id	sid	tid	token	lemma	pos
ejpam-702	1	1	5_702_ozen.dvi	5_702_ozen.dvi	NUM
ejpam-702	1	2	european	european	ADJ
ejpam-702	1	3	journal	journal	NOUN
ejpam-702	1	4	of	of	ADP
ejpam-702	1	5	pure	pure	ADJ
ejpam-702	1	6	and	and	CCONJ
ejpam-702	1	7	applied	apply	VERB
ejpam-702	1	8	mathematics	mathematic	NOUN
ejpam-702	1	9	vol	vol	NOUN
ejpam-702	1	10	.	.	PUNCT
ejpam-702	2	1	3	3	NUM
ejpam-702	2	2	,	,	PUNCT
ejpam-702	2	3	no	no	INTJ
ejpam-702	2	4	.	.	NOUN
ejpam-702	2	5	4	4	NUM
ejpam-702	2	6	,	,	PUNCT
ejpam-702	2	7	2010	2010	NUM
ejpam-702	2	8	,	,	PUNCT
ejpam-702	2	9	670	670	NUM
ejpam-702	2	10	-	-	SYM
ejpam-702	2	11	677	677	NUM
ejpam-702	2	12	issn	issn	PROPN
ejpam-702	2	13	1307	1307	NUM
ejpam-702	2	14	-	-	SYM
ejpam-702	2	15	5543	5543	NUM
ejpam-702	2	16	–	–	PUNCT
ejpam-702	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-702	2	18	codes	code	VERB
ejpam-702	2	19	over	over	ADP
ejpam-702	2	20	quaternion	quaternion	NOUN
ejpam-702	2	21	integers	integer	NOUN
ejpam-702	2	22	mehmet	mehmet	PROPN
ejpam-702	2	23	özen∗	özen∗	PROPN
ejpam-702	2	24	,	,	PUNCT
ejpam-702	2	25	murat	murat	PROPN
ejpam-702	2	26	güzeltepe	güzeltepe	PROPN
ejpam-702	2	27	department	department	PROPN
ejpam-702	2	28	of	of	ADP
ejpam-702	2	29	mathematics	mathematic	NOUN
ejpam-702	2	30	,	,	PUNCT
ejpam-702	2	31	sakarya	sakarya	NOUN
ejpam-702	2	32	university	university	NOUN
ejpam-702	2	33	,	,	PUNCT
ejpam-702	2	34	tr54187	tr54187	NOUN
ejpam-702	2	35	sakarya	sakarya	NOUN
ejpam-702	2	36	,	,	PUNCT
ejpam-702	2	37	turkey	turkey	NOUN
ejpam-702	2	38	abstract	abstract	NOUN
ejpam-702	2	39	.	.	PUNCT
ejpam-702	3	1	in	in	ADP
ejpam-702	3	2	this	this	DET
ejpam-702	3	3	paper	paper	NOUN
ejpam-702	3	4	,	,	PUNCT
ejpam-702	3	5	we	we	PRON
ejpam-702	3	6	study	study	VERB
ejpam-702	3	7	codes	code	NOUN
ejpam-702	3	8	over	over	ADP
ejpam-702	3	9	some	some	DET
ejpam-702	3	10	finite	finite	ADJ
ejpam-702	3	11	fields	field	NOUN
ejpam-702	3	12	by	by	ADP
ejpam-702	3	13	using	use	VERB
ejpam-702	3	14	quaternion	quaternion	NOUN
ejpam-702	3	15	integers	integer	NOUN
ejpam-702	3	16	.	.	PUNCT
ejpam-702	4	1	also	also	ADV
ejpam-702	4	2	,	,	PUNCT
ejpam-702	4	3	we	we	PRON
ejpam-702	4	4	obtain	obtain	VERB
ejpam-702	4	5	the	the	DET
ejpam-702	4	6	decoding	decode	VERB
ejpam-702	4	7	procedure	procedure	NOUN
ejpam-702	4	8	of	of	ADP
ejpam-702	4	9	these	these	DET
ejpam-702	4	10	codes	code	NOUN
ejpam-702	4	11	.	.	PUNCT
ejpam-702	5	1	2000	2000	NUM
ejpam-702	5	2	mathematics	mathematic	NOUN
ejpam-702	5	3	subject	subject	NOUN
ejpam-702	5	4	classifications	classification	NOUN
ejpam-702	5	5	:	:	PUNCT
ejpam-702	5	6	94b05	94b05	NUM
ejpam-702	5	7	,	,	PUNCT
ejpam-702	5	8	94b15	94b15	NUM
ejpam-702	5	9	,	,	PUNCT
ejpam-702	5	10	94b35	94b35	NUM
ejpam-702	5	11	,	,	PUNCT
ejpam-702	5	12	94b60	94b60	NUM
ejpam-702	5	13	key	key	ADJ
ejpam-702	5	14	words	word	NOUN
ejpam-702	5	15	and	and	CCONJ
ejpam-702	5	16	phrases	phrase	NOUN
ejpam-702	5	17	:	:	PUNCT
ejpam-702	5	18	block	block	NOUN
ejpam-702	5	19	codes	code	NOUN
ejpam-702	5	20	,	,	PUNCT
ejpam-702	5	21	mannheim	mannheim	NOUN
ejpam-702	5	22	distance	distance	NOUN
ejpam-702	5	23	,	,	PUNCT
ejpam-702	5	24	cyclic	cyclic	ADJ
ejpam-702	5	25	codes	code	NOUN
ejpam-702	5	26	,	,	PUNCT
ejpam-702	5	27	syndrome	syndrome	NOUN
ejpam-702	5	28	decoding	decode	VERB
ejpam-702	5	29	1	1	NUM
ejpam-702	5	30	.	.	PUNCT
ejpam-702	6	1	introduction	introduction	NOUN
ejpam-702	6	2	in	in	ADP
ejpam-702	6	3	this	this	DET
ejpam-702	6	4	study	study	NOUN
ejpam-702	6	5	,	,	PUNCT
ejpam-702	6	6	we	we	PRON
ejpam-702	6	7	consider	consider	VERB
ejpam-702	6	8	codes	code	NOUN
ejpam-702	6	9	over	over	ADP
ejpam-702	6	10	finite	finite	ADJ
ejpam-702	6	11	fields	field	NOUN
ejpam-702	6	12	equipped	equip	VERB
ejpam-702	6	13	with	with	ADP
ejpam-702	6	14	a	a	DET
ejpam-702	6	15	quaternion	quaternion	ADJ
ejpam-702	6	16	mannheim	mannheim	NOUN
ejpam-702	6	17	metric	metric	NOUN
ejpam-702	6	18	.	.	PUNCT
ejpam-702	7	1	mannheim	mannheim	PROPN
ejpam-702	7	2	distance	distance	NOUN
ejpam-702	7	3	and	and	CCONJ
ejpam-702	7	4	codes	code	NOUN
ejpam-702	7	5	over	over	ADP
ejpam-702	7	6	gaussian	gaussian	ADJ
ejpam-702	7	7	integers	integer	NOUN
ejpam-702	7	8	with	with	ADP
ejpam-702	7	9	respect	respect	NOUN
ejpam-702	7	10	to	to	ADP
ejpam-702	7	11	the	the	DET
ejpam-702	7	12	mannheim	mannheim	PROPN
ejpam-702	7	13	metric	metric	NOUN
ejpam-702	7	14	which	which	PRON
ejpam-702	7	15	are	be	AUX
ejpam-702	7	16	suitable	suitable	ADJ
ejpam-702	7	17	for	for	ADP
ejpam-702	7	18	quadrature	quadrature	NOUN
ejpam-702	7	19	amplitude	amplitude	NOUN
ejpam-702	7	20	modulation	modulation	NOUN
ejpam-702	7	21	(	(	PUNCT
ejpam-702	7	22	qam)-type	qam)-type	PROPN
ejpam-702	7	23	were	be	AUX
ejpam-702	7	24	introduced	introduce	VERB
ejpam-702	7	25	by	by	ADP
ejpam-702	7	26	huber	huber	PROPN
ejpam-702	7	27	in	in	ADP
ejpam-702	7	28	[	[	X
ejpam-702	7	29	1	1	NUM
ejpam-702	7	30	,	,	PUNCT
ejpam-702	7	31	2	2	NUM
ejpam-702	7	32	]	]	PUNCT
ejpam-702	7	33	.	.	PUNCT
ejpam-702	8	1	a	a	DET
ejpam-702	8	2	mannheim	mannheim	NOUN
ejpam-702	8	3	metric	metric	NOUN
ejpam-702	8	4	is	be	AUX
ejpam-702	8	5	a	a	DET
ejpam-702	8	6	manhattan	manhattan	PROPN
ejpam-702	8	7	metric	metric	PROPN
ejpam-702	8	8	modulo	modulo	PROPN
ejpam-702	8	9	a	a	DET
ejpam-702	8	10	two	two	NUM
ejpam-702	8	11	-	-	PUNCT
ejpam-702	8	12	dimensional	dimensional	ADJ
ejpam-702	8	13	(	(	PUNCT
ejpam-702	8	14	2	2	NUM
ejpam-702	8	15	-	-	SYM
ejpam-702	8	16	d	d	NOUN
ejpam-702	8	17	)	)	PUNCT
ejpam-702	8	18	grid	grid	NOUN
ejpam-702	8	19	.	.	PUNCT
ejpam-702	9	1	two	two	NUM
ejpam-702	9	2	classes	class	NOUN
ejpam-702	9	3	of	of	ADP
ejpam-702	9	4	codes	code	NOUN
ejpam-702	9	5	over	over	ADP
ejpam-702	9	6	gaussian	gaussian	ADJ
ejpam-702	9	7	integers	integer	NOUN
ejpam-702	9	8	z	z	VERB
ejpam-702	10	1	[	[	X
ejpam-702	10	2	i	i	X
ejpam-702	10	3	]	]	PUNCT
ejpam-702	10	4	were	be	AUX
ejpam-702	10	5	considered	consider	VERB
ejpam-702	10	6	in	in	ADP
ejpam-702	10	7	[	[	X
ejpam-702	10	8	1	1	NUM
ejpam-702	10	9	]	]	PUNCT
ejpam-702	10	10	,	,	PUNCT
ejpam-702	10	11	namely	namely	ADV
ejpam-702	10	12	,	,	PUNCT
ejpam-702	10	13	the	the	DET
ejpam-702	10	14	one	one	NUM
ejpam-702	10	15	mannheim	mannheim	NOUN
ejpam-702	10	16	error	error	NOUN
ejpam-702	10	17	-	-	PUNCT
ejpam-702	10	18	correcting	correct	VERB
ejpam-702	10	19	(	(	PUNCT
ejpam-702	10	20	omec	omec	ADJ
ejpam-702	10	21	)	)	PUNCT
ejpam-702	10	22	codes	code	NOUN
ejpam-702	10	23	,	,	PUNCT
ejpam-702	10	24	and	and	CCONJ
ejpam-702	10	25	codes	code	NOUN
ejpam-702	10	26	having	have	VERB
ejpam-702	10	27	minimum	minimum	NOUN
ejpam-702	10	28	mannheim	mannheim	NOUN
ejpam-702	10	29	distance	distance	NOUN
ejpam-702	10	30	greater	great	ADJ
ejpam-702	10	31	than	than	ADP
ejpam-702	10	32	3	3	NUM
ejpam-702	10	33	.	.	PUNCT
ejpam-702	11	1	in	in	ADP
ejpam-702	11	2	[	[	X
ejpam-702	11	3	2	2	NUM
ejpam-702	11	4	]	]	PUNCT
ejpam-702	11	5	,	,	PUNCT
ejpam-702	11	6	most	most	ADJ
ejpam-702	11	7	of	of	ADP
ejpam-702	11	8	the	the	DET
ejpam-702	11	9	proposed	propose	VERB
ejpam-702	11	10	codes	code	NOUN
ejpam-702	11	11	were	be	AUX
ejpam-702	11	12	shown	show	VERB
ejpam-702	11	13	to	to	PART
ejpam-702	11	14	be	be	AUX
ejpam-702	11	15	i−cyclic	i−cyclic	PROPN
ejpam-702	11	16	in	in	ADP
ejpam-702	11	17	the	the	DET
ejpam-702	11	18	sense	sense	NOUN
ejpam-702	11	19	of	of	ADP
ejpam-702	11	20	constacyclic	constacyclic	ADJ
ejpam-702	11	21	codes	code	NOUN
ejpam-702	11	22	,	,	PUNCT
ejpam-702	11	23	as	as	SCONJ
ejpam-702	11	24	defined	define	VERB
ejpam-702	11	25	by	by	ADP
ejpam-702	11	26	berlekamp	berlekamp	NOUN
ejpam-702	11	27	.	.	PUNCT
ejpam-702	12	1	in	in	ADP
ejpam-702	12	2	a	a	DET
ejpam-702	12	3	similar	similar	ADJ
ejpam-702	12	4	technique	technique	NOUN
ejpam-702	12	5	,	,	PUNCT
ejpam-702	12	6	in	in	ADP
ejpam-702	12	7	[	[	PUNCT
ejpam-702	12	8	3	3	NUM
ejpam-702	12	9	]	]	PUNCT
ejpam-702	12	10	,	,	PUNCT
ejpam-702	12	11	codes	code	NOUN
ejpam-702	12	12	over	over	ADP
ejpam-702	12	13	eisenstein	eisenstein	PROPN
ejpam-702	12	14	-	-	PUNCT
ejpam-702	12	15	jacobi	jacobi	PROPN
ejpam-702	12	16	integers	integer	NOUN
ejpam-702	12	17	z	z	NOUN
ejpam-702	13	1	[	[	X
ejpam-702	13	2	w	w	X
ejpam-702	13	3	]	]	X
ejpam-702	13	4	were	be	AUX
ejpam-702	13	5	presented	present	VERB
ejpam-702	13	6	together	together	ADV
ejpam-702	13	7	with	with	ADP
ejpam-702	13	8	a	a	DET
ejpam-702	13	9	decoding	decode	VERB
ejpam-702	13	10	algorithm	algorithm	NOUN
ejpam-702	13	11	for	for	ADP
ejpam-702	13	12	a	a	DET
ejpam-702	13	13	proper	proper	ADJ
ejpam-702	13	14	mannheim	mannheim	NOUN
ejpam-702	13	15	metric	metric	NOUN
ejpam-702	13	16	.	.	PUNCT
ejpam-702	14	1	in	in	ADP
ejpam-702	14	2	section	section	NOUN
ejpam-702	14	3	2	2	NUM
ejpam-702	14	4	,	,	PUNCT
ejpam-702	14	5	quaternion	quaternion	NOUN
ejpam-702	14	6	integers	integer	NOUN
ejpam-702	14	7	and	and	CCONJ
ejpam-702	14	8	quaternion	quaternion	NOUN
ejpam-702	14	9	mannheim	mannheim	PROPN
ejpam-702	14	10	distance	distance	NOUN
ejpam-702	14	11	have	have	AUX
ejpam-702	14	12	been	be	AUX
ejpam-702	14	13	considered	consider	VERB
ejpam-702	14	14	.	.	PUNCT
ejpam-702	15	1	also	also	ADV
ejpam-702	15	2	,	,	PUNCT
ejpam-702	15	3	we	we	PRON
ejpam-702	15	4	present	present	VERB
ejpam-702	15	5	some	some	DET
ejpam-702	15	6	fundamental	fundamental	ADJ
ejpam-702	15	7	algebraic	algebraic	ADJ
ejpam-702	15	8	concepts	concept	NOUN
ejpam-702	15	9	.	.	PUNCT
ejpam-702	16	1	in	in	ADP
ejpam-702	16	2	section	section	NOUN
ejpam-702	16	3	3	3	NUM
ejpam-702	16	4	,	,	PUNCT
ejpam-702	16	5	we	we	PRON
ejpam-702	16	6	are	be	AUX
ejpam-702	16	7	interested	interested	ADJ
ejpam-702	16	8	in	in	ADP
ejpam-702	16	9	constructing	construct	VERB
ejpam-702	16	10	perfect	perfect	ADJ
ejpam-702	16	11	codes	code	NOUN
ejpam-702	16	12	which	which	PRON
ejpam-702	16	13	are	be	AUX
ejpam-702	16	14	able	able	ADJ
ejpam-702	16	15	to	to	PART
ejpam-702	16	16	correct	correct	VERB
ejpam-702	16	17	errors	error	NOUN
ejpam-702	16	18	of	of	ADP
ejpam-702	16	19	quaternion	quaternion	NOUN
ejpam-702	16	20	mannheim	mannheim	NOUN
ejpam-702	16	21	weight	weight	NOUN
ejpam-702	16	22	one	one	NOUN
ejpam-702	16	23	.	.	PUNCT
ejpam-702	17	1	in	in	ADP
ejpam-702	17	2	section	section	NOUN
ejpam-702	17	3	4	4	NUM
ejpam-702	17	4	,	,	PUNCT
ejpam-702	17	5	double	double	ADJ
ejpam-702	17	6	error	error	NOUN
ejpam-702	17	7	correcting	correct	VERB
ejpam-702	17	8	codes	code	NOUN
ejpam-702	17	9	which	which	PRON
ejpam-702	17	10	have	have	VERB
ejpam-702	17	11	minimum	minimum	ADJ
ejpam-702	17	12	distance	distance	NOUN
ejpam-702	17	13	four	four	NUM
ejpam-702	17	14	or	or	CCONJ
ejpam-702	17	15	more	more	ADJ
ejpam-702	17	16	are	be	AUX
ejpam-702	17	17	constructed	construct	VERB
ejpam-702	17	18	and	and	CCONJ
ejpam-702	17	19	decoding	decode	VERB
ejpam-702	17	20	procedure	procedure	NOUN
ejpam-702	17	21	for	for	ADP
ejpam-702	17	22	these	these	DET
ejpam-702	17	23	codes	code	NOUN
ejpam-702	17	24	is	be	AUX
ejpam-702	17	25	given	give	VERB
ejpam-702	17	26	.	.	PUNCT
ejpam-702	18	1	∗corresponding	∗corresponde	VERB
ejpam-702	18	2	author	author	NOUN
ejpam-702	18	3	.	.	PUNCT
ejpam-702	19	1	email	email	NOUN
ejpam-702	19	2	addresses	address	NOUN
ejpam-702	19	3	:	:	PUNCT
ejpam-702	19	4	ozen�sakarya.edu.tr	ozen�sakarya.edu.tr	PROPN
ejpam-702	19	5	(	(	PUNCT
ejpam-702	19	6	m.	m.	NOUN
ejpam-702	19	7	özen	özen	NOUN
ejpam-702	19	8	)	)	PUNCT
ejpam-702	19	9	,	,	PUNCT
ejpam-702	19	10	mguzeltepe�sakarya.edu.tr	mguzeltepe�sakarya.edu.tr	PROPN
ejpam-702	19	11	(	(	PUNCT
ejpam-702	19	12	m.güzeltepe	m.güzeltepe	NOUN
ejpam-702	19	13	)	)	PUNCT
ejpam-702	19	14	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-702	20	1	670	670	NUM
ejpam-702	20	2	c	c	X
ejpam-702	20	3	©	©	PROPN
ejpam-702	20	4	2010	2010	NUM
ejpam-702	20	5	ejpam	ejpam	NOUN
ejpam-702	20	6	all	all	DET
ejpam-702	20	7	rights	right	NOUN
ejpam-702	20	8	reserved	reserve	VERB
