15	we	we	PRON
ejpam-6409	937	16	have	have	VERB
ejpam-6409	937	17	only	only	ADV
ejpam-6409	937	18	two	two	NUM
ejpam-6409	937	19	nonlinear	nonlinear	ADJ
ejpam-6409	937	20	z2s	z2	NOUN
ejpam-6409	938	1	[	[	X
ejpam-6409	938	2	ω]-linear	ω]-linear	ADJ
ejpam-6409	938	3	hadamard	hadamard	ADJ
ejpam-6409	938	4	codes	code	NOUN
ejpam-6409	938	5	,	,	PUNCT
ejpam-6409	938	6	namely	namely	ADV
ejpam-6409	938	7	h(3,0	h(3,0	VERB
ejpam-6409	938	8	)	)	PUNCT
ejpam-6409	938	9	and	and	CCONJ
ejpam-6409	938	10	h(2,0,0	h(2,0,0	NOUN
ejpam-6409	938	11	)	)	PUNCT
ejpam-6409	938	12	.	.	PUNCT
ejpam-6409	939	1	based	base	VERB
ejpam-6409	939	2	on	on	ADP
ejpam-6409	939	3	example	example	NOUN
ejpam-6409	939	4	7.3	7.3	NUM
ejpam-6409	939	5	,	,	PUNCT
ejpam-6409	939	6	the	the	DET
ejpam-6409	939	7	muhammad	muhammad	PROPN
ejpam-6409	939	8	sajjad	sajjad	PROPN
ejpam-6409	939	9	et	et	PROPN
ejpam-6409	939	10	al	al	PROPN
ejpam-6409	939	11	.	.	PUNCT
ejpam-6409	939	12	/	/	SYM
ejpam-6409	939	13	eur	eur	PROPN
ejpam-6409	939	14	.	.	PUNCT
ejpam-6409	940	1	j.	j.	PROPN
ejpam-6409	940	2	pure	pure	PROPN
ejpam-6409	940	3	appl	appl	PROPN
ejpam-6409	940	4	.	.	PROPN
ejpam-6409	940	5	math	math	PROPN
ejpam-6409	940	6	,	,	PUNCT
ejpam-6409	940	7	18	18	NUM
ejpam-6409	940	8	(	(	PUNCT
ejpam-6409	940	9	3	3	NUM
ejpam-6409	940	10	)	)	PUNCT
ejpam-6409	940	11	(	(	PUNCT
ejpam-6409	940	12	2025	2025	NUM
ejpam-6409	940	13	)	)	PUNCT
ejpam-6409	940	14	,	,	PUNCT
ejpam-6409	940	15	6409	6409	NUM
ejpam-6409	940	16	28	28	NUM
ejpam-6409	940	17	of	of	ADP
ejpam-6409	940	18	32	32	NUM
ejpam-6409	940	19	table	table	NOUN
ejpam-6409	940	20	3	3	NUM
ejpam-6409	940	21	:	:	PUNCT
ejpam-6409	940	22	number	number	NOUN
ejpam-6409	940	23	at	at	ADP
ejpam-6409	940	24	,	,	PUNCT
ejpam-6409	940	25	s	s	VERB
ejpam-6409	940	26	of	of	ADP
ejpam-6409	940	27	nonequivalent	nonequivalent	ADJ
ejpam-6409	940	28	z2s	z2s	PROPN
ejpam-6409	941	1	[	[	X
ejpam-6409	941	2	ω]-linear	ω]-linear	ADJ
ejpam-6409	941	3	hadamard	hadamard	ADJ
ejpam-6409	941	4	codes	code	NOUN
ejpam-6409	941	5	of	of	ADP
ejpam-6409	941	6	length	length	NOUN
ejpam-6409	941	7	22	22	NUM
ejpam-6409	941	8	t	t	NOUN
ejpam-6409	941	9	t	t	NOUN
ejpam-6409	941	10	3	3	NUM
ejpam-6409	941	11	4	4	NUM
ejpam-6409	941	12	5	5	NUM
ejpam-6409	941	13	6	6	NUM
ejpam-6409	941	14	7	7	NUM
ejpam-6409	941	15	8	8	NUM
ejpam-6409	941	16	9	9	NUM
ejpam-6409	941	17	10	10	NUM
ejpam-6409	941	18	11	11	NUM
ejpam-6409	941	19	z4[ω	z4[ω	NOUN
ejpam-6409	941	20	]	]	X
ejpam-6409	941	21	1	1	NUM
ejpam-6409	941	22	1	1	NUM
ejpam-6409	941	23	2	2	NUM
ejpam-6409	941	24	2	2	NUM
ejpam-6409	941	25	3	3	NUM
ejpam-6409	941	26	3	3	NUM
ejpam-6409	941	27	4	4	NUM
ejpam-6409	941	28	4	4	NUM
ejpam-6409	941	29	5	5	NUM
ejpam-6409	941	30	z8[ω	z8[ω	PROPN
ejpam-6409	941	31	]	]	X
ejpam-6409	941	32	1	1	NUM
ejpam-6409	941	33	1	1	NUM
ejpam-6409	941	34	2	2	NUM
ejpam-6409	941	35	3	3	NUM
ejpam-6409	941	36	4	4	NUM
ejpam-6409	941	37	6	6	NUM
ejpam-6409	941	38	7	7	NUM
ejpam-6409	941	39	9	9	NUM
ejpam-6409	941	40	11	11	NUM
ejpam-6409	941	41	z16[ω	z16[ω	NOUN
ejpam-6409	941	42	]	]	X
ejpam-6409	941	43	1	1	NUM
ejpam-6409	941	44	1	1	NUM
ejpam-6409	941	45	1	1	NUM
ejpam-6409	941	46	2	2	NUM
ejpam-6409	941	47	4	4	NUM
ejpam-6409	941	48	5	5	NUM
ejpam-6409	941	49	8	8	NUM
ejpam-6409	941	50	10	10	NUM
ejpam-6409	941	51	14	14	NUM
ejpam-6409	941	52	z32[ω	z32[ω	NOUN
ejpam-6409	941	53	]	]	PUNCT
ejpam-6409	941	54	0	0	NUM
ejpam-6409	941	55	1	1	NUM
ejpam-6409	941	56	1	1	NUM
ejpam-6409	941	57	1	1	NUM
ejpam-6409	941	58	2	2	NUM
ejpam-6409	941	59	4	4	NUM
ejpam-6409	941	60	6	6	NUM
ejpam-6409	941	61	9	9	NUM
ejpam-6409	941	62	12	12	NUM
ejpam-6409	941	63	z64[ω	z64[ω	NOUN
ejpam-6409	941	64	]	]	PUNCT
ejpam-6409	941	65	0	0	SYM
ejpam-6409	941	66	0	0	NUM
ejpam-6409	941	67	1	1	NUM
ejpam-6409	941	68	1	1	NUM
ejpam-6409	941	69	1	1	NUM
ejpam-6409	941	70	2	2	NUM
ejpam-6409	941	71	4	4	NUM
ejpam-6409	941	72	6	6	NUM
ejpam-6409	941	73	10	10	NUM
ejpam-6409	941	74	z128[ω	z128[ω	NUM
ejpam-6409	941	75	]	]	PUNCT
ejpam-6409	941	76	0	0	NUM
ejpam-6409	941	77	0	0	NUM
ejpam-6409	941	78	0	0	NUM
ejpam-6409	941	79	1	1	NUM
ejpam-6409	941	80	1	1	NUM
ejpam-6409	941	81	1	1	NUM
ejpam-6409	941	82	2	2	NUM
ejpam-6409	941	83	4	4	NUM
ejpam-6409	941	84	6	6	NUM
ejpam-6409	941	85	z256[ω	z256[ω	NUM
ejpam-6409	941	86	]	]	X
ejpam-6409	941	87	0	0	NUM
ejpam-6409	941	88	0	0	NUM
ejpam-6409	941	89	0	0	NUM
ejpam-6409	941	90	0	0	NUM
ejpam-6409	941	91	1	1	NUM
ejpam-6409	941	92	1	1	NUM
ejpam-6409	941	93	1	1	NUM
ejpam-6409	941	94	2	2	NUM
ejpam-6409	941	95	4	4	NUM
ejpam-6409	941	96	z512[ω	z512[ω	ADV
ejpam-6409	941	97	]	]	PUNCT
ejpam-6409	941	98	0	0	NUM
ejpam-6409	941	99	0	0	NUM
ejpam-6409	941	100	0	0	NUM
ejpam-6409	941	101	0	0	NUM
ejpam-6409	941	102	0	0	NUM
ejpam-6409	941	103	1	1	NUM
ejpam-6409	941	104	1	1	NUM
ejpam-6409	941	105	1	1	NUM
ejpam-6409	941	106	2	2	NUM
ejpam-6409	941	107	codes	code	NOUN
ejpam-6409	941	108	are	be	AUX
ejpam-6409	941	109	shown	show	VERB
ejpam-6409	941	110	to	to	PART
ejpam-6409	941	111	be	be	AUX
ejpam-6409	941	112	distinct	distinct	ADJ
ejpam-6409	941	113	because	because	SCONJ
ejpam-6409	941	114	they	they	PRON
ejpam-6409	941	115	differ	differ	VERB
ejpam-6409	941	116	in	in	ADP
ejpam-6409	941	117	the	the	DET
ejpam-6409	941	118	kernel	kernel	NOUN
ejpam-6409	941	119	’s	’s	PART
ejpam-6409	941	120	dimension	dimension	NOUN
ejpam-6409	941	121	.	.	PUNCT
ejpam-6409	942	1	more	more	ADJ
ejpam-6409	942	2	examples	example	NOUN
ejpam-6409	942	3	can	can	AUX
ejpam-6409	942	4	be	be	AUX
ejpam-6409	942	5	seen	see	VERB
ejpam-6409	942	6	when	when	SCONJ
ejpam-6409	942	7	t	t	PROPN
ejpam-6409	942	8	is	be	AUX
ejpam-6409	942	9	an	an	DET
ejpam-6409	942	10	odd	odd	ADJ
ejpam-6409	942	11	number	number	NOUN
ejpam-6409	942	12	.	.	PUNCT
ejpam-6409	943	1	for	for	ADP
ejpam-6409	943	2	example	example	NOUN
ejpam-6409	943	3	,	,	PUNCT
ejpam-6409	943	4	based	base	VERB
ejpam-6409	943	5	on	on	ADP
ejpam-6409	943	6	tables	table	NOUN
ejpam-6409	943	7	2	2	NUM
ejpam-6409	943	8	and	and	CCONJ
ejpam-6409	943	9	5	5	NUM
ejpam-6409	943	10	,	,	PUNCT
ejpam-6409	943	11	when	when	SCONJ
ejpam-6409	943	12	t	t	NOUN
ejpam-6409	943	13	=	=	SYM
ejpam-6409	943	14	7	7	NUM
ejpam-6409	943	15	,	,	PUNCT
ejpam-6409	943	16	t	t	NOUN
ejpam-6409	943	17	=	=	SYM
ejpam-6409	943	18	9	9	NUM
ejpam-6409	943	19	,	,	PUNCT
ejpam-6409	943	20	and	and	CCONJ
ejpam-6409	943	21	t	t	X
ejpam-6409	943	22	=	=	SYM
ejpam-6409	943	23	11	11	NUM
ejpam-6409	943	24	,	,	PUNCT
ejpam-6409	943	25	we	we	PRON
ejpam-6409	943	26	observe	observe	VERB
ejpam-6409	943	27	that	that	SCONJ
ejpam-6409	943	28	the	the	DET
ejpam-6409	943	29	z4[ω]-linear	z4[ω]-linear	PROPN
ejpam-6409	943	30	hadamard	hadamard	PROPN
ejpam-6409	943	31	codes	code	NOUN
ejpam-6409	943	32	h(4,0	h(4,0	PRON
ejpam-6409	943	33	)	)	PUNCT
ejpam-6409	943	34	,	,	PUNCT
ejpam-6409	943	35	h(5,0	h(5,0	NOUN
ejpam-6409	943	36	)	)	PUNCT
ejpam-6409	943	37	,	,	PUNCT
ejpam-6409	943	38	and	and	CCONJ
ejpam-6409	943	39	h(6,0	h(6,0	AUX
ejpam-6409	943	40	)	)	PUNCT
ejpam-6409	943	41	do	do	AUX
ejpam-6409	943	42	not	not	PART
ejpam-6409	943	43	coincide	coincide	VERB
ejpam-6409	943	44	in	in	ADP
ejpam-6409	943	45	equivalence	equivalence	NOUN
ejpam-6409	943	46	with	with	ADP
ejpam-6409	943	47	any	any	DET
ejpam-6409	943	48	z2s	z2s	PROPN
ejpam-6409	943	49	[	[	X
ejpam-6409	943	50	ω]-linear	ω]-linear	ADJ
ejpam-6409	943	51	hadamard	hadamard	ADJ
ejpam-6409	943	52	codes	code	NOUN
ejpam-6409	943	53	of	of	ADP
ejpam-6409	943	54	the	the	DET
ejpam-6409	943	55	same	same	ADJ
ejpam-6409	943	56	length	length	NOUN
ejpam-6409	943	57	and	and	CCONJ
ejpam-6409	943	58	s	s	X
ejpam-6409	943	59	>	>	X
ejpam-6409	943	60	2	2	NUM
ejpam-6409	943	61	,	,	PUNCT
ejpam-6409	943	62	under	under	ADP
ejpam-6409	943	63	both	both	CCONJ
ejpam-6409	943	64	the	the	DET
ejpam-6409	943	65	rank	rank	NOUN
ejpam-6409	943	66	and	and	CCONJ
ejpam-6409	943	67	the	the	DET
ejpam-6409	943	68	dimension	dimension	NOUN
ejpam-6409	943	69	of	of	ADP
ejpam-6409	943	70	the	the	DET
ejpam-6409	943	71	kernel	kernel	NOUN
ejpam-6409	943	72	.	.	PUNCT
ejpam-6409	944	1	it	it	PRON
ejpam-6409	944	2	has	have	AUX
ejpam-6409	944	3	been	be	AUX
ejpam-6409	944	4	established	establish	VERB
ejpam-6409	944	5	that	that	SCONJ
ejpam-6409	944	6	for	for	ADP
ejpam-6409	944	7	z2	z2	PROPN
ejpam-6409	944	8	-	-	PUNCT
ejpam-6409	944	9	linear	linear	ADJ
ejpam-6409	944	10	hadamard	hadamard	ADJ
ejpam-6409	944	11	codes	code	NOUN
ejpam-6409	944	12	,	,	PUNCT
ejpam-6409	944	13	the	the	DET
ejpam-6409	944	14	lower	low	ADJ
ejpam-6409	944	15	bounds	bound	NOUN
ejpam-6409	944	16	k	k	X
ejpam-6409	944	17	(	(	PUNCT
ejpam-6409	944	18	the	the	DET
ejpam-6409	944	19	kernel	kernel	PROPN
ejpam-6409	944	20	dimension	dimension	PROPN
ejpam-6409	944	21	)	)	PUNCT
ejpam-6409	944	22	and	and	CCONJ
ejpam-6409	944	23	rk	rk	PROPN
ejpam-6409	944	24	(	(	PUNCT
ejpam-6409	944	25	the	the	DET
ejpam-6409	944	26	kernel	kernel	PROPN
ejpam-6409	944	27	dimension	dimension	NOUN
ejpam-6409	944	28	and	and	CCONJ
ejpam-6409	944	29	the	the	DET
ejpam-6409	944	30	rank	rank	NOUN
ejpam-6409	944	31	)	)	PUNCT
ejpam-6409	944	32	are	be	AUX
ejpam-6409	944	33	known	know	VERB
ejpam-6409	944	34	.	.	PUNCT
ejpam-6409	945	1	in	in	ADP
ejpam-6409	945	2	this	this	DET
ejpam-6409	945	3	paper	paper	NOUN
ejpam-6409	945	4	,	,	PUNCT
ejpam-6409	945	5	we	we	PRON
ejpam-6409	945	6	have	have	AUX
ejpam-6409	945	7	shown	show	VERB
ejpam-6409	945	8	that	that	SCONJ
ejpam-6409	945	9	for	for	ADP
ejpam-6409	945	10	both	both	DET
ejpam-6409	945	11	z2sand	z2sand	NOUN
ejpam-6409	945	12	z2s	z2s	X
ejpam-6409	946	1	[	[	X
ejpam-6409	946	2	ω]-linear	ω]-linear	ADJ
ejpam-6409	946	3	hadamard	hadamard	ADJ
ejpam-6409	946	4	codes	code	NOUN
ejpam-6409	946	5	,	,	PUNCT
ejpam-6409	946	6	the	the	DET
ejpam-6409	946	7	upper	upper	ADJ
ejpam-6409	946	8	bounds	bound	NOUN
ejpam-6409	946	9	k	k	PROPN
ejpam-6409	946	10	and	and	CCONJ
ejpam-6409	946	11	rk	rk	PROPN
ejpam-6409	946	12	are	be	AUX
ejpam-6409	946	13	the	the	DET
ejpam-6409	946	14	same	same	ADJ
ejpam-6409	946	15	.	.	PUNCT
ejpam-6409	947	1	the	the	DET
ejpam-6409	947	2	table	table	NOUN
ejpam-6409	947	3	4	4	NUM
ejpam-6409	947	4	contains	contain	VERB
ejpam-6409	947	5	bounds	bound	NOUN
ejpam-6409	947	6	for	for	ADP
ejpam-6409	947	7	the	the	DET
ejpam-6409	947	8	range	range	NOUN
ejpam-6409	947	9	3	3	NUM
ejpam-6409	947	10	≤	≤	NOUN
ejpam-6409	947	11	t	t	PROPN
ejpam-6409	947	12	≤	≤	NUM
ejpam-6409	947	13	11	11	NUM
ejpam-6409	947	14	.	.	PUNCT
ejpam-6409	947	15	table	table	NOUN
ejpam-6409	947	16	4	4	NUM
ejpam-6409	947	17	:	:	PUNCT
ejpam-6409	947	18	bounds	bound	VERB
ejpam-6409	947	19	for	for	ADP
ejpam-6409	947	20	the	the	DET
ejpam-6409	947	21	number	number	NOUN
ejpam-6409	947	22	at	at	ADP
ejpam-6409	947	23	of	of	ADP
ejpam-6409	947	24	nonequivalent	nonequivalent	ADJ
ejpam-6409	947	25	z2s	z2s	PROPN
ejpam-6409	948	1	[	[	X
ejpam-6409	948	2	ω]-linear	ω]-linear	ADJ
ejpam-6409	948	3	hadamard	hadamard	ADJ
ejpam-6409	948	4	codes	code	NOUN
ejpam-6409	948	5	of	of	ADP
ejpam-6409	948	6	length	length	NOUN
ejpam-6409	948	7	22	22	NUM
ejpam-6409	948	8	t	t	NOUN
ejpam-6409	948	9	t	t	NOUN
ejpam-6409	948	10	3	3	NUM
ejpam-6409	948	11	4	4	NUM
ejpam-6409	948	12	5	5	NUM
ejpam-6409	948	13	6	6	NUM
ejpam-6409	948	14	7	7	NUM
ejpam-6409	948	15	8	8	NUM
ejpam-6409	948	16	9	9	NUM
ejpam-6409	948	17	10	10	NUM