ejpam-702	20	9	.	.	PUNCT
ejpam-702	21	1	m.	m.	NOUN
ejpam-702	21	2	özen	özen	NOUN
ejpam-702	21	3	,	,	PUNCT
ejpam-702	21	4	m.	m.	NOUN
ejpam-702	21	5	güzeltepe	güzeltepe	PROPN
ejpam-702	21	6	/	/	SYM
ejpam-702	21	7	eur	eur	NOUN
ejpam-702	21	8	.	.	PUNCT
ejpam-702	22	1	j.	j.	PROPN
ejpam-702	22	2	pure	pure	PROPN
ejpam-702	22	3	appl	appl	PROPN
ejpam-702	22	4	.	.	PROPN
ejpam-702	22	5	math	math	PROPN
ejpam-702	22	6	,	,	PUNCT
ejpam-702	22	7	3	3	NUM
ejpam-702	22	8	(	(	PUNCT
ejpam-702	22	9	2010	2010	NUM
ejpam-702	22	10	)	)	PUNCT
ejpam-702	22	11	,	,	PUNCT
ejpam-702	22	12	670	670	NUM
ejpam-702	22	13	-	-	SYM
ejpam-702	22	14	677	677	NUM
ejpam-702	22	15	671	671	NUM
ejpam-702	22	16	2	2	NUM
ejpam-702	22	17	.	.	PUNCT
ejpam-702	23	1	quaternion	quaternion	NOUN
ejpam-702	23	2	integers	integer	NOUN
ejpam-702	23	3	and	and	CCONJ
ejpam-702	23	4	quaternion	quaternion	NOUN
ejpam-702	23	5	mannheim	mannheim	PROPN
ejpam-702	23	6	distance	distance	NOUN
ejpam-702	23	7	definition	definition	NOUN
ejpam-702	23	8	1	1	NUM
ejpam-702	23	9	.	.	PUNCT
ejpam-702	24	1	the	the	DET
ejpam-702	24	2	hamilton	hamilton	PROPN
ejpam-702	24	3	quaternion	quaternion	PROPN
ejpam-702	24	4	algebra	algebra	NOUN
ejpam-702	24	5	over	over	ADP
ejpam-702	24	6	the	the	DET
ejpam-702	24	7	set	set	NOUN
ejpam-702	24	8	of	of	ADP
ejpam-702	24	9	the	the	DET
ejpam-702	24	10	real	real	ADJ
ejpam-702	24	11	numbers	number	NOUN
ejpam-702	24	12	(	(	PUNCT
ejpam-702	24	13	r	r	NOUN
ejpam-702	24	14	)	)	PUNCT
ejpam-702	24	15	,	,	PUNCT
ejpam-702	24	16	denoted	denote	VERB
ejpam-702	24	17	by	by	ADP
ejpam-702	24	18	h(r	h(r	PROPN
ejpam-702	24	19	)	)	PUNCT
ejpam-702	24	20	,	,	PUNCT
ejpam-702	24	21	is	be	AUX
ejpam-702	24	22	the	the	DET
ejpam-702	24	23	associative	associative	ADJ
ejpam-702	24	24	unital	unital	ADJ
ejpam-702	24	25	algebra	algebra	NOUN
ejpam-702	24	26	given	give	VERB
ejpam-702	24	27	by	by	ADP
ejpam-702	24	28	the	the	DET
ejpam-702	24	29	following	follow	VERB
ejpam-702	24	30	representation	representation	NOUN
ejpam-702	24	31	:	:	PUNCT
ejpam-702	24	32	i	i	NOUN
ejpam-702	24	33	)	)	PUNCT
ejpam-702	24	34	h(r	h(r	NOUN
ejpam-702	24	35	)	)	PUNCT
ejpam-702	24	36	is	be	AUX
ejpam-702	24	37	the	the	DET
ejpam-702	24	38	free	free	ADJ
ejpam-702	24	39	r	r	NOUN
ejpam-702	24	40	module	module	NOUN
ejpam-702	24	41	over	over	ADP
ejpam-702	24	42	the	the	DET
ejpam-702	24	43	symbols	symbol	NOUN
ejpam-702	24	44	1	1	NUM
ejpam-702	24	45	,	,	PUNCT
ejpam-702	24	46	i	i	PRON
ejpam-702	24	47	,	,	PUNCT
ejpam-702	24	48	j	j	PROPN
ejpam-702	24	49	,	,	PUNCT
ejpam-702	24	50	k	k	PROPN
ejpam-702	24	51	,	,	PUNCT
ejpam-702	24	52	that	that	ADV
ejpam-702	24	53	is	is	ADV
ejpam-702	24	54	,	,	PUNCT
ejpam-702	24	55	h(r	h(r	NOUN
ejpam-702	24	56	)	)	PUNCT
ejpam-702	24	57	=	=	PRON
ejpam-702	24	58	{	{	PUNCT
ejpam-702	24	59	a0	a0	NOUN
ejpam-702	24	60	+	+	CCONJ
ejpam-702	24	61	a1i+	a1i+	NOUN
ejpam-702	24	62	a2	a2	PROPN
ejpam-702	24	63	j	j	PROPN
ejpam-702	25	1	+	+	CCONJ
ejpam-702	25	2	a3k	a3k	PROPN
ejpam-702	25	3	:	:	PUNCT
ejpam-702	25	4	a0	a0	PROPN
ejpam-702	25	5	,	,	PUNCT
ejpam-702	25	6	a1	a1	PROPN
ejpam-702	25	7	,	,	PUNCT
ejpam-702	25	8	a2	a2	PROPN
ejpam-702	25	9	,	,	PUNCT
ejpam-702	25	10	a3	a3	NOUN
ejpam-702	25	11	∈	∈	PROPN
ejpam-702	25	12	r	r	PROPN
ejpam-702	25	13	}	}	PUNCT
ejpam-702	25	14	;	;	PUNCT
ejpam-702	25	15	ii	ii	X
ejpam-702	25	16	)	)	PUNCT
ejpam-702	25	17	1	1	NUM
ejpam-702	25	18	is	be	AUX
ejpam-702	25	19	the	the	DET
ejpam-702	25	20	multiplicative	multiplicative	ADJ
ejpam-702	25	21	identity	identity	NOUN
ejpam-702	25	22	;	;	PUNCT
ejpam-702	25	23	iii	iii	X
ejpam-702	25	24	)	)	PUNCT
ejpam-702	25	25	i2	i2	PROPN
ejpam-702	25	26	=	=	PROPN
ejpam-702	25	27	j2	j2	PROPN
ejpam-702	25	28	=	=	SYM
ejpam-702	25	29	k2	k2	PROPN
ejpam-702	25	30	=	=	PROPN
ejpam-702	25	31	−1	−1	NOUN
ejpam-702	25	32	;	;	PUNCT
ejpam-702	25	33	iv	iv	X
ejpam-702	25	34	)	)	PUNCT
ejpam-702	26	1	i	i	PRON
ejpam-702	26	2	j	j	NOUN
ejpam-702	26	3	=	=	PUNCT
ejpam-702	27	1	−	−	PROPN
ejpam-702	27	2	ji	ji	NOUN
ejpam-702	27	3	=	=	SYM
ejpam-702	27	4	k	k	PROPN
ejpam-702	27	5	,	,	PUNCT
ejpam-702	27	6	ik	ik	PROPN
ejpam-702	27	7	=	=	SYM
ejpam-702	27	8	−ki	−ki	PROPN
ejpam-702	27	9	=	=	SYM
ejpam-702	27	10	j	j	PROPN
ejpam-702	27	11	,	,	PUNCT
ejpam-702	27	12	jk	jk	X
ejpam-702	27	13	=	=	PUNCT
ejpam-702	27	14	−k	−k	PROPN
ejpam-702	27	15	j	j	PROPN
ejpam-702	28	1	=	=	SYM
ejpam-702	28	2	i	i	PRON
ejpam-702	28	3	[	[	X
ejpam-702	28	4	4	4	NUM
ejpam-702	28	5	]	]	PUNCT
ejpam-702	28	6	.	.	PUNCT
ejpam-702	29	1	the	the	DET
ejpam-702	29	2	set	set	VERB
ejpam-702	29	3	h(z	h(z	PROPN
ejpam-702	29	4	)	)	PUNCT
ejpam-702	29	5	,	,	PUNCT
ejpam-702	29	6	which	which	PRON
ejpam-702	29	7	is	be	AUX
ejpam-702	29	8	defined	define	VERB
ejpam-702	29	9	by	by	ADP
ejpam-702	29	10	h(z	h(z	NOUN
ejpam-702	29	11	)	)	PUNCT
ejpam-702	29	12	=	=	SYM
ejpam-702	29	13	�	�	PROPN
ejpam-702	29	14	a0	a0	PROPN
ejpam-702	29	15	+	+	CCONJ
ejpam-702	29	16	a1i	a1i	PROPN
ejpam-702	29	17	+	+	NUM
ejpam-702	29	18	a2	a2	PROPN
ejpam-702	29	19	j+	j+	VERB
ejpam-702	29	20	a3k	a3k	PROPN
ejpam-702	29	21	:	:	PUNCT
ejpam-702	29	22	a0	a0	PROPN
ejpam-702	29	23	,	,	PUNCT
ejpam-702	29	24	a1	a1	PROPN
ejpam-702	29	25	,	,	PUNCT
ejpam-702	29	26	a2	a2	PROPN
ejpam-702	29	27	,	,	PUNCT
ejpam-702	29	28	a3	a3	NOUN
ejpam-702	29	29	∈	∈	PROPN
ejpam-702	29	30	z	z	PROPN
ejpam-702	29	31	is	be	AUX
ejpam-702	29	32	a	a	DET
ejpam-702	29	33	subset	subset	NOUN
ejpam-702	29	34	of	of	ADP
ejpam-702	29	35	h(r	h(r	NOUN
ejpam-702	29	36	)	)	PUNCT
ejpam-702	29	37	,	,	PUNCT
ejpam-702	29	38	where	where	SCONJ
ejpam-702	29	39	z	z	NOUN
ejpam-702	29	40	is	be	AUX
ejpam-702	29	41	the	the	DET
ejpam-702	29	42	set	set	NOUN
ejpam-702	29	43	of	of	ADP
ejpam-702	29	44	all	all	DET
ejpam-702	29	45	integers	integer	NOUN
ejpam-702	29	46	.	.	PUNCT
ejpam-702	30	1	more	more	ADJ
ejpam-702	30	2	information	information	NOUN
ejpam-702	30	3	which	which	PRON
ejpam-702	30	4	is	be	AUX
ejpam-702	30	5	related	relate	VERB
ejpam-702	30	6	to	to	ADP
ejpam-702	30	7	the	the	DET
ejpam-702	30	8	arithmetic	arithmetic	ADJ
ejpam-702	30	9	properties	property	NOUN
ejpam-702	30	10	of	of	ADP
ejpam-702	30	11	h(z	h(z	NOUN
ejpam-702	30	12	)	)	PUNCT
ejpam-702	30	13	can	can	AUX
ejpam-702	30	14	be	be	AUX
ejpam-702	30	15	found	find	VERB
ejpam-702	30	16	in	in	ADP
ejpam-702	30	17	[	[	X
ejpam-702	30	18	4	4	NUM
ejpam-702	30	19	,	,	PUNCT
ejpam-702	30	20	pp	pp	ADJ
ejpam-702	30	21	.	.	PUNCT
ejpam-702	31	1	57	57	NUM
ejpam-702	31	2	-	-	SYM
ejpam-702	31	3	71	71	NUM
ejpam-702	31	4	,	,	PUNCT
ejpam-702	31	5	]	]	PUNCT
ejpam-702	31	6	.	.	PUNCT
ejpam-702	32	1	if	if	SCONJ
ejpam-702	32	2	q	q	NOUN
ejpam-702	32	3	=	=	SYM
ejpam-702	32	4	a0	a0	NOUN
ejpam-702	32	5	+	+	CCONJ
ejpam-702	32	6	a1i	a1i	PROPN
ejpam-702	32	7	+	+	NUM
ejpam-702	32	8	a2	a2	PROPN
ejpam-702	32	9	j	j	PROPN
ejpam-702	32	10	+	+	CCONJ
ejpam-702	32	11	a3k	a3k	ADV
ejpam-702	32	12	is	be	AUX
ejpam-702	32	13	a	a	DET
ejpam-702	32	14	quaternion	quaternion	NOUN
ejpam-702	32	15	integer	integer	NOUN
ejpam-702	32	16	,	,	PUNCT
ejpam-702	32	17	its	its	PRON
ejpam-702	32	18	conjugate	conjugate	ADJ
ejpam-702	32	19	quaternion	quaternion	NOUN
ejpam-702	32	20	is	be	AUX
ejpam-702	32	21	q	q	NOUN
ejpam-702	32	22	=	=	SYM
ejpam-702	32	23	a0	a0	PROPN
ejpam-702	32	24	−	−	PROPN
ejpam-702	32	25	(	(	PUNCT
ejpam-702	32	26	a1i	a1i	VERB
ejpam-702	32	27	+	+	NUM
ejpam-702	32	28	a2	a2	PROPN
ejpam-702	32	29	j+	j+	VERB
ejpam-702	32	30	a3k	a3k	NOUN
ejpam-702	32	31	)	)	PUNCT
ejpam-702	32	32	.	.	PUNCT
ejpam-702	33	1	the	the	DET
ejpam-702	33	2	norm	norm	NOUN
ejpam-702	33	3	of	of	ADP
ejpam-702	33	4	q	q	PROPN
ejpam-702	33	5	is	be	AUX
ejpam-702	33	6	n(q	n(q	PROPN
ejpam-702	33	7	)	)	PUNCT
ejpam-702	34	1	=	=	PUNCT
ejpam-702	34	2	qq	qq	X
ejpam-702	34	3	=	=	PROPN
ejpam-702	34	4	a2	a2	PROPN
ejpam-702	34	5	0	0	PUNCT
ejpam-702	35	1	+	+	NUM
ejpam-702	35	2	a2	a2	PROPN
ejpam-702	35	3	1	1	NUM
ejpam-702	35	4	+	+	NUM
ejpam-702	35	5	a2	a2	PROPN
ejpam-702	35	6	2	2	NUM
ejpam-702	35	7	+	+	NUM
ejpam-702	35	8	a2	a2	PROPN
ejpam-702	35	9	3	3	NUM
ejpam-702	35	10	.	.	PUNCT
ejpam-702	36	1	a	a	DET
ejpam-702	36	2	quaternion	quaternion	NOUN
ejpam-702	36	3	integer	integer	NOUN
ejpam-702	36	4	consists	consist	VERB
ejpam-702	36	5	of	of	ADP
ejpam-702	36	6	two	two	NUM
ejpam-702	36	7	parts	part	NOUN
ejpam-702	36	8	which	which	PRON
ejpam-702	36	9	are	be	AUX
ejpam-702	36	10	the	the	DET
ejpam-702	36	11	real	real	ADJ
ejpam-702	36	12	part	part	NOUN
ejpam-702	36	13	and	and	CCONJ
ejpam-702	36	14	the	the	DET
ejpam-702	36	15	imaginary	imaginary	ADJ
ejpam-702	36	16	part	part	NOUN
ejpam-702	36	17	.	.	PUNCT
ejpam-702	37	1	let	let	VERB
ejpam-702	37	2	q	q	NOUN
ejpam-702	37	3	=	=	SYM
ejpam-702	37	4	a0	a0	NOUN
ejpam-702	37	5	+	+	CCONJ
ejpam-702	37	6	a1i	a1i	PROPN
ejpam-702	37	7	+	+	NUM
ejpam-702	37	8	a2	a2	PROPN
ejpam-702	37	9	j	j	PROPN
ejpam-702	38	1	+	+	CCONJ
ejpam-702	38	2	a3k	a3k	ADV
ejpam-702	38	3	be	be	VERB
ejpam-702	38	4	a	a	DET
ejpam-702	38	5	quaternion	quaternion	NOUN
ejpam-702	38	6	integer	integer	NOUN
ejpam-702	38	7	.	.	PUNCT
ejpam-702	39	1	then	then	ADV
ejpam-702	39	2	its	its	PRON
ejpam-702	39	3	real	real	ADJ
ejpam-702	39	4	part	part	NOUN
ejpam-702	39	5	is	be	AUX
ejpam-702	39	6	a0	a0	NOUN
ejpam-702	39	7	and	and	CCONJ
ejpam-702	39	8	its	its	PRON
ejpam-702	39	9	imaginary	imaginary	ADJ
ejpam-702	39	10	part	part	NOUN
ejpam-702	39	11	is	be	AUX
ejpam-702	39	12	a1i	a1i	VERB
ejpam-702	39	13	+	+	NUM
ejpam-702	39	14	a2	a2	PROPN
ejpam-702	39	15	j	j	PROPN
ejpam-702	39	16	+	+	CCONJ
ejpam-702	39	17	a3k	a3k	ADV
ejpam-702	39	18	.	.	PUNCT
ejpam-702	40	1	in	in	ADP
ejpam-702	40	2	the	the	DET
ejpam-702	40	3	rest	rest	NOUN
ejpam-702	40	4	of	of	ADP
ejpam-702	40	5	this	this	DET
ejpam-702	40	6	paper	paper	NOUN
ejpam-702	40	7	v	v	NOUN
ejpam-702	40	8	denotes	denote	VERB
ejpam-702	40	9	the	the	DET
ejpam-702	40	10	imaginary	imaginary	ADJ
ejpam-702	40	11	part	part	NOUN
ejpam-702	40	12	of	of	ADP
ejpam-702	40	13	a	a	DET
ejpam-702	40	14	quaternion	quaternion	NOUN
ejpam-702	40	15	integer	integer	NOUN
ejpam-702	40	16	.	.	PUNCT
ejpam-702	41	1	the	the	DET
ejpam-702	41	2	commutative	commutative	ADJ
ejpam-702	41	3	property	property	NOUN
ejpam-702	41	4	of	of	ADP
ejpam-702	41	5	multiplication	multiplication	NOUN
ejpam-702	41	6	does	do	AUX
ejpam-702	41	7	not	not	PART
ejpam-702	41	8	hold	hold	VERB
ejpam-702	41	9	for	for	ADP
ejpam-702	41	10	quaternion	quaternion	NOUN
ejpam-702	41	11	integers	integer	NOUN
ejpam-702	41	12	.	.	PUNCT
ejpam-702	42	1	however	however	ADV
ejpam-702	42	2	,	,	PUNCT
ejpam-702	42	3	if	if	SCONJ
ejpam-702	42	4	the	the	DET
ejpam-702	42	5	imaginary	imaginary	ADJ
ejpam-702	42	6	parts	part	NOUN
ejpam-702	42	7	of	of	ADP
ejpam-702	42	8	quaternion	quaternion	NOUN
ejpam-702	42	9	integers	integer	NOUN
ejpam-702	42	10	are	be	AUX
ejpam-702	42	11	parallel	parallel	ADJ
ejpam-702	42	12	to	to	ADP
ejpam-702	42	13	each	each	DET
ejpam-702	42	14	other	other	ADJ
ejpam-702	42	15	,	,	PUNCT
ejpam-702	42	16	then	then	ADV
ejpam-702	42	17	their	their	PRON
ejpam-702	42	18	product	product	NOUN
ejpam-702	42	19	is	be	AUX
ejpam-702	42	20	commutative	commutative	ADJ
ejpam-702	42	21	.	.	PUNCT
ejpam-702	43	1	define	define	VERB
ejpam-702	43	2	r	r	NOUN
ejpam-702	43	3	as	as	SCONJ
ejpam-702	43	4	follows	follow	VERB
ejpam-702	43	5	:	:	PUNCT
ejpam-702	43	6	r=	r=	ADJ
ejpam-702	43	7	{	{	PUNCT
ejpam-702	43	8	a+	a+	X
ejpam-702	43	9	bv	bv	PROPN
ejpam-702	43	10	:	:	PUNCT
ejpam-702	43	11	a	a	X
ejpam-702	43	12	,	,	PUNCT
ejpam-702	43	13	b	b	PROPN
ejpam-702	43	14	∈	∈	PROPN
ejpam-702	43	15	z	z	PROPN
ejpam-702	43	16	}	}	PUNCT
ejpam-702	43	17	which	which	PRON
ejpam-702	43	18	is	be	AUX
ejpam-702	43	19	a	a	DET
ejpam-702	43	20	subring	subring	NOUN
ejpam-702	43	21	of	of	ADP
ejpam-702	43	22	the	the	DET
ejpam-702	43	23	quaternion	quaternion	NOUN
ejpam-702	43	24	integer	integer	NOUN