ejpam-6409	948	18	11	11	NUM
ejpam-6409	948	19	lower	lower	ADV
ejpam-6409	948	20	bound	bind	VERB
ejpam-6409	948	21	k	k	PROPN
ejpam-6409	948	22	1	1	NUM
ejpam-6409	948	23	1	1	NUM
ejpam-6409	948	24	3	3	NUM
ejpam-6409	948	25	3	3	NUM
ejpam-6409	948	26	5	5	NUM
ejpam-6409	948	27	5	5	NUM
ejpam-6409	948	28	7	7	NUM
ejpam-6409	948	29	7	7	NUM
ejpam-6409	948	30	9	9	NUM
ejpam-6409	948	31	lower	lower	ADV
ejpam-6409	948	32	bound	bind	VERB
ejpam-6409	948	33	rk	rk	NOUN
ejpam-6409	948	34	1	1	NUM
ejpam-6409	948	35	1	1	NUM
ejpam-6409	948	36	3	3	NUM
ejpam-6409	948	37	3	3	NUM
ejpam-6409	948	38	6	6	NUM
ejpam-6409	948	39	7	7	NUM
ejpam-6409	948	40	11	11	NUM
ejpam-6409	948	41	13	13	NUM
ejpam-6409	948	42	20	20	NUM
ejpam-6409	948	43	upper	upper	ADV
ejpam-6409	948	44	bound	bind	VERB
ejpam-6409	948	45	1	1	NUM
ejpam-6409	948	46	1	1	NUM
ejpam-6409	948	47	3	3	NUM
ejpam-6409	948	48	5	5	NUM
ejpam-6409	948	49	10	10	NUM
ejpam-6409	948	50	16	16	NUM
ejpam-6409	948	51	26	26	NUM
ejpam-6409	948	52	38	38	NUM
ejpam-6409	948	53	57	57	NUM
ejpam-6409	948	54	by	by	ADP
ejpam-6409	948	55	studying	study	VERB
ejpam-6409	948	56	all	all	DET
ejpam-6409	948	57	nonequivalent	nonequivalent	ADJ
ejpam-6409	948	58	z2s	z2	NOUN
ejpam-6409	949	1	[	[	X
ejpam-6409	949	2	ω]-linear	ω]-linear	ADJ
ejpam-6409	949	3	hadamard	hadamard	ADJ
ejpam-6409	949	4	codes	code	NOUN
ejpam-6409	949	5	of	of	ADP
ejpam-6409	949	6	length	length	NOUN
ejpam-6409	949	7	22	22	NUM
ejpam-6409	949	8	t	t	PROPN
ejpam-6409	949	9	,	,	PUNCT
ejpam-6409	949	10	it	it	PRON
ejpam-6409	949	11	is	be	AUX
ejpam-6409	949	12	possible	possible	ADJ
ejpam-6409	949	13	to	to	PART
ejpam-6409	949	14	produce	produce	VERB
ejpam-6409	949	15	an	an	DET
ejpam-6409	949	16	easy	easy	ADJ
ejpam-6409	949	17	upper	upper	ADJ
ejpam-6409	949	18	bound	bind	VERB
ejpam-6409	949	19	calculation	calculation	NOUN
ejpam-6409	949	20	if	if	SCONJ
ejpam-6409	949	21	t	t	PROPN
ejpam-6409	949	22	and	and	CCONJ
ejpam-6409	949	23	s	s	PRON
ejpam-6409	949	24	are	be	AUX
ejpam-6409	949	25	given	give	VERB
ejpam-6409	949	26	.	.	PUNCT
ejpam-6409	950	1	table	table	NOUN
ejpam-6409	950	2	4	4	NUM
ejpam-6409	950	3	shows	show	VERB
ejpam-6409	950	4	the	the	DET
ejpam-6409	950	5	values	value	NOUN
ejpam-6409	950	6	for	for	ADP
ejpam-6409	950	7	all	all	DET
ejpam-6409	950	8	3	3	NUM
ejpam-6409	950	9	≤	≤	NOUN
ejpam-6409	950	10	t	t	NOUN
ejpam-6409	950	11	≤	≤	NUM
ejpam-6409	950	12	11	11	NUM
ejpam-6409	950	13	.	.	PUNCT
ejpam-6409	951	1	theorem	theorem	VERB
ejpam-6409	951	2	7.3	7.3	NUM
ejpam-6409	951	3	:	:	PUNCT
ejpam-6409	951	4	let	let	VERB
ejpam-6409	951	5	a(t	a(t	VERB
ejpam-6409	951	6	,	,	PUNCT
ejpam-6409	951	7	s	s	PART
ejpam-6409	951	8	)	)	PUNCT
ejpam-6409	951	9	be	be	AUX
ejpam-6409	951	10	defined	define	VERB
ejpam-6409	951	11	as	as	ADP
ejpam-6409	951	12	the	the	DET
ejpam-6409	951	13	number	number	NOUN
ejpam-6409	951	14	of	of	ADP
ejpam-6409	951	15	nonequivalent	nonequivalent	ADJ
ejpam-6409	951	16	z2s	z2s	PROPN
ejpam-6409	952	1	[	[	X
ejpam-6409	952	2	ω]-linear	ω]-linear	ADJ
ejpam-6409	952	3	hadamard	hadamard	ADJ
ejpam-6409	952	4	codes	code	NOUN
ejpam-6409	952	5	of	of	ADP
ejpam-6409	952	6	length	length	NOUN
ejpam-6409	952	7	22	22	NUM
ejpam-6409	952	8	t.	t.	NOUN
ejpam-6409	952	9	let	let	VERB
ejpam-6409	952	10	at	at	PART
ejpam-6409	952	11	denote	denote	VERB
ejpam-6409	952	12	the	the	DET
ejpam-6409	952	13	total	total	ADJ
ejpam-6409	952	14	number	number	NOUN
ejpam-6409	952	15	of	of	ADP
ejpam-6409	952	16	distinct	distinct	ADJ
ejpam-6409	952	17	z2s	z2	NOUN
ejpam-6409	953	1	[	[	X
ejpam-6409	953	2	ω]-linear	ω]-linear	ADJ
ejpam-6409	953	3	hadamard	hadamard	ADJ
ejpam-6409	953	4	codes	code	NOUN
ejpam-6409	953	5	of	of	ADP
ejpam-6409	953	6	length	length	NOUN
ejpam-6409	953	7	22	22	NUM
ejpam-6409	953	8	t	t	NOUN
ejpam-6409	953	9	for	for	ADP
ejpam-6409	953	10	s	s	PRON
ejpam-6409	953	11	≥	≥	NOUN
ejpam-6409	953	12	2	2	NUM
ejpam-6409	953	13	.	.	PUNCT
ejpam-6409	954	1	then	then	ADV
ejpam-6409	954	2	,	,	PUNCT
ejpam-6409	954	3	at	at	ADP
ejpam-6409	954	4	≤	≤	NOUN
ejpam-6409	954	5	t−2∑	t−2∑	CCONJ
ejpam-6409	954	6	s=2	s=2	PRON
ejpam-6409	954	7	(	(	PUNCT
ejpam-6409	954	8	a(t	a(t	PROPN
ejpam-6409	954	9	,	,	PUNCT
ejpam-6409	954	10	s	s	NOUN
ejpam-6409	954	11	)	)	PUNCT
ejpam-6409	954	12	−	−	PROPN
ejpam-6409	954	13	1	1	NUM
ejpam-6409	954	14	)	)	PUNCT
ejpam-6409	955	1	+	+	CCONJ
ejpam-6409	955	2	1	1	X
ejpam-6409	955	3	.	.	X
ejpam-6409	955	4	theorem	theorem	VERB
ejpam-6409	955	5	7.4	7.4	NUM
ejpam-6409	955	6	:	:	PUNCT
ejpam-6409	955	7	for	for	ADP
ejpam-6409	955	8	lengths	length	NOUN
ejpam-6409	955	9	22	22	NUM
ejpam-6409	955	10	t	t	PROPN
ejpam-6409	955	11	,	,	PUNCT
ejpam-6409	955	12	where	where	SCONJ
ejpam-6409	955	13	t	t	NOUN
ejpam-6409	955	14	=	=	SYM
ejpam-6409	955	15	3	3	NUM
ejpam-6409	955	16	,	,	PUNCT
ejpam-6409	955	17	4	4	NUM
ejpam-6409	955	18	,	,	PUNCT
ejpam-6409	955	19	5	5	NUM
ejpam-6409	955	20	,	,	PUNCT
ejpam-6409	955	21	6	6	NUM
ejpam-6409	955	22	and	and	CCONJ
ejpam-6409	955	23	7	7	NUM
ejpam-6409	955	24	,	,	PUNCT
ejpam-6409	955	25	there	there	PRON
ejpam-6409	955	26	exist	exist	VERB
ejpam-6409	955	27	exactly	exactly	ADV
ejpam-6409	955	28	1	1	NUM
ejpam-6409	955	29	,	,	PUNCT
ejpam-6409	955	30	1	1	NUM
ejpam-6409	955	31	,	,	PUNCT
ejpam-6409	955	32	3	3	NUM
ejpam-6409	955	33	,	,	PUNCT
ejpam-6409	955	34	3	3	NUM
ejpam-6409	955	35	,	,	PUNCT
ejpam-6409	955	36	and	and	CCONJ
ejpam-6409	955	37	6	6	NUM
ejpam-6409	955	38	distinct	distinct	ADJ
ejpam-6409	955	39	nonequivalent	nonequivalent	ADJ
ejpam-6409	955	40	z2s	z2	NOUN
ejpam-6409	956	1	[	[	X
ejpam-6409	956	2	ω]-linear	ω]-linear	ADJ
ejpam-6409	956	3	hadamard	hadamard	ADJ
ejpam-6409	956	4	codes	code	NOUN
ejpam-6409	956	5	,	,	PUNCT
ejpam-6409	956	6	respectively	respectively	ADV
ejpam-6409	956	7	.	.	PUNCT
ejpam-6409	956	8	8	8	NUM
ejpam-6409	956	9	.	.	X
ejpam-6409	956	10	conclusion	conclusion	NOUN
ejpam-6409	956	11	and	and	CCONJ
ejpam-6409	956	12	future	future	ADJ
ejpam-6409	956	13	directions	direction	NOUN
ejpam-6409	956	14	our	our	PRON
ejpam-6409	956	15	research	research	NOUN
ejpam-6409	956	16	created	create	VERB
ejpam-6409	956	17	and	and	CCONJ
ejpam-6409	956	18	studied	study	VERB
ejpam-6409	956	19	generalized	generalized	ADJ
ejpam-6409	956	20	hadamard	hadamard	NOUN
ejpam-6409	956	21	(	(	PUNCT
ejpam-6409	956	22	gh	gh	PROPN
ejpam-6409	956	23	)	)	PUNCT
ejpam-6409	956	24	codes	code	NOUN
ejpam-6409	956	25	using	use	VERB
ejpam-6409	956	26	eisenstein	eisenstein	PROPN
ejpam-6409	956	27	local	local	ADJ
ejpam-6409	956	28	rings	ring	NOUN
ejpam-6409	956	29	z2s	z2s	PROPN
ejpam-6409	957	1	[	[	X
ejpam-6409	957	2	ω	ω	X
ejpam-6409	957	3	]	]	X
ejpam-6409	957	4	to	to	PART
ejpam-6409	957	5	develop	develop	VERB
ejpam-6409	957	6	their	their	PRON
ejpam-6409	957	7	complete	complete	ADJ
ejpam-6409	957	8	set	set	NOUN
ejpam-6409	957	9	of	of	ADP
ejpam-6409	957	10	algebraic	algebraic	ADJ
ejpam-6409	957	11	and	and	CCONJ
ejpam-6409	957	12	combinatorial	combinatorial	ADJ
ejpam-6409	957	13	properties	property	NOUN
ejpam-6409	957	14	.	.	PUNCT
ejpam-6409	958	1	muhammad	muhammad	PROPN
ejpam-6409	958	2	sajjad	sajjad	PROPN
ejpam-6409	958	3	et	et	PROPN
ejpam-6409	958	4	al	al	PROPN
ejpam-6409	958	5	.	.	PUNCT
ejpam-6409	958	6	/	/	SYM
ejpam-6409	958	7	eur	eur	PROPN
ejpam-6409	958	8	.	.	PUNCT
ejpam-6409	959	1	j.	j.	PROPN
ejpam-6409	959	2	pure	pure	PROPN
ejpam-6409	959	3	appl	appl	PROPN
ejpam-6409	959	4	.	.	PROPN
ejpam-6409	959	5	math	math	PROPN
ejpam-6409	959	6	,	,	PUNCT
ejpam-6409	959	7	18	18	NUM
ejpam-6409	959	8	(	(	PUNCT
ejpam-6409	959	9	3	3	NUM
ejpam-6409	959	10	)	)	PUNCT
ejpam-6409	959	11	(	(	PUNCT
ejpam-6409	959	12	2025	2025	NUM
ejpam-6409	959	13	)	)	PUNCT
ejpam-6409	959	14	,	,	PUNCT
ejpam-6409	959	15	6409	6409	NUM
ejpam-6409	959	16	29	29	NUM
ejpam-6409	959	17	of	of	ADP
ejpam-6409	959	18	32	32	NUM
ejpam-6409	959	19	our	our	PRON
ejpam-6409	959	20	use	use	NOUN
ejpam-6409	959	21	of	of	ADP
ejpam-6409	959	22	eisenstein	eisenstein	NOUN
ejpam-6409	959	23	integers	integer	NOUN
ejpam-6409	959	24	established	establish	VERB
ejpam-6409	959	25	a	a	DET
ejpam-6409	959	26	gray	gray	ADJ
ejpam-6409	959	27	map	map	NOUN
ejpam-6409	959	28	to	to	PART
ejpam-6409	959	29	switch	switch	VERB
ejpam-6409	959	30	from	from	ADP
ejpam-6409	959	31	these	these	DET
ejpam-6409	959	32	codes	code	NOUN
ejpam-6409	959	33	to	to	ADP
ejpam-6409	959	34	binary	binary	ADJ
ejpam-6409	959	35	numbers	number	NOUN
ejpam-6409	959	36	for	for	ADP
ejpam-6409	959	37	expert	expert	NOUN
ejpam-6409	959	38	study	study	NOUN
ejpam-6409	959	39	.	.	PUNCT
ejpam-6409	960	1	our	our	PRON
ejpam-6409	960	2	research	research	NOUN
ejpam-6409	960	3	revealed	reveal	VERB
ejpam-6409	960	4	specific	specific	ADJ
ejpam-6409	960	5	rules	rule	NOUN
ejpam-6409	960	6	to	to	PART
ejpam-6409	960	7	distinguish	distinguish	VERB
ejpam-6409	960	8	z2s	z2	NOUN
ejpam-6409	960	9	[	[	X
ejpam-6409	960	10	ω]-linear	ω]-linear	ADJ
ejpam-6409	960	11	gh	gh	PROPN
ejpam-6409	960	12	codes	code	NOUN
ejpam-6409	960	13	that	that	PRON
ejpam-6409	960	14	are	be	AUX
ejpam-6409	960	15	linear	linear	ADJ
ejpam-6409	960	16	from	from	ADP
ejpam-6409	960	17	those	those	PRON
ejpam-6409	960	18	that	that	PRON
ejpam-6409	960	19	are	be	AUX
ejpam-6409	960	20	not	not	PART
ejpam-6409	960	21	.	.	PUNCT
ejpam-6409	961	1	our	our	PRON
ejpam-6409	961	2	research	research	NOUN
ejpam-6409	961	3	into	into	ADP
ejpam-6409	961	4	the	the	DET
ejpam-6409	961	5	kernel	kernel	NOUN
ejpam-6409	961	6	properties	property	NOUN
ejpam-6409	961	7	helps	helps	AUX
ejpam-6409	961	8	reveal	reveal	VERB
ejpam-6409	961	9	how	how	SCONJ
ejpam-6409	961	10	these	these	DET
ejpam-6409	961	11	codes	code	NOUN
ejpam-6409	961	12	differ	differ	VERB
ejpam-6409	961	13	from	from	ADP
ejpam-6409	961	14	one	one	NUM
ejpam-6409	961	15	another	another	DET
ejpam-6409	961	16	.	.	PUNCT
ejpam-6409	962	1	the	the	DET
ejpam-6409	962	2	theory	theory	NOUN
ejpam-6409	962	3	proves	prove	VERB
ejpam-6409	962	4	that	that	SCONJ
ejpam-6409	962	5	eisenstein	eisenstein	PROPN
ejpam-6409	962	6	fields	field	NOUN
ejpam-6409	962	7	offer	offer	VERB
ejpam-6409	962	8	effective	effective	ADJ
ejpam-6409	962	9	ways	way	NOUN
ejpam-6409	962	10	to	to	PART
ejpam-6409	962	11	build	build	VERB
ejpam-6409	962	12	useful	useful	ADJ
ejpam-6409	962	13	and	and	CCONJ
ejpam-6409	962	14	space	space	NOUN
ejpam-6409	962	15	-	-	PUNCT
ejpam-6409	962	16	saving	save	VERB
ejpam-6409	962	17	error	error	NOUN
ejpam-6409	962	18	correction	correction	NOUN
ejpam-6409	962	19	codes	code	NOUN
ejpam-6409	962	20	.	.	PUNCT
ejpam-6409	963	1	the	the	DET
ejpam-6409	963	2	future	future	NOUN
ejpam-6409	963	3	of	of	ADP
ejpam-6409	963	4	research	research	NOUN
ejpam-6409	963	5	should	should	AUX
ejpam-6409	963	6	study	study	VERB
ejpam-6409	963	7	different	different	ADJ
ejpam-6409	963	8	families	family	NOUN
ejpam-6409	963	9	of	of	ADP
ejpam-6409	963	10	error	error	NOUN
ejpam-6409	963	11	-	-	PUNCT
ejpam-6409	963	12	correcting	correct	VERB
ejpam-6409	963	13	codes	code	NOUN
ejpam-6409	963	14	using	use	VERB
ejpam-6409	963	15	eisenstein	eisenstein	PROPN
ejpam-6409	963	16	algebraic	algebraic	PROPN
ejpam-6409	963	17	rings	ring	NOUN
ejpam-6409	963	18	and	and	CCONJ
ejpam-6409	963	19	their	their	PRON
ejpam-6409	963	20	relatives	relative	NOUN
ejpam-6409	963	21	,	,	PUNCT
ejpam-6409	963	22	such	such	ADJ
ejpam-6409	963	23	as	as	ADP
ejpam-6409	963	24	cyclic	cyclic	ADJ
ejpam-6409	963	25	,	,	PUNCT
ejpam-6409	963	26	quasi	quasi	ADJ
ejpam-6409	963	27	-	-	ADJ
ejpam-6409	963	28	cyclic	cyclic	ADJ
ejpam-6409	963	29	,	,	PUNCT
ejpam-6409	963	30	and	and	CCONJ
ejpam-6409	963	31	consta	consta	ADJ
ejpam-6409	963	32	-	-	PUNCT
ejpam-6409	963	33	cyclic	cyclic	ADJ
ejpam-6409	963	34	codes	code	NOUN
ejpam-6409	963	35	.	.	PUNCT
ejpam-6409	964	1	studying	study	VERB
ejpam-6409	964	2	automorphism	automorphism	NOUN
ejpam-6409	964	3	groups	group	NOUN
ejpam-6409	964	4	and	and	CCONJ
ejpam-6409	964	5	decoding	decode	VERB
ejpam-6409	964	6	methods	method	NOUN
ejpam-6409	964	7	for	for	ADP
ejpam-6409	964	8	gh	gh	PROPN
ejpam-6409	964	9	codes	code	NOUN
ejpam-6409	964	10	with	with	ADP
ejpam-6409	964	11	z2s	z2s	PROPN
ejpam-6409	964	12	[	[	X
ejpam-6409	964	13	ω	ω	X
ejpam-6409	964	14	]	]	X
ejpam-6409	964	15	as	as	SCONJ
ejpam-6409	964	16	field	field	NOUN
ejpam-6409	964	17	structure	structure	NOUN
ejpam-6409	964	18	will	will	AUX
ejpam-6409	964	19	lead	lead	VERB
ejpam-6409	964	20	to	to	ADP
ejpam-6409	964	21	theoretical	theoretical	ADJ
ejpam-6409	964	22	advancement	advancement	NOUN
ejpam-6409	964	23	and	and	CCONJ
ejpam-6409	964	24	practical	practical	ADJ
ejpam-6409	964	25	applications	application	NOUN
ejpam-6409	964	26	.	.	PUNCT
ejpam-6409	965	1	acknowledgements	acknowledgement	NOUN
ejpam-6409	965	2	this	this	DET
ejpam-6409	965	3	research	research	NOUN
ejpam-6409	965	4	is	be	AUX
ejpam-6409	965	5	supported	support	VERB
ejpam-6409	965	6	by	by	ADP
ejpam-6409	965	7	universidad	universidad	PROPN
ejpam-6409	965	8	pedagógica	pedagógica	PROPN
ejpam-6409	965	9	y	y	PROPN
ejpam-6409	965	10	tecnológica	tecnológica	PROPN
ejpam-6409	965	11	de	de	PROPN
ejpam-6409	965	12	colombia	colombia	PROPN
ejpam-6409	965	13	578	578	NUM
ejpam-6409	965	14	(	(	PUNCT
ejpam-6409	965	15	sgi	sgi	PROPN
ejpam-6409	965	16	3725	3725	NUM
ejpam-6409	965	17	)	)	PUNCT
ejpam-6409	965	18	and	and	CCONJ
ejpam-6409	965	19	minciencias	minciencia	NOUN
ejpam-6409	965	20	(	(	PUNCT
ejpam-6409	965	21	conv	conv	ADJ
ejpam-6409	965	22	.	.	PROPN
ejpam-6409	965	23	934	934	NUM
ejpam-6409	965	24	)	)	PUNCT
ejpam-6409	965	25	.	.	PUNCT
ejpam-6409	966	1	tribute	tribute	NOUN
ejpam-6409	966	2	we	we	PRON
ejpam-6409	966	3	would	would	AUX
ejpam-6409	966	4	like	like	VERB
ejpam-6409	966	5	to	to	PART
ejpam-6409	966	6	express	express	VERB
ejpam-6409	966	7	our	our	PRON
ejpam-6409	966	8	heartfelt	heartfelt	ADJ
ejpam-6409	966	9	gratitude	gratitude	NOUN
ejpam-6409	966	10	to	to	ADP
ejpam-6409	966	11	our	our	PRON
ejpam-6409	966	12	beloved	beloved	ADJ
ejpam-6409	966	13	supervisor	supervisor	NOUN
ejpam-6409	966	14	,	,	PUNCT
ejpam-6409	966	15	professor	professor	PROPN
ejpam-6409	966	16	dr	dr	PROPN
ejpam-6409	966	17	.	.	PROPN
ejpam-6409	966	18	tariq	tariq	PROPN
ejpam-6409	966	19	shah	shah	PROPN
ejpam-6409	966	20	(	(	PUNCT
ejpam-6409	966	21	late	late	ADJ
ejpam-6409	966	22	)	)	PUNCT
ejpam-6409	966	23	,	,	PUNCT
ejpam-6409	966	24	whose	whose	DET
ejpam-6409	966	25	exceptional	exceptional	ADJ
ejpam-6409	966	26	guidance	guidance	NOUN
ejpam-6409	966	27	,	,	PUNCT
ejpam-6409	966	28	profound	profound	ADJ
ejpam-6409	966	29	expertise	expertise	NOUN
ejpam-6409	966	30	,	,	PUNCT
ejpam-6409	966	31	and	and	CCONJ
ejpam-6409	966	32	steadfast	steadfast	ADJ
ejpam-6409	966	33	support	support	NOUN
ejpam-6409	966	34	were	be	AUX
ejpam-6409	966	35	instrumental	instrumental	ADJ
ejpam-6409	966	36	in	in	ADP
ejpam-6409	966	37	shaping	shape	VERB
ejpam-6409	966	38	our	our	PRON
ejpam-6409	966	39	academic	academic	ADJ
ejpam-6409	966	40	journey	journey	NOUN
ejpam-6409	966	41	.	.	PUNCT
ejpam-6409	967	1	his	his	PRON
ejpam-6409	967	2	mentorship	mentorship	NOUN
ejpam-6409	967	3	not	not	PART
ejpam-6409	967	4	only	only	ADV
ejpam-6409	967	5	nurtured	nurture	VERB
ejpam-6409	967	6	our	our	PRON
ejpam-6409	967	7	growth	growth	NOUN
ejpam-6409	967	8	as	as	SCONJ
ejpam-6409	967	9	researchers	researcher	NOUN
ejpam-6409	967	10	in	in	ADP
ejpam-6409	967	11	algebra	algebra	NOUN
ejpam-6409	967	12	,	,	PUNCT
ejpam-6409	967	13	number	number	NOUN
ejpam-6409	967	14	theory	theory	NOUN
ejpam-6409	967	15	,	,	PUNCT
ejpam-6409	967	16	coding	code	VERB
ejpam-6409	967	17	theory	theory	NOUN
ejpam-6409	967	18	,	,	PUNCT
ejpam-6409	967	19	and	and	CCONJ
ejpam-6409	967	20	cryptography	cryptography	NOUN
ejpam-6409	967	21	but	but	CCONJ
ejpam-6409	967	22	also	also	ADV
ejpam-6409	967	23	profoundly	profoundly	ADV
ejpam-6409	967	24	influenced	influence	VERB
ejpam-6409	967	25	our	our	PRON
ejpam-6409	967	26	personal	personal	ADJ
ejpam-6409	967	27	and	and	CCONJ
ejpam-6409	967	28	professional	professional	ADJ
ejpam-6409	967	29	development	development	NOUN
ejpam-6409	967	30	.	.	PUNCT
ejpam-6409	968	1	his	his	PRON
ejpam-6409	968	2	legacy	legacy	NOUN
ejpam-6409	968	3	of	of	ADP
ejpam-6409	968	4	wisdom	wisdom	NOUN
ejpam-6409	968	5	,	,	PUNCT
ejpam-6409	968	6	integrity	integrity	NOUN
ejpam-6409	968	7	,	,	PUNCT
ejpam-6409	968	8	and	and	CCONJ
ejpam-6409	968	9	inspiration	inspiration	NOUN
ejpam-6409	968	10	continues	continue	VERB
ejpam-6409	968	11	to	to	PART
ejpam-6409	968	12	guide	guide	VERB
ejpam-6409	968	13	us	we	PRON
ejpam-6409	968	14	.	.	PUNCT
ejpam-6409	969	1	may	may	AUX
ejpam-6409	969	2	his	his	PRON
ejpam-6409	969	3	soul	soul	NOUN
ejpam-6409	969	4	rest	rest	VERB
ejpam-6409	969	5	in	in	ADP
ejpam-6409	969	6	eternal	eternal	ADJ
ejpam-6409	969	7	peace	peace	NOUN
ejpam-6409	969	8	.	.	PUNCT
ejpam-6409	970	1	figure	figure	NOUN
ejpam-6409	970	2	1	1	NUM
ejpam-6409	970	3	:	:	PUNCT
ejpam-6409	970	4	prof	prof	PROPN
ejpam-6409	970	5	.	.	PUNCT
ejpam-6409	971	1	dr	dr	PROPN
ejpam-6409	971	2	.	.	PROPN
ejpam-6409	971	3	tariq	tariq	PROPN
ejpam-6409	971	4	shah	shah	PROPN
ejpam-6409	971	5	muhammad	muhammad	PROPN
ejpam-6409	971	6	sajjad	sajjad	PROPN
ejpam-6409	971	7	et	et	PROPN
ejpam-6409	971	8	al	al	PROPN
ejpam-6409	971	9	.	.	PUNCT
ejpam-6409	971	10	/	/	SYM
ejpam-6409	971	11	eur	eur	PROPN
ejpam-6409	971	12	.	.	PUNCT
ejpam-6409	972	1	j.	j.	PROPN
ejpam-6409	972	2	pure	pure	PROPN
ejpam-6409	972	3	appl	appl	PROPN
ejpam-6409	972	4	.	.	PROPN
ejpam-6409	972	5	math	math	PROPN
ejpam-6409	972	6	,	,	PUNCT
ejpam-6409	972	7	18	18	NUM
ejpam-6409	972	8	(	(	PUNCT
ejpam-6409	972	9	3	3	NUM
ejpam-6409	972	10	)	)	PUNCT
ejpam-6409	972	11	(	(	PUNCT
ejpam-6409	972	12	2025	2025	NUM
ejpam-6409	972	13	)	)	PUNCT
ejpam-6409	972	14	,	,	PUNCT
ejpam-6409	972	15	6409	6409	NUM
ejpam-6409	972	16	30	30	NUM
ejpam-6409	972	17	of	of	ADP
ejpam-6409	972	18	32	32	NUM
ejpam-6409	972	19	data	datum	NOUN
ejpam-6409	972	20	availability	availability	NOUN
ejpam-6409	972	21	all	all	DET
ejpam-6409	972	22	the	the	DET
ejpam-6409	972	23	data	datum	NOUN
ejpam-6409	972	24	is	be	AUX
ejpam-6409	972	25	given	give	VERB
ejpam-6409	972	26	in	in	ADP
ejpam-6409	972	27	this	this	DET
ejpam-6409	972	28	study	study	NOUN
ejpam-6409	972	29	.	.	PUNCT
ejpam-6409	973	1	references	reference	NOUN
ejpam-6409	973	2	[	[	X
ejpam-6409	973	3	1	1	X
ejpam-6409	973	4	]	]	PUNCT
ejpam-6409	973	5	e.	e.	PROPN
ejpam-6409	973	6	f.	f.	PROPN
ejpam-6409	973	7	assmus	assmus	PROPN
ejpam-6409	973	8	and	and	CCONJ
ejpam-6409	973	9	j.	j.	PROPN
ejpam-6409	973	10	d.	d.	PROPN
ejpam-6409	973	11	key	key	PROPN
ejpam-6409	973	12	.	.	PUNCT
ejpam-6409	974	1	designs	design	NOUN
ejpam-6409	974	2	and	and	CCONJ
ejpam-6409	974	3	their	their	PRON
ejpam-6409	974	4	codes	code	NOUN
ejpam-6409	974	5	.	.	PUNCT
ejpam-6409	975	1	number	number	NOUN
ejpam-6409	975	2	103	103	NUM
ejpam-6409	975	3	.	.	PUNCT
ejpam-6409	976	1	cambridge	cambridge	PROPN
ejpam-6409	976	2	university	university	PROPN
ejpam-6409	976	3	press	press	NOUN
ejpam-6409	976	4	,	,	PUNCT
ejpam-6409	976	5	1994	1994	NUM
ejpam-6409	976	6	.	.	PUNCT
ejpam-6409	977	1	[	[	X
ejpam-6409	977	2	2	2	NUM
ejpam-6409	977	3	]	]	PUNCT
ejpam-6409	977	4	h.	h.	PROPN
ejpam-6409	977	5	bauer	bauer	PROPN
ejpam-6409	977	6	,	,	PUNCT
ejpam-6409	977	7	b.	b.	PROPN
ejpam-6409	977	8	ganter	ganter	PROPN
ejpam-6409	977	9	,	,	PUNCT
ejpam-6409	977	10	and	and	CCONJ
ejpam-6409	977	11	f.	f.	PROPN
ejpam-6409	977	12	hergert	hergert	PROPN
ejpam-6409	977	13	.	.	PUNCT
ejpam-6409	978	1	algebraic	algebraic	ADJ
ejpam-6409	978	2	techniques	technique	NOUN
ejpam-6409	978	3	for	for	ADP
ejpam-6409	978	4	nonlinear	nonlinear	ADJ
ejpam-6409	978	5	codes	code	NOUN
ejpam-6409	978	6	.	.	PUNCT
ejpam-6409	979	1	combinatorica	combinatorica	PROPN
ejpam-6409	979	2	,	,	PUNCT
ejpam-6409	979	3	3:21–33	3:21–33	NUM
ejpam-6409	979	4	,	,	PUNCT
ejpam-6409	979	5	1983	1983	NUM
ejpam-6409	979	6	.	.	PUNCT
ejpam-6409	980	1	[	[	X
ejpam-6409	980	2	3	3	NUM
ejpam-6409	980	3	]	]	X
ejpam-6409	980	4	r.	r.	PROPN
ejpam-6409	980	5	c.	c.	PROPN
ejpam-6409	980	6	bose	bose	PROPN
ejpam-6409	980	7	and	and	CCONJ
ejpam-6409	980	8	k.	k.	PROPN
ejpam-6409	980	9	a.	a.	PROPN
ejpam-6409	980	10	bush	bush	PROPN
ejpam-6409	980	11	.	.	PUNCT
ejpam-6409	981	1	orthogonal	orthogonal	ADJ
ejpam-6409	981	2	arrays	array	NOUN
ejpam-6409	981	3	of	of	ADP
ejpam-6409	981	4	strength	strength	NOUN
ejpam-6409	981	5	two	two	NUM
ejpam-6409	981	6	and	and	CCONJ
ejpam-6409	981	7	three	three	NUM
ejpam-6409	981	8	.	.	PUNCT
ejpam-6409	982	1	the	the	DET
ejpam-6409	982	2	annals	annal	NOUN
ejpam-6409	982	3	of	of	ADP
ejpam-6409	982	4	mathematical	mathematical	ADJ
ejpam-6409	982	5	statistics	statistic	NOUN
ejpam-6409	982	6	,	,	PUNCT
ejpam-6409	982	7	23(4):508–524	23(4):508–524	PROPN
ejpam-6409	982	8	,	,	PUNCT
ejpam-6409	982	9	1952	1952	NUM
ejpam-6409	982	10	.	.	PUNCT
ejpam-6409	983	1	[	[	X
ejpam-6409	983	2	4	4	X
ejpam-6409	983	3	]	]	PUNCT
ejpam-6409	983	4	m.	m.	NOUN
ejpam-6409	983	5	sajjad	sajjad	PROPN
ejpam-6409	983	6	and	and	CCONJ
ejpam-6409	983	7	t.	t.	PROPN
ejpam-6409	983	8	shah	shah	NOUN
ejpam-6409	983	9	.	.	PUNCT
ejpam-6409	984	1	decoding	decode	VERB
ejpam-6409	984	2	of	of	ADP
ejpam-6409	984	3	cyclic	cyclic	ADJ
ejpam-6409	984	4	codes	code	NOUN
ejpam-6409	984	5	over	over	ADP
ejpam-6409	984	6	quaternion	quaternion	NOUN
ejpam-6409	984	7	integers	integer	NOUN
ejpam-6409	984	8	by	by	ADP
ejpam-6409	984	9	modified	modified	ADJ
ejpam-6409	984	10	berlekamp	berlekamp	NOUN
ejpam-6409	984	11	–	–	PUNCT
ejpam-6409	984	12	massey	massey	NOUN
ejpam-6409	984	13	algorithm	algorithm	NOUN
ejpam-6409	984	14	.	.	PUNCT
ejpam-6409	985	1	computational	computational	ADJ
ejpam-6409	985	2	and	and	CCONJ
ejpam-6409	985	3	applied	applied	ADJ
ejpam-6409	985	4	mathematics	mathematic	NOUN
ejpam-6409	985	5	,	,	PUNCT
ejpam-6409	985	6	43(2):102	43(2):102	NUM
ejpam-6409	985	7	,	,	PUNCT
ejpam-6409	985	8	2024	2024	NUM
ejpam-6409	985	9	.	.	PUNCT
ejpam-6409	986	1	[	[	X
ejpam-6409	986	2	5	5	NUM
ejpam-6409	986	3	]	]	PUNCT
ejpam-6409	986	4	c.	c.	PROPN
ejpam-6409	986	5	carlet	carlet	PROPN
ejpam-6409	986	6	.	.	PUNCT
ejpam-6409	987	1	z2k	z2k	NOUN
ejpam-6409	987	2	-	-	PUNCT
ejpam-6409	987	3	linear	linear	ADJ
ejpam-6409	987	4	codes	code	NOUN
ejpam-6409	987	5	.	.	PUNCT
ejpam-6409	988	1	ieee	ieee	NOUN
ejpam-6409	988	2	transactions	transaction	NOUN
ejpam-6409	988	3	on	on	ADP
ejpam-6409	988	4	information	information	NOUN
ejpam-6409	988	5	theory	theory	NOUN
ejpam-6409	988	6	,	,	PUNCT
ejpam-6409	988	7	44(4):1543	44(4):1543	PROPN
ejpam-6409	988	8	–	–	PUNCT
ejpam-6409	988	9	1547	1547	NUM