ejpam-702	43	25	ring	ring	NOUN
ejpam-702	43	26	.	.	PUNCT
ejpam-702	44	1	the	the	DET
ejpam-702	44	2	commutative	commutative	ADJ
ejpam-702	44	3	property	property	NOUN
ejpam-702	44	4	of	of	ADP
ejpam-702	44	5	multiplication	multiplication	NOUN
ejpam-702	44	6	holds	hold	VERB
ejpam-702	44	7	over	over	ADP
ejpam-702	44	8	r.	r.	PROPN
ejpam-702	44	9	note	note	PROPN
ejpam-702	44	10	that	that	SCONJ
ejpam-702	44	11	the	the	DET
ejpam-702	44	12	structure	structure	NOUN
ejpam-702	44	13	of	of	ADP
ejpam-702	44	14	the	the	DET
ejpam-702	44	15	ring	ring	NOUN
ejpam-702	44	16	r	r	NOUN
ejpam-702	44	17	=	=	PUNCT
ejpam-702	44	18	{	{	PUNCT
ejpam-702	44	19	a+	a+	X
ejpam-702	44	20	bv	bv	PROPN
ejpam-702	44	21	:	:	PUNCT
ejpam-702	44	22	a	a	X
ejpam-702	44	23	,	,	PUNCT
ejpam-702	44	24	b	b	X
ejpam-702	44	25	∈	∈	PROPN
ejpam-702	44	26	z	z	AUX
ejpam-702	44	27	}	}	PUNCT
ejpam-702	44	28	is	be	AUX
ejpam-702	44	29	related	relate	VERB
ejpam-702	44	30	to	to	ADP
ejpam-702	44	31	the	the	DET
ejpam-702	44	32	prime	prime	ADJ
ejpam-702	44	33	π=	π=	NUM
ejpam-702	44	34	m+	m+	NUM
ejpam-702	44	35	nv	nv	PROPN
ejpam-702	44	36	,	,	PUNCT
ejpam-702	44	37	where	where	SCONJ
ejpam-702	44	38	m	m	VERB
ejpam-702	44	39	,	,	PUNCT
ejpam-702	44	40	n	n	PROPN
ejpam-702	44	41	∈	∈	PROPN
ejpam-702	44	42	z	z	X
ejpam-702	44	43	.	.	PUNCT
ejpam-702	45	1	theorem	theorem	NOUN
ejpam-702	45	2	1	1	NUM
ejpam-702	45	3	.	.	PUNCT
ejpam-702	46	1	for	for	ADP
ejpam-702	46	2	every	every	DET
ejpam-702	46	3	odd	odd	ADJ
ejpam-702	46	4	,	,	PUNCT
ejpam-702	46	5	rational	rational	ADJ
ejpam-702	46	6	prime	prime	NOUN
ejpam-702	46	7	p	p	NOUN
ejpam-702	46	8	in	in	ADP
ejpam-702	46	9	the	the	DET
ejpam-702	46	10	set	set	NOUN
ejpam-702	46	11	of	of	ADP
ejpam-702	46	12	natural	natural	ADJ
ejpam-702	46	13	numbers	number	NOUN
ejpam-702	46	14	n	n	CCONJ
ejpam-702	46	15	,	,	PUNCT
ejpam-702	46	16	there	there	PRON
ejpam-702	46	17	exists	exist	VERB
ejpam-702	46	18	a	a	DET
ejpam-702	46	19	prime	prime	NOUN
ejpam-702	46	20	π	π	PROPN
ejpam-702	46	21	∈	∈	PROPN
ejpam-702	46	22	h(z	h(z	PROPN
ejpam-702	46	23	)	)	PUNCT
ejpam-702	46	24	,	,	PUNCT
ejpam-702	46	25	such	such	ADJ
ejpam-702	46	26	that	that	SCONJ
ejpam-702	46	27	n(π	n(π	NOUN
ejpam-702	46	28	)	)	PUNCT
ejpam-702	47	1	=	=	PUNCT
ejpam-702	48	1	p	p	NOUN
ejpam-702	48	2	=	=	SYM
ejpam-702	48	3	ππ	ππ	PROPN
ejpam-702	48	4	.	.	PROPN
ejpam-702	48	5	in	in	ADP
ejpam-702	48	6	particular	particular	ADJ
ejpam-702	48	7	,	,	PUNCT
ejpam-702	48	8	p	p	PRON
ejpam-702	48	9	is	be	AUX
ejpam-702	48	10	not	not	PART
ejpam-702	48	11	prime	prime	ADJ
ejpam-702	48	12	in	in	ADP
ejpam-702	48	13	h(z	h(z	NOUN
ejpam-702	48	14	)	)	PUNCT
ejpam-702	49	1	[	[	X
ejpam-702	49	2	4	4	NUM
ejpam-702	49	3	,	,	PUNCT
ejpam-702	49	4	p.	p.	NOUN
ejpam-702	49	5	66	66	NUM
ejpam-702	49	6	]	]	PUNCT
ejpam-702	49	7	.	.	PUNCT
ejpam-702	50	1	corollary	corollary	ADJ
ejpam-702	50	2	1	1	NUM
ejpam-702	50	3	.	.	PUNCT
ejpam-702	51	1	π	π	PROPN
ejpam-702	51	2	∈	∈	PROPN
ejpam-702	51	3	h(z	h(z	PROPN
ejpam-702	51	4	)	)	PUNCT
ejpam-702	51	5	is	be	AUX
ejpam-702	51	6	prime	prime	ADJ
ejpam-702	51	7	in	in	ADP
ejpam-702	51	8	h(z	h(z	NOUN
ejpam-702	51	9	)	)	PUNCT
ejpam-702	52	1	if	if	SCONJ
ejpam-702	52	2	and	and	CCONJ
ejpam-702	52	3	only	only	ADV
ejpam-702	52	4	if	if	SCONJ
ejpam-702	52	5	n(π	n(π	NOUN
ejpam-702	52	6	)	)	PUNCT
ejpam-702	52	7	is	be	AUX
ejpam-702	52	8	prime	prime	ADJ
ejpam-702	52	9	in	in	ADP
ejpam-702	52	10	z	z	NOUN
ejpam-702	52	11	[	[	X
ejpam-702	52	12	4	4	NUM
ejpam-702	52	13	,	,	PUNCT
ejpam-702	52	14	p.	p.	NOUN
ejpam-702	52	15	66	66	NUM
ejpam-702	52	16	]	]	PUNCT
ejpam-702	52	17	.	.	PUNCT
ejpam-702	53	1	definition	definition	NOUN
ejpam-702	53	2	2	2	NUM
ejpam-702	53	3	.	.	PUNCT
ejpam-702	54	1	let	let	VERB
ejpam-702	54	2	rπ	rπ	NOUN
ejpam-702	54	3	be	be	AUX
ejpam-702	54	4	the	the	DET
ejpam-702	54	5	residue	residue	NOUN
ejpam-702	54	6	class	class	NOUN
ejpam-702	54	7	of	of	ADP
ejpam-702	54	8	r	r	NOUN
ejpam-702	54	9	modulo	modulo	PROPN
ejpam-702	54	10	π	π	NOUN
ejpam-702	54	11	,	,	PUNCT
ejpam-702	54	12	where	where	SCONJ
ejpam-702	54	13	π	π	PROPN
ejpam-702	54	14	=	=	PUNCT
ejpam-702	54	15	m+nv	m+nv	PROPN
ejpam-702	54	16	is	be	AUX
ejpam-702	54	17	a	a	DET
ejpam-702	54	18	prime	prime	ADJ
ejpam-702	54	19	quaternion	quaternion	NOUN
ejpam-702	54	20	integer	integer	NOUN
ejpam-702	54	21	.	.	PUNCT
ejpam-702	55	1	then	then	ADV
ejpam-702	55	2	the	the	DET
ejpam-702	55	3	modulo	modulo	PROPN
ejpam-702	55	4	function	function	NOUN
ejpam-702	55	5	µ	µ	NOUN
ejpam-702	55	6	:	:	PUNCT
ejpam-702	55	7	r=	r=	ADJ
ejpam-702	55	8	{	{	PUNCT
ejpam-702	55	9	a+	a+	X
ejpam-702	55	10	bv	bv	PROPN
ejpam-702	55	11	:	:	PUNCT
ejpam-702	55	12	a	a	X
ejpam-702	55	13	,	,	PUNCT
ejpam-702	55	14	b	b	PROPN
ejpam-702	55	15	∈	∈	PROPN
ejpam-702	55	16	z}→	z}→	NUM
ejpam-702	55	17	rπ	rπ	NOUN
ejpam-702	55	18	is	be	AUX
ejpam-702	55	19	defined	define	VERB
ejpam-702	55	20	by	by	ADP
ejpam-702	55	21	µ(q	µ(q	NOUN
ejpam-702	55	22	)	)	PUNCT
ejpam-702	55	23	=	=	SYM
ejpam-702	55	24	z	z	NOUN
ejpam-702	55	25	modπ=	modπ=	NUM
ejpam-702	55	26	q−	q−	PROPN
ejpam-702	55	27	[	[	PUNCT
ejpam-702	55	28	qπ	qπ	NOUN
ejpam-702	55	29	ππ	ππ	ADP
ejpam-702	55	30	]	]	X
ejpam-702	55	31	π	π	X
ejpam-702	55	32	,	,	PUNCT
ejpam-702	55	33	(	(	PUNCT
ejpam-702	55	34	1	1	X
ejpam-702	55	35	)	)	PUNCT
ejpam-702	55	36	where	where	SCONJ
ejpam-702	55	37	z	z	PROPN
ejpam-702	55	38	∈	∈	PROPN
ejpam-702	55	39	rπ	rπ	NOUN
ejpam-702	55	40	.	.	PUNCT
ejpam-702	56	1	in	in	ADP
ejpam-702	56	2	(	(	PUNCT
ejpam-702	56	3	1	1	NUM
ejpam-702	56	4	)	)	PUNCT
ejpam-702	56	5	,	,	PUNCT
ejpam-702	56	6	the	the	DET
ejpam-702	56	7	symbol	symbol	NOUN
ejpam-702	56	8	of	of	ADP
ejpam-702	56	9	[	[	X
ejpam-702	56	10	·	·	PUNCT
ejpam-702	56	11	]	]	X
ejpam-702	56	12	is	be	AUX
ejpam-702	56	13	rounding	round	VERB
ejpam-702	56	14	to	to	ADP
ejpam-702	56	15	the	the	DET
ejpam-702	56	16	closest	close	ADJ
ejpam-702	56	17	integer	integer	NOUN
ejpam-702	56	18	.	.	PUNCT
ejpam-702	57	1	the	the	DET
ejpam-702	57	2	rounding	rounding	NOUN
ejpam-702	57	3	of	of	ADP
ejpam-702	57	4	a	a	DET
ejpam-702	57	5	quaternion	quaternion	NOUN
ejpam-702	57	6	integer	integer	NOUN
ejpam-702	57	7	can	can	AUX
ejpam-702	57	8	be	be	AUX
ejpam-702	57	9	done	do	VERB
ejpam-702	57	10	by	by	ADP
ejpam-702	57	11	rounding	round	VERB
ejpam-702	57	12	the	the	DET
ejpam-702	57	13	real	real	ADJ
ejpam-702	57	14	part	part	NOUN
ejpam-702	57	15	and	and	CCONJ
ejpam-702	57	16	coefficients	coefficient	NOUN
ejpam-702	57	17	of	of	ADP
ejpam-702	57	18	the	the	DET
ejpam-702	57	19	imaginary	imaginary	ADJ
ejpam-702	57	20	part	part	NOUN
ejpam-702	57	21	separately	separately	ADV
ejpam-702	57	22	to	to	ADP
ejpam-702	57	23	the	the	DET
ejpam-702	57	24	closest	close	ADJ
ejpam-702	57	25	integer	integer	NOUN
ejpam-702	57	26	.	.	PUNCT
ejpam-702	58	1	m.	m.	NOUN
ejpam-702	58	2	özen	özen	NOUN
ejpam-702	58	3	,	,	PUNCT
ejpam-702	58	4	m.	m.	NOUN
ejpam-702	58	5	güzeltepe	güzeltepe	PROPN
ejpam-702	58	6	/	/	SYM
ejpam-702	58	7	eur	eur	NOUN
ejpam-702	58	8	.	.	PUNCT
ejpam-702	59	1	j.	j.	PROPN
ejpam-702	59	2	pure	pure	PROPN
ejpam-702	59	3	appl	appl	PROPN
ejpam-702	59	4	.	.	PROPN
ejpam-702	59	5	math	math	PROPN
ejpam-702	59	6	,	,	PUNCT
ejpam-702	59	7	3	3	NUM
ejpam-702	59	8	(	(	PUNCT
ejpam-702	59	9	2010	2010	NUM
ejpam-702	59	10	)	)	PUNCT
ejpam-702	59	11	,	,	PUNCT
ejpam-702	59	12	670	670	NUM
ejpam-702	59	13	-	-	SYM
ejpam-702	59	14	677	677	NUM
ejpam-702	59	15	672	672	NUM
ejpam-702	59	16	we	we	PRON
ejpam-702	59	17	can	can	AUX
ejpam-702	59	18	employ	employ	VERB
ejpam-702	59	19	the	the	DET
ejpam-702	59	20	extended	extended	ADJ
ejpam-702	59	21	euclidean	euclidean	ADJ
ejpam-702	59	22	algorithm	algorithm	NOUN
ejpam-702	59	23	for	for	ADP
ejpam-702	59	24	the	the	DET
ejpam-702	59	25	ring	ring	NOUN
ejpam-702	59	26	r	r	NOUN
ejpam-702	59	27	to	to	PART
ejpam-702	59	28	compute	compute	VERB
ejpam-702	59	29	u	u	NOUN
ejpam-702	59	30	and	and	CCONJ
ejpam-702	59	31	v	v	ADP
ejpam-702	59	32	which	which	PRON
ejpam-702	59	33	satisfy	satisfy	VERB
ejpam-702	59	34	1=	1=	NUM
ejpam-702	59	35	uπ+	uπ+	ADJ
ejpam-702	59	36	vπ	vπ	PROPN
ejpam-702	59	37	.	.	PUNCT
ejpam-702	60	1	the	the	DET
ejpam-702	60	2	table	table	NOUN
ejpam-702	60	3	1	1	NUM
ejpam-702	60	4	gives	give	VERB
ejpam-702	60	5	π	π	PROPN
ejpam-702	60	6	,	,	PUNCT
ejpam-702	60	7	u	u	NOUN
ejpam-702	60	8	,	,	PUNCT
ejpam-702	60	9	and	and	CCONJ
ejpam-702	60	10	v	v	NOUN
ejpam-702	60	11	for	for	ADP
ejpam-702	60	12	the	the	DET
ejpam-702	60	13	primes	prime	NOUN
ejpam-702	60	14	in	in	ADP
ejpam-702	60	15	z	z	PROPN
ejpam-702	60	16	.	.	PUNCT
ejpam-702	61	1	let	let	VERB
ejpam-702	61	2	π	π	PRON
ejpam-702	61	3	be	be	AUX
ejpam-702	61	4	a	a	DET
ejpam-702	61	5	prime	prime	ADJ
ejpam-702	61	6	quaternion	quaternion	NOUN
ejpam-702	61	7	integer	integer	NOUN
ejpam-702	61	8	and	and	CCONJ
ejpam-702	61	9	let	let	VERB
ejpam-702	61	10	p	p	NOUN
ejpam-702	61	11	=	=	NOUN
ejpam-702	61	12	ππ	ππ	X
ejpam-702	61	13	.	.	PUNCT
ejpam-702	61	14	let	let	VERB
ejpam-702	61	15	us	we	PRON
ejpam-702	61	16	define	define	VERB
ejpam-702	61	17	a	a	DET
ejpam-702	61	18	function	function	NOUN
ejpam-702	61	19	f	f	NOUN
ejpam-702	61	20	:	:	PUNCT
ejpam-702	61	21	zp→	zp→	PUNCT
ejpam-702	61	22	rπ	rπ	VERB
ejpam-702	61	23	by	by	ADP
ejpam-702	61	24	z	z	PROPN
ejpam-702	61	25	=	=	SYM
ejpam-702	61	26	f	f	PROPN
ejpam-702	61	27	(	(	PUNCT
ejpam-702	61	28	r	r	NOUN
ejpam-702	61	29	)	)	PUNCT
ejpam-702	61	30	=	=	SYM
ejpam-702	61	31	µ(r	µ(r	ADP
ejpam-702	61	32	+	+	PROPN
ejpam-702	61	33	π	π	X
ejpam-702	61	34	)	)	PUNCT
ejpam-702	61	35	.	.	PUNCT
ejpam-702	62	1	the	the	DET
ejpam-702	62	2	function	function	NOUN
ejpam-702	62	3	f	f	PROPN
ejpam-702	62	4	defines	define	VERB
ejpam-702	62	5	a	a	DET
ejpam-702	62	6	bijective	bijective	ADJ
ejpam-702	62	7	mapping	mapping	NOUN
ejpam-702	62	8	from	from	ADP
ejpam-702	62	9	zp	zp	PROPN
ejpam-702	62	10	into	into	ADP
ejpam-702	62	11	rπ	rπ	NOUN
ejpam-702	62	12	.	.	PUNCT
ejpam-702	63	1	using	use	VERB
ejpam-702	63	2	the	the	DET
ejpam-702	63	3	equation	equation	NOUN
ejpam-702	63	4	1=	1=	NUM
ejpam-702	63	5	uπ+	uπ+	ADJ
ejpam-702	63	6	vπ	vπ	PROPN
ejpam-702	63	7	,	,	PUNCT
ejpam-702	63	8	we	we	PRON
ejpam-702	63	9	get	get	VERB
ejpam-702	63	10	the	the	DET
ejpam-702	63	11	inverse	inverse	NOUN
ejpam-702	63	12	mapping	mapping	NOUN
ejpam-702	63	13	f	f	NOUN
ejpam-702	63	14	−1	−1	NOUN
ejpam-702	63	15	as	as	ADP
ejpam-702	63	16	r	r	NOUN
ejpam-702	63	17	=	=	SYM
ejpam-702	63	18	f	f	PROPN
ejpam-702	63	19	−1(z)≡	−1(z)≡	PROPN
ejpam-702	63	20	z(vπ	z(vπ	NUM
ejpam-702	63	21	)	)	PUNCT
ejpam-702	64	1	+	+	CCONJ
ejpam-702	64	2	z(uπ)mod	z(uπ)mod	PROPN
ejpam-702	64	3	p.	p.	NOUN
ejpam-702	64	4	if	if	SCONJ
ejpam-702	64	5	r	r	NOUN
ejpam-702	64	6	is	be	AUX
ejpam-702	64	7	an	an	DET
ejpam-702	64	8	element	element	NOUN
ejpam-702	64	9	of	of	ADP
ejpam-702	64	10	zp	zp	PROPN
ejpam-702	64	11	,	,	PUNCT
ejpam-702	64	12	then	then	ADV
ejpam-702	64	13	r	r	NOUN
ejpam-702	64	14	=	=	SYM
ejpam-702	64	15	kπ+	kπ+	PROPN
ejpam-702	64	16	z	z	PROPN
ejpam-702	64	17	and	and	CCONJ
ejpam-702	64	18	r	r	NOUN
ejpam-702	64	19	=	=	SYM
ejpam-702	64	20	r	r	NOUN
ejpam-702	64	21	=	=	SYM
ejpam-702	64	22	kπ+	kπ+	PROPN
ejpam-702	64	23	z	z	PROPN
ejpam-702	64	24	,	,	PUNCT
ejpam-702	64	25	hence	hence	ADV
ejpam-702	64	26	,	,	PUNCT
ejpam-702	64	27	z(vπ	z(vπ	NUM
ejpam-702	64	28	)	)	PUNCT
ejpam-702	64	29	+	+	CCONJ
ejpam-702	64	30	z(uπ	z(uπ	NOUN
ejpam-702	64	31	)	)	PUNCT
ejpam-702	64	32	=	=	SYM
ejpam-702	64	33	(	(	PUNCT
ejpam-702	64	34	r	r	NOUN
ejpam-702	64	35	−	−	PROPN
ejpam-702	64	36	kπ)(vπ	kπ)(vπ	PROPN
ejpam-702	64	37	)	)	PUNCT
ejpam-702	65	1	+	+	CCONJ
ejpam-702	65	2	(	(	PUNCT
ejpam-702	65	3	r	r	NOUN
ejpam-702	65	4	−	−	PROPN
ejpam-702	65	5	kπ)(uπ	kπ)(uπ	PROPN
ejpam-702	65	6	)	)	PUNCT
ejpam-702	65	7	≡	≡	PROPN
ejpam-702	65	8	r(vπ+	r(vπ+	VERB
ejpam-702	65	9	uπ	uπ	PROPN
ejpam-702	65	10	)	)	PUNCT
ejpam-702	65	11	mod	mod	NOUN
ejpam-702	65	12	p	p	PROPN
ejpam-702	65	13	≡	≡	PROPN
ejpam-702	65	14	r	r	PROPN
ejpam-702	65	15	mod	mod	PROPN
ejpam-702	65	16	p.	p.	PROPN
ejpam-702	65	17	f	f	PROPN
ejpam-702	65	18	defines	define	VERB
ejpam-702	65	19	an	an	DET
ejpam-702	65	20	isomorphism	isomorphism	NOUN
ejpam-702	65	21	,	,	PUNCT
ejpam-702	65	22	namely	namely	ADV
ejpam-702	65	23	,	,	PUNCT
ejpam-702	65	24	f	f	PROPN
ejpam-702	65	25	(	(	PUNCT
ejpam-702	65	26	z1	z1	X
ejpam-702	65	27	+	+	CCONJ
ejpam-702	65	28	z2	z2	NUM
ejpam-702	65	29	)	)	PUNCT
ejpam-702	65	30	=	=	SYM
ejpam-702	65	31	f	f	PROPN
ejpam-702	65	32	(	(	PUNCT
ejpam-702	65	33	z1	z1	PROPN
ejpam-702	65	34	)	)	PUNCT
ejpam-702	65	35	+	+	NUM
ejpam-702	65	36	f	f	X
ejpam-702	65	37	(	(	PUNCT
ejpam-702	65	38	z2	z2	PROPN
ejpam-702	65	39	)	)	PUNCT
ejpam-702	65	40	and	and	CCONJ
ejpam-702	65	41	f	f	PROPN
ejpam-702	65	42	(	(	PUNCT
ejpam-702	65	43	z1z2	z1z2	PROPN
ejpam-702	65	44	)	)	PUNCT
ejpam-702	65	45	=	=	SYM
ejpam-702	65	46	f	f	PROPN
ejpam-702	65	47	(	(	PUNCT
ejpam-702	65	48	z1	z1	PROPN
ejpam-702	65	49	)	)	PUNCT
ejpam-702	65	50	f	f	PROPN
ejpam-702	65	51	(	(	PUNCT
ejpam-702	65	52	z2	z2	PROPN
ejpam-702	65	53	)	)	PUNCT
ejpam-702	65	54	.	.	PUNCT
ejpam-702	66	1	since	since	SCONJ
ejpam-702	66	2	the	the	DET
ejpam-702	66	3	remainder	remainder	NOUN
ejpam-702	66	4	from	from	ADP
ejpam-702	66	5	dividing	divide	VERB
ejpam-702	66	6	the	the	DET
ejpam-702	66	7	element	element	NOUN
ejpam-702	66	8	q	q	NOUN
ejpam-702	66	9	by	by	ADP
ejpam-702	66	10	the	the	DET
ejpam-702	66	11	element	element	NOUN
ejpam-702	66	12	π	π	PROPN
ejpam-702	66	13	is	be	AUX
ejpam-702	66	14	z	z	NOUN
ejpam-702	66	15	=	=	SYM
ejpam-702	66	16	µ(q	µ(q	PROPN
ejpam-702	66	17	)	)	PUNCT
ejpam-702	66	18	the	the	DET
ejpam-702	66	19	norm	norm	NOUN
ejpam-702	66	20	of	of	ADP
ejpam-702	66	21	the	the	DET
ejpam-702	66	22	element	element	NOUN
ejpam-702	66	23	obtained	obtain	VERB
ejpam-702	66	24	by	by	ADP
ejpam-702	66	25	the	the	DET
ejpam-702	66	26	function	function	NOUN
ejpam-702	66	27	µ	µ	NOUN
ejpam-702	66	28	is	be	AUX
ejpam-702	66	29	minimum	minimum	ADJ
ejpam-702	66	30	[	[	X
ejpam-702	66	31	4	4	NUM
ejpam-702	66	32	,	,	PUNCT
ejpam-702	66	33	p.	p.	NOUN
ejpam-702	66	34	61	61	NUM
ejpam-702	66	35	]	]	PUNCT
ejpam-702	66	36	.	.	PUNCT
ejpam-702	67	1	let	let	VERB
ejpam-702	67	2	us	we	PRON
ejpam-702	67	3	introduce	introduce	VERB
ejpam-702	67	4	the	the	DET
ejpam-702	67	5	quaternion	quaternion	NOUN
ejpam-702	67	6	mannheim	mannheim	NOUN
ejpam-702	67	7	distance	distance	NOUN
ejpam-702	67	8	.	.	PUNCT
ejpam-702	68	1	let	let	VERB
ejpam-702	68	2	α	α	PRON
ejpam-702	68	3	,	,	PUNCT
ejpam-702	68	4	β	β	X
ejpam-702	68	5	∈	∈	PROPN
ejpam-702	68	6	rπ	rπ	NOUN
ejpam-702	68	7	and	and	CCONJ
ejpam-702	68	8	γ	γ	X
ejpam-702	68	9	=	=	SYM
ejpam-702	68	10	β	β	NOUN
ejpam-702	68	11	−	−	NOUN
ejpam-702	68	12	α	α	PROPN
ejpam-702	68	13	=	=	SYM
ejpam-702	68	14	a0	a0	PROPN
ejpam-702	68	15	+	+	CCONJ
ejpam-702	68	16	a1i	a1i	PROPN
ejpam-702	68	17	+	+	NUM
ejpam-702	68	18	a2	a2	PROPN
ejpam-702	68	19	j	j	PROPN
ejpam-702	69	1	+	+	CCONJ
ejpam-702	69	2	a3k	a3k	PROPN
ejpam-702	69	3	(	(	PUNCT
ejpam-702	69	4	mod	mod	PROPN
ejpam-702	69	5	π	π	PROPN
ejpam-702	69	6	)	)	PUNCT
ejpam-702	69	7	,	,	PUNCT
ejpam-702	69	8	where	where	SCONJ
ejpam-702	69	9	a	a	DET
ejpam-702	69	10	proper	proper	ADJ
ejpam-702	69	11	π	π	NOUN
ejpam-702	69	12	is	be	AUX
ejpam-702	69	13	a	a	DET
ejpam-702	69	14	prime	prime	ADJ
ejpam-702	69	15	quaternion	quaternion	NOUN
ejpam-702	69	16	integer	integer	NOUN
ejpam-702	69	17	.	.	PUNCT
ejpam-702	70	1	let	let	VERB
ejpam-702	70	2	the	the	DET
ejpam-702	70	3	quaternion	quaternion	NOUN
ejpam-702	70	4	mannheim	mannheim	NOUN
ejpam-702	70	5	weight	weight	NOUN
ejpam-702	70	6	of	of	ADP
ejpam-702	70	7	γ	γ	NOUN
ejpam-702	70	8	be	be	AUX
ejpam-702	70	9	defined	define	VERB
ejpam-702	70	10	as	as	ADP
ejpam-702	70	11	wqm	wqm	NOUN
ejpam-702	70	12	(	(	PUNCT
ejpam-702	70	13	γ	γ	NOUN
ejpam-702	70	14	)	)	PUNCT
ejpam-702	70	15	=	=	SYM
ejpam-702	70	16	�	�	PROPN
ejpam-702	70	17	�	�	PROPN
ejpam-702	70	18	a0	a0	PROPN
ejpam-702	70	19	�	�	PROPN
ejpam-702	70	20	�	�	PROPN
ejpam-702	70	21	+	+	CCONJ
ejpam-702	70	22	�	�	PROPN
ejpam-702	70	23	�	�	PROPN
ejpam-702	70	24	a1	a1	PROPN
ejpam-702	70	25	�	�	PROPN
ejpam-702	70	26	�	�	PROPN
ejpam-702	70	27	+	+	PROPN
ejpam-702	70	28	�	�	PROPN
ejpam-702	70	29	�	�	PROPN
ejpam-702	70	30	a2	a2	PROPN
ejpam-702	70	31	�	�	PROPN
ejpam-702	70	32	�	�	PROPN
ejpam-702	70	33	+	+	CCONJ
ejpam-702	70	34	�	�	PROPN
ejpam-702	70	35	�	�	PROPN
ejpam-702	70	36	a3	a3	PROPN
ejpam-702	70	37	�	�	PROPN
ejpam-702	70	38	�	�	PROPN
ejpam-702	70	39	.	.	PUNCT
ejpam-702	71	1	the	the	DET
ejpam-702	71	2	quaternion	quaternion	PROPN
ejpam-702	71	3	mannheim	mannheim	PROPN
ejpam-702	71	4	distance	distance	NOUN
ejpam-702	71	5	dqm	dqm	NOUN
ejpam-702	71	6	between	between	ADP
ejpam-702	71	7	α	α	PROPN
ejpam-702	71	8	and	and	CCONJ
ejpam-702	71	9	β	β	X
ejpam-702	71	10	is	be	AUX
ejpam-702	71	11	defined	define	VERB
ejpam-702	71	12	as	as	ADP
ejpam-702	71	13	dqm	dqm	PROPN
ejpam-702	71	14	(	(	PUNCT
ejpam-702	71	15	α	α	NOUN
ejpam-702	71	16	,	,	PUNCT
ejpam-702	71	17	β	β	NOUN
ejpam-702	71	18	)	)	PUNCT
ejpam-702	71	19	=	=	PUNCT
ejpam-702	71	20	wqm	wqm	NOUN
ejpam-702	71	21	(	(	PUNCT
ejpam-702	71	22	γ	γ	NOUN
ejpam-702	71	23	)	)	PUNCT
ejpam-702	71	24	.	.	PUNCT
ejpam-702	72	1	indeed	indeed	ADV
ejpam-702	72	2	,	,	PUNCT
ejpam-702	72	3	dqm	dqm	PROPN
ejpam-702	72	4	is	be	AUX
ejpam-702	72	5	a	a	DET
ejpam-702	72	6	metric	metric	NOUN
ejpam-702	72	7	.	.	PUNCT
ejpam-702	73	1	let	let	VERB
ejpam-702	73	2	α	α	PRON
ejpam-702	73	3	,	,	PUNCT
ejpam-702	73	4	β	β	X
ejpam-702	73	5	and	and	CCONJ
ejpam-702	73	6	γ	γ	PROPN
ejpam-702	73	7	be	be	VERB
ejpam-702	73	8	any	any	DET
ejpam-702	73	9	three	three	NUM
ejpam-702	73	10	elements	element	NOUN
ejpam-702	73	11	of	of	ADP
ejpam-702	73	12	rπ	rπ	NOUN
ejpam-702	73	13	.	.	PUNCT
ejpam-702	74	1	we	we	PRON
ejpam-702	74	2	have	have	VERB
ejpam-702	74	3	i	i	PRON
ejpam-702	74	4	)	)	PUNCT
ejpam-702	74	5	dqm	dqm	PROPN
ejpam-702	74	6	(	(	PUNCT
ejpam-702	74	7	α	α	NOUN
ejpam-702	74	8	,	,	PUNCT
ejpam-702	74	9	β	β	NOUN
ejpam-702	74	10	)	)	PUNCT
ejpam-702	75	1	=	=	NOUN
ejpam-702	75	2	wqm	wqm	NOUN
ejpam-702	75	3	(	(	PUNCT
ejpam-702	75	4	δ1	δ1	NOUN
ejpam-702	75	5	)	)	PUNCT
ejpam-702	75	6	,	,	PUNCT
ejpam-702	75	7	with	with	ADP
ejpam-702	75	8	α−	α−	ADP
ejpam-702	75	9	β	β	X
ejpam-702	75	10	≡	≡	PROPN
ejpam-702	75	11	δ1	δ1	NOUN
ejpam-702	75	12	(	(	PUNCT
ejpam-702	75	13	modπ),δ1	modπ),δ1	NOUN
ejpam-702	75	14	∈	∈	PROPN
ejpam-702	75	15	rπ	rπ	NOUN
ejpam-702	75	16	and	and	CCONJ
ejpam-702	75	17	n(δ1	n(δ1	NOUN
ejpam-702	75	18	)	)	PUNCT
ejpam-702	75	19	minimum	minimum	NOUN
ejpam-702	75	20	,	,	PUNCT
ejpam-702	75	21	ii	ii	NOUN
ejpam-702	75	22	)	)	PUNCT
ejpam-702	75	23	dqm	dqm	NOUN
ejpam-702	75	24	(	(	PUNCT
ejpam-702	75	25	α	α	NOUN
ejpam-702	75	26	,	,	PUNCT
ejpam-702	75	27	γ	γ	NOUN
ejpam-702	75	28	)	)	PUNCT
ejpam-702	75	29	=	=	NOUN
ejpam-702	75	30	wqm	wqm	NOUN
ejpam-702	75	31	(	(	PUNCT
ejpam-702	75	32	δ2	δ2	ADV
ejpam-702	75	33	)	)	PUNCT
ejpam-702	75	34	,	,	PUNCT
ejpam-702	75	35	with	with	ADP
ejpam-702	75	36	α−	α−	ADP
ejpam-702	75	37	γ≡	γ≡	ADV
ejpam-702	75	38	δ2	δ2	VERB
ejpam-702	75	39	(	(	PUNCT
ejpam-702	75	40	modπ),δ2	modπ),δ2	PROPN
ejpam-702	75	41	∈	∈	PROPN
ejpam-702	75	42	rπ	rπ	NOUN
ejpam-702	75	43	and	and	CCONJ
ejpam-702	75	44	n(δ2	n(δ2	ADJ
ejpam-702	75	45	)	)	PUNCT
ejpam-702	75	46	minimum	minimum	ADJ
ejpam-702	75	47	,	,	PUNCT
ejpam-702	75	48	iii	iii	NOUN
ejpam-702	75	49	)	)	PUNCT
ejpam-702	75	50	dqm	dqm	NOUN
ejpam-702	75	51	(	(	PUNCT
ejpam-702	75	52	β	β	X
ejpam-702	75	53	,	,	PUNCT
ejpam-702	75	54	γ	γ	NOUN
ejpam-702	75	55	)	)	PUNCT
ejpam-702	75	56	=	=	NOUN
ejpam-702	75	57	wqm	wqm	NOUN
ejpam-702	75	58	(	(	PUNCT
ejpam-702	75	59	δ3	δ3	PROPN
ejpam-702	75	60	)	)	PUNCT
ejpam-702	75	61	,	,	PUNCT
ejpam-702	75	62	with	with	ADP
ejpam-702	75	63	γ−	γ−	NUM
ejpam-702	75	64	β	β	X
ejpam-702	75	65	≡	≡	PROPN
ejpam-702	75	66	δ3	δ3	PROPN
ejpam-702	75	67	(	(	PUNCT
ejpam-702	75	68	modπ),δ3	modπ),δ3	PROPN
ejpam-702	75	69	∈	∈	PROPN
ejpam-702	75	70	rπ	rπ	NOUN
ejpam-702	75	71	and	and	CCONJ
ejpam-702	75	72	n(δ3	n(δ3	PROPN
ejpam-702	75	73	)	)	PUNCT
ejpam-702	75	74	minimum	minimum	NOUN
ejpam-702	75	75	.	.	PUNCT
ejpam-702	76	1	thus	thus	ADV
ejpam-702	76	2	,	,	PUNCT
ejpam-702	76	3	α−	α−	ADP
ejpam-702	76	4	β	β	PROPN
ejpam-702	76	5	≡	≡	PROPN
ejpam-702	76	6	δ2	δ2	PROPN
ejpam-702	76	7	+	+	CCONJ
ejpam-702	76	8	δ3	δ3	PROPN
ejpam-702	76	9	(	(	PUNCT
ejpam-702	76	10	modπ	modπ	PROPN
ejpam-702	76	11	)	)	PUNCT
ejpam-702	76	12	.	.	PUNCT
ejpam-702	77	1	notwithstanding	notwithstanding	ADP
ejpam-702	77	2	,	,	PUNCT
ejpam-702	77	3	n(δ2	n(δ2	ADJ
ejpam-702	77	4	+	+	CCONJ
ejpam-702	77	5	δ3)≥	δ3)≥	NOUN
ejpam-702	77	6	n(δ1	n(δ1	NOUN
ejpam-702	77	7	)	)	PUNCT
ejpam-702	77	8	,	,	PUNCT
ejpam-702	77	9	because	because	SCONJ
ejpam-702	77	10	α	α	DET
ejpam-702	77	11	−	−	PROPN
ejpam-702	77	12	β	β	X
ejpam-702	77	13	≡	≡	PROPN
ejpam-702	77	14	δ1	δ1	PROPN
ejpam-702	77	15	(	(	PUNCT
ejpam-702	77	16	modπ	modπ	PROPN
ejpam-702	77	17	)	)	PUNCT
ejpam-702	77	18	and	and	CCONJ
ejpam-702	77	19	n(δ1	n(δ1	NOUN
ejpam-702	77	20	)	)	PUNCT
ejpam-702	77	21	is	be	AUX
ejpam-702	77	22	minimum	minimum	ADJ
ejpam-702	77	23	.	.	PUNCT
ejpam-702	78	1	therefore	therefore	ADV
ejpam-702	78	2	,	,	PUNCT
ejpam-702	78	3	dqm	dqm	PROPN
ejpam-702	78	4	(	(	PUNCT
ejpam-702	78	5	α	α	NOUN
ejpam-702	78	6	,	,	PUNCT
ejpam-702	78	7	β	β	NOUN
ejpam-702	78	8	)	)	PUNCT
ejpam-702	78	9	≤	≤	NOUN
ejpam-702	78	10	dqm	dqm	X
ejpam-702	78	11	(	(	PUNCT
ejpam-702	78	12	α	α	NOUN
ejpam-702	78	13	,	,	PUNCT
ejpam-702	78	14	γ	γ	NOUN
ejpam-702	78	15	)	)	PUNCT
ejpam-702	78	16	+	+	CCONJ
ejpam-702	78	17	dqm	dqm	X
ejpam-702	78	18	(	(	PUNCT
ejpam-702	78	19	γ	γ	X
ejpam-702	78	20	,	,	PUNCT
ejpam-702	78	21	β	β	NOUN
ejpam-702	78	22	)	)	PUNCT
ejpam-702	78	23	.	.	PUNCT
ejpam-702	79	1	other	other	ADJ
ejpam-702	79	2	metric	metric	ADJ
ejpam-702	79	3	conditions	condition	NOUN
ejpam-702	79	4	are	be	AUX
ejpam-702	79	5	clear	clear	ADJ
ejpam-702	79	6	.	.	PUNCT
ejpam-702	80	1	m.	m.	NOUN
ejpam-702	80	2	özen	özen	NOUN
ejpam-702	80	3	,	,	PUNCT
ejpam-702	80	4	m.	m.	NOUN
ejpam-702	80	5	güzeltepe	güzeltepe	PROPN
ejpam-702	80	6	/	/	SYM
ejpam-702	80	7	eur	eur	NOUN
ejpam-702	80	8	.	.	PUNCT
ejpam-702	81	1	j.	j.	PROPN
ejpam-702	81	2	pure	pure	PROPN
ejpam-702	81	3	appl	appl	PROPN
ejpam-702	81	4	.	.	PROPN
ejpam-702	81	5	math	math	PROPN
ejpam-702	81	6	,	,	PUNCT
ejpam-702	81	7	3	3	NUM
ejpam-702	81	8	(	(	PUNCT
ejpam-702	81	9	2010	2010	NUM
ejpam-702	81	10	)	)	PUNCT
ejpam-702	81	11	,	,	PUNCT
ejpam-702	81	12	670	670	NUM
ejpam-702	81	13	-	-	SYM
ejpam-702	81	14	677	677	NUM
ejpam-702	81	15	673	673	NUM
ejpam-702	81	16	3	3	NUM
ejpam-702	81	17	.	.	PUNCT
ejpam-702	81	18	one	one	NUM
ejpam-702	81	19	quaternion	quaternion	NOUN
ejpam-702	81	20	mannheim	mannheim	NOUN
ejpam-702	81	21	error	error	NOUN
ejpam-702	81	22	correcting	correct	VERB
ejpam-702	81	23	codes	code	NOUN
ejpam-702	81	24	let	let	VERB
ejpam-702	81	25	α	α	PRON
ejpam-702	81	26	be	be	AUX
ejpam-702	81	27	a	a	DET
ejpam-702	81	28	primitive	primitive	ADJ
ejpam-702	81	29	element	element	NOUN
ejpam-702	81	30	of	of	ADP
ejpam-702	81	31	rπ	rπ	NOUN
ejpam-702	81	32	and	and	CCONJ
ejpam-702	81	33	let	let	VERB
ejpam-702	81	34	p	p	PRON
ejpam-702	81	35	be	be	AUX
ejpam-702	81	36	a	a	DET
ejpam-702	81	37	prime	prime	ADJ
ejpam-702	81	38	inz	inz	PROPN
ejpam-702	81	39	,	,	PUNCT
ejpam-702	81	40	where	where	SCONJ
ejpam-702	81	41	π=	π=	NUM
ejpam-702	81	42	a0+a1i+a2	a0+a1i+a2	NOUN
ejpam-702	81	43	j+a3k	j+a3k	ADV
ejpam-702	81	44	is	be	AUX
ejpam-702	81	45	a	a	DET
ejpam-702	81	46	quaternion	quaternion	ADJ
ejpam-702	81	47	prime	prime	ADJ
ejpam-702	81	48	number	number	NOUN
ejpam-702	81	49	and	and	CCONJ