ejpam-6409	988	10	,	,	PUNCT
ejpam-6409	988	11	1998	1998	NUM
ejpam-6409	988	12	.	.	PUNCT
ejpam-6409	989	1	[	[	X
ejpam-6409	989	2	6	6	NUM
ejpam-6409	989	3	]	]	PUNCT
ejpam-6409	989	4	s.	s.	PROPN
ejpam-6409	989	5	t.	t.	PROPN
ejpam-6409	989	6	dougherty	dougherty	PROPN
ejpam-6409	989	7	,	,	PUNCT
ejpam-6409	989	8	j.	j.	PROPN
ejpam-6409	989	9	rifà	rifà	PROPN
ejpam-6409	989	10	,	,	PUNCT
ejpam-6409	989	11	and	and	CCONJ
ejpam-6409	989	12	m.	m.	PROPN
ejpam-6409	989	13	villanueva	villanueva	PROPN
ejpam-6409	989	14	.	.	PROPN
ejpam-6409	989	15	ranks	rank	VERB
ejpam-6409	989	16	and	and	CCONJ
ejpam-6409	989	17	kernels	kernel	NOUN
ejpam-6409	989	18	of	of	ADP
ejpam-6409	989	19	codes	code	NOUN
ejpam-6409	989	20	from	from	ADP
ejpam-6409	989	21	generalized	generalized	ADJ
ejpam-6409	989	22	hadamard	hadamard	ADJ
ejpam-6409	989	23	matrices	matrix	NOUN
ejpam-6409	989	24	.	.	PUNCT
ejpam-6409	990	1	ieee	ieee	NOUN
ejpam-6409	990	2	transactions	transaction	NOUN
ejpam-6409	990	3	on	on	ADP
ejpam-6409	990	4	information	information	NOUN
ejpam-6409	990	5	theory	theory	NOUN
ejpam-6409	990	6	,	,	PUNCT
ejpam-6409	990	7	62(2):687–694	62(2):687–694	PROPN
ejpam-6409	990	8	,	,	PUNCT
ejpam-6409	990	9	2015	2015	NUM
ejpam-6409	990	10	.	.	PUNCT
ejpam-6409	991	1	[	[	X
ejpam-6409	991	2	7	7	NUM
ejpam-6409	991	3	]	]	PUNCT
ejpam-6409	991	4	a.	a.	PROPN
ejpam-6409	991	5	r.	r.	PROPN
ejpam-6409	991	6	hammons	hammons	PROPN
ejpam-6409	991	7	,	,	PUNCT
ejpam-6409	991	8	p.	p.	NOUN
ejpam-6409	991	9	v.	v.	PROPN
ejpam-6409	991	10	kumar	kumar	PROPN
ejpam-6409	991	11	,	,	PUNCT
ejpam-6409	991	12	a.	a.	PROPN
ejpam-6409	991	13	r.	r.	PROPN
ejpam-6409	991	14	calderbank	calderbank	PROPN
ejpam-6409	991	15	,	,	PUNCT
ejpam-6409	991	16	n.	n.	PROPN
ejpam-6409	991	17	j.	j.	PROPN
ejpam-6409	991	18	sloane	sloane	PROPN
ejpam-6409	991	19	,	,	PUNCT
ejpam-6409	991	20	and	and	CCONJ
ejpam-6409	991	21	p.	p.	NOUN
ejpam-6409	991	22	solé.	solé.	PROPN
ejpam-6409	992	1	the	the	DET
ejpam-6409	992	2	z4	z4	PROPN
ejpam-6409	992	3	-	-	PUNCT
ejpam-6409	992	4	linearity	linearity	NOUN
ejpam-6409	992	5	of	of	ADP
ejpam-6409	992	6	kerdock	kerdock	NOUN
ejpam-6409	992	7	,	,	PUNCT
ejpam-6409	992	8	preparata	preparata	NOUN
ejpam-6409	992	9	,	,	PUNCT
ejpam-6409	992	10	goethals	goethal	NOUN
ejpam-6409	992	11	,	,	PUNCT
ejpam-6409	992	12	and	and	CCONJ
ejpam-6409	992	13	related	related	ADJ
ejpam-6409	992	14	codes	code	NOUN
ejpam-6409	992	15	.	.	PUNCT
ejpam-6409	993	1	ieee	ieee	NOUN
ejpam-6409	993	2	transactions	transaction	NOUN
ejpam-6409	993	3	on	on	ADP
ejpam-6409	993	4	information	information	NOUN
ejpam-6409	993	5	theory	theory	NOUN
ejpam-6409	993	6	,	,	PUNCT
ejpam-6409	993	7	40(2):301–319	40(2):301–319	PROPN
ejpam-6409	993	8	,	,	PUNCT
ejpam-6409	993	9	1994	1994	NUM
ejpam-6409	993	10	.	.	PUNCT
ejpam-6409	994	1	[	[	X
ejpam-6409	994	2	8	8	NUM
ejpam-6409	994	3	]	]	X
ejpam-6409	994	4	m.	m.	NOUN
ejpam-6409	994	5	greferath	greferath	NOUN
ejpam-6409	994	6	and	and	CCONJ
ejpam-6409	994	7	s.	s.	PROPN
ejpam-6409	994	8	e.	e.	PROPN
ejpam-6409	994	9	schmidt	schmidt	PROPN
ejpam-6409	994	10	.	.	PUNCT
ejpam-6409	995	1	gray	gray	ADJ
ejpam-6409	995	2	isometries	isometry	NOUN
ejpam-6409	995	3	for	for	ADP
ejpam-6409	995	4	finite	finite	ADJ
ejpam-6409	995	5	chain	chain	NOUN
ejpam-6409	995	6	rings	ring	NOUN
ejpam-6409	995	7	and	and	CCONJ
ejpam-6409	995	8	a	a	DET
ejpam-6409	995	9	nonlinear	nonlinear	ADJ
ejpam-6409	995	10	ternary	ternary	NOUN
ejpam-6409	995	11	(	(	PUNCT
ejpam-6409	995	12	36	36	NUM
ejpam-6409	995	13	,	,	PUNCT
ejpam-6409	995	14	3	3	NUM
ejpam-6409	995	15	/	/	SYM
ejpam-6409	995	16	sup	sup	NOUN
ejpam-6409	995	17	12/	12/	NUM
ejpam-6409	995	18	,	,	PUNCT
ejpam-6409	995	19	15	15	NUM
ejpam-6409	995	20	)	)	PUNCT
ejpam-6409	995	21	code	code	NOUN
ejpam-6409	995	22	.	.	PUNCT
ejpam-6409	996	1	ieee	ieee	NOUN
ejpam-6409	996	2	transactions	transaction	NOUN
ejpam-6409	996	3	on	on	ADP
ejpam-6409	996	4	information	information	NOUN
ejpam-6409	996	5	theory	theory	NOUN
ejpam-6409	996	6	,	,	PUNCT
ejpam-6409	996	7	45(7):2522–2524	45(7):2522–2524	PROPN
ejpam-6409	996	8	,	,	PUNCT
ejpam-6409	996	9	1999	1999	NUM
ejpam-6409	996	10	.	.	PUNCT
ejpam-6409	997	1	[	[	X
ejpam-6409	997	2	9	9	NUM
ejpam-6409	997	3	]	]	PUNCT
ejpam-6409	997	4	m.	m.	NOUN
ejpam-6409	997	5	shi	shi	PROPN
ejpam-6409	997	6	,	,	PUNCT
ejpam-6409	997	7	z.	z.	PROPN
ejpam-6409	997	8	sepasdar	sepasdar	PROPN
ejpam-6409	997	9	,	,	PUNCT
ejpam-6409	997	10	a.	a.	NOUN
ejpam-6409	997	11	alahmadi	alahmadi	PROPN
ejpam-6409	997	12	,	,	PUNCT
ejpam-6409	997	13	and	and	CCONJ
ejpam-6409	997	14	p.	p.	NOUN
ejpam-6409	997	15	solé.	solé.	PROPN
ejpam-6409	997	16	on	on	ADP
ejpam-6409	997	17	two	two	NUM
ejpam-6409	997	18	-	-	PUNCT
ejpam-6409	997	19	weight	weight	NOUN
ejpam-6409	997	20	z2k	z2k	NOUN
ejpam-6409	997	21	-	-	NOUN
ejpam-6409	997	22	codes	code	NOUN
ejpam-6409	997	23	.	.	PUNCT
ejpam-6409	998	1	designs	design	NOUN
ejpam-6409	998	2	,	,	PUNCT
ejpam-6409	998	3	codes	code	NOUN
ejpam-6409	998	4	and	and	CCONJ
ejpam-6409	998	5	cryptography	cryptography	NOUN
ejpam-6409	998	6	,	,	PUNCT
ejpam-6409	998	7	86(6):1201–1209	86(6):1201–1209	NUM
ejpam-6409	998	8	,	,	PUNCT
ejpam-6409	998	9	2018	2018	NUM
ejpam-6409	998	10	.	.	PUNCT
ejpam-6409	999	1	[	[	X
ejpam-6409	999	2	10	10	NUM
ejpam-6409	999	3	]	]	PUNCT
ejpam-6409	999	4	m.	m.	NOUN
ejpam-6409	999	5	sajjad	sajjad	PROPN
ejpam-6409	999	6	,	,	PUNCT
ejpam-6409	999	7	t.	t.	NOUN
ejpam-6409	999	8	shah	shah	PROPN
ejpam-6409	999	9	,	,	PUNCT
ejpam-6409	999	10	m.	m.	NOUN
ejpam-6409	999	11	alammari	alammari	PROPN
ejpam-6409	999	12	,	,	PUNCT
ejpam-6409	999	13	and	and	CCONJ
ejpam-6409	999	14	h.	h.	PROPN
ejpam-6409	999	15	alsaud	alsaud	PROPN
ejpam-6409	999	16	.	.	PUNCT
ejpam-6409	1000	1	construction	construction	NOUN
ejpam-6409	1000	2	and	and	CCONJ
ejpam-6409	1000	3	decoding	decoding	NOUN
ejpam-6409	1000	4	of	of	ADP
ejpam-6409	1000	5	bch	bch	NOUN
ejpam-6409	1000	6	-	-	PUNCT
ejpam-6409	1000	7	codes	code	NOUN
ejpam-6409	1000	8	over	over	ADP
ejpam-6409	1000	9	the	the	DET
ejpam-6409	1000	10	gaussian	gaussian	ADJ
ejpam-6409	1000	11	field	field	NOUN
ejpam-6409	1000	12	.	.	PUNCT
ejpam-6409	1001	1	ieee	ieee	NOUN
ejpam-6409	1001	2	access	access	NOUN
ejpam-6409	1001	3	,	,	PUNCT
ejpam-6409	1001	4	11:71972–71980	11:71972–71980	NUM
ejpam-6409	1001	5	,	,	PUNCT
ejpam-6409	1001	6	2023	2023	NUM
ejpam-6409	1001	7	.	.	PUNCT
ejpam-6409	1002	1	[	[	X
ejpam-6409	1002	2	11	11	NUM
ejpam-6409	1002	3	]	]	PUNCT
ejpam-6409	1002	4	m.	m.	NOUN
ejpam-6409	1002	5	sajjad	sajjad	PROPN
ejpam-6409	1002	6	,	,	PUNCT
ejpam-6409	1002	7	t.	t.	NOUN
ejpam-6409	1002	8	shah	shah	PROPN
ejpam-6409	1002	9	,	,	PUNCT
ejpam-6409	1002	10	m.	m.	NOUN
ejpam-6409	1002	11	m.	m.	PROPN
ejpam-6409	1002	12	hazzazi	hazzazi	PROPN
ejpam-6409	1002	13	,	,	PUNCT
ejpam-6409	1002	14	a.	a.	PROPN
ejpam-6409	1002	15	r.	r.	PROPN
ejpam-6409	1002	16	alharbi	alharbi	PROPN
ejpam-6409	1002	17	,	,	PUNCT
ejpam-6409	1002	18	and	and	CCONJ
ejpam-6409	1002	19	i.	i.	PROPN
ejpam-6409	1002	20	hussain	hussain	PROPN
ejpam-6409	1002	21	.	.	PUNCT
ejpam-6409	1003	1	quaternion	quaternion	NOUN
ejpam-6409	1003	2	integers	integer	NOUN
ejpam-6409	1003	3	based	base	VERB
ejpam-6409	1003	4	higher	high	ADJ
ejpam-6409	1003	5	length	length	NOUN
ejpam-6409	1003	6	cyclic	cyclic	NOUN
ejpam-6409	1003	7	codes	code	NOUN
ejpam-6409	1003	8	and	and	CCONJ
ejpam-6409	1003	9	their	their	PRON
ejpam-6409	1003	10	decoding	decode	VERB
ejpam-6409	1003	11	algorithm	algorithm	NOUN
ejpam-6409	1003	12	.	.	PUNCT
ejpam-6409	1004	1	computers	computer	NOUN
ejpam-6409	1004	2	,	,	PUNCT
ejpam-6409	1004	3	materials	material	NOUN
ejpam-6409	1004	4	&	&	CCONJ
ejpam-6409	1004	5	continua	continua	PROPN
ejpam-6409	1004	6	,	,	PUNCT
ejpam-6409	1004	7	73:1177–1194	73:1177–1194	NUM
ejpam-6409	1004	8	,	,	PUNCT
ejpam-6409	1004	9	2022	2022	NUM
ejpam-6409	1004	10	.	.	PUNCT
ejpam-6409	1005	1	[	[	X
ejpam-6409	1005	2	12	12	NUM
ejpam-6409	1005	3	]	]	PUNCT
ejpam-6409	1005	4	m.	m.	NOUN
ejpam-6409	1005	5	sajjad	sajjad	PROPN
ejpam-6409	1005	6	,	,	PUNCT
ejpam-6409	1005	7	t.	t.	NOUN
ejpam-6409	1005	8	shah	shah	PROPN
ejpam-6409	1005	9	,	,	PUNCT
ejpam-6409	1005	10	q.	q.	PROPN
ejpam-6409	1005	11	xin	xin	PROPN
ejpam-6409	1005	12	,	,	PUNCT
ejpam-6409	1005	13	and	and	CCONJ
ejpam-6409	1005	14	b.	b.	PROPN
ejpam-6409	1005	15	almutairi	almutairi	PROPN
ejpam-6409	1005	16	.	.	PUNCT
ejpam-6409	1006	1	eisenstein	eisenstein	PROPN
ejpam-6409	1006	2	field	field	PROPN
ejpam-6409	1006	3	bch	bch	PROPN
ejpam-6409	1006	4	codes	code	VERB
ejpam-6409	1006	5	construction	construction	NOUN
ejpam-6409	1006	6	and	and	CCONJ
ejpam-6409	1006	7	decoding	decoding	NOUN
ejpam-6409	1006	8	.	.	PUNCT
ejpam-6409	1007	1	aims	aim	VERB
ejpam-6409	1007	2	mathematics	mathematic	NOUN
ejpam-6409	1007	3	,	,	PUNCT
ejpam-6409	1007	4	8(12):29453–29473	8(12):29453–29473	NUM
ejpam-6409	1007	5	,	,	PUNCT
ejpam-6409	1007	6	2023	2023	NUM
ejpam-6409	1007	7	.	.	PUNCT
ejpam-6409	1008	1	[	[	X
ejpam-6409	1008	2	13	13	NUM
ejpam-6409	1008	3	]	]	X
ejpam-6409	1008	4	d.	d.	PROPN
ejpam-6409	1008	5	jungnickel	jungnickel	PROPN
ejpam-6409	1008	6	.	.	PUNCT
ejpam-6409	1009	1	on	on	ADP
ejpam-6409	1009	2	difference	difference	NOUN
ejpam-6409	1009	3	matrices	matrix	NOUN
ejpam-6409	1009	4	,	,	PUNCT
ejpam-6409	1009	5	resolvable	resolvable	ADJ
ejpam-6409	1009	6	transversal	transversal	NOUN
ejpam-6409	1009	7	designs	design	NOUN
ejpam-6409	1009	8	and	and	CCONJ
ejpam-6409	1009	9	generalized	generalize	VERB
ejpam-6409	1009	10	hadamard	hadamard	ADJ
ejpam-6409	1009	11	matrices	matrix	NOUN
ejpam-6409	1009	12	.	.	PUNCT
ejpam-6409	1010	1	unpublished	unpublished	ADJ
ejpam-6409	1010	2	,	,	PUNCT
ejpam-6409	1010	3	1979	1979	NUM
ejpam-6409	1010	4	.	.	PUNCT
ejpam-6409	1011	1	[	[	X
ejpam-6409	1011	2	14	14	NUM
ejpam-6409	1011	3	]	]	PUNCT
ejpam-6409	1011	4	k.	k.	PROPN
ejpam-6409	1011	5	t.	t.	PROPN
ejpam-6409	1011	6	phelps	phelps	PROPN
ejpam-6409	1011	7	,	,	PUNCT
ejpam-6409	1011	8	j.	j.	PROPN
ejpam-6409	1011	9	rifa	rifa	PROPN
ejpam-6409	1011	10	,	,	PUNCT
ejpam-6409	1011	11	and	and	CCONJ
ejpam-6409	1011	12	m.	m.	PROPN
ejpam-6409	1011	13	villanueva	villanueva	PROPN
ejpam-6409	1011	14	.	.	PUNCT
ejpam-6409	1012	1	kernels	kernels	PROPN
ejpam-6409	1012	2	and	and	CCONJ
ejpam-6409	1012	3	p	p	NOUN
ejpam-6409	1012	4	-	-	PUNCT
ejpam-6409	1012	5	kernels	kernel	NOUN
ejpam-6409	1012	6	of	of	ADP
ejpam-6409	1012	7	pr	pr	NOUN
ejpam-6409	1012	8	-	-	NOUN
ejpam-6409	1012	9	ary	ary	ADJ
ejpam-6409	1012	10	1	1	NUM
ejpam-6409	1012	11	-	-	PUNCT
ejpam-6409	1012	12	perfect	perfect	ADJ
ejpam-6409	1012	13	codes	code	NOUN
ejpam-6409	1012	14	.	.	PUNCT
ejpam-6409	1013	1	designs	design	NOUN
ejpam-6409	1013	2	,	,	PUNCT
ejpam-6409	1013	3	codes	code	NOUN
ejpam-6409	1013	4	and	and	CCONJ
ejpam-6409	1013	5	cryptography	cryptography	NOUN
ejpam-6409	1013	6	,	,	PUNCT
ejpam-6409	1013	7	37(2):243–261	37(2):243–261	NUM
ejpam-6409	1013	8	,	,	PUNCT
ejpam-6409	1013	9	2005	2005	NUM
ejpam-6409	1013	10	.	.	PUNCT
ejpam-6409	1014	1	[	[	X
ejpam-6409	1014	2	15	15	NUM
ejpam-6409	1014	3	]	]	X
ejpam-6409	1014	4	d.	d.	PROPN
ejpam-6409	1014	5	k.	k.	PROPN
ejpam-6409	1014	6	bhunia	bhunia	PROPN
ejpam-6409	1014	7	,	,	PUNCT
ejpam-6409	1014	8	c.	c.	PROPN
ejpam-6409	1014	9	fernández	fernández	PROPN
ejpam-6409	1014	10	-	-	PUNCT