ejpam-702	81	50	p	p	NOUN
ejpam-702	81	51	=	=	X
ejpam-702	81	52	ππ̄.	ππ̄.	X
ejpam-702	82	1	then	then	ADV
ejpam-702	82	2	,	,	PUNCT
ejpam-702	82	3	αp−1	αp−1	PROPN
ejpam-702	82	4	=	=	PROPN
ejpam-702	82	5	1	1	NUM
ejpam-702	82	6	and	and	CCONJ
ejpam-702	82	7	the	the	DET
ejpam-702	82	8	parity	parity	NOUN
ejpam-702	82	9	check	check	NOUN
ejpam-702	82	10	matrix	matrix	NOUN
ejpam-702	82	11	h	h	NOUN
ejpam-702	82	12	and	and	CCONJ
ejpam-702	82	13	the	the	DET
ejpam-702	82	14	generator	generator	NOUN
ejpam-702	82	15	matrix	matrix	NOUN
ejpam-702	82	16	g	g	NOUN
ejpam-702	82	17	by	by	ADP
ejpam-702	82	18	using	use	VERB
ejpam-702	82	19	the	the	DET
ejpam-702	82	20	primitive	primitive	ADJ
ejpam-702	82	21	element	element	NOUN
ejpam-702	82	22	α	α	NOUN
ejpam-702	82	23	are	be	AUX
ejpam-702	82	24	obtained	obtain	VERB
ejpam-702	82	25	as	as	ADP
ejpam-702	82	26	follows	follow	NOUN
ejpam-702	82	27	,	,	PUNCT
ejpam-702	82	28	respectively	respectively	ADV
ejpam-702	82	29	;	;	PUNCT
ejpam-702	82	30	h	h	NOUN
ejpam-702	82	31	=	=	SYM
ejpam-702	82	32	�	�	PROPN
ejpam-702	82	33	α0	α0	ADJ
ejpam-702	82	34	α1	α1	PROPN
ejpam-702	82	35	·	·	PUNCT
ejpam-702	82	36	·	·	PUNCT
ejpam-702	82	37	·	·	PUNCT
ejpam-702	83	1	α(p−1)/2−1	α(p−1)/2−1	DET
ejpam-702	83	2	�	�	PROPN
ejpam-702	83	3	,	,	PUNCT
ejpam-702	83	4	g	g	NOUN
ejpam-702	83	5	=	=	PUNCT
ejpam-702	83	6			PROPN
ejpam-702	83	7			NOUN
ejpam-702	83	8			NOUN
ejpam-702	83	9			NOUN
ejpam-702	83	10			NOUN
ejpam-702	83	11			NOUN
ejpam-702	83	12	−α1	−α1	VERB
ejpam-702	83	13	1	1	NUM
ejpam-702	83	14	0	0	NUM
ejpam-702	83	15	·	·	PUNCT
ejpam-702	83	16	·	·	PUNCT
ejpam-702	83	17	·	·	PUNCT
ejpam-702	83	18	0	0	NUM
ejpam-702	84	1	−α2	−α2	NOUN
ejpam-702	84	2	0	0	NUM
ejpam-702	84	3	1	1	NUM
ejpam-702	84	4	·	·	PUNCT
ejpam-702	84	5	·	·	PUNCT
ejpam-702	84	6	·	·	PUNCT
ejpam-702	84	7	0	0	NUM
ejpam-702	84	8	...	...	PUNCT
ejpam-702	84	9	...	...	PUNCT
ejpam-702	84	10	.	.	PUNCT
ejpam-702	84	11	.	.	PUNCT
ejpam-702	84	12	.	.	PUNCT
ejpam-702	85	1	...	...	PUNCT
ejpam-702	85	2	−α(p−1)/2−1	−α(p−1)/2−1	X
ejpam-702	85	3	0	0	NUM
ejpam-702	85	4	0	0	NUM
ejpam-702	85	5	1	1	NUM
ejpam-702	85	6			NOUN
ejpam-702	85	7			NOUN
ejpam-702	85	8			VERB
ejpam-702	85	9			NOUN
ejpam-702	85	10			NOUN
ejpam-702	85	11			PUNCT
ejpam-702	85	12	.	.	PUNCT
ejpam-702	86	1	hence	hence	ADV
ejpam-702	86	2	,	,	PUNCT
ejpam-702	86	3	the	the	DET
ejpam-702	86	4	one	one	NUM
ejpam-702	86	5	quaternion	quaternion	NOUN
ejpam-702	86	6	mannheim	mannheim	NOUN
ejpam-702	86	7	error	error	NOUN
ejpam-702	86	8	correcting	correct	VERB
ejpam-702	86	9	codes	code	NOUN
ejpam-702	86	10	of	of	ADP
ejpam-702	86	11	length	length	NOUN
ejpam-702	86	12	n	n	NOUN
ejpam-702	86	13	=	=	PUNCT
ejpam-702	86	14	(	(	PUNCT
ejpam-702	86	15	p−	p−	NOUN
ejpam-702	86	16	1	1	NUM
ejpam-702	86	17	)	)	PUNCT
ejpam-702	86	18	�	�	PROPN
ejpam-702	86	19	2	2	NUM
ejpam-702	86	20	can	can	AUX
ejpam-702	86	21	be	be	AUX
ejpam-702	86	22	constructed	construct	VERB
ejpam-702	86	23	by	by	ADP
ejpam-702	86	24	the	the	DET
ejpam-702	86	25	parity	parity	NOUN
ejpam-702	86	26	check	check	NOUN
ejpam-702	86	27	matrix	matrix	NOUN
ejpam-702	86	28	h.	h.	NOUN
ejpam-702	87	1	then	then	ADV
ejpam-702	87	2	the	the	DET
ejpam-702	87	3	code	code	NOUN
ejpam-702	87	4	c	c	NOUN
ejpam-702	87	5	defined	define	VERB
ejpam-702	87	6	by	by	ADP
ejpam-702	87	7	the	the	DET
ejpam-702	87	8	above	above	ADJ
ejpam-702	87	9	parity	parity	NOUN
ejpam-702	87	10	check	check	NOUN
ejpam-702	87	11	matrix	matrix	NOUN
ejpam-702	87	12	h	h	NOUN
ejpam-702	87	13	is	be	AUX
ejpam-702	87	14	able	able	ADJ
ejpam-702	87	15	to	to	PART
ejpam-702	87	16	correct	correct	VERB
ejpam-702	87	17	any	any	DET
ejpam-702	87	18	quaternion	quaternion	NOUN
ejpam-702	87	19	mannheim	mannheim	NOUN
ejpam-702	87	20	error	error	NOUN
ejpam-702	87	21	of	of	ADP
ejpam-702	87	22	weight	weight	NOUN
ejpam-702	87	23	one	one	NUM
ejpam-702	87	24	.	.	PUNCT
ejpam-702	88	1	the	the	DET
ejpam-702	88	2	value	value	NOUN
ejpam-702	88	3	of	of	ADP
ejpam-702	88	4	one	one	NUM
ejpam-702	88	5	quaternion	quaternion	NOUN
ejpam-702	88	6	mannheim	mannheim	NOUN
ejpam-702	88	7	error	error	NOUN
ejpam-702	88	8	is	be	AUX
ejpam-702	88	9	1	1	NUM
ejpam-702	88	10	or	or	CCONJ
ejpam-702	88	11	-1	-1	NOUN
ejpam-702	88	12	.	.	PUNCT
ejpam-702	89	1	the	the	DET
ejpam-702	89	2	decoding	decode	VERB
ejpam-702	89	3	algorithm	algorithm	NOUN
ejpam-702	89	4	for	for	ADP
ejpam-702	89	5	these	these	DET
ejpam-702	89	6	codes	code	NOUN
ejpam-702	89	7	is	be	AUX
ejpam-702	89	8	clear	clear	ADJ
ejpam-702	89	9	.	.	PUNCT
ejpam-702	90	1	let	let	VERB
ejpam-702	90	2	the	the	DET
ejpam-702	90	3	received	received	ADJ
ejpam-702	90	4	vector	vector	NOUN
ejpam-702	90	5	be	be	AUX
ejpam-702	90	6	r	r	NOUN
ejpam-702	90	7	=	=	PUNCT
ejpam-702	90	8	c+	c+	NOUN
ejpam-702	90	9	e	e	NOUN
ejpam-702	90	10	,	,	PUNCT
ejpam-702	90	11	where	where	SCONJ
ejpam-702	90	12	the	the	DET
ejpam-702	90	13	weight	weight	NOUN
ejpam-702	90	14	of	of	ADP
ejpam-702	90	15	the	the	DET
ejpam-702	90	16	error	error	NOUN
ejpam-702	90	17	vector	vector	NOUN
ejpam-702	90	18	e	e	NOUN
ejpam-702	90	19	is	be	AUX
ejpam-702	90	20	1	1	NUM
ejpam-702	90	21	and	and	CCONJ
ejpam-702	90	22	the	the	DET
ejpam-702	90	23	vector	vector	NOUN
ejpam-702	90	24	c	c	PROPN
ejpam-702	90	25	is	be	AUX
ejpam-702	90	26	a	a	DET
ejpam-702	90	27	codeword	codeword	NOUN
ejpam-702	90	28	.	.	PUNCT
ejpam-702	91	1	then	then	ADV
ejpam-702	91	2	the	the	DET
ejpam-702	91	3	syndrome	syndrome	NOUN
ejpam-702	91	4	of	of	ADP
ejpam-702	91	5	the	the	DET
ejpam-702	91	6	received	receive	VERB
ejpam-702	91	7	vector	vector	NOUN
ejpam-702	91	8	r	r	NOUN
ejpam-702	91	9	is	be	AUX
ejpam-702	91	10	computed	compute	VERB
ejpam-702	91	11	by	by	ADP
ejpam-702	91	12	s(r	s(r	NOUN
ejpam-702	91	13	)	)	PUNCT
ejpam-702	92	1	=	=	SYM
ejpam-702	92	2	hr	hr	NOUN
ejpam-702	92	3	t	t	NOUN
ejpam-702	92	4	r	r	NOUN
ejpam-702	92	5	,	,	PUNCT
ejpam-702	92	6	where	where	SCONJ
ejpam-702	92	7	r	r	NOUN
ejpam-702	92	8	t	t	NOUN
ejpam-702	92	9	r	r	NOUN
ejpam-702	92	10	denote	denote	VERB
ejpam-702	92	11	the	the	DET
ejpam-702	92	12	transpose	transpose	NOUN
ejpam-702	92	13	of	of	ADP
ejpam-702	92	14	the	the	DET
ejpam-702	92	15	received	receive	VERB
ejpam-702	92	16	vector	vector	NOUN
ejpam-702	92	17	r.	r.	NOUN
ejpam-702	92	18	the	the	DET
ejpam-702	92	19	value	value	NOUN
ejpam-702	92	20	of	of	ADP
ejpam-702	92	21	the	the	DET
ejpam-702	92	22	error	error	NOUN
ejpam-702	92	23	is	be	AUX
ejpam-702	92	24	computed	compute	VERB
ejpam-702	92	25	by	by	ADP
ejpam-702	92	26	sα−l	sα−l	NOUN
ejpam-702	92	27	,	,	PUNCT
ejpam-702	92	28	where	where	SCONJ
ejpam-702	92	29	l	l	NOUN
ejpam-702	92	30	(	(	PUNCT
ejpam-702	92	31	mod	mod	PROPN
ejpam-702	92	32	n	n	CCONJ
ejpam-702	92	33	)	)	PUNCT
ejpam-702	92	34	helps	help	VERB
ejpam-702	92	35	to	to	PART
ejpam-702	92	36	find	find	VERB
ejpam-702	92	37	the	the	DET
ejpam-702	92	38	location	location	NOUN
ejpam-702	92	39	of	of	ADP
ejpam-702	92	40	the	the	DET
ejpam-702	92	41	error	error	NOUN
ejpam-702	92	42	,	,	PUNCT
ejpam-702	92	43	l	l	NOUN
ejpam-702	92	44	is	be	AUX
ejpam-702	92	45	a	a	DET
ejpam-702	92	46	nonnegative	nonnegative	ADJ
ejpam-702	92	47	integer	integer	NOUN
ejpam-702	92	48	,	,	PUNCT
ejpam-702	92	49	and	and	CCONJ
ejpam-702	92	50	n	n	PRON
ejpam-702	92	51	is	be	AUX
ejpam-702	92	52	equal	equal	ADJ
ejpam-702	92	53	to	to	ADP
ejpam-702	92	54	(	(	PUNCT
ejpam-702	92	55	p−	p−	NOUN
ejpam-702	92	56	1	1	NUM
ejpam-702	92	57	)	)	PUNCT
ejpam-702	92	58	�	�	PROPN
ejpam-702	92	59	2	2	NUM
ejpam-702	92	60	.	.	PUNCT
ejpam-702	93	1	we	we	PRON
ejpam-702	93	2	now	now	ADV
ejpam-702	93	3	consider	consider	VERB
ejpam-702	93	4	a	a	DET
ejpam-702	93	5	basic	basic	ADJ
ejpam-702	93	6	example	example	NOUN
ejpam-702	93	7	with	with	ADP
ejpam-702	93	8	regard	regard	NOUN
ejpam-702	93	9	to	to	ADP
ejpam-702	93	10	the	the	DET
ejpam-702	93	11	one	one	NUM
ejpam-702	93	12	quaternion	quaternion	NOUN
ejpam-702	93	13	mannheim	mannheim	NOUN
ejpam-702	93	14	error	error	NOUN
ejpam-702	93	15	correcting	correct	VERB
ejpam-702	93	16	codes	code	NOUN
ejpam-702	93	17	.	.	PUNCT
ejpam-702	94	1	example	example	NOUN
ejpam-702	95	1	1	1	NUM
ejpam-702	95	2	.	.	PUNCT
ejpam-702	96	1	let	let	VERB
ejpam-702	96	2	π=	π=	NUM
ejpam-702	96	3	2	2	NUM
ejpam-702	96	4	+	+	NUM
ejpam-702	96	5	i+	i+	NUM
ejpam-702	96	6	j+	j+	PROPN
ejpam-702	96	7	k	k	PROPN
ejpam-702	96	8	and	and	CCONJ
ejpam-702	96	9	α=	α=	PROPN
ejpam-702	96	10	1−	1−	NUM
ejpam-702	96	11	i−	i−	PROPN
ejpam-702	96	12	j−	j−	PROPN
ejpam-702	96	13	k.	k.	PROPN
ejpam-702	96	14	then	then	ADV
ejpam-702	96	15	,	,	PUNCT
ejpam-702	96	16	we	we	PRON
ejpam-702	96	17	obtain	obtain	VERB
ejpam-702	96	18	the	the	DET
ejpam-702	96	19	parity	parity	NOUN
ejpam-702	96	20	check	check	NOUN
ejpam-702	96	21	matrix	matrix	NOUN
ejpam-702	96	22	h	h	NOUN
ejpam-702	96	23	and	and	CCONJ
ejpam-702	96	24	the	the	DET
ejpam-702	96	25	generator	generator	NOUN
ejpam-702	96	26	matrix	matrix	NOUN
ejpam-702	96	27	g	g	NOUN
ejpam-702	96	28	by	by	ADP
ejpam-702	96	29	using	use	VERB
ejpam-702	96	30	the	the	DET
ejpam-702	96	31	primitive	primitive	ADJ
ejpam-702	96	32	element	element	NOUN
ejpam-702	96	33	α	α	NOUN
ejpam-702	96	34	of	of	ADP
ejpam-702	96	35	rπ	rπ	NOUN
ejpam-702	96	36	as	as	SCONJ
ejpam-702	96	37	follows	follow	VERB
ejpam-702	96	38	,	,	PUNCT
ejpam-702	96	39	respectively	respectively	ADV
ejpam-702	96	40	;	;	PUNCT
ejpam-702	96	41	h	h	NOUN
ejpam-702	96	42	=	=	SYM
ejpam-702	96	43	�	�	PROPN
ejpam-702	96	44	α0	α0	ADJ
ejpam-702	96	45	,	,	PUNCT
ejpam-702	96	46	α1	α1	PROPN
ejpam-702	96	47	,	,	PUNCT
ejpam-702	96	48	·	·	PUNCT
ejpam-702	96	49	·	·	PUNCT
ejpam-702	97	1	·	·	PUNCT
ejpam-702	97	2	,	,	PUNCT
ejpam-702	97	3	α(p−1)/2−1	α(p−1)/2−1	DET
ejpam-702	97	4	�	�	PROPN
ejpam-702	97	5	,	,	PUNCT
ejpam-702	97	6	g	g	NOUN
ejpam-702	97	7	=	=	PUNCT
ejpam-702	97	8			PROPN
ejpam-702	97	9			NOUN
ejpam-702	97	10			NOUN
ejpam-702	97	11			NOUN
ejpam-702	97	12			NOUN
ejpam-702	97	13			NOUN
ejpam-702	97	14	−α1	−α1	NOUN
ejpam-702	97	15	,	,	PUNCT
ejpam-702	97	16	1	1	NUM
ejpam-702	97	17	,	,	PUNCT
ejpam-702	97	18	0	0	NUM
ejpam-702	97	19	,	,	PUNCT
ejpam-702	97	20	·	·	PUNCT
ejpam-702	97	21	·	·	PUNCT
ejpam-702	97	22	·	·	PUNCT
ejpam-702	97	23	,	,	PUNCT
ejpam-702	97	24	0	0	NUM
ejpam-702	97	25	−α2	−α2	PROPN
ejpam-702	97	26	,	,	PUNCT
ejpam-702	97	27	0	0	NUM
ejpam-702	97	28	,	,	PUNCT
ejpam-702	97	29	1	1	NUM
ejpam-702	97	30	,	,	PUNCT
ejpam-702	97	31	·	·	PUNCT
ejpam-702	97	32	·	·	PUNCT
ejpam-702	97	33	·	·	PUNCT
ejpam-702	97	34	,	,	PUNCT
ejpam-702	97	35	0	0	NUM
ejpam-702	97	36	...	...	PUNCT
ejpam-702	97	37	...	...	PUNCT
ejpam-702	97	38	.	.	PUNCT
ejpam-702	97	39	.	.	PUNCT
ejpam-702	97	40	.	.	PUNCT
ejpam-702	98	1	...	...	PUNCT
ejpam-702	99	1	−α(p−1)/2−1	−α(p−1)/2−1	NUM
ejpam-702	99	2	,	,	PUNCT
ejpam-702	99	3	0	0	NUM
ejpam-702	99	4	,	,	PUNCT
ejpam-702	99	5	0	0	NUM
ejpam-702	99	6	,	,	PUNCT
ejpam-702	99	7	1	1	NUM
ejpam-702	99	8			NOUN
ejpam-702	99	9			NOUN
ejpam-702	99	10			VERB
ejpam-702	99	11			NOUN
ejpam-702	99	12			NOUN
ejpam-702	99	13			PUNCT
ejpam-702	99	14	.	.	PUNCT
ejpam-702	100	1	let	let	VERB
ejpam-702	100	2	the	the	DET
ejpam-702	100	3	received	receive	VERB
ejpam-702	100	4	vector	vector	NOUN
ejpam-702	100	5	r	r	NOUN
ejpam-702	100	6	be	be	VERB
ejpam-702	100	7	�	�	PROPN
ejpam-702	100	8	1−	1−	NUM
ejpam-702	101	1	i	i	PRON
ejpam-702	101	2	−	−	PROPN
ejpam-702	101	3	j−	j−	VERB
ejpam-702	101	4	k	k	PROPN
ejpam-702	101	5	,	,	PUNCT
ejpam-702	101	6	0	0	NUM
ejpam-702	101	7	,	,	PUNCT
ejpam-702	101	8	−1	−1	NOUN
ejpam-702	102	1	+	+	SYM
ejpam-702	102	2	i	i	PRON
ejpam-702	102	3	+	+	CCONJ
ejpam-702	102	4	j+	j+	NUM
ejpam-702	102	5	k	k	PROPN
ejpam-702	102	6	�	�	PROPN
ejpam-702	102	7	,	,	PUNCT
ejpam-702	102	8	then	then	ADV
ejpam-702	102	9	s(r	s(r	X
ejpam-702	102	10	)	)	PUNCT
ejpam-702	103	1	=	=	SYM
ejpam-702	103	2	hr	hr	NOUN
ejpam-702	103	3	t	t	NOUN
ejpam-702	103	4	r	r	NOUN
ejpam-702	103	5	=	=	SYM
ejpam-702	103	6	4	4	NUM
ejpam-702	103	7	≡	≡	PROPN
ejpam-702	103	8	−1	−1	NOUN
ejpam-702	104	1	+	+	CCONJ
ejpam-702	104	2	i	i	PRON
ejpam-702	104	3	+	+	NUM
ejpam-702	104	4	j	j	PROPN
ejpam-702	105	1	+	+	CCONJ
ejpam-702	105	2	k	k	PROPN
ejpam-702	105	3	≡	≡	PROPN
ejpam-702	105	4	α4(modπ	α4(modπ	PROPN
ejpam-702	105	5	)	)	PUNCT
ejpam-702	106	1	(	(	PUNCT
ejpam-702	106	2	see	see	VERB
ejpam-702	106	3	table	table	NOUN
ejpam-702	106	4	2	2	NUM
ejpam-702	106	5	)	)	PUNCT
ejpam-702	106	6	,	,	PUNCT
ejpam-702	106	7	and	and	CCONJ
ejpam-702	106	8	the	the	DET
ejpam-702	106	9	location	location	NOUN
ejpam-702	106	10	of	of	ADP
ejpam-702	106	11	the	the	DET
ejpam-702	106	12	error	error	NOUN
ejpam-702	106	13	is	be	AUX
ejpam-702	106	14	found	find	VERB
ejpam-702	106	15	4	4	NUM
ejpam-702	106	16	≡	≡	PROPN
ejpam-702	106	17	1(mod3	1(mod3	NUM
ejpam-702	106	18	)	)	PUNCT
ejpam-702	106	19	.	.	PUNCT
ejpam-702	107	1	the	the	DET
ejpam-702	107	2	value	value	NOUN
ejpam-702	107	3	of	of	ADP
ejpam-702	107	4	the	the	DET
ejpam-702	107	5	error	error	NOUN
ejpam-702	107	6	is	be	AUX
ejpam-702	107	7	computed	compute	VERB
ejpam-702	107	8	as	as	ADP
ejpam-702	107	9	sα−l	sα−l	NOUN
ejpam-702	107	10	=	=	SYM
ejpam-702	107	11	−1	−1	NOUN
ejpam-702	107	12	.	.	PUNCT
ejpam-702	108	1	so	so	ADV
ejpam-702	108	2	,	,	PUNCT
ejpam-702	108	3	the	the	DET
ejpam-702	108	4	received	receive	VERB
ejpam-702	108	5	vector	vector	NOUN
ejpam-702	108	6	r	r	NOUN
ejpam-702	108	7	is	be	AUX
ejpam-702	108	8	corrected	correct	VERB
ejpam-702	108	9	as	as	ADP
ejpam-702	108	10	c	c	NOUN
ejpam-702	108	11	=	=	SYM
ejpam-702	108	12	r	r	NOUN
ejpam-702	108	13	−	−	NOUN
ejpam-702	108	14	e	e	NOUN
ejpam-702	108	15	=	=	SYM
ejpam-702	108	16	�	�	PROPN
ejpam-702	108	17	1−	1−	NUM
ejpam-702	109	1	i	i	PRON
ejpam-702	109	2	−	−	PROPN
ejpam-702	109	3	j−	j−	PROPN
ejpam-702	109	4	k	k	PROPN
ejpam-702	109	5	,	,	PUNCT
ejpam-702	109	6	1	1	NUM
ejpam-702	109	7	,	,	PUNCT
ejpam-702	109	8	−1	−1	NOUN
ejpam-702	110	1	+	+	SYM
ejpam-702	110	2	i	i	PRON
ejpam-702	110	3	+	+	CCONJ
ejpam-702	110	4	j+	j+	NUM
ejpam-702	110	5	k	k	PROPN
ejpam-702	110	6	�	�	PROPN
ejpam-702	110	7	.	.	PUNCT
ejpam-702	111	1	the	the	DET
ejpam-702	111	2	code	code	NOUN
ejpam-702	111	3	,	,	PUNCT
ejpam-702	111	4	of	of	ADP
ejpam-702	111	5	which	which	PRON
ejpam-702	111	6	the	the	DET
ejpam-702	111	7	parity	parity	NOUN
ejpam-702	111	8	check	check	NOUN
ejpam-702	111	9	matrix	matrix	NOUN
ejpam-702	111	10	is	be	AUX
ejpam-702	111	11	h	h	NOUN
ejpam-702	111	12	=	=	SYM
ejpam-702	111	13	�	�	PROPN
ejpam-702	111	14	α0	α0	ADJ
ejpam-702	111	15	,	,	PUNCT
ejpam-702	111	16	α1	α1	PROPN
ejpam-702	111	17	,	,	PUNCT
ejpam-702	111	18	·	·	PUNCT
ejpam-702	111	19	·	·	PUNCT
ejpam-702	111	20	·	·	PUNCT
ejpam-702	111	21	,	,	PUNCT
ejpam-702	111	22	α(p−1)/2−1	α(p−1)/2−1	DET
ejpam-702	111	23	�	�	PROPN
ejpam-702	111	24	can	can	AUX
ejpam-702	111	25	be	be	AUX
ejpam-702	111	26	generalized	generalize	VERB
ejpam-702	111	27	as	as	ADP
ejpam-702	111	28	n=	n=	ADJ
ejpam-702	111	29	(	(	PUNCT
ejpam-702	111	30	pr	pr	NOUN
ejpam-702	111	31	−	−	NOUN
ejpam-702	111	32	1	1	X
ejpam-702	111	33	)	)	PUNCT
ejpam-702	111	34	�	�	PROPN
ejpam-702	111	35	2	2	NUM
ejpam-702	111	36	.	.	PUNCT
ejpam-702	112	1	in	in	ADP
ejpam-702	112	2	this	this	DET
ejpam-702	112	3	situation	situation	NOUN
ejpam-702	112	4	,	,	PUNCT
ejpam-702	112	5	the	the	DET
ejpam-702	112	6	parity	parity	NOUN
ejpam-702	112	7	check	check	NOUN
ejpam-702	112	8	matrix	matrix	NOUN
ejpam-702	112	9	would	would	AUX
ejpam-702	112	10	be	be	AUX
ejpam-702	112	11	h	h	NOUN
ejpam-702	112	12	=	=	SYM
ejpam-702	112	13	�	�	PROPN
ejpam-702	112	14	α0	α0	ADJ
ejpam-702	112	15	,	,	PUNCT
ejpam-702	112	16	α1	α1	PROPN
ejpam-702	112	17	,	,	PUNCT
ejpam-702	112	18	·	·	PUNCT
ejpam-702	112	19	·	·	PUNCT
ejpam-702	112	20	·	·	PUNCT
ejpam-702	112	21	,	,	PUNCT
ejpam-702	112	22	α(p	α(p	PROPN
ejpam-702	112	23	r−1)/2−1	r−1)/2−1	NOUN
ejpam-702	112	24	�	�	PROPN
ejpam-702	112	25	.	.	PUNCT
ejpam-702	113	1	(	(	PUNCT
ejpam-702	113	2	2	2	X
ejpam-702	113	3	)	)	PUNCT
ejpam-702	113	4	the	the	DET
ejpam-702	113	5	codes	code	NOUN
ejpam-702	113	6	defined	define	VERB
ejpam-702	113	7	by	by	ADP
ejpam-702	113	8	(	(	PUNCT
ejpam-702	113	9	2	2	X
ejpam-702	113	10	)	)	PUNCT
ejpam-702	113	11	are	be	AUX
ejpam-702	113	12	perfect	perfect	ADJ
ejpam-702	113	13	since	since	SCONJ
ejpam-702	113	14	we	we	PRON
ejpam-702	113	15	have	have	AUX
ejpam-702	113	16	pn−r(2n+1	pn−r(2n+1	NOUN
ejpam-702	113	17	)	)	PUNCT
ejpam-702	114	1	=	=	SYM
ejpam-702	114	2	pn−r	pn−r	NOUN
ejpam-702	114	3	pr	pr	NOUN
ejpam-702	114	4	=	=	PUNCT
ejpam-702	114	5	pn	pn	PROPN
ejpam-702	114	6	by	by	ADP
ejpam-702	114	7	the	the	DET
ejpam-702	114	8	sphere	sphere	NOUN
ejpam-702	114	9	packing	pack	VERB
ejpam-702	114	10	bound	bind	VERB
ejpam-702	114	11	.	.	PUNCT
ejpam-702	115	1	m.	m.	NOUN
ejpam-702	115	2	özen	özen	NOUN
ejpam-702	115	3	,	,	PUNCT
ejpam-702	115	4	m.	m.	NOUN
ejpam-702	115	5	güzeltepe	güzeltepe	PROPN
ejpam-702	115	6	/	/	SYM
ejpam-702	115	7	eur	eur	NOUN
ejpam-702	115	8	.	.	PUNCT
ejpam-702	116	1	j.	j.	PROPN
ejpam-702	116	2	pure	pure	PROPN
ejpam-702	116	3	appl	appl	PROPN
ejpam-702	116	4	.	.	PROPN
ejpam-702	116	5	math	math	PROPN
ejpam-702	116	6	,	,	PUNCT
ejpam-702	116	7	3	3	NUM
ejpam-702	116	8	(	(	PUNCT
ejpam-702	116	9	2010	2010	NUM
ejpam-702	116	10	)	)	PUNCT
ejpam-702	116	11	,	,	PUNCT
ejpam-702	116	12	670	670	NUM
ejpam-702	116	13	-	-	SYM
ejpam-702	116	14	677	677	NUM
ejpam-702	116	15	674	674	NUM
ejpam-702	116	16	4	4	NUM
ejpam-702	116	17	.	.	NOUN
ejpam-702	116	18	double	double	ADJ
ejpam-702	116	19	error	error	NOUN
ejpam-702	116	20	correcting	correct	VERB
ejpam-702	116	21	codes	code	NOUN
ejpam-702	116	22	let	let	VERB
ejpam-702	116	23	p	p	PRON
ejpam-702	116	24	be	be	AUX
ejpam-702	116	25	a	a	DET
ejpam-702	116	26	prime	prime	NOUN
ejpam-702	116	27	in	in	ADP
ejpam-702	116	28	z	z	NOUN
ejpam-702	116	29	which	which	PRON
ejpam-702	116	30	is	be	AUX
ejpam-702	116	31	factored	factor	VERB
ejpam-702	116	32	in	in	ADP
ejpam-702	116	33	h(z	h(z	NOUN
ejpam-702	116	34	)	)	PUNCT
ejpam-702	116	35	as	as	ADP
ejpam-702	116	36	ππ	ππ	NOUN
ejpam-702	116	37	,	,	PUNCT
ejpam-702	116	38	where	where	SCONJ
ejpam-702	116	39	π	π	PROPN
ejpam-702	116	40	is	be	AUX
ejpam-702	116	41	a	a	DET
ejpam-702	116	42	prime	prime	NOUN
ejpam-702	116	43	in	in	ADP
ejpam-702	116	44	h(z	h(z	NOUN
ejpam-702	116	45	)	)	PUNCT
ejpam-702	116	46	.	.	PUNCT
ejpam-702	117	1	let	let	VERB
ejpam-702	117	2	β	β	PRON
ejpam-702	117	3	denote	denote	VERB
ejpam-702	117	4	an	an	DET
ejpam-702	117	5	element	element	NOUN
ejpam-702	117	6	of	of	ADP
ejpam-702	117	7	rπ	rπ	NOUN
ejpam-702	117	8	of	of	ADP
ejpam-702	117	9	order	order	NOUN
ejpam-702	117	10	2n	2n	NUM
ejpam-702	117	11	.	.	PUNCT
ejpam-702	118	1	thus	thus	ADV
ejpam-702	118	2	βn	βn	ADJ
ejpam-702	118	3	=	=	SYM
ejpam-702	118	4	−1	−1	NOUN
ejpam-702	118	5	,	,	PUNCT
ejpam-702	118	6	and	and	CCONJ
ejpam-702	118	7	we	we	PRON
ejpam-702	118	8	can	can	AUX
ejpam-702	118	9	write	write	VERB
ejpam-702	118	10	rπ	rπ	NOUN
ejpam-702	118	11	=	=	PUNCT
ejpam-702	118	12	β	β	X
ejpam-702	118	13	�	�	PROPN
ejpam-702	118	14	∪	∪	X
ejpam-702	118	15	{	{	PUNCT
ejpam-702	118	16	0	0	NUM
ejpam-702	118	17	}	}	PUNCT
ejpam-702	118	18	since	since	SCONJ
ejpam-702	118	19	β	β	NOUN
ejpam-702	118	20	is	be	AUX
ejpam-702	118	21	a	a	DET
ejpam-702	118	22	primitive	primitive	ADJ
ejpam-702	118	23	element	element	NOUN
ejpam-702	118	24	of	of	ADP
ejpam-702	118	25	rπ	rπ	NOUN
ejpam-702	118	26	,	,	PUNCT
ejpam-702	118	27	where	where	SCONJ
ejpam-702	118	28	〈	〈	PROPN
ejpam-702	118	29	.	.	PROPN
ejpam-702	118	30	〉	〉	NOUN
ejpam-702	118	31	denotes	denote	VERB
ejpam-702	118	32	an	an	DET
ejpam-702	118	33	ideal	ideal	NOUN
ejpam-702	118	34	.	.	PUNCT
ejpam-702	119	1	therefore	therefore	ADV
ejpam-702	119	2	,	,	PUNCT
ejpam-702	119	3	we	we	PRON
ejpam-702	119	4	consider	consider	VERB
ejpam-702	119	5	the	the	DET
ejpam-702	119	6	code	code	NOUN
ejpam-702	119	7	c	c	NOUN
ejpam-702	119	8	defined	define	VERB
ejpam-702	119	9	by	by	ADP
ejpam-702	119	10	the	the	DET
ejpam-702	119	11	following	follow	VERB
ejpam-702	119	12	parity	parity	NOUN
ejpam-702	119	13	check	check	NOUN
ejpam-702	119	14	matrix	matrix	NOUN
ejpam-702	119	15	h	h	NOUN
ejpam-702	119	16	:	:	PUNCT
ejpam-702	119	17	h	h	NOUN
ejpam-702	119	18	=	=	SYM
ejpam-702	119	19			PROPN
ejpam-702	119	20			NOUN
ejpam-702	119	21			NOUN
ejpam-702	119	22			NOUN
ejpam-702	119	23			NOUN
ejpam-702	119	24			PROPN
ejpam-702	119	25	β0	β0	NOUN
ejpam-702	119	26	,	,	PUNCT
ejpam-702	119	27	β1	β1	PROPN
ejpam-702	119	28	,	,	PUNCT
ejpam-702	119	29	β2	β2	NOUN
ejpam-702	119	30	,	,	PUNCT
ejpam-702	119	31	·	·	PUNCT
ejpam-702	119	32	·	·	PUNCT
ejpam-702	119	33	·	·	PUNCT
ejpam-702	119	34	,	,	PUNCT
ejpam-702	119	35	βn−1	βn−1	PROPN
ejpam-702	119	36	β0	β0	PROPN
ejpam-702	119	37	,	,	PUNCT
ejpam-702	119	38	β3	β3	ADJ
ejpam-702	119	39	,	,	PUNCT
ejpam-702	119	40	β6	β6	PROPN
ejpam-702	119	41	,	,	PUNCT
ejpam-702	119	42	·	·	PUNCT
ejpam-702	119	43	·	·	PUNCT
ejpam-702	119	44	·	·	PUNCT
ejpam-702	119	45	,	,	PUNCT
ejpam-702	119	46	β3(n−1	β3(n−1	NUM
ejpam-702	119	47	)	)	PUNCT
ejpam-702	119	48	...	...	PUNCT
ejpam-702	120	1	β0	β0	ADJ
ejpam-702	120	2	,	,	PUNCT
ejpam-702	120	3	β2t+1	β2t+1	PROPN
ejpam-702	120	4	,	,	PUNCT
ejpam-702	120	5	β2(2t+1	β2(2t+1	PROPN
ejpam-702	120	6	)	)	PUNCT
ejpam-702	120	7	,	,	PUNCT
ejpam-702	120	8	·	·	PUNCT
ejpam-702	120	9	·	·	PUNCT
ejpam-702	120	10	·	·	PUNCT
ejpam-702	120	11	,	,	PUNCT
ejpam-702	120	12	β	β	X
ejpam-702	120	13	(	(	PUNCT
ejpam-702	120	14	n−1)(2t+1	n−1)(2t+1	X
ejpam-702	120	15	)	)	PUNCT
ejpam-702	120	16			PROPN
ejpam-702	120	17			NOUN
ejpam-702	120	18			VERB
ejpam-702	120	19			NOUN
ejpam-702	120	20			NOUN
ejpam-702	120	21			PUNCT
ejpam-702	120	22	,	,	PUNCT
ejpam-702	120	23	(	(	PUNCT
ejpam-702	120	24	3	3	X
ejpam-702	120	25	)	)	PUNCT
ejpam-702	120	26	where	where	SCONJ
ejpam-702	120	27	t	t	PROPN
ejpam-702	120	28	<	<	X
ejpam-702	120	29	n	n	X
ejpam-702	120	30	is	be	AUX
ejpam-702	120	31	a	a	DET
ejpam-702	120	32	nonnegative	nonnegative	ADJ
ejpam-702	120	33	integer	integer	NOUN
ejpam-702	120	34	.	.	PUNCT
ejpam-702	121	1	a	a	DET
ejpam-702	121	2	word	word	NOUN
ejpam-702	121	3	c	c	NOUN
ejpam-702	121	4	=	=	SYM
ejpam-702	121	5	�	�	PROPN
ejpam-702	121	6	c0	c0	PROPN
ejpam-702	121	7	,	,	PUNCT
ejpam-702	121	8	c1	c1	PROPN
ejpam-702	121	9	,	,	PUNCT
ejpam-702	121	10	·	·	PUNCT
ejpam-702	121	11	·	·	PUNCT
ejpam-702	121	12	·	·	PUNCT
ejpam-702	121	13	,	,	PUNCT
ejpam-702	121	14	cn−1	cn−1	PROPN
ejpam-702	121	15	�	�	PROPN
ejpam-702	121	16	∈	∈	PROPN
ejpam-702	121	17	rn	rn	PROPN
ejpam-702	121	18	π	π	PROPN
ejpam-702	121	19	is	be	AUX
ejpam-702	121	20	a	a	DET
ejpam-702	121	21	codeword	codeword	NOUN
ejpam-702	121	22	of	of	ADP
ejpam-702	121	23	c	c	PROPN
ejpam-702	121	24	if	if	SCONJ
ejpam-702	122	1	and	and	CCONJ
ejpam-702	122	2	only	only	ADV
ejpam-702	122	3	if	if	SCONJ
ejpam-702	122	4	hc	hc	PROPN
ejpam-702	122	5	t	t	NOUN
ejpam-702	122	6	r	r	NOUN
ejpam-702	122	7	=	=	SYM
ejpam-702	122	8	0	0	NUM
ejpam-702	122	9	.	.	PUNCT
ejpam-702	123	1	if	if	SCONJ
ejpam-702	123	2	c(x	c(x	NOUN
ejpam-702	123	3	)	)	PUNCT
ejpam-702	123	4	=	=	SYM
ejpam-702	123	5	∑n−1	∑n−1	ADP
ejpam-702	123	6	i=0	i=0	PROPN
ejpam-702	123	7	ci	ci	PROPN
ejpam-702	124	1	x	x	X
ejpam-702	124	2	i	i	PRON
ejpam-702	124	3	is	be	AUX
ejpam-702	124	4	a	a	DET
ejpam-702	124	5	codeword	codeword	NOUN
ejpam-702	124	6	polynomial	polynomial	NOUN
ejpam-702	124	7	,	,	PUNCT
ejpam-702	124	8	we	we	PRON
ejpam-702	124	9	get	get	VERB
ejpam-702	124	10	c(β2	c(β2	NOUN
ejpam-702	124	11	j+1	j+1	NUM
ejpam-702	124	12	)	)	PUNCT
ejpam-702	124	13	=	=	SYM
ejpam-702	124	14	0	0	NUM
ejpam-702	124	15	,	,	PUNCT
ejpam-702	124	16	for	for	ADP
ejpam-702	124	17	j	j	PROPN
ejpam-702	124	18	=	=	SYM
ejpam-702	124	19	0,1	0,1	NUM