ejpam-6409	1014	11	córdoba	córdoba	PROPN
ejpam-6409	1014	12	,	,	PUNCT
ejpam-6409	1014	13	c.	c.	PROPN
ejpam-6409	1014	14	vela	vela	PROPN
ejpam-6409	1014	15	,	,	PUNCT
ejpam-6409	1014	16	and	and	CCONJ
ejpam-6409	1014	17	m.	m.	PROPN
ejpam-6409	1014	18	villanueva	villanueva	PROPN
ejpam-6409	1014	19	.	.	PUNCT
ejpam-6409	1015	1	on	on	ADP
ejpam-6409	1015	2	the	the	DET
ejpam-6409	1015	3	equivalence	equivalence	NOUN
ejpam-6409	1015	4	of	of	ADP
ejpam-6409	1015	5	zps	zps	PROPN
ejpam-6409	1015	6	-	-	PUNCT
ejpam-6409	1015	7	linear	linear	PROPN
ejpam-6409	1015	8	generalized	generalized	ADJ
ejpam-6409	1015	9	hadamard	hadamard	ADJ
ejpam-6409	1015	10	codes	code	NOUN
ejpam-6409	1015	11	.	.	PUNCT
ejpam-6409	1016	1	designs	design	NOUN
ejpam-6409	1016	2	,	,	PUNCT
ejpam-6409	1016	3	codes	code	NOUN
ejpam-6409	1016	4	and	and	CCONJ
ejpam-6409	1016	5	cryptography	cryptography	NOUN
ejpam-6409	1016	6	,	,	PUNCT
ejpam-6409	1016	7	92(4):999–1022	92(4):999–1022	NOUN
ejpam-6409	1016	8	,	,	PUNCT
ejpam-6409	1016	9	2024	2024	NUM
ejpam-6409	1016	10	.	.	PUNCT
ejpam-6409	1017	1	[	[	X
ejpam-6409	1017	2	16	16	NUM
ejpam-6409	1017	3	]	]	X
ejpam-6409	1017	4	d.	d.	PROPN
ejpam-6409	1017	5	k.	k.	PROPN
ejpam-6409	1017	6	bhunia	bhunia	PROPN
ejpam-6409	1017	7	,	,	PUNCT
ejpam-6409	1017	8	c.	c.	PROPN
ejpam-6409	1017	9	fernández	fernández	PROPN
ejpam-6409	1017	10	-	-	PUNCT
ejpam-6409	1017	11	córdoba	córdoba	PROPN
ejpam-6409	1017	12	,	,	PUNCT
ejpam-6409	1017	13	and	and	CCONJ
ejpam-6409	1017	14	m.	m.	PROPN
ejpam-6409	1017	15	villanueva	villanueva	PROPN
ejpam-6409	1017	16	.	.	PUNCT
ejpam-6409	1018	1	on	on	ADP
ejpam-6409	1018	2	the	the	DET
ejpam-6409	1018	3	linearity	linearity	NOUN
ejpam-6409	1018	4	and	and	CCONJ
ejpam-6409	1018	5	classimuhammad	classimuhammad	PROPN
ejpam-6409	1018	6	sajjad	sajjad	PROPN
ejpam-6409	1018	7	et	et	PROPN
ejpam-6409	1018	8	al	al	PROPN
ejpam-6409	1018	9	.	.	PUNCT
ejpam-6409	1018	10	/	/	SYM
ejpam-6409	1018	11	eur	eur	PROPN
ejpam-6409	1018	12	.	.	PUNCT
ejpam-6409	1019	1	j.	j.	PROPN
ejpam-6409	1019	2	pure	pure	PROPN
ejpam-6409	1019	3	appl	appl	PROPN
ejpam-6409	1019	4	.	.	PROPN
ejpam-6409	1019	5	math	math	PROPN
ejpam-6409	1019	6	,	,	PUNCT
ejpam-6409	1019	7	18	18	NUM
ejpam-6409	1019	8	(	(	PUNCT
ejpam-6409	1019	9	3	3	NUM
ejpam-6409	1019	10	)	)	PUNCT
ejpam-6409	1019	11	(	(	PUNCT
ejpam-6409	1019	12	2025	2025	NUM
ejpam-6409	1019	13	)	)	PUNCT
ejpam-6409	1019	14	,	,	PUNCT
ejpam-6409	1019	15	6409	6409	NUM
ejpam-6409	1019	16	31	31	NUM
ejpam-6409	1019	17	of	of	ADP
ejpam-6409	1019	18	32	32	NUM
ejpam-6409	1019	19	fication	fication	NOUN
ejpam-6409	1019	20	of	of	ADP
ejpam-6409	1019	21	z	z	PROPN
ejpam-6409	1019	22	ps	ps	ADJ
ejpam-6409	1019	23	-	-	PUNCT
ejpam-6409	1019	24	linear	linear	ADJ
ejpam-6409	1019	25	generalized	generalized	ADJ
ejpam-6409	1019	26	hadamard	hadamard	ADJ
ejpam-6409	1019	27	codes	code	NOUN
ejpam-6409	1019	28	.	.	PUNCT
ejpam-6409	1020	1	designs	design	NOUN
ejpam-6409	1020	2	,	,	PUNCT
ejpam-6409	1020	3	codes	code	NOUN
ejpam-6409	1020	4	and	and	CCONJ
ejpam-6409	1020	5	cryptography	cryptography	NOUN
ejpam-6409	1020	6	,	,	PUNCT
ejpam-6409	1020	7	90(4):1037–1058	90(4):1037–1058	NUM
ejpam-6409	1020	8	,	,	PUNCT
ejpam-6409	1020	9	2022	2022	NUM
ejpam-6409	1020	10	.	.	PUNCT
ejpam-6409	1021	1	[	[	X
ejpam-6409	1021	2	17	17	NUM
ejpam-6409	1021	3	]	]	X
ejpam-6409	1021	4	c.	c.	PROPN
ejpam-6409	1021	5	fernández	fernández	PROPN
ejpam-6409	1021	6	-	-	PUNCT
ejpam-6409	1021	7	córdoba	córdoba	PROPN
ejpam-6409	1021	8	,	,	PUNCT
ejpam-6409	1021	9	c.	c.	PROPN
ejpam-6409	1021	10	vela	vela	PROPN
ejpam-6409	1021	11	,	,	PUNCT
ejpam-6409	1021	12	and	and	CCONJ
ejpam-6409	1021	13	m.	m.	PROPN
ejpam-6409	1021	14	villanueva	villanueva	PROPN
ejpam-6409	1021	15	.	.	PUNCT
ejpam-6409	1022	1	on	on	ADP
ejpam-6409	1022	2	z2s	z2	NOUN
ejpam-6409	1022	3	-	-	PUNCT
ejpam-6409	1022	4	linear	linear	ADJ
ejpam-6409	1022	5	hadamard	hadamard	ADJ
ejpam-6409	1022	6	codes	code	NOUN
ejpam-6409	1022	7	:	:	PUNCT
ejpam-6409	1022	8	kernel	kernel	NOUN
ejpam-6409	1022	9	and	and	CCONJ
ejpam-6409	1022	10	partial	partial	ADJ
ejpam-6409	1022	11	classification	classification	NOUN
ejpam-6409	1022	12	.	.	PUNCT
ejpam-6409	1023	1	designs	design	NOUN
ejpam-6409	1023	2	,	,	PUNCT
ejpam-6409	1023	3	codes	code	NOUN
ejpam-6409	1023	4	and	and	CCONJ
ejpam-6409	1023	5	cryptography	cryptography	NOUN
ejpam-6409	1023	6	,	,	PUNCT
ejpam-6409	1023	7	87(2):417–435	87(2):417–435	NOUN
ejpam-6409	1023	8	,	,	PUNCT
ejpam-6409	1023	9	2019	2019	NUM
ejpam-6409	1023	10	.	.	PUNCT
ejpam-6409	1024	1	[	[	X
ejpam-6409	1024	2	18	18	NUM
ejpam-6409	1024	3	]	]	PUNCT
ejpam-6409	1024	4	k.	k.	PROPN
ejpam-6409	1024	5	t.	t.	PROPN
ejpam-6409	1024	6	phelps	phelps	PROPN
ejpam-6409	1024	7	,	,	PUNCT
ejpam-6409	1024	8	j.	j.	PROPN
ejpam-6409	1024	9	rifà	rifà	PROPN
ejpam-6409	1024	10	,	,	PUNCT
ejpam-6409	1024	11	and	and	CCONJ
ejpam-6409	1024	12	m.	m.	PROPN
ejpam-6409	1024	13	villanueva	villanueva	PROPN
ejpam-6409	1024	14	.	.	PUNCT
ejpam-6409	1025	1	on	on	ADP
ejpam-6409	1025	2	the	the	DET
ejpam-6409	1025	3	additive	additive	NOUN
ejpam-6409	1025	4	(	(	PUNCT
ejpam-6409	1025	5	z4	z4	PROPN
ejpam-6409	1025	6	-	-	PUNCT
ejpam-6409	1025	7	linear	linear	ADJ
ejpam-6409	1025	8	and	and	CCONJ
ejpam-6409	1025	9	non	non	ADJ
ejpam-6409	1025	10	-	-	ADJ
ejpam-6409	1025	11	z4linear	z4linear	ADJ
ejpam-6409	1025	12	)	)	PUNCT
ejpam-6409	1025	13	hadamard	hadamard	ADJ
ejpam-6409	1025	14	codes	code	NOUN
ejpam-6409	1025	15	:	:	PUNCT
ejpam-6409	1025	16	rank	rank	NOUN
ejpam-6409	1025	17	and	and	CCONJ
ejpam-6409	1025	18	kernel	kernel	PROPN
ejpam-6409	1025	19	.	.	PUNCT
ejpam-6409	1026	1	ieee	ieee	NOUN
ejpam-6409	1026	2	transactions	transaction	NOUN
ejpam-6409	1026	3	on	on	ADP
ejpam-6409	1026	4	information	information	NOUN
ejpam-6409	1026	5	theory	theory	NOUN
ejpam-6409	1026	6	,	,	PUNCT
ejpam-6409	1026	7	52(1):316–319	52(1):316–319	PROPN
ejpam-6409	1026	8	,	,	PUNCT
ejpam-6409	1026	9	2006	2006	NUM
ejpam-6409	1026	10	.	.	PUNCT
ejpam-6409	1027	1	[	[	X
ejpam-6409	1027	2	19	19	NUM
ejpam-6409	1027	3	]	]	X
ejpam-6409	1027	4	j.	j.	PROPN
ejpam-6409	1027	5	borges	borges	PROPN
ejpam-6409	1027	6	,	,	PUNCT
ejpam-6409	1027	7	c.	c.	PROPN
ejpam-6409	1027	8	fernández	fernández	PROPN
ejpam-6409	1027	9	,	,	PUNCT
ejpam-6409	1027	10	and	and	CCONJ
ejpam-6409	1027	11	j.	j.	PROPN
ejpam-6409	1027	12	rifà.	rifà.	PROPN
ejpam-6409	1028	1	every	every	DET
ejpam-6409	1028	2	z2	z2	PROPN
ejpam-6409	1028	3	-	-	PUNCT
ejpam-6409	1028	4	code	code	NOUN
ejpam-6409	1028	5	is	be	AUX
ejpam-6409	1028	6	a	a	DET
ejpam-6409	1028	7	binary	binary	ADJ
ejpam-6409	1028	8	prope	prope	PROPN
ejpam-6409	1028	9	linear	linear	PROPN
ejpam-6409	1028	10	code	code	PROPN
ejpam-6409	1028	11	.	.	PUNCT
ejpam-6409	1029	1	comb’01	comb’01	PROPN
ejpam-6409	1029	2	electronic	electronic	ADJ
ejpam-6409	1029	3	notes	note	NOUN
ejpam-6409	1029	4	in	in	ADP
ejpam-6409	1029	5	discrete	discrete	ADJ
ejpam-6409	1029	6	mathematics	mathematic	NOUN
ejpam-6409	1029	7	,	,	PUNCT
ejpam-6409	1029	8	10:100–102	10:100–102	NUM
ejpam-6409	1029	9	,	,	PUNCT
ejpam-6409	1029	10	2001	2001	NUM
ejpam-6409	1029	11	.	.	PUNCT
ejpam-6409	1030	1	[	[	X
ejpam-6409	1030	2	20	20	NUM
ejpam-6409	1030	3	]	]	PUNCT
ejpam-6409	1030	4	a.	a.	NOUN
ejpam-6409	1030	5	t.	t.	PROPN
ejpam-6409	1030	6	butson	butson	PROPN
ejpam-6409	1030	7	.	.	PUNCT
ejpam-6409	1031	1	generalized	generalize	VERB
ejpam-6409	1031	2	hadamard	hadamard	ADJ
ejpam-6409	1031	3	matrices	matrix	NOUN
ejpam-6409	1031	4	.	.	PUNCT
ejpam-6409	1031	5	proceedings	proceeding	NOUN
ejpam-6409	1031	6	of	of	ADP
ejpam-6409	1031	7	the	the	DET
ejpam-6409	1031	8	american	american	PROPN
ejpam-6409	1031	9	mathematical	mathematical	PROPN
ejpam-6409	1031	10	society	society	NOUN
ejpam-6409	1031	11	,	,	PUNCT
ejpam-6409	1031	12	13(6):894–898	13(6):894–898	PROPN
ejpam-6409	1031	13	,	,	PUNCT
ejpam-6409	1031	14	1962	1962	NUM
ejpam-6409	1031	15	.	.	PUNCT
ejpam-6409	1032	1	[	[	X
ejpam-6409	1032	2	21	21	NUM
ejpam-6409	1032	3	]	]	X
ejpam-6409	1032	4	d.	d.	PROPN
ejpam-6409	1032	5	s.	s.	PROPN
ejpam-6409	1032	6	krotov	krotov	PROPN
ejpam-6409	1032	7	.	.	PUNCT
ejpam-6409	1033	1	z4	z4	PROPN
ejpam-6409	1033	2	-	-	PUNCT
ejpam-6409	1033	3	linear	linear	PROPN
ejpam-6409	1033	4	hadamard	hadamard	NOUN
ejpam-6409	1033	5	and	and	CCONJ
ejpam-6409	1033	6	extended	extend	VERB
ejpam-6409	1033	7	perfect	perfect	ADJ
ejpam-6409	1033	8	codes	code	NOUN
ejpam-6409	1033	9	.	.	PUNCT
ejpam-6409	1034	1	electronic	electronic	ADJ
ejpam-6409	1034	2	notes	note	NOUN
ejpam-6409	1034	3	in	in	ADP
ejpam-6409	1034	4	discrete	discrete	ADJ
ejpam-6409	1034	5	mathematics	mathematic	NOUN
ejpam-6409	1034	6	,	,	PUNCT
ejpam-6409	1034	7	6:107–112	6:107–112	PROPN
ejpam-6409	1034	8	,	,	PUNCT
ejpam-6409	1034	9	2001	2001	NUM
ejpam-6409	1034	10	.	.	PUNCT
ejpam-6409	1035	1	[	[	X
ejpam-6409	1035	2	22	22	NUM
ejpam-6409	1035	3	]	]	X
ejpam-6409	1035	4	d.	d.	PROPN
ejpam-6409	1035	5	s.	s.	PROPN
ejpam-6409	1035	6	krotov	krotov	PROPN
ejpam-6409	1035	7	.	.	PUNCT
ejpam-6409	1036	1	on	on	ADP
ejpam-6409	1036	2	z	z	PROPN
ejpam-6409	1036	3	2k−dualbinarycodes.ieeetransactionsoninformationtheory	2k−dualbinarycodes.ieeetransactionsoninformationtheory	NUM
ejpam-6409	1036	4	,	,	PUNCT
ejpam-6409	1036	5	53(4	53(4	NOUN
ejpam-6409	1036	6	)	)	PUNCT
ejpam-6409	1036	7	:	:	PUNCT
ejpam-6409	1037	1	1532−−1537	1532−−1537	NUM
ejpam-6409	1037	2	,	,	PUNCT
ejpam-6409	1037	3	2007	2007	NUM
ejpam-6409	1037	4	.	.	PUNCT
ejpam-6409	1038	1	[	[	X
ejpam-6409	1038	2	23	23	NUM
ejpam-6409	1038	3	]	]	PUNCT
ejpam-6409	1038	4	m.	m.	PROPN
ejpam-6409	1038	5	shi	shi	PROPN
ejpam-6409	1038	6	,	,	PUNCT
ejpam-6409	1038	7	r.	r.	PROPN
ejpam-6409	1038	8	wu	wu	PROPN
ejpam-6409	1038	9	,	,	PUNCT
ejpam-6409	1038	10	and	and	CCONJ
ejpam-6409	1038	11	d.	d.	PROPN
ejpam-6409	1038	12	s.	s.	PROPN
ejpam-6409	1038	13	krotov	krotov	PROPN
ejpam-6409	1038	14	.	.	PUNCT
ejpam-6409	1039	1	on	on	ADP
ejpam-6409	1039	2	zp	zp	PROPN
ejpam-6409	1039	3	zpk	zpk	NOUN
ejpam-6409	1039	4	-	-	PUNCT
ejpam-6409	1039	5	additive	additive	ADJ
ejpam-6409	1039	6	codes	code	NOUN
ejpam-6409	1039	7	and	and	CCONJ
ejpam-6409	1039	8	their	their	PRON
ejpam-6409	1039	9	duality	duality	NOUN
ejpam-6409	1039	10	.	.	PUNCT
ejpam-6409	1040	1	ieee	ieee	NOUN
ejpam-6409	1040	2	transactions	transaction	NOUN
ejpam-6409	1040	3	on	on	ADP
ejpam-6409	1040	4	information	information	NOUN
ejpam-6409	1040	5	theory	theory	NOUN
ejpam-6409	1040	6	,	,	PUNCT
ejpam-6409	1040	7	65(6):3841–3847	65(6):3841–3847	NUM
ejpam-6409	1040	8	,	,	PUNCT
ejpam-6409	1040	9	2018	2018	NUM
ejpam-6409	1040	10	.	.	PUNCT
ejpam-6409	1041	1	[	[	X
ejpam-6409	1041	2	24	24	NUM
ejpam-6409	1041	3	]	]	PUNCT
ejpam-6409	1041	4	m.	m.	NOUN
ejpam-6409	1041	5	sajjad	sajjad	PROPN
ejpam-6409	1041	6	,	,	PUNCT
ejpam-6409	1041	7	t.	t.	NOUN
ejpam-6409	1041	8	shah	shah	PROPN
ejpam-6409	1041	9	,	,	PUNCT
ejpam-6409	1041	10	m.	m.	NOUN
ejpam-6409	1041	11	abbas	abbas	PROPN
ejpam-6409	1041	12	,	,	PUNCT
ejpam-6409	1041	13	m.	m.	NOUN
ejpam-6409	1041	14	alammari	alammari	PROPN
ejpam-6409	1041	15	,	,	PUNCT
ejpam-6409	1041	16	and	and	CCONJ
ejpam-6409	1041	17	r.	r.	PROPN
ejpam-6409	1041	18	j.	j.	PROPN
ejpam-6409	1041	19	serna	serna	PROPN
ejpam-6409	1041	20	.	.	PUNCT
ejpam-6409	1042	1	the	the	DET
ejpam-6409	1042	2	impact	impact	NOUN