ejpam-702	124	20	,	,	PUNCT
ejpam-702	124	21	·	·	PUNCT
ejpam-702	124	22	·	·	PUNCT
ejpam-702	124	23	·	·	PUNCT
ejpam-702	124	24	,	,	PUNCT
ejpam-702	124	25	t.	t.	X
ejpam-702	124	26	the	the	DET
ejpam-702	124	27	polynomial	polynomial	ADJ
ejpam-702	124	28	g(x	g(x	NOUN
ejpam-702	124	29	)	)	PUNCT
ejpam-702	125	1	=	=	PUNCT
ejpam-702	125	2	(	(	PUNCT
ejpam-702	125	3	x	x	X
ejpam-702	125	4	−	−	NOUN
ejpam-702	125	5	β)(x	β)(x	PUNCT
ejpam-702	125	6	−	−	NOUN
ejpam-702	125	7	β3	β3	ADJ
ejpam-702	125	8	)	)	PUNCT
ejpam-702	125	9	·	·	PUNCT
ejpam-702	125	10	·	·	PUNCT
ejpam-702	125	11	·	·	PUNCT
ejpam-702	126	1	(	(	PUNCT
ejpam-702	126	2	x	x	X
ejpam-702	126	3	−	−	NOUN
ejpam-702	126	4	β2t+1	β2t+1	NUM
ejpam-702	126	5	)	)	PUNCT
ejpam-702	126	6	is	be	AUX
ejpam-702	126	7	the	the	DET
ejpam-702	126	8	generator	generator	NOUN
ejpam-702	126	9	polynomial	polynomial	NOUN
ejpam-702	126	10	of	of	ADP
ejpam-702	126	11	c	c	PROPN
ejpam-702	126	12	,	,	PUNCT
ejpam-702	126	13	and	and	CCONJ
ejpam-702	126	14	c	c	NOUN
ejpam-702	126	15	=	=	SYM
ejpam-702	126	16	g(x	g(x	NOUN
ejpam-702	126	17	)	)	PUNCT
ejpam-702	126	18	�	�	PROPN
ejpam-702	126	19	is	be	AUX
ejpam-702	126	20	a	a	DET
ejpam-702	126	21	principal	principal	ADJ
ejpam-702	126	22	ideal	ideal	NOUN
ejpam-702	126	23	of	of	ADP
ejpam-702	126	24	rπ[x	rπ[x	PROPN
ejpam-702	126	25	]	]	PUNCT
ejpam-702	126	26	�	�	PROPN
ejpam-702	126	27	〈	〈	PROPN
ejpam-702	126	28	xn+	xn+	PROPN
ejpam-702	126	29	1	1	NUM
ejpam-702	126	30	〉	〉	NOUN
ejpam-702	126	31	.	.	PUNCT
ejpam-702	127	1	if	if	SCONJ
ejpam-702	127	2	we	we	PRON
ejpam-702	127	3	multiply	multiply	VERB
ejpam-702	127	4	the	the	DET
ejpam-702	127	5	code	code	NOUN
ejpam-702	127	6	polynomial	polynomial	ADJ
ejpam-702	127	7	c(x	c(x	NOUN
ejpam-702	127	8	)	)	PUNCT
ejpam-702	127	9	by	by	ADP
ejpam-702	127	10	x(mod(xn+	x(mod(xn+	PROPN
ejpam-702	127	11	1	1	NUM
ejpam-702	127	12	)	)	PUNCT
ejpam-702	127	13	)	)	PUNCT
ejpam-702	127	14	,	,	PUNCT
ejpam-702	127	15	then	then	ADV
ejpam-702	127	16	we	we	PRON
ejpam-702	127	17	get	get	VERB
ejpam-702	127	18	xc(x	xc(x	PRON
ejpam-702	127	19	)	)	PUNCT
ejpam-702	128	1	=	=	SYM
ejpam-702	128	2	c0	c0	NOUN
ejpam-702	128	3	x	x	PUNCT
ejpam-702	128	4	+	+	CCONJ
ejpam-702	128	5	c1	c1	PROPN
ejpam-702	128	6	x2	x2	PROPN
ejpam-702	128	7	+	+	PROPN
ejpam-702	128	8	·	·	PUNCT
ejpam-702	128	9	·	·	PUNCT
ejpam-702	128	10	·	·	PUNCT
ejpam-702	128	11	+	+	NUM
ejpam-702	128	12	cn−1	cn−1	PROPN
ejpam-702	128	13	xn	xn	PROPN
ejpam-702	128	14	.	.	PUNCT
ejpam-702	129	1	but	but	CCONJ
ejpam-702	129	2	we	we	PRON
ejpam-702	129	3	know	know	VERB
ejpam-702	129	4	that	that	PRON
ejpam-702	129	5	xn	xn	PUNCT
ejpam-702	130	1	=	=	PUNCT
ejpam-702	130	2	−1	−1	NOUN
ejpam-702	130	3	.	.	PUNCT
ejpam-702	131	1	therefore	therefore	ADV
ejpam-702	131	2	,	,	PUNCT
ejpam-702	131	3	if	if	SCONJ
ejpam-702	131	4	c(x	c(x	NOUN
ejpam-702	131	5	)	)	PUNCT
ejpam-702	131	6	∈	∈	PROPN
ejpam-702	131	7	c	c	NOUN
ejpam-702	131	8	,	,	PUNCT
ejpam-702	131	9	then	then	ADV
ejpam-702	131	10	xc(x	xc(x	PUNCT
ejpam-702	131	11	)	)	PUNCT
ejpam-702	132	1	∈	∈	PROPN
ejpam-702	132	2	c	c	NOUN
ejpam-702	132	3	.	.	PUNCT
ejpam-702	133	1	thus	thus	ADV
ejpam-702	133	2	,	,	PUNCT
ejpam-702	133	3	multiplying	multiply	VERB
ejpam-702	133	4	c(x	c(x	NOUN
ejpam-702	133	5	)	)	PUNCT
ejpam-702	133	6	by	by	ADP
ejpam-702	133	7	x	x	SYM
ejpam-702	133	8	(	(	PUNCT
ejpam-702	133	9	mod	mod	X
ejpam-702	133	10	(	(	PUNCT
ejpam-702	133	11	xn+	xn+	PROPN
ejpam-702	133	12	1	1	NUM
ejpam-702	133	13	)	)	PUNCT
ejpam-702	133	14	)	)	PUNCT
ejpam-702	133	15	means	mean	VERB
ejpam-702	133	16	the	the	DET
ejpam-702	133	17	following	follow	VERB
ejpam-702	133	18	:	:	PUNCT
ejpam-702	133	19	i	i	PRON
ejpam-702	133	20	)	)	PUNCT
ejpam-702	133	21	shifting	shift	VERB
ejpam-702	133	22	c(x	c(x	NOUN
ejpam-702	133	23	)	)	PUNCT
ejpam-702	133	24	cyclically	cyclically	ADV
ejpam-702	133	25	one	one	NUM
ejpam-702	133	26	position	position	NOUN
ejpam-702	133	27	to	to	ADP
ejpam-702	133	28	the	the	DET
ejpam-702	133	29	right	right	NOUN
ejpam-702	133	30	;	;	PUNCT
ejpam-702	133	31	ii	ii	X
ejpam-702	133	32	)	)	PUNCT
ejpam-702	133	33	rotating	rotate	VERB
ejpam-702	133	34	the	the	DET
ejpam-702	133	35	coefficient	coefficient	NOUN
ejpam-702	133	36	cn−1	cn−1	ADV
ejpam-702	133	37	by	by	ADP
ejpam-702	133	38	π	π	PROPN
ejpam-702	133	39	radians	radian	NOUN
ejpam-702	133	40	and	and	CCONJ
ejpam-702	133	41	locating	locate	VERB
ejpam-702	133	42	it	it	PRON
ejpam-702	133	43	for	for	ADP
ejpam-702	133	44	the	the	DET
ejpam-702	133	45	first	first	ADJ
ejpam-702	133	46	symbol	symbol	NOUN
ejpam-702	133	47	of	of	ADP
ejpam-702	133	48	the	the	DET
ejpam-702	133	49	new	new	ADJ
ejpam-702	133	50	codeword	codeword	NOUN
ejpam-702	133	51	.	.	PUNCT
ejpam-702	134	1	theorem	theorem	NOUN
ejpam-702	134	2	2	2	NUM
ejpam-702	134	3	.	.	PUNCT
ejpam-702	135	1	let	let	VERB
ejpam-702	135	2	c	c	NOUN
ejpam-702	135	3	be	be	AUX
ejpam-702	135	4	the	the	DET
ejpam-702	135	5	code	code	NOUN
ejpam-702	135	6	defined	define	VERB
ejpam-702	135	7	by	by	ADP
ejpam-702	135	8	the	the	DET
ejpam-702	135	9	parity	parity	NOUN
ejpam-702	135	10	check	check	NOUN
ejpam-702	135	11	matrix	matrix	NOUN
ejpam-702	135	12	as	as	ADP
ejpam-702	135	13	in	in	ADP
ejpam-702	135	14	(	(	PUNCT
ejpam-702	135	15	3	3	NUM
ejpam-702	135	16	)	)	PUNCT
ejpam-702	135	17	.	.	PUNCT
ejpam-702	136	1	then	then	ADV
ejpam-702	136	2	c	c	PROPN
ejpam-702	136	3	is	be	AUX
ejpam-702	136	4	able	able	ADJ
ejpam-702	136	5	to	to	PART
ejpam-702	136	6	correct	correct	VERB
ejpam-702	136	7	any	any	DET
ejpam-702	136	8	error	error	NOUN
ejpam-702	136	9	pattern	pattern	NOUN
ejpam-702	136	10	of	of	ADP
ejpam-702	136	11	the	the	DET
ejpam-702	136	12	form	form	NOUN
ejpam-702	136	13	e(x	e(x	NUM
ejpam-702	136	14	)	)	PUNCT
ejpam-702	137	1	=	=	PUNCT
ejpam-702	138	1	ei	ei	NOUN
ejpam-702	138	2	x	x	PUNCT
ejpam-702	139	1	i	i	PRON
ejpam-702	139	2	+	+	CCONJ
ejpam-702	139	3	e	e	PROPN
ejpam-702	139	4	j	j	PROPN
ejpam-702	139	5	x	x	SYM
ejpam-702	139	6	j	j	PROPN
ejpam-702	139	7	,	,	PUNCT
ejpam-702	139	8	where	where	SCONJ
ejpam-702	139	9	0≤	0≤	NUM
ejpam-702	139	10	wqm	wqm	NOUN
ejpam-702	139	11	(	(	PUNCT
ejpam-702	139	12	ei	ei	NOUN
ejpam-702	139	13	)	)	PUNCT
ejpam-702	139	14	,	,	PUNCT
ejpam-702	139	15	wqm	wqm	NOUN
ejpam-702	139	16	(	(	PUNCT
ejpam-702	139	17	e	e	NOUN
ejpam-702	139	18	j)≤	j)≤	NOUN
ejpam-702	139	19	1	1	X
ejpam-702	139	20	.	.	X
ejpam-702	139	21	proof	proof	NOUN
ejpam-702	139	22	.	.	PUNCT
ejpam-702	140	1	suppose	suppose	VERB
ejpam-702	140	2	that	that	SCONJ
ejpam-702	140	3	double	double	ADJ
ejpam-702	140	4	error	error	NOUN
ejpam-702	140	5	occurs	occur	VERB
ejpam-702	140	6	at	at	ADP
ejpam-702	140	7	two	two	NUM
ejpam-702	140	8	different	different	ADJ
ejpam-702	140	9	components	component	NOUN
ejpam-702	140	10	l1	l1	PROPN
ejpam-702	140	11	,	,	PUNCT
ejpam-702	140	12	l2	l2	NOUN
ejpam-702	140	13	of	of	ADP
ejpam-702	140	14	the	the	DET
ejpam-702	140	15	received	receive	VERB
ejpam-702	140	16	vector	vector	NOUN
ejpam-702	140	17	r	r	NOUN
ejpam-702	140	18	.	.	PUNCT
ejpam-702	141	1	let	let	VERB
ejpam-702	141	2	the	the	DET
ejpam-702	141	3	error	error	NOUN
ejpam-702	141	4	vectors	vector	NOUN
ejpam-702	141	5	be	be	VERB
ejpam-702	141	6	e1	e1	PROPN
ejpam-702	141	7	,	,	PUNCT
ejpam-702	141	8	e2	e2	PROPN
ejpam-702	141	9	,	,	PUNCT
ejpam-702	141	10	where	where	SCONJ
ejpam-702	141	11	0	0	NUM
ejpam-702	141	12	≤	≤	NUM
ejpam-702	141	13	wqm	wqm	NOUN
ejpam-702	141	14	(	(	PUNCT
ejpam-702	141	15	e1	e1	PROPN
ejpam-702	141	16	)	)	PUNCT
ejpam-702	141	17	,	,	PUNCT
ejpam-702	141	18	wqm	wqm	NOUN
ejpam-702	141	19	(	(	PUNCT
ejpam-702	141	20	e2	e2	PROPN
ejpam-702	141	21	)	)	PUNCT
ejpam-702	141	22	≤	≤	NOUN
ejpam-702	141	23	1	1	NUM
ejpam-702	141	24	,	,	PUNCT
ejpam-702	141	25	respectively	respectively	ADV
ejpam-702	141	26	.	.	PUNCT
ejpam-702	142	1	first	first	ADV
ejpam-702	142	2	we	we	PRON
ejpam-702	142	3	compute	compute	VERB
ejpam-702	142	4	the	the	DET
ejpam-702	142	5	syndrome	syndrome	NOUN
ejpam-702	142	6	s	s	VERB
ejpam-702	142	7	of	of	ADP
ejpam-702	142	8	r	r	NOUN
ejpam-702	142	9	:	:	PUNCT
ejpam-702	142	10	s(r	s(r	X
ejpam-702	142	11	)	)	PUNCT
ejpam-702	143	1	=	=	SYM
ejpam-702	143	2	hr	hr	NOUN
ejpam-702	143	3	t	t	NOUN
ejpam-702	143	4	r	r	NOUN
ejpam-702	143	5	=	=	SYM
ejpam-702	143	6	�	�	PROPN
ejpam-702	143	7	s1	s1	PROPN
ejpam-702	143	8	s3	s3	PROPN
ejpam-702	143	9	�	�	PROPN
ejpam-702	143	10	.	.	PUNCT
ejpam-702	144	1	(	(	PUNCT
ejpam-702	144	2	4	4	X
ejpam-702	144	3	)	)	PUNCT
ejpam-702	144	4	the	the	DET
ejpam-702	144	5	polynomial	polynomial	ADJ
ejpam-702	144	6	σ(z	σ(z	PROPN
ejpam-702	144	7	)	)	PUNCT
ejpam-702	144	8	,	,	PUNCT
ejpam-702	144	9	which	which	PRON
ejpam-702	144	10	helps	help	VERB
ejpam-702	144	11	us	we	PRON
ejpam-702	144	12	to	to	PART
ejpam-702	144	13	find	find	VERB
ejpam-702	144	14	the	the	DET
ejpam-702	144	15	errors	error	NOUN
ejpam-702	144	16	location	location	NOUN
ejpam-702	144	17	and	and	CCONJ
ejpam-702	144	18	the	the	DET
ejpam-702	144	19	value	value	NOUN
ejpam-702	144	20	of	of	ADP
ejpam-702	144	21	the	the	DET
ejpam-702	144	22	errors	error	NOUN
ejpam-702	144	23	,	,	PUNCT
ejpam-702	144	24	is	be	AUX
ejpam-702	144	25	computed	compute	VERB
ejpam-702	144	26	as	as	SCONJ
ejpam-702	144	27	follows	follow	VERB
ejpam-702	144	28	.	.	PUNCT
ejpam-702	145	1	σ(z	σ(z	NOUN
ejpam-702	145	2	)	)	PUNCT
ejpam-702	145	3	=	=	PUNCT
ejpam-702	146	1	(	(	PUNCT
ejpam-702	146	2	z	z	NOUN
ejpam-702	146	3	−	−	NOUN
ejpam-702	146	4	β	β	NOUN
ejpam-702	146	5	l1)(z−	l1)(z−	NOUN
ejpam-702	146	6	β	β	X
ejpam-702	146	7	l2	l2	NOUN
ejpam-702	146	8	)	)	PUNCT
ejpam-702	146	9	=	=	SYM
ejpam-702	146	10	z2	z2	PROPN
ejpam-702	146	11	−	−	PROPN
ejpam-702	146	12	(	(	PUNCT
ejpam-702	146	13	β	β	X
ejpam-702	146	14	l1	l1	PROPN
ejpam-702	146	15	+	+	CCONJ
ejpam-702	146	16	β	β	X
ejpam-702	146	17	l2)z	l2)z	NOUN
ejpam-702	146	18	+	+	CCONJ
ejpam-702	146	19	β	β	X
ejpam-702	146	20	l1.β	l1.β	ADJ
ejpam-702	146	21	l2	l2	NOUN
ejpam-702	146	22	=	=	SYM
ejpam-702	146	23	z2	z2	PROPN
ejpam-702	146	24	−	−	PROPN
ejpam-702	147	1	(	(	PUNCT
ejpam-702	147	2	s1)z+	s1)z+	NOUN
ejpam-702	147	3	ǫ	ǫ	X
ejpam-702	147	4	,	,	PUNCT
ejpam-702	147	5	(	(	PUNCT
ejpam-702	147	6	5	5	X
ejpam-702	147	7	)	)	PUNCT
ejpam-702	147	8	m.	m.	NOUN
ejpam-702	147	9	özen	özen	NOUN
ejpam-702	147	10	,	,	PUNCT
ejpam-702	147	11	m.	m.	NOUN
ejpam-702	147	12	güzeltepe	güzeltepe	PROPN
ejpam-702	147	13	/	/	SYM
ejpam-702	147	14	eur	eur	NOUN
ejpam-702	147	15	.	.	PUNCT
ejpam-702	148	1	j.	j.	PROPN
ejpam-702	148	2	pure	pure	PROPN
ejpam-702	148	3	appl	appl	PROPN
ejpam-702	148	4	.	.	PROPN
ejpam-702	148	5	math	math	PROPN
ejpam-702	148	6	,	,	PUNCT
ejpam-702	148	7	3	3	NUM
ejpam-702	148	8	(	(	PUNCT
ejpam-702	148	9	2010	2010	NUM
ejpam-702	148	10	)	)	PUNCT
ejpam-702	148	11	,	,	PUNCT
ejpam-702	148	12	670	670	NUM
ejpam-702	148	13	-	-	SYM
ejpam-702	148	14	677	677	NUM
ejpam-702	148	15	675	675	NUM
ejpam-702	148	16	where	where	SCONJ
ejpam-702	148	17	ǫ	ǫ	PRON
ejpam-702	148	18	is	be	AUX
ejpam-702	148	19	determined	determine	VERB
ejpam-702	148	20	from	from	ADP
ejpam-702	148	21	the	the	DET
ejpam-702	148	22	syndromes	syndrome	NOUN
ejpam-702	148	23	.	.	PUNCT
ejpam-702	149	1	from	from	ADP
ejpam-702	149	2	s1	s1	NOUN
ejpam-702	149	3	=	=	PUNCT
ejpam-702	149	4	β	β	X
ejpam-702	149	5	l1+β	l1+β	PROPN
ejpam-702	149	6	l2	l2	NOUN
ejpam-702	149	7	,	,	PUNCT
ejpam-702	149	8	s3	s3	PROPN
ejpam-702	149	9	=	=	PUNCT
ejpam-702	149	10	β	β	NOUN
ejpam-702	149	11	3l1+β3l2	3l1+β3l2	NUM
ejpam-702	149	12	and	and	CCONJ
ejpam-702	149	13	ǫ	ǫ	PRON
ejpam-702	149	14	=	=	NOUN
ejpam-702	149	15	β	β	X
ejpam-702	149	16	l1+l2	l1+l2	NOUN
ejpam-702	149	17	we	we	PRON
ejpam-702	149	18	get	get	VERB
ejpam-702	149	19	s3	s3	PROPN
ejpam-702	149	20	1	1	NUM
ejpam-702	149	21	−	−	PROPN
ejpam-702	149	22	s3	s3	PROPN
ejpam-702	149	23	=	=	PROPN
ejpam-702	149	24	3ǫβ	3ǫβ	PROPN
ejpam-702	149	25	l1	l1	PROPN