ejpam-6409	1042	3	of	of	ADP
ejpam-6409	1042	4	alternant	alternant	ADJ
ejpam-6409	1042	5	codes	code	NOUN
ejpam-6409	1042	6	over	over	ADP
ejpam-6409	1042	7	eisenstein	eisenstein	NOUN
ejpam-6409	1042	8	integers	integer	NOUN
ejpam-6409	1042	9	on	on	ADP
ejpam-6409	1042	10	modern	modern	ADJ
ejpam-6409	1042	11	technology	technology	NOUN
ejpam-6409	1042	12	.	.	PUNCT
ejpam-6409	1043	1	computational	computational	ADJ
ejpam-6409	1043	2	and	and	CCONJ
ejpam-6409	1043	3	applied	applied	ADJ
ejpam-6409	1043	4	mathematics	mathematic	NOUN
ejpam-6409	1043	5	,	,	PUNCT
ejpam-6409	1043	6	44(1):95	44(1):95	NOUN
ejpam-6409	1043	7	,	,	PUNCT
ejpam-6409	1043	8	2025	2025	NUM
ejpam-6409	1043	9	.	.	PUNCT
ejpam-6409	1044	1	[	[	X
ejpam-6409	1044	2	25	25	NUM
ejpam-6409	1044	3	]	]	PUNCT
ejpam-6409	1044	4	m.	m.	NOUN
ejpam-6409	1044	5	villanueva	villanueva	PROPN
ejpam-6409	1044	6	,	,	PUNCT
ejpam-6409	1044	7	v.	v.	ADP
ejpam-6409	1044	8	a.	a.	NOUN
ejpam-6409	1044	9	zinoviev	zinoviev	PROPN
ejpam-6409	1044	10	,	,	PUNCT
ejpam-6409	1044	11	and	and	CCONJ
ejpam-6409	1044	12	d.	d.	PROPN
ejpam-6409	1044	13	a.	a.	PROPN
ejpam-6409	1044	14	zinoviev	zinoviev	PROPN
ejpam-6409	1044	15	.	.	PUNCT
ejpam-6409	1045	1	on	on	ADP
ejpam-6409	1045	2	one	one	NUM
ejpam-6409	1045	3	construction	construction	NOUN
ejpam-6409	1045	4	method	method	NOUN
ejpam-6409	1045	5	for	for	ADP
ejpam-6409	1045	6	hadamard	hadamard	ADJ
ejpam-6409	1045	7	matrices	matrix	NOUN
ejpam-6409	1045	8	.	.	PUNCT
ejpam-6409	1046	1	problems	problem	NOUN
ejpam-6409	1046	2	of	of	ADP
ejpam-6409	1046	3	information	information	NOUN
ejpam-6409	1046	4	transmission	transmission	NOUN
ejpam-6409	1046	5	,	,	PUNCT
ejpam-6409	1046	6	58(4):306–328	58(4):306–328	PROPN
ejpam-6409	1046	7	,	,	PUNCT
ejpam-6409	1046	8	2022	2022	NUM
ejpam-6409	1046	9	.	.	PUNCT
ejpam-6409	1047	1	[	[	X
ejpam-6409	1047	2	26	26	NUM
ejpam-6409	1047	3	]	]	PUNCT
ejpam-6409	1047	4	v.	v.	ADP
ejpam-6409	1047	5	a.	a.	PROPN
ejpam-6409	1047	6	zinoviev	zinoviev	PROPN
ejpam-6409	1047	7	and	and	CCONJ
ejpam-6409	1047	8	d.	d.	PROPN
ejpam-6409	1047	9	v.	v.	PROPN
ejpam-6409	1047	10	zinoviev	zinoviev	PROPN
ejpam-6409	1047	11	.	.	PUNCT
ejpam-6409	1048	1	on	on	ADP
ejpam-6409	1048	2	the	the	DET
ejpam-6409	1048	3	generalized	generalize	VERB
ejpam-6409	1048	4	concatenated	concatenate	VERB
ejpam-6409	1048	5	construction	construction	NOUN
ejpam-6409	1048	6	for	for	ADP
ejpam-6409	1048	7	codes	code	NOUN
ejpam-6409	1048	8	in	in	ADP
ejpam-6409	1048	9	and	and	CCONJ
ejpam-6409	1048	10	lee	lee	PROPN
ejpam-6409	1048	11	metrics	metric	NOUN
ejpam-6409	1048	12	.	.	PUNCT
ejpam-6409	1049	1	problems	problem	NOUN
ejpam-6409	1049	2	of	of	ADP
ejpam-6409	1049	3	information	information	NOUN
ejpam-6409	1049	4	transmission	transmission	NOUN
ejpam-6409	1049	5	,	,	PUNCT
ejpam-6409	1049	6	57(1):70–83	57(1):70–83	NUM
ejpam-6409	1049	7	,	,	PUNCT
ejpam-6409	1049	8	2021	2021	NUM
ejpam-6409	1049	9	.	.	PUNCT
ejpam-6409	1050	1	[	[	X
ejpam-6409	1050	2	27	27	NUM
ejpam-6409	1050	3	]	]	X
ejpam-6409	1050	4	c.	c.	PROPN
ejpam-6409	1050	5	fernández	fernández	PROPN
ejpam-6409	1050	6	-	-	PUNCT
ejpam-6409	1050	7	córdoba	córdoba	PROPN
ejpam-6409	1050	8	,	,	PUNCT
ejpam-6409	1050	9	c.	c.	PROPN
ejpam-6409	1050	10	vela	vela	PROPN
ejpam-6409	1050	11	,	,	PUNCT
ejpam-6409	1050	12	and	and	CCONJ
ejpam-6409	1050	13	m.	m.	PROPN
ejpam-6409	1050	14	villanueva	villanueva	PROPN
ejpam-6409	1050	15	.	.	PUNCT
ejpam-6409	1050	16	equivalences	equivalence	VERB
ejpam-6409	1050	17	among	among	ADP
ejpam-6409	1050	18	z2s	z2	NOUN
ejpam-6409	1050	19	-	-	PUNCT
ejpam-6409	1050	20	linear	linear	ADJ
ejpam-6409	1050	21	hadamard	hadamard	ADJ
ejpam-6409	1050	22	codes	code	NOUN
ejpam-6409	1050	23	.	.	PUNCT
ejpam-6409	1051	1	discrete	discrete	ADJ
ejpam-6409	1051	2	mathematics	mathematic	NOUN
ejpam-6409	1051	3	,	,	PUNCT
ejpam-6409	1051	4	343(3):111721	343(3):111721	PROPN
ejpam-6409	1051	5	,	,	PUNCT
ejpam-6409	1051	6	2020	2020	NUM
ejpam-6409	1051	7	.	.	PUNCT
ejpam-6409	1052	1	muhammad	muhammad	PROPN
ejpam-6409	1052	2	sajjad	sajjad	PROPN
ejpam-6409	1052	3	et	et	PROPN
ejpam-6409	1052	4	al	al	PROPN
ejpam-6409	1052	5	.	.	PUNCT
ejpam-6409	1052	6	/	/	SYM
ejpam-6409	1052	7	eur	eur	PROPN
ejpam-6409	1052	8	.	.	PUNCT
ejpam-6409	1053	1	j.	j.	PROPN
ejpam-6409	1053	2	pure	pure	PROPN
ejpam-6409	1053	3	appl	appl	PROPN
ejpam-6409	1053	4	.	.	PROPN
ejpam-6409	1053	5	math	math	PROPN
ejpam-6409	1053	6	,	,	PUNCT
ejpam-6409	1053	7	18	18	NUM
ejpam-6409	1053	8	(	(	PUNCT
ejpam-6409	1053	9	3	3	NUM
ejpam-6409	1053	10	)	)	PUNCT
ejpam-6409	1053	11	(	(	PUNCT
ejpam-6409	1053	12	2025	2025	NUM
ejpam-6409	1053	13	)	)	PUNCT
ejpam-6409	1053	14	,	,	PUNCT
ejpam-6409	1053	15	6409	6409	NUM
ejpam-6409	1053	16	32	32	NUM
ejpam-6409	1053	17	of	of	ADP
ejpam-6409	1053	18	32	32	NUM
ejpam-6409	1053	19	table	table	NOUN
ejpam-6409	1053	20	5	5	NUM
ejpam-6409	1053	21	:	:	PUNCT
ejpam-6409	1053	22	rank	rank	NOUN
ejpam-6409	1053	23	and	and	CCONJ
ejpam-6409	1053	24	kernel	kernel	NOUN
ejpam-6409	1053	25	for	for	ADP
ejpam-6409	1053	26	all	all	DET
ejpam-6409	1053	27	nonlinear	nonlinear	ADJ
ejpam-6409	1053	28	z2s	z2	NOUN
ejpam-6409	1054	1	[	[	X
ejpam-6409	1054	2	ω]-linear	ω]-linear	ADJ
ejpam-6409	1054	3	hadamard	hadamard	ADJ
ejpam-6409	1054	4	codes	code	NOUN
ejpam-6409	1054	5	of	of	ADP
ejpam-6409	1054	6	length	length	NOUN
ejpam-6409	1054	7	2	2	NUM
ejpam-6409	1054	8	t	t	NOUN
ejpam-6409	1054	9	t	t	NOUN
ejpam-6409	1054	10	=	=	SYM
ejpam-6409	1054	11	8	8	NUM
ejpam-6409	1054	12	t	t	NOUN
ejpam-6409	1054	13	=	=	SYM
ejpam-6409	1054	14	9	9	NUM
ejpam-6409	1054	15	t	t	NOUN
ejpam-6409	1054	16	=	=	SYM
ejpam-6409	1054	17	10	10	NUM
ejpam-6409	1054	18	(	(	PUNCT
ejpam-6409	1054	19	t1	t1	NOUN
ejpam-6409	1054	20	,	,	PUNCT
ejpam-6409	1054	21	.	.	PUNCT
ejpam-6409	1054	22	.	.	PUNCT
ejpam-6409	1054	23	.	.	PUNCT
ejpam-6409	1055	1	,	,	PUNCT
ejpam-6409	1055	2	ts	ts	NOUN
ejpam-6409	1055	3	)	)	PUNCT
ejpam-6409	1055	4	(	(	PUNCT
ejpam-6409	1055	5	r	r	NOUN
ejpam-6409	1055	6	,	,	PUNCT
ejpam-6409	1055	7	k	k	NOUN
ejpam-6409	1055	8	)	)	PUNCT
ejpam-6409	1055	9	(	(	PUNCT
ejpam-6409	1055	10	t1	t1	NOUN
ejpam-6409	1055	11	,	,	PUNCT
ejpam-6409	1055	12	.	.	PUNCT
ejpam-6409	1055	13	.	.	PUNCT
ejpam-6409	1055	14	.	.	PUNCT
ejpam-6409	1056	1	,	,	PUNCT
ejpam-6409	1056	2	ts	ts	NOUN
ejpam-6409	1056	3	)	)	PUNCT
ejpam-6409	1056	4	(	(	PUNCT
ejpam-6409	1056	5	r	r	NOUN
ejpam-6409	1056	6	,	,	PUNCT
ejpam-6409	1056	7	k	k	NOUN
ejpam-6409	1056	8	)	)	PUNCT
ejpam-6409	1056	9	(	(	PUNCT
ejpam-6409	1056	10	t1	t1	NOUN
ejpam-6409	1056	11	,	,	PUNCT
ejpam-6409	1056	12	.	.	PUNCT
ejpam-6409	1056	13	.	.	PUNCT
ejpam-6409	1056	14	.	.	PUNCT
ejpam-6409	1057	1	,	,	PUNCT
ejpam-6409	1057	2	ts	ts	NOUN
ejpam-6409	1057	3	)	)	PUNCT
ejpam-6409	1057	4	(	(	PUNCT
ejpam-6409	1057	5	r	r	NOUN
ejpam-6409	1057	6	,	,	PUNCT
ejpam-6409	1057	7	k	k	NOUN
ejpam-6409	1057	8	)	)	PUNCT
ejpam-6409	1057	9	z4[ω	z4[ω	NOUN
ejpam-6409	1057	10	]	]	PUNCT
ejpam-6409	1057	11	(	(	PUNCT
ejpam-6409	1057	12	3,3	3,3	NUM
ejpam-6409	1057	13	)	)	PUNCT
ejpam-6409	1057	14	(	(	PUNCT
ejpam-6409	1057	15	10,7	10,7	NUM
ejpam-6409	1057	16	)	)	PUNCT
ejpam-6409	1057	17	(	(	PUNCT
ejpam-6409	1057	18	3,4	3,4	NUM
ejpam-6409	1057	19	)	)	PUNCT
ejpam-6409	1057	20	(	(	PUNCT
ejpam-6409	1057	21	11,8	11,8	NUM
ejpam-6409	1057	22	)	)	PUNCT
ejpam-6409	1057	23	(	(	PUNCT
ejpam-6409	1057	24	3,5	3,5	NUM
ejpam-6409	1057	25	)	)	PUNCT
ejpam-6409	1057	26	(	(	PUNCT
ejpam-6409	1057	27	12,9	12,9	NOUN
ejpam-6409	1057	28	)	)	PUNCT
ejpam-6409	1057	29	(	(	PUNCT
ejpam-6409	1057	30	4	4	NUM
ejpam-6409	1057	31	,	,	PUNCT
ejpam-6409	1057	32	1	1	NUM
ejpam-6409	1057	33	)	)	PUNCT
ejpam-6409	1057	34	(	(	PUNCT
ejpam-6409	1057	35	12	12	NUM
ejpam-6409	1057	36	,	,	PUNCT
ejpam-6409	1057	37	6	6	NUM
ejpam-6409	1057	38	)	)	PUNCT
ejpam-6409	1057	39	(	(	PUNCT
ejpam-6409	1057	40	4	4	NUM
ejpam-6409	1057	41	,	,	PUNCT
ejpam-6409	1057	42	2	2	NUM
ejpam-6409	1057	43	)	)	PUNCT
ejpam-6409	1057	44	(	(	PUNCT
ejpam-6409	1057	45	13	13	NUM
ejpam-6409	1057	46	,	,	PUNCT
ejpam-6409	1057	47	7	7	NUM
ejpam-6409	1057	48	)	)	PUNCT
ejpam-6409	1057	49	(	(	PUNCT
ejpam-6409	1057	50	4,3	4,3	NUM
ejpam-6409	1057	51	)	)	PUNCT
ejpam-6409	1057	52	(	(	PUNCT
ejpam-6409	1057	53	14,8	14,8	NUM
ejpam-6409	1057	54	)	)	PUNCT
ejpam-6409	1057	55	(	(	PUNCT
ejpam-6409	1057	56	5	5	NUM
ejpam-6409	1057	57	,	,	PUNCT
ejpam-6409	1057	58	0	0	NUM
ejpam-6409	1057	59	)	)	PUNCT
ejpam-6409	1057	60	(	(	PUNCT
ejpam-6409	1057	61	16	16	NUM
ejpam-6409	1057	62	,	,	PUNCT
ejpam-6409	1057	63	6	6	NUM
ejpam-6409	1057	64	)	)	PUNCT
ejpam-6409	1057	65	(	(	PUNCT
ejpam-6409	1057	66	5,1	5,1	NUM
ejpam-6409	1057	67	)	)	PUNCT
ejpam-6409	1057	68	(	(	PUNCT
ejpam-6409	1057	69	17,7	17,7	NOUN
ejpam-6409	1057	70	)	)	PUNCT
ejpam-6409	1057	71	z8[ω	z8[ω	PROPN
ejpam-6409	1057	72	]	]	PUNCT
ejpam-6409	1057	73	(	(	PUNCT
ejpam-6409	1057	74	1,2,2	1,2,2	NUM
ejpam-6409	1057	75	)	)	PUNCT
ejpam-6409	1057	76	(	(	PUNCT
ejpam-6409	1057	77	10,7	10,7	NUM
ejpam-6409	1057	78	)	)	PUNCT
ejpam-6409	1057	79	(	(	PUNCT
ejpam-6409	1057	80	1,2,3	1,2,3	NOUN
ejpam-6409	1057	81	)	)	PUNCT
ejpam-6409	1057	82	(	(	PUNCT
ejpam-6409	1057	83	11,8	11,8	NUM
ejpam-6409	1057	84	)	)	PUNCT
ejpam-6409	1057	85	(	(	PUNCT
ejpam-6409	1057	86	1,2,4	1,2,4	NUM
ejpam-6409	1057	87	)	)	PUNCT
ejpam-6409	1057	88	(	(	PUNCT
ejpam-6409	1057	89	12,9	12,9	NOUN
ejpam-6409	1057	90	)	)	PUNCT
ejpam-6409	1057	91	(	(	PUNCT
ejpam-6409	1057	92	1	1	NUM
ejpam-6409	1057	93	,	,	PUNCT
ejpam-6409	1057	94	3	3	NUM
ejpam-6409	1057	95	,	,	PUNCT
ejpam-6409	1057	96	0	0	NUM
ejpam-6409	1057	97	)	)	PUNCT
ejpam-6409	1057	98	(	(	PUNCT
ejpam-6409	1057	99	12	12	NUM
ejpam-6409	1057	100	,	,	PUNCT
ejpam-6409	1057	101	6	6	NUM
ejpam-6409	1057	102	)	)	PUNCT
ejpam-6409	1057	103	(	(	PUNCT
ejpam-6409	1057	104	1,3,1	1,3,1	NUM
ejpam-6409	1057	105	)	)	PUNCT
ejpam-6409	1057	106	(	(	PUNCT
ejpam-6409	1057	107	13,7	13,7	NUM
ejpam-6409	1057	108	)	)	PUNCT
ejpam-6409	1057	109	(	(	PUNCT
ejpam-6409	1057	110	1,3,2	1,3,2	NUM
ejpam-6409	1057	111	)	)	PUNCT
ejpam-6409	1057	112	(	(	PUNCT
ejpam-6409	1057	113	14,8	14,8	NUM
ejpam-6409	1057	114	)	)	PUNCT
ejpam-6409	1057	115	(	(	PUNCT
ejpam-6409	1057	116	2	2	NUM
ejpam-6409	1057	117	,	,	PUNCT
ejpam-6409	1057	118	0	0	NUM
ejpam-6409	1057	119	,	,	PUNCT
ejpam-6409	1057	120	3	3	NUM
ejpam-6409	1057	121	)	)	PUNCT
ejpam-6409	1057	122	(	(	PUNCT
ejpam-6409	1057	123	11	11	NUM
ejpam-6409	1057	124	,	,	PUNCT
ejpam-6409	1057	125	6	6	NUM
ejpam-6409	1057	126	)	)	PUNCT
ejpam-6409	1057	127	(	(	PUNCT
ejpam-6409	1057	128	2	2	NUM
ejpam-6409	1057	129	,	,	PUNCT
ejpam-6409	1057	130	0	0	NUM
ejpam-6409	1057	131	,	,	PUNCT
ejpam-6409	1057	132	4	4	NUM
ejpam-6409	1057	133	)	)	PUNCT
ejpam-6409	1057	134	(	(	PUNCT
ejpam-6409	1057	135	12	12	NUM
ejpam-6409	1057	136	,	,	PUNCT
ejpam-6409	1057	137	7	7	NUM
ejpam-6409	1057	138	)	)	PUNCT
ejpam-6409	1057	139	(	(	PUNCT
ejpam-6409	1057	140	1	1	NUM
ejpam-6409	1057	141	,	,	PUNCT
ejpam-6409	1057	142	4	4	NUM
ejpam-6409	1057	143	,	,	PUNCT
ejpam-6409	1057	144	0	0	NUM
ejpam-6409	1057	145	)	)	PUNCT
ejpam-6409	1057	146	(	(	PUNCT
ejpam-6409	1057	147	17,7	17,7	NUM
ejpam-6409	1057	148	)	)	PUNCT
ejpam-6409	1057	149	(	(	PUNCT
ejpam-6409	1057	150	2,1,1	2,1,1	NUM
ejpam-6409	1057	151	)	)	PUNCT
ejpam-6409	1057	152	(	(	PUNCT
ejpam-6409	1057	153	13,5	13,5	NUM
ejpam-6409	1057	154	)	)	PUNCT
ejpam-6409	1057	155	(	(	PUNCT
ejpam-6409	1057	156	2,1,2	2,1,2	NUM
ejpam-6409	1057	157	)	)	PUNCT
ejpam-6409	1057	158	(	(	PUNCT
ejpam-6409	1057	159	14,6	14,6	NOUN
ejpam-6409	1057	160	)	)	PUNCT
ejpam-6409	1057	161	(	(	PUNCT
ejpam-6409	1057	162	2,0,5	2,0,5	NUM
ejpam-6409	1057	163	)	)	PUNCT
ejpam-6409	1057	164	(	(	PUNCT
ejpam-6409	1057	165	13,8	13,8	NUM
ejpam-6409	1057	166	)	)	PUNCT
ejpam-6409	1057	167	(	(	PUNCT
ejpam-6409	1057	168	3,0,0	3,0,0	NUM
ejpam-6409	1057	169	)	)	PUNCT
ejpam-6409	1057	170	(	(	PUNCT
ejpam-6409	1057	171	17,4	17,4	NUM
ejpam-6409	1057	172	)	)	PUNCT
ejpam-6409	1057	173	(	(	PUNCT
ejpam-6409	1057	174	2,2,0	2,2,0	NUM
ejpam-6409	1057	175	)	)	PUNCT
ejpam-6409	1057	176	(	(	PUNCT
ejpam-6409	1057	177	17,5	17,5	NUM
ejpam-6409	1057	178	)	)	PUNCT
ejpam-6409	1057	179	(	(	PUNCT
ejpam-6409	1057	180	2,1,3	2,1,3	NUM
ejpam-6409	1057	181	)	)	PUNCT
ejpam-6409	1057	182	(	(	PUNCT
ejpam-6409	1057	183	15,7	15,7	NUM
ejpam-6409	1057	184	)	)	PUNCT
ejpam-6409	1057	185	(	(	PUNCT
ejpam-6409	1057	186	3,0,1	3,0,1	NOUN
ejpam-6409	1057	187	)	)	PUNCT
ejpam-6409	1057	188	(	(	PUNCT
ejpam-6409	1057	189	18,5	18,5	NUM
ejpam-6409	1057	190	)	)	PUNCT
ejpam-6409	1057	191	(	(	PUNCT
ejpam-6409	1057	192	2,2,1	2,2,1	NUM
ejpam-6409	1057	193	)	)	PUNCT
ejpam-6409	1057	194	(	(	PUNCT
ejpam-6409	1057	195	18,6	18,6	NUM
ejpam-6409	1057	196	)	)	PUNCT
ejpam-6409	1057	197	(	(	PUNCT
ejpam-6409	1057	198	3,0,2	3,0,2	NUM
ejpam-6409	1057	199	)	)	PUNCT
ejpam-6409	1057	200	(	(	PUNCT
ejpam-6409	1057	201	19,6	19,6	NUM
ejpam-6409	1057	202	)	)	PUNCT
ejpam-6409	1057	203	(	(	PUNCT
ejpam-6409	1057	204	3,1,0	3,1,0	NUM
ejpam-6409	1057	205	)	)	PUNCT
ejpam-6409	1057	206	(	(	PUNCT
ejpam-6409	1057	207	24,5	24,5	NUM
ejpam-6409	1057	208	)	)	PUNCT
ejpam-6409	1057	209	z16[ω	z16[ω	NOUN
ejpam-6409	1057	210	]	]	PUNCT
ejpam-6409	1057	211	(	(	PUNCT
ejpam-6409	1057	212	1	1	NUM
ejpam-6409	1057	213	,	,	PUNCT
ejpam-6409	1057	214	0	0	NUM
ejpam-6409	1057	215	,	,	PUNCT
ejpam-6409	1057	216	2	2	NUM
ejpam-6409	1057	217	,	,	PUNCT
ejpam-6409	1057	218	1	1	NUM
ejpam-6409	1057	219	)	)	PUNCT
ejpam-6409	1057	220	(	(	PUNCT
ejpam-6409	1057	221	10	10	NUM
ejpam-6409	1057	222	,	,	PUNCT
ejpam-6409	1057	223	7	7	NUM
ejpam-6409	1057	224	)	)	PUNCT
ejpam-6409	1057	225	(	(	PUNCT
ejpam-6409	1057	226	1,0,2,2	1,0,2,2	NUM
ejpam-6409	1057	227	)	)	PUNCT
ejpam-6409	1057	228	(	(	PUNCT
ejpam-6409	1057	229	11,8	11,8	NUM
ejpam-6409	1057	230	)	)	PUNCT
ejpam-6409	1057	231	(	(	PUNCT
ejpam-6409	1057	232	1,0,2,3	1,0,2,3	NUM
ejpam-6409	1057	233	)	)	PUNCT
ejpam-6409	1057	234	(	(	PUNCT
ejpam-6409	1057	235	12,9	12,9	NUM
ejpam-6409	1057	236	)	)	PUNCT
ejpam-6409	1057	237	(	(	PUNCT
ejpam-6409	1057	238	1	1	NUM
ejpam-6409	1057	239	,	,	PUNCT
ejpam-6409	1057	240	1	1	NUM
ejpam-6409	1057	241	,	,	PUNCT
ejpam-6409	1057	242	0	0	NUM
ejpam-6409	1057	243	,	,	PUNCT
ejpam-6409	1057	244	2	2	NUM
ejpam-6409	1057	245	)	)	PUNCT
ejpam-6409	1057	246	(	(	PUNCT
ejpam-6409	1057	247	11	11	NUM
ejpam-6409	1057	248	,	,	PUNCT
ejpam-6409	1057	249	6	6	NUM
ejpam-6409	1057	250	)	)	PUNCT
ejpam-6409	1057	251	(	(	PUNCT
ejpam-6409	1057	252	1,0,3,0	1,0,3,0	NUM
ejpam-6409	1057	253	)	)	PUNCT
ejpam-6409	1057	254	(	(	PUNCT
ejpam-6409	1057	255	17,7	17,7	NUM
ejpam-6409	1057	256	)	)	PUNCT
ejpam-6409	1057	257	(	(	PUNCT
ejpam-6409	1057	258	1,0,3,1	1,0,3,1	NUM
ejpam-6409	1057	259	)	)	PUNCT
ejpam-6409	1057	260	(	(	PUNCT
ejpam-6409	1057	261	14,8	14,8	NUM
ejpam-6409	1057	262	)	)	PUNCT
ejpam-6409	1057	263	(	(	PUNCT
ejpam-6409	1057	264	1	1	NUM
ejpam-6409	1057	265	,	,	PUNCT
ejpam-6409	1057	266	1	1	NUM
ejpam-6409	1057	267	,	,	PUNCT
ejpam-6409	1057	268	1	1	NUM
ejpam-6409	1057	269	,	,	PUNCT
ejpam-6409	1057	270	0	0	NUM
ejpam-6409	1057	271	)	)	PUNCT
ejpam-6409	1057	272	(	(	PUNCT
ejpam-6409	1057	273	13	13	NUM
ejpam-6409	1057	274	,	,	PUNCT
ejpam-6409	1057	275	5	5	NUM
ejpam-6409	1057	276	)	)	PUNCT
ejpam-6409	1057	277	(	(	PUNCT
ejpam-6409	1057	278	1,2,0,0	1,2,0,0	NUM
ejpam-6409	1057	279	)	)	PUNCT
ejpam-6409	1057	280	(	(	PUNCT
ejpam-6409	1057	281	18,5	18,5	NUM
ejpam-6409	1057	282	)	)	PUNCT
ejpam-6409	1057	283	(	(	PUNCT
ejpam-6409	1057	284	1,1,0,4	1,1,0,4	NUM
ejpam-6409	1057	285	)	)	PUNCT
ejpam-6409	1057	286	(	(	PUNCT
ejpam-6409	1057	287	13,8	13,8	NUM
ejpam-6409	1057	288	)	)	PUNCT
ejpam-6409	1057	289	(	(	PUNCT
ejpam-6409	1057	290	2	2	NUM
ejpam-6409	1057	291	,	,	PUNCT
ejpam-6409	1057	292	0	0	NUM
ejpam-6409	1057	293	,	,	PUNCT
ejpam-6409	1057	294	0	0	NUM
ejpam-6409	1057	295	,	,	PUNCT
ejpam-6409	1057	296	1	1	NUM
ejpam-6409	1057	297	)	)	PUNCT
ejpam-6409	1057	298	(	(	PUNCT
ejpam-6409	1057	299	15	15	NUM
ejpam-6409	1057	300	,	,	PUNCT
ejpam-6409	1057	301	4	4	NUM
ejpam-6409	1057	302	)	)	PUNCT
ejpam-6409	1057	303	(	(	PUNCT
ejpam-6409	1057	304	1,1,0,3	1,1,0,3	NUM
ejpam-6409	1057	305	)	)	PUNCT
ejpam-6409	1057	306	(	(	PUNCT
ejpam-6409	1057	307	12,7	12,7	NUM
ejpam-6409	1057	308	)	)	PUNCT
ejpam-6409	1057	309	(	(	PUNCT
ejpam-6409	1057	310	1,1,1,2	1,1,1,2	NUM
ejpam-6409	1057	311	)	)	PUNCT
ejpam-6409	1057	312	(	(	PUNCT
ejpam-6409	1057	313	15,7	15,7	NUM
ejpam-6409	1057	314	)	)	PUNCT
ejpam-6409	1057	315	(	(	PUNCT
ejpam-6409	1057	316	1	1	NUM
ejpam-6409	1057	317	,	,	PUNCT
ejpam-6409	1057	318	1	1	NUM
ejpam-6409	1057	319	,	,	PUNCT
ejpam-6409	1057	320	1	1	NUM
ejpam-6409	1057	321	,	,	PUNCT
ejpam-6409	1057	322	1	1	NUM
ejpam-6409	1057	323	)	)	PUNCT
ejpam-6409	1057	324	(	(	PUNCT
ejpam-6409	1057	325	14,6	14,6	NOUN
ejpam-6409	1057	326	)	)	PUNCT
ejpam-6409	1057	327	(	(	PUNCT
ejpam-6409	1057	328	1,1,2,0	1,1,2,0	NUM
ejpam-6409	1057	329	)	)	PUNCT
ejpam-6409	1057	330	(	(	PUNCT
ejpam-6409	1057	331	18,6	18,6	NUM
ejpam-6409	1057	332	)	)	PUNCT
ejpam-6409	1057	333	(	(	PUNCT
ejpam-6409	1057	334	2	2	NUM
ejpam-6409	1057	335	,	,	PUNCT
ejpam-6409	1057	336	0	0	NUM
ejpam-6409	1057	337	,	,	PUNCT
ejpam-6409	1057	338	0	0	NUM
ejpam-6409	1057	339	,	,	PUNCT
ejpam-6409	1057	340	2	2	NUM
ejpam-6409	1057	341	)	)	PUNCT
ejpam-6409	1057	342	(	(	PUNCT
ejpam-6409	1057	343	16,5	16,5	NUM
ejpam-6409	1057	344	)	)	PUNCT
ejpam-6409	1057	345	(	(	PUNCT
ejpam-6409	1057	346	1,2,0,1	1,2,0,1	NUM
ejpam-6409	1057	347	)	)	PUNCT
ejpam-6409	1057	348	(	(	PUNCT
ejpam-6409	1057	349	19,6	19,6	NUM
ejpam-6409	1057	350	)	)	PUNCT
ejpam-6409	1057	351	(	(	PUNCT
ejpam-6409	1057	352	2,0,1,0	2,0,1,0	NUM
ejpam-6409	1057	353	)	)	PUNCT
ejpam-6409	1057	354	(	(	PUNCT
ejpam-6409	1057	355	20,4	20,4	NOUN
ejpam-6409	1057	356	)	)	PUNCT
ejpam-6409	1057	357	(	(	PUNCT
ejpam-6409	1057	358	2,0,0,3	2,0,0,3	NUM
ejpam-6409	1057	359	)	)	PUNCT
ejpam-6409	1057	360	(	(	PUNCT
ejpam-6409	1057	361	17,6	17,6	NUM
ejpam-6409	1057	362	)	)	PUNCT
ejpam-6409	1057	363	(	(	PUNCT
ejpam-6409	1057	364	2	2	NUM
ejpam-6409	1057	365	,	,	PUNCT
ejpam-6409	1057	366	0	0	NUM
ejpam-6409	1057	367	,	,	PUNCT
ejpam-6409	1057	368	1	1	NUM
ejpam-6409	1057	369	,	,	PUNCT
ejpam-6409	1057	370	1	1	NUM
ejpam-6409	1057	371	)	)	PUNCT
ejpam-6409	1057	372	(	(	PUNCT
ejpam-6409	1057	373	21,5	21,5	NUM
ejpam-6409	1057	374	)	)	PUNCT
ejpam-6409	1057	375	(	(	PUNCT
ejpam-6409	1057	376	2,1,0,0	2,1,0,0	NUM
ejpam-6409	1057	377	)	)	PUNCT
ejpam-6409	1057	378	(	(	PUNCT
ejpam-6409	1057	379	28,4	28,4	NUM
ejpam-6409	1057	380	)	)	PUNCT
ejpam-6409	1057	381	z32[ω	z32[ω	NOUN
ejpam-6409	1057	382	]	]	PUNCT
ejpam-6409	1057	383	(	(	PUNCT
ejpam-6409	1057	384	1	1	NUM
ejpam-6409	1057	385	,	,	PUNCT
ejpam-6409	1057	386	0	0	NUM
ejpam-6409	1057	387	,	,	PUNCT
ejpam-6409	1057	388	0	0	NUM
ejpam-6409	1057	389	,	,	PUNCT
ejpam-6409	1057	390	2	2	NUM
ejpam-6409	1057	391	,	,	PUNCT
ejpam-6409	1057	392	0	0	NUM
ejpam-6409	1057	393	)	)	PUNCT
ejpam-6409	1057	394	(	(	PUNCT
ejpam-6409	1057	395	10	10	NUM
ejpam-6409	1057	396	,	,	PUNCT
ejpam-6409	1057	397	7	7	NUM
ejpam-6409	1057	398	)	)	PUNCT
ejpam-6409	1057	399	(	(	PUNCT
ejpam-6409	1057	400	1	1	NUM
ejpam-6409	1057	401	,	,	PUNCT
ejpam-6409	1057	402	0	0	NUM
ejpam-6409	1057	403	,	,	PUNCT
ejpam-6409	1057	404	0	0	NUM
ejpam-6409	1057	405	,	,	PUNCT
ejpam-6409	1057	406	2	2	NUM
ejpam-6409	1057	407	,	,	PUNCT
ejpam-6409	1057	408	1	1	NUM
ejpam-6409	1057	409	)	)	PUNCT
ejpam-6409	1057	410	(	(	PUNCT
ejpam-6409	1057	411	11	11	NUM
ejpam-6409	1057	412	,	,	PUNCT
ejpam-6409	1057	413	8)	8)	NUM
ejpam-6409	1057	414	(	(	PUNCT
ejpam-6409	1057	415	1	1	NUM
ejpam-6409	1057	416	,	,	PUNCT
ejpam-6409	1057	417	0	0	NUM
ejpam-6409	1057	418	,	,	PUNCT
ejpam-6409	1057	419	0	0	NUM
ejpam-6409	1057	420	,	,	PUNCT
ejpam-6409	1057	421	2	2	NUM
ejpam-6409	1057	422	,	,	PUNCT
ejpam-6409	1057	423	2	2	NUM
ejpam-6409	1057	424	)	)	PUNCT
ejpam-6409	1057	425	(	(	PUNCT
ejpam-6409	1057	426	12	12	NUM
ejpam-6409	1057	427	,	,	PUNCT
ejpam-6409	1057	428	9	9	NUM
ejpam-6409	1057	429	)	)	PUNCT
ejpam-6409	1057	430	(	(	PUNCT
ejpam-6409	1057	431	1	1	NUM
ejpam-6409	1057	432	,	,	PUNCT
ejpam-6409	1057	433	0	0	NUM
ejpam-6409	1057	434	,	,	PUNCT
ejpam-6409	1057	435	1	1	NUM
ejpam-6409	1057	436	,	,	PUNCT
ejpam-6409	1057	437	0	0	NUM
ejpam-6409	1057	438	,	,	PUNCT
ejpam-6409	1057	439	1	1	NUM
ejpam-6409	1057	440	)	)	PUNCT
ejpam-6409	1057	441	(	(	PUNCT
ejpam-6409	1057	442	11	11	NUM
ejpam-6409	1057	443	,	,	PUNCT
ejpam-6409	1057	444	6	6	NUM
ejpam-6409	1057	445	)	)	PUNCT
ejpam-6409	1057	446	(	(	PUNCT
ejpam-6409	1057	447	1	1	NUM
ejpam-6409	1057	448	,	,	PUNCT
ejpam-6409	1057	449	0	0	NUM
ejpam-6409	1057	450	,	,	PUNCT
ejpam-6409	1057	451	1	1	NUM
ejpam-6409	1057	452	,	,	PUNCT
ejpam-6409	1057	453	0	0	NUM
ejpam-6409	1057	454	,	,	PUNCT
ejpam-6409	1057	455	2	2	NUM
ejpam-6409	1057	456	)	)	PUNCT
ejpam-6409	1057	457	(	(	PUNCT
ejpam-6409	1057	458	12	12	NUM
ejpam-6409	1057	459	,	,	PUNCT
ejpam-6409	1057	460	7	7	NUM
ejpam-6409	1057	461	)	)	PUNCT
ejpam-6409	1057	462	(	(	PUNCT
ejpam-6409	1057	463	1	1	NUM
ejpam-6409	1057	464	,	,	PUNCT
ejpam-6409	1057	465	0	0	NUM
ejpam-6409	1057	466	,	,	PUNCT
ejpam-6409	1057	467	0	0	NUM
ejpam-6409	1057	468	,	,	PUNCT
ejpam-6409	1057	469	3	3	NUM
ejpam-6409	1057	470	,	,	PUNCT
ejpam-6409	1057	471	0	0	NUM
ejpam-6409	1057	472	)	)	PUNCT
ejpam-6409	1057	473	(	(	PUNCT
ejpam-6409	1057	474	14,8	14,8	NUM
ejpam-6409	1057	475	)	)	PUNCT
ejpam-6409	1057	476	(	(	PUNCT
ejpam-6409	1057	477	1	1	NUM
ejpam-6409	1057	478	,	,	PUNCT
ejpam-6409	1057	479	1	1	NUM
ejpam-6409	1057	480	,	,	PUNCT
ejpam-6409	1057	481	0	0	NUM
ejpam-6409	1057	482	,	,	PUNCT
ejpam-6409	1057	483	0	0	NUM
ejpam-6409	1057	484	,	,	PUNCT
ejpam-6409	1057	485	0	0	NUM
ejpam-6409	1057	486	)	)	PUNCT
ejpam-6409	1057	487	(	(	PUNCT
ejpam-6409	1057	488	15	15	NUM
ejpam-6409	1057	489	,	,	PUNCT
ejpam-6409	1057	490	4	4	NUM
ejpam-6409	1057	491	)	)	PUNCT
ejpam-6409	1057	492	(	(	PUNCT
ejpam-6409	1057	493	1	1	NUM
ejpam-6409	1057	494	,	,	PUNCT
ejpam-6409	1057	495	0	0	NUM
ejpam-6409	1057	496	,	,	PUNCT
ejpam-6409	1057	497	1	1	NUM
ejpam-6409	1057	498	,	,	PUNCT
ejpam-6409	1057	499	1	1	NUM
ejpam-6409	1057	500	,	,	PUNCT
ejpam-6409	1057	501	0	0	NUM
ejpam-6409	1057	502	)	)	PUNCT
ejpam-6409	1057	503	(	(	PUNCT
ejpam-6409	1057	504	14	14	NUM
ejpam-6409	1057	505	,	,	PUNCT
ejpam-6409	1057	506	6	6	NUM
ejpam-6409	1057	507	)	)	PUNCT
ejpam-6409	1057	508	(	(	PUNCT
ejpam-6409	1057	509	1	1	NUM
ejpam-6409	1057	510	,	,	PUNCT
ejpam-6409	1057	511	0	0	NUM
ejpam-6409	1057	512	,	,	PUNCT
ejpam-6409	1057	513	1	1	NUM
ejpam-6409	1057	514	,	,	PUNCT
ejpam-6409	1057	515	0	0	NUM
ejpam-6409	1057	516	,	,	PUNCT
ejpam-6409	1057	517	3	3	NUM
ejpam-6409	1057	518	)	)	PUNCT
ejpam-6409	1057	519	(	(	PUNCT
ejpam-6409	1057	520	13,8	13,8	NUM
ejpam-6409	1057	521	)	)	PUNCT
ejpam-6409	1057	522	(	(	PUNCT
ejpam-6409	1057	523	1	1	NUM
ejpam-6409	1057	524	,	,	PUNCT
ejpam-6409	1057	525	1	1	NUM
ejpam-6409	1057	526	,	,	PUNCT
ejpam-6409	1057	527	0	0	NUM
ejpam-6409	1057	528	,	,	PUNCT
ejpam-6409	1057	529	0	0	NUM
ejpam-6409	1057	530	,	,	PUNCT
ejpam-6409	1057	531	1	1	NUM
ejpam-6409	1057	532	)	)	PUNCT
ejpam-6409	1057	533	(	(	PUNCT
ejpam-6409	1057	534	16	16	NUM
ejpam-6409	1057	535	,	,	PUNCT
ejpam-6409	1057	536	5	5	NUM
ejpam-6409	1057	537	)	)	PUNCT
ejpam-6409	1057	538	(	(	PUNCT
ejpam-6409	1057	539	1	1	NUM
ejpam-6409	1057	540	,	,	PUNCT
ejpam-6409	1057	541	0	0	NUM
ejpam-6409	1057	542	,	,	PUNCT
ejpam-6409	1057	543	1	1	NUM
ejpam-6409	1057	544	,	,	PUNCT
ejpam-6409	1057	545	1	1	NUM
ejpam-6409	1057	546	,	,	PUNCT
ejpam-6409	1057	547	1	1	NUM
ejpam-6409	1057	548	)	)	PUNCT
ejpam-6409	1057	549	(	(	PUNCT
ejpam-6409	1057	550	15,7	15,7	NUM
ejpam-6409	1057	551	)	)	PUNCT
ejpam-6409	1057	552	(	(	PUNCT
ejpam-6409	1057	553	2	2	NUM
ejpam-6409	1057	554	,	,	PUNCT
ejpam-6409	1057	555	0	0	NUM
ejpam-6409	1057	556	,	,	PUNCT
ejpam-6409	1057	557	0	0	NUM
ejpam-6409	1057	558	,	,	PUNCT
ejpam-6409	1057	559	0	0	NUM
ejpam-6409	1057	560	,	,	PUNCT
ejpam-6409	1057	561	0	0	NUM
ejpam-6409	1057	562	)	)	PUNCT
ejpam-6409	1057	563	(	(	PUNCT
ejpam-6409	1057	564	26	26	NUM
ejpam-6409	1057	565	,	,	PUNCT
ejpam-6409	1057	566	3	3	NUM
ejpam-6409	1057	567	)	)	PUNCT
ejpam-6409	1057	568	(	(	PUNCT
ejpam-6409	1057	569	1	1	NUM
ejpam-6409	1057	570	,	,	PUNCT
ejpam-6409	1057	571	0	0	NUM
ejpam-6409	1057	572	,	,	PUNCT
ejpam-6409	1057	573	2	2	NUM
ejpam-6409	1057	574	,	,	PUNCT
ejpam-6409	1057	575	0	0	NUM
ejpam-6409	1057	576	,	,	PUNCT
ejpam-6409	1057	577	0	0	NUM
ejpam-6409	1057	578	)	)	PUNCT
ejpam-6409	1057	579	(	(	PUNCT
ejpam-6409	1057	580	19,6	19,6	NUM
ejpam-6409	1057	581	)	)	PUNCT
ejpam-6409	1057	582	(	(	PUNCT
ejpam-6409	1057	583	1	1	NUM
ejpam-6409	1057	584	,	,	PUNCT
ejpam-6409	1057	585	1	1	NUM
ejpam-6409	1057	586	,	,	PUNCT
ejpam-6409	1057	587	0	0	NUM
ejpam-6409	1057	588	,	,	PUNCT
ejpam-6409	1057	589	0	0	NUM
ejpam-6409	1057	590	,	,	PUNCT
ejpam-6409	1057	591	2	2	NUM
ejpam-6409	1057	592	)	)	PUNCT
ejpam-6409	1057	593	(	(	PUNCT
ejpam-6409	1057	594	17,6	17,6	NUM
ejpam-6409	1057	595	)	)	PUNCT
ejpam-6409	1057	596	(	(	PUNCT
ejpam-6409	1057	597	1	1	NUM
ejpam-6409	1057	598	,	,	PUNCT
ejpam-6409	1057	599	1	1	NUM
ejpam-6409	1057	600	,	,	PUNCT
ejpam-6409	1057	601	0	0	NUM
ejpam-6409	1057	602	,	,	PUNCT
ejpam-6409	1057	603	1	1	NUM
ejpam-6409	1057	604	,	,	PUNCT
ejpam-6409	1057	605	0	0	NUM
ejpam-6409	1057	606	)	)	PUNCT
ejpam-6409	1057	607	(	(	PUNCT
ejpam-6409	1057	608	21	21	NUM
ejpam-6409	1057	609	,	,	PUNCT
ejpam-6409	1057	610	5	5	NUM
ejpam-6409	1057	611	)	)	PUNCT
ejpam-6409	1057	612	(	(	PUNCT
ejpam-6409	1057	613	2	2	NUM
ejpam-6409	1057	614	,	,	PUNCT
ejpam-6409	1057	615	0	0	NUM
ejpam-6409	1057	616	,	,	PUNCT
ejpam-6409	1057	617	0	0	NUM
ejpam-6409	1057	618	,	,	PUNCT
ejpam-6409	1057	619	0	0	NUM
ejpam-6409	1057	620	,	,	PUNCT
ejpam-6409	1057	621	1	1	NUM
ejpam-6409	1057	622	)	)	PUNCT
ejpam-6409	1057	623	(	(	PUNCT
ejpam-6409	1057	624	27	27	NUM
ejpam-6409	1057	625	,	,	PUNCT
ejpam-6409	1057	626	4	4	NUM
ejpam-6409	1057	627	)	)	PUNCT
ejpam-6409	1057	628	z64[ω	z64[ω	NOUN
ejpam-6409	1057	629	]	]	PUNCT
ejpam-6409	1057	630	(	(	PUNCT
ejpam-6409	1057	631	1	1	NUM
ejpam-6409	1057	632	,	,	PUNCT
ejpam-6409	1057	633	0	0	NUM
ejpam-6409	1057	634	,	,	PUNCT
ejpam-6409	1057	635	0	0	NUM
ejpam-6409	1057	636	,	,	PUNCT
ejpam-6409	1057	637	1	1	NUM
ejpam-6409	1057	638	,	,	PUNCT
ejpam-6409	1057	639	0	0	NUM
ejpam-6409	1057	640	,	,	PUNCT
ejpam-6409	1057	641	0	0	NUM
ejpam-6409	1057	642	)	)	PUNCT
ejpam-6409	1057	643	(	(	PUNCT
ejpam-6409	1057	644	11	11	NUM
ejpam-6409	1057	645	,	,	PUNCT
ejpam-6409	1057	646	6	6	NUM
ejpam-6409	1057	647	)	)	PUNCT
ejpam-6409	1057	648	(	(	PUNCT
ejpam-6409	1057	649	1	1	NUM
ejpam-6409	1057	650	,	,	PUNCT
ejpam-6409	1057	651	0	0	NUM
ejpam-6409	1057	652	,	,	PUNCT
ejpam-6409	1057	653	0	0	NUM
ejpam-6409	1057	654	,	,	PUNCT
ejpam-6409	1057	655	0	0	NUM
ejpam-6409	1057	656	,	,	PUNCT
ejpam-6409	1057	657	2	2	NUM
ejpam-6409	1057	658	,	,	PUNCT
ejpam-6409	1057	659	0	0	NUM
ejpam-6409	1057	660	)	)	PUNCT
ejpam-6409	1057	661	(	(	PUNCT
ejpam-6409	1057	662	11	11	NUM
ejpam-6409	1057	663	,	,	PUNCT
ejpam-6409	1057	664	8)	8)	NUM
ejpam-6409	1057	665	(	(	PUNCT
ejpam-6409	1057	666	1	1	NUM
ejpam-6409	1057	667	,	,	PUNCT
ejpam-6409	1057	668	0	0	NUM
ejpam-6409	1057	669	,	,	PUNCT
ejpam-6409	1057	670	0	0	NUM
ejpam-6409	1057	671	,	,	PUNCT
ejpam-6409	1057	672	0	0	NUM
ejpam-6409	1057	673	,	,	PUNCT
ejpam-6409	1057	674	2	2	NUM
ejpam-6409	1057	675	,	,	PUNCT
ejpam-6409	1057	676	1	1	NUM
ejpam-6409	1057	677	)	)	PUNCT
ejpam-6409	1057	678	(	(	PUNCT
ejpam-6409	1057	679	12	12	NUM
ejpam-6409	1057	680	,	,	PUNCT
ejpam-6409	1057	681	9	9	NUM
ejpam-6409	1057	682	)	)	PUNCT
ejpam-6409	1057	683	(	(	PUNCT
ejpam-6409	1057	684	1	1	NUM
ejpam-6409	1057	685	,	,	PUNCT
ejpam-6409	1057	686	0	0	NUM
ejpam-6409	1057	687	,	,	PUNCT
ejpam-6409	1057	688	0	0	NUM
ejpam-6409	1057	689	,	,	PUNCT
ejpam-6409	1057	690	1	1	NUM
ejpam-6409	1057	691	,	,	PUNCT
ejpam-6409	1057	692	0	0	NUM
ejpam-6409	1057	693	,	,	PUNCT
ejpam-6409	1057	694	1	1	NUM
ejpam-6409	1057	695	)	)	PUNCT
ejpam-6409	1057	696	(	(	PUNCT
ejpam-6409	1057	697	12	12	NUM
ejpam-6409	1057	698	,	,	PUNCT
ejpam-6409	1057	699	7	7	NUM
ejpam-6409	1057	700	)	)	PUNCT
ejpam-6409	1057	701	(	(	PUNCT
ejpam-6409	1057	702	1	1	NUM
ejpam-6409	1057	703	,	,	PUNCT
ejpam-6409	1057	704	0	0	NUM
ejpam-6409	1057	705	,	,	PUNCT
ejpam-6409	1057	706	0	0	NUM
ejpam-6409	1057	707	,	,	PUNCT
ejpam-6409	1057	708	1	1	NUM
ejpam-6409	1057	709	,	,	PUNCT
ejpam-6409	1057	710	0	0	NUM
ejpam-6409	1057	711	,	,	PUNCT
ejpam-6409	1057	712	2	2	NUM
ejpam-6409	1057	713	)	)	PUNCT
ejpam-6409	1057	714	(	(	PUNCT
ejpam-6409	1057	715	13	13	NUM
ejpam-6409	1057	716	,	,	PUNCT
ejpam-6409	1057	717	8)	8)	NUM
ejpam-6409	1057	718	(	(	PUNCT
ejpam-6409	1057	719	1	1	NUM
ejpam-6409	1057	720	,	,	PUNCT
ejpam-6409	1057	721	0	0	NUM
ejpam-6409	1057	722	,	,	PUNCT
ejpam-6409	1057	723	1	1	NUM
ejpam-6409	1057	724	,	,	PUNCT
ejpam-6409	1057	725	0	0	NUM
ejpam-6409	1057	726	,	,	PUNCT
ejpam-6409	1057	727	0	0	NUM
ejpam-6409	1057	728	,	,	PUNCT
ejpam-6409	1057	729	0	0	NUM
ejpam-6409	1057	730	)	)	PUNCT
ejpam-6409	1057	731	(	(	PUNCT
ejpam-6409	1057	732	16	16	NUM
ejpam-6409	1057	733	,	,	PUNCT
ejpam-6409	1057	734	5	5	NUM
ejpam-6409	1057	735	)	)	PUNCT
ejpam-6409	1057	736	(	(	PUNCT
ejpam-6409	1057	737	1	1	NUM
ejpam-6409	1057	738	,	,	PUNCT
ejpam-6409	1057	739	0	0	NUM
ejpam-6409	1057	740	,	,	PUNCT
ejpam-6409	1057	741	0	0	NUM
ejpam-6409	1057	742	,	,	PUNCT
ejpam-6409	1057	743	1	1	NUM
ejpam-6409	1057	744	,	,	PUNCT
ejpam-6409	1057	745	1	1	NUM
ejpam-6409	1057	746	,	,	PUNCT
ejpam-6409	1057	747	0	0	NUM
ejpam-6409	1057	748	)	)	PUNCT
ejpam-6409	1057	749	(	(	PUNCT
ejpam-6409	1057	750	15	15	NUM
ejpam-6409	1057	751	,	,	PUNCT
ejpam-6409	1057	752	7	7	NUM
ejpam-6409	1057	753	)	)	PUNCT
ejpam-6409	1057	754	(	(	PUNCT
ejpam-6409	1057	755	1	1	NUM
ejpam-6409	1057	756	,	,	PUNCT
ejpam-6409	1057	757	0	0	NUM
ejpam-6409	1057	758	,	,	PUNCT
ejpam-6409	1057	759	1	1	NUM
ejpam-6409	1057	760	,	,	PUNCT
ejpam-6409	1057	761	0	0	NUM
ejpam-6409	1057	762	,	,	PUNCT
ejpam-6409	1057	763	0	0	NUM
ejpam-6409	1057	764	,	,	PUNCT
ejpam-6409	1057	765	1	1	NUM
ejpam-6409	1057	766	)	)	PUNCT
ejpam-6409	1057	767	(	(	PUNCT
ejpam-6409	1057	768	17	17	NUM
ejpam-6409	1057	769	,	,	PUNCT
ejpam-6409	1057	770	6	6	NUM
ejpam-6409	1057	771	)	)	PUNCT
ejpam-6409	1057	772	(	(	PUNCT
ejpam-6409	1057	773	1	1	NUM
ejpam-6409	1057	774	,	,	PUNCT
ejpam-6409	1057	775	1	1	NUM
ejpam-6409	1057	776	,	,	PUNCT
ejpam-6409	1057	777	0	0	NUM
ejpam-6409	1057	778	,	,	PUNCT
ejpam-6409	1057	779	0	0	NUM
ejpam-6409	1057	780	,	,	PUNCT
ejpam-6409	1057	781	0	0	NUM
ejpam-6409	1057	782	,	,	PUNCT
ejpam-6409	1057	783	0	0	NUM
ejpam-6409	1057	784	)	)	PUNCT
ejpam-6409	1057	785	(	(	PUNCT
ejpam-6409	1057	786	27	27	NUM
ejpam-6409	1057	787	,	,	PUNCT
ejpam-6409	1057	788	4	4	NUM
ejpam-6409	1057	789	)	)	PUNCT
ejpam-6409	1057	790	z128[ω	z128[ω	NUM
ejpam-6409	1057	791	]	]	PUNCT
ejpam-6409	1057	792	(	(	PUNCT
ejpam-6409	1057	793	1	1	NUM
ejpam-6409	1057	794	,	,	PUNCT
ejpam-6409	1057	795	0	0	NUM
ejpam-6409	1057	796	,	,	PUNCT
ejpam-6409	1057	797	0	0	NUM
ejpam-6409	1057	798	,	,	PUNCT
ejpam-6409	1057	799	0	0	NUM
ejpam-6409	1057	800	,	,	PUNCT
ejpam-6409	1057	801	1	1	NUM
ejpam-6409	1057	802	,	,	PUNCT
ejpam-6409	1057	803	0	0	NUM
ejpam-6409	1057	804	,	,	PUNCT
ejpam-6409	1057	805	0	0	NUM
ejpam-6409	1057	806	)	)	PUNCT
ejpam-6409	1057	807	(	(	PUNCT
ejpam-6409	1057	808	12	12	NUM
ejpam-6409	1057	809	,	,	PUNCT
ejpam-6409	1057	810	7	7	NUM
ejpam-6409	1057	811	)	)	PUNCT
ejpam-6409	1057	812	(	(	PUNCT
ejpam-6409	1057	813	1	1	NUM
ejpam-6409	1057	814	,	,	PUNCT
ejpam-6409	1057	815	0	0	NUM
ejpam-6409	1057	816	,	,	PUNCT
ejpam-6409	1057	817	0	0	NUM
ejpam-6409	1057	818	,	,	PUNCT
ejpam-6409	1057	819	0	0	NUM
ejpam-6409	1057	820	,	,	PUNCT
ejpam-6409	1057	821	0	0	NUM
ejpam-6409	1057	822	,	,	PUNCT
ejpam-6409	1057	823	2	2	NUM
ejpam-6409	1057	824	,	,	PUNCT
ejpam-6409	1057	825	0	0	NUM
ejpam-6409	1057	826	)	)	PUNCT
ejpam-6409	1057	827	(	(	PUNCT
ejpam-6409	1057	828	19	19	NUM
ejpam-6409	1057	829	,	,	PUNCT
ejpam-6409	1057	830	2	2	NUM
ejpam-6409	1057	831	)	)	PUNCT
ejpam-6409	1057	832	(	(	PUNCT
ejpam-6409	1057	833	1	1	NUM
ejpam-6409	1057	834	,	,	PUNCT
ejpam-6409	1057	835	0	0	NUM
ejpam-6409	1057	836	,	,	PUNCT
ejpam-6409	1057	837	0	0	NUM
ejpam-6409	1057	838	,	,	PUNCT
ejpam-6409	1057	839	0	0	NUM
ejpam-6409	1057	840	,	,	PUNCT
ejpam-6409	1057	841	1	1	NUM
ejpam-6409	1057	842	,	,	PUNCT
ejpam-6409	1057	843	0	0	NUM
ejpam-6409	1057	844	,	,	PUNCT
ejpam-6409	1057	845	1	1	NUM
ejpam-6409	1057	846	)	)	PUNCT
ejpam-6409	1057	847	(	(	PUNCT
ejpam-6409	1057	848	13	13	NUM
ejpam-6409	1057	849	,	,	PUNCT
ejpam-6409	1057	850	8)	8)	NUM
ejpam-6409	1057	851	(	(	PUNCT
ejpam-6409	1057	852	1	1	NUM
ejpam-6409	1057	853	,	,	PUNCT
ejpam-6409	1057	854	0	0	NUM
ejpam-6409	1057	855	,	,	PUNCT
ejpam-6409	1057	856	0	0	NUM
ejpam-6409	1057	857	,	,	PUNCT
ejpam-6409	1057	858	1	1	NUM
ejpam-6409	1057	859	,	,	PUNCT
ejpam-6409	1057	860	0	0	NUM
ejpam-6409	1057	861	,	,	PUNCT
ejpam-6409	1057	862	0	0	NUM
ejpam-6409	1057	863	,	,	PUNCT
ejpam-6409	1057	864	0	0	NUM
ejpam-6409	1057	865	)	)	PUNCT
ejpam-6409	1057	866	(	(	PUNCT
ejpam-6409	1057	867	17	17	NUM
ejpam-6409	1057	868	,	,	PUNCT
ejpam-6409	1057	869	6	6	NUM
ejpam-6409	1057	870	)	)	PUNCT
ejpam-6409	1057	871	z256[ω	z256[ω	NUM
ejpam-6409	1057	872	]	]	PUNCT
ejpam-6409	1057	873	(	(	PUNCT
ejpam-6409	1057	874	1	1	NUM
ejpam-6409	1057	875	,	,	PUNCT
ejpam-6409	1057	876	0	0	NUM
ejpam-6409	1057	877	,	,	PUNCT
ejpam-6409	1057	878	0	0	NUM
ejpam-6409	1057	879	,	,	PUNCT
ejpam-6409	1057	880	0	0	NUM
ejpam-6409	1057	881	,	,	PUNCT
ejpam-6409	1057	882	0	0	NUM
ejpam-6409	1057	883	,	,	PUNCT
ejpam-6409	1057	884	1	1	NUM
ejpam-6409	1057	885	,	,	PUNCT
ejpam-6409	1057	886	0	0	NUM
ejpam-6409	1057	887	,	,	PUNCT
ejpam-6409	1057	888	0	0	NUM
ejpam-6409	1057	889	)	)	PUNCT
ejpam-6409	1057	890	(	(	PUNCT
ejpam-6409	1057	891	13	13	NUM
ejpam-6409	1057	892	,	,	PUNCT
ejpam-6409	1057	893	8)	8)	NUM