ejpam-702	149	26	+	+	CCONJ
ejpam-702	149	27	3ǫβ	3ǫβ	ADJ
ejpam-702	149	28	l2	l2	NOUN
ejpam-702	149	29	+	+	CCONJ
ejpam-702	149	30	β3l2	β3l2	PUNCT
ejpam-702	149	31	+	+	NUM
ejpam-702	149	32	β3l1	β3l1	ADP
ejpam-702	149	33	−	−	PROPN
ejpam-702	149	34	(	(	PUNCT
ejpam-702	149	35	β3l2	β3l2	SYM
ejpam-702	149	36	+	+	NUM
ejpam-702	149	37	β3l1	β3l1	X
ejpam-702	149	38	)	)	PUNCT
ejpam-702	149	39	=	=	SYM
ejpam-702	149	40	3ǫ(β	3ǫ(β	NUM
ejpam-702	149	41	l1	l1	PROPN
ejpam-702	149	42	+	+	CCONJ
ejpam-702	149	43	β	β	X
ejpam-702	149	44	l2	l2	NOUN
ejpam-702	149	45	)	)	PUNCT
ejpam-702	149	46	(	(	PUNCT
ejpam-702	149	47	6	6	NUM
ejpam-702	149	48	)	)	PUNCT
ejpam-702	149	49	from	from	ADP
ejpam-702	149	50	which	which	PRON
ejpam-702	149	51	we	we	PRON
ejpam-702	149	52	obtain	obtain	VERB
ejpam-702	149	53	s3	s3	PROPN
ejpam-702	149	54	1	1	NUM
ejpam-702	149	55	−	−	PROPN
ejpam-702	149	56	s3	s3	PROPN
ejpam-702	149	57	3s1	3s1	NUM
ejpam-702	149	58	=	=	SYM
ejpam-702	149	59	3ǫ(β	3ǫ(β	NUM
ejpam-702	149	60	l1	l1	NOUN
ejpam-702	149	61	+	+	CCONJ
ejpam-702	149	62	β	β	X
ejpam-702	149	63	l2	l2	NOUN
ejpam-702	149	64	)	)	PUNCT
ejpam-702	149	65	3(β	3(β	NUM
ejpam-702	149	66	l1	l1	NOUN
ejpam-702	149	67	+	+	CCONJ
ejpam-702	149	68	β	β	X
ejpam-702	149	69	l2	l2	NOUN
ejpam-702	149	70	)	)	PUNCT
ejpam-702	150	1	=	=	SYM
ejpam-702	150	2	ǫ	ǫ	PROPN
ejpam-702	150	3	(	(	PUNCT
ejpam-702	150	4	modπ	modπ	NOUN
ejpam-702	150	5	)	)	PUNCT
ejpam-702	150	6	.	.	PUNCT
ejpam-702	151	1	(	(	PUNCT
ejpam-702	151	2	7	7	NUM
ejpam-702	151	3	)	)	PUNCT
ejpam-702	151	4	thus	thus	ADV
ejpam-702	151	5	,	,	PUNCT
ejpam-702	151	6	the	the	DET
ejpam-702	151	7	roots	root	NOUN
ejpam-702	151	8	of	of	ADP
ejpam-702	151	9	the	the	DET
ejpam-702	151	10	polynomial	polynomial	ADJ
ejpam-702	151	11	σ(z	σ(z	NOUN
ejpam-702	151	12	)	)	PUNCT
ejpam-702	151	13	lead	lead	VERB
ejpam-702	151	14	us	we	PRON
ejpam-702	151	15	to	to	PART
ejpam-702	151	16	find	find	VERB
ejpam-702	151	17	the	the	DET
ejpam-702	151	18	locations	location	NOUN
ejpam-702	151	19	of	of	ADP
ejpam-702	151	20	the	the	DET
ejpam-702	151	21	errors	error	NOUN
ejpam-702	151	22	and	and	CCONJ
ejpam-702	151	23	their	their	PRON
ejpam-702	151	24	values	value	NOUN
ejpam-702	151	25	.	.	PUNCT
ejpam-702	152	1	if	if	SCONJ
ejpam-702	152	2	β	β	PROPN
ejpam-702	152	3	l1	l1	PROPN
ejpam-702	152	4	1	1	NUM
ejpam-702	152	5	and	and	CCONJ
ejpam-702	152	6	β	β	X
ejpam-702	152	7	l2	l2	NOUN
ejpam-702	152	8	2	2	NUM
ejpam-702	152	9	are	be	AUX
ejpam-702	152	10	the	the	DET
ejpam-702	152	11	roots	root	NOUN
ejpam-702	152	12	of	of	ADP
ejpam-702	152	13	the	the	DET
ejpam-702	152	14	polynomial	polynomial	ADJ
ejpam-702	152	15	σ(z	σ(z	NOUN
ejpam-702	152	16	)	)	PUNCT
ejpam-702	152	17	,	,	PUNCT
ejpam-702	152	18	then	then	ADV
ejpam-702	152	19	l1	l1	PROPN
ejpam-702	152	20	and	and	CCONJ
ejpam-702	152	21	l2	l2	PROPN
ejpam-702	152	22	(	(	PUNCT
ejpam-702	152	23	mod	mod	NOUN
ejpam-702	152	24	n	n	CCONJ
ejpam-702	152	25	)	)	PUNCT
ejpam-702	152	26	are	be	AUX
ejpam-702	152	27	locations	location	NOUN
ejpam-702	152	28	of	of	ADP
ejpam-702	152	29	the	the	DET
ejpam-702	152	30	errors	error	NOUN
ejpam-702	152	31	.	.	PUNCT
ejpam-702	153	1	thus	thus	ADV
ejpam-702	153	2	,	,	PUNCT
ejpam-702	153	3	we	we	PRON
ejpam-702	153	4	can	can	AUX
ejpam-702	153	5	distinguish	distinguish	VERB
ejpam-702	153	6	three	three	NUM
ejpam-702	153	7	situations	situation	NOUN
ejpam-702	153	8	:	:	PUNCT
ejpam-702	153	9	no	no	DET
ejpam-702	153	10	error	error	NOUN
ejpam-702	153	11	,	,	PUNCT
ejpam-702	153	12	single	single	ADJ
ejpam-702	153	13	error	error	NOUN
ejpam-702	153	14	,	,	PUNCT
ejpam-702	153	15	and	and	CCONJ
ejpam-702	153	16	two	two	NUM
ejpam-702	153	17	errors	error	NOUN
ejpam-702	153	18	.	.	PUNCT
ejpam-702	154	1	i	i	PRON
ejpam-702	154	2	)	)	PUNCT
ejpam-702	154	3	no	no	DET
ejpam-702	154	4	error	error	NOUN
ejpam-702	154	5	:	:	PUNCT
ejpam-702	154	6	s1	s1	NOUN
ejpam-702	154	7	=	=	SYM
ejpam-702	154	8	s3	s3	PROPN
ejpam-702	154	9	=	=	SYM
ejpam-702	154	10	0	0	NUM
ejpam-702	154	11	,	,	PUNCT
ejpam-702	154	12	ı8tem[ii	ı8tem[ii	PROPN
ejpam-702	154	13	)	)	PUNCT
ejpam-702	154	14	]	]	PUNCT
ejpam-702	155	1	one	one	NUM
ejpam-702	155	2	error	error	NOUN
ejpam-702	155	3	:	:	PUNCT
ejpam-702	155	4	s3	s3	PROPN
ejpam-702	155	5	1	1	NUM
ejpam-702	155	6	=	=	SYM
ejpam-702	155	7	s3	s3	PROPN
ejpam-702	155	8	6=	6=	PROPN
ejpam-702	155	9	0	0	NUM
ejpam-702	155	10	,	,	PUNCT
ejpam-702	155	11	iii	iii	X
ejpam-702	155	12	)	)	PUNCT
ejpam-702	155	13	two	two	NUM
ejpam-702	155	14	errors	error	NOUN
ejpam-702	155	15	:	:	PUNCT
ejpam-702	155	16	s3	s3	PROPN
ejpam-702	155	17	1	1	NUM
ejpam-702	155	18	6=	6=	ADP
ejpam-702	155	19	s3	s3	PROPN
ejpam-702	155	20	and	and	CCONJ
ejpam-702	155	21	s1	s1	PROPN
ejpam-702	155	22	6=	6=	ADP
ejpam-702	155	23	0	0	X
ejpam-702	155	24	.	.	PUNCT
ejpam-702	156	1	we	we	PRON
ejpam-702	156	2	illustrate	illustrate	VERB
ejpam-702	156	3	the	the	DET
ejpam-702	156	4	decoding	decode	VERB
ejpam-702	156	5	procedure	procedure	NOUN
ejpam-702	156	6	with	with	ADP
ejpam-702	156	7	the	the	DET
ejpam-702	156	8	following	follow	VERB
ejpam-702	156	9	example	example	NOUN
ejpam-702	156	10	.	.	PUNCT
ejpam-702	157	1	example	example	NOUN
ejpam-702	158	1	2	2	NUM
ejpam-702	158	2	.	.	PUNCT
ejpam-702	158	3	let	let	VERB
ejpam-702	158	4	π	π	NOUN
ejpam-702	158	5	=	=	SYM
ejpam-702	158	6	1	1	NUM
ejpam-702	158	7	+	+	NUM
ejpam-702	158	8	2i	2i	NUM
ejpam-702	158	9	+	+	CCONJ
ejpam-702	158	10	2	2	NUM
ejpam-702	158	11	j	j	NOUN
ejpam-702	158	12	+	+	CCONJ
ejpam-702	158	13	2k	2k	NUM
ejpam-702	158	14	and	and	CCONJ
ejpam-702	158	15	let	let	VERB
ejpam-702	158	16	β	β	X
ejpam-702	158	17	=	=	SYM
ejpam-702	158	18	2	2	X
ejpam-702	158	19	.	.	PUNCT
ejpam-702	159	1	let	let	VERB
ejpam-702	159	2	c	c	NOUN
ejpam-702	159	3	be	be	AUX
ejpam-702	159	4	the	the	DET
ejpam-702	159	5	code	code	NOUN
ejpam-702	159	6	defined	define	VERB
ejpam-702	159	7	by	by	ADP
ejpam-702	159	8	the	the	DET
ejpam-702	159	9	parity	parity	NOUN
ejpam-702	159	10	check	check	NOUN
ejpam-702	159	11	matrix	matrix	NOUN
ejpam-702	159	12	h	h	NOUN
ejpam-702	159	13	=	=	PUNCT
ejpam-702	159	14	�	�	PROPN
ejpam-702	159	15	1	1	NUM
ejpam-702	159	16	,	,	PUNCT
ejpam-702	159	17	2	2	NUM
ejpam-702	159	18	,	,	PUNCT
ejpam-702	159	19	−2	−2	NOUN
ejpam-702	160	1	+	+	CCONJ
ejpam-702	160	2	i	i	PRON
ejpam-702	160	3	+	+	CCONJ
ejpam-702	160	4	j+	j+	NUM
ejpam-702	160	5	k	k	PROPN
ejpam-702	160	6	,	,	PUNCT
ejpam-702	160	7	1−	1−	NUM
ejpam-702	161	1	i	i	PRON
ejpam-702	161	2	−	−	PROPN
ejpam-702	161	3	j−	j−	PROPN
ejpam-702	161	4	k	k	PROPN
ejpam-702	161	5	,	,	PUNCT
ejpam-702	161	6	3	3	NUM
ejpam-702	161	7	,	,	PUNCT
ejpam-702	161	8	i	i	PRON
ejpam-702	161	9	+	+	CCONJ
ejpam-702	162	1	j+	j+	NUM
ejpam-702	162	2	k	k	PROPN
ejpam-702	162	3	1	1	NUM
ejpam-702	162	4	,	,	PUNCT
ejpam-702	162	5	1−	1−	NUM
ejpam-702	162	6	i−	i−	PROPN
ejpam-702	162	7	j−	j−	PROPN
ejpam-702	162	8	k	k	PROPN
ejpam-702	162	9	,	,	PUNCT
ejpam-702	162	10	−1	−1	NOUN
ejpam-702	162	11	,	,	PUNCT
ejpam-702	162	12	−1	−1	PROPN
ejpam-702	162	13	+	+	SYM
ejpam-702	162	14	i+	i+	NUM
ejpam-702	162	15	j+	j+	NUM
ejpam-702	162	16	k	k	PROPN
ejpam-702	162	17	,	,	PUNCT
ejpam-702	162	18	1	1	NUM
ejpam-702	162	19	,	,	PUNCT
ejpam-702	162	20	1−	1−	NUM
ejpam-702	163	1	i	i	PRON
ejpam-702	163	2	−	−	PROPN
ejpam-702	163	3	j−	j−	PROPN
ejpam-702	163	4	k	k	PROPN
ejpam-702	163	5	�	�	PROPN
ejpam-702	163	6	.	.	PUNCT
ejpam-702	163	7	suppose	suppose	VERB
ejpam-702	163	8	that	that	SCONJ
ejpam-702	163	9	the	the	DET
ejpam-702	163	10	received	receive	VERB
ejpam-702	163	11	vector	vector	NOUN
ejpam-702	163	12	is	be	AUX
ejpam-702	163	13	r	r	NOUN
ejpam-702	163	14	=	=	SYM
ejpam-702	163	15	�	�	PROPN
ejpam-702	163	16	3	3	NUM
ejpam-702	163	17	,	,	PUNCT
ejpam-702	163	18	3	3	NUM
ejpam-702	163	19	,	,	PUNCT
ejpam-702	163	20	1	1	NUM
ejpam-702	163	21	,	,	PUNCT
ejpam-702	163	22	0	0	NUM
ejpam-702	163	23	,	,	PUNCT
ejpam-702	163	24	−1	−1	NOUN
ejpam-702	163	25	,	,	PUNCT
ejpam-702	163	26	1	1	NUM
ejpam-702	163	27	�	�	PROPN
ejpam-702	163	28	.	.	PUNCT
ejpam-702	164	1	we	we	PRON
ejpam-702	164	2	now	now	ADV
ejpam-702	164	3	apply	apply	VERB
ejpam-702	164	4	the	the	DET
ejpam-702	164	5	decoding	decode	VERB
ejpam-702	164	6	procedure	procedure	NOUN
ejpam-702	164	7	for	for	ADP
ejpam-702	164	8	the	the	DET
ejpam-702	164	9	code	code	NOUN
ejpam-702	164	10	in	in	ADP
ejpam-702	164	11	theorem	theorem	NOUN
ejpam-702	164	12	2	2	NUM
ejpam-702	164	13	.	.	NOUN
ejpam-702	164	14	1	1	NUM
ejpam-702	164	15	)	)	PUNCT
ejpam-702	164	16	calculating	calculate	VERB
ejpam-702	164	17	the	the	DET
ejpam-702	164	18	syndrome	syndrome	NOUN
ejpam-702	164	19	:	:	PUNCT
ejpam-702	164	20	s	s	X
ejpam-702	164	21	=	=	PUNCT
ejpam-702	164	22	hr	hr	NOUN
ejpam-702	164	23	t	t	NOUN
ejpam-702	164	24	r	r	NOUN
ejpam-702	164	25	=	=	SYM
ejpam-702	164	26	�	�	PROPN
ejpam-702	164	27	s1	s1	PROPN
ejpam-702	164	28	s3	s3	PROPN
ejpam-702	164	29	�	�	PROPN
ejpam-702	164	30	=	=	SYM
ejpam-702	164	31	�	�	PROPN
ejpam-702	164	32	3	3	NUM
ejpam-702	164	33	−i	−i	NOUN
ejpam-702	164	34	−	−	PROPN
ejpam-702	164	35	j−	j−	PROPN
ejpam-702	165	1	k	k	PROPN
ejpam-702	165	2	�	�	PROPN
ejpam-702	165	3	mod	mod	PROPN
ejpam-702	165	4	π	π	PROPN
ejpam-702	165	5	.	.	PUNCT
ejpam-702	166	1	one	one	PRON
ejpam-702	166	2	can	can	AUX
ejpam-702	166	3	verify	verify	VERB
ejpam-702	166	4	that	that	DET
ejpam-702	166	5	s3	s3	PROPN
ejpam-702	166	6	6=	6=	PROPN
ejpam-702	166	7	s3	s3	PROPN
ejpam-702	166	8	1	1	NUM
ejpam-702	166	9	,	,	PUNCT
ejpam-702	166	10	which	which	PRON
ejpam-702	166	11	shows	show	VERB
ejpam-702	166	12	that	that	SCONJ
ejpam-702	166	13	two	two	NUM
ejpam-702	166	14	errors	error	NOUN
ejpam-702	166	15	have	have	AUX
ejpam-702	166	16	occurred	occur	VERB
ejpam-702	166	17	.	.	PUNCT
ejpam-702	167	1	2	2	X
ejpam-702	167	2	)	)	PUNCT
ejpam-702	167	3	calculating	calculate	VERB
ejpam-702	167	4	the	the	DET
ejpam-702	167	5	error	error	NOUN
ejpam-702	167	6	locations	location	NOUN
ejpam-702	167	7	and	and	CCONJ
ejpam-702	167	8	the	the	DET
ejpam-702	167	9	value	value	NOUN
ejpam-702	167	10	of	of	ADP
ejpam-702	167	11	errors	error	NOUN
ejpam-702	167	12	:	:	PUNCT
ejpam-702	167	13	using	use	VERB
ejpam-702	167	14	the	the	DET
ejpam-702	167	15	formulas	formula	NOUN
ejpam-702	167	16	(	(	PUNCT
ejpam-702	167	17	6	6	NUM
ejpam-702	167	18	)	)	PUNCT
ejpam-702	167	19	and	and	CCONJ
ejpam-702	167	20	(	(	PUNCT
ejpam-702	167	21	7	7	NUM
ejpam-702	167	22	)	)	PUNCT
ejpam-702	167	23	,	,	PUNCT
ejpam-702	167	24	we	we	PRON
ejpam-702	167	25	obtain	obtain	VERB
ejpam-702	167	26	ǫ	ǫ	PRON
ejpam-702	167	27	≡	≡	PROPN
ejpam-702	167	28	1−	1−	NUM
ejpam-702	167	29	i−	i−	PROPN
ejpam-702	167	30	j−k	j−k	NOUN
ejpam-702	167	31	(	(	PUNCT
ejpam-702	167	32	mod	mod	PROPN
ejpam-702	167	33	π	π	PROPN
ejpam-702	167	34	)	)	PUNCT
ejpam-702	167	35	and	and	CCONJ
ejpam-702	167	36	the	the	DET
ejpam-702	167	37	roots	root	NOUN
ejpam-702	167	38	of	of	ADP
ejpam-702	167	39	the	the	DET
ejpam-702	167	40	polynomial	polynomial	ADJ
ejpam-702	167	41	σ(z	σ(z	NOUN
ejpam-702	167	42	)	)	PUNCT
ejpam-702	167	43	are	be	AUX
ejpam-702	167	44	β5,β10	β5,β10	PRON
ejpam-702	167	45	(	(	PUNCT
ejpam-702	167	46	see	see	VERB
ejpam-702	167	47	table	table	NOUN
ejpam-702	167	48	3	3	NUM
ejpam-702	167	49	)	)	PUNCT
ejpam-702	167	50	.	.	PUNCT
ejpam-702	168	1	therefore	therefore	ADV
ejpam-702	168	2	,	,	PUNCT
ejpam-702	168	3	the	the	DET
ejpam-702	168	4	locations	location	NOUN
ejpam-702	168	5	of	of	ADP
ejpam-702	168	6	the	the	DET
ejpam-702	168	7	errors	error	NOUN
ejpam-702	168	8	are	be	AUX
ejpam-702	168	9	l1	l1	PROPN
ejpam-702	168	10	=	=	PROPN
ejpam-702	169	1	5≡	5≡	NUM
ejpam-702	169	2	5	5	NUM
ejpam-702	169	3	(	(	PUNCT
ejpam-702	169	4	mod	mod	PROPN
ejpam-702	169	5	6	6	NUM
ejpam-702	169	6	)	)	PUNCT
ejpam-702	169	7	and	and	CCONJ
ejpam-702	169	8	its	its	PRON
ejpam-702	169	9	value	value	NOUN
ejpam-702	169	10	is	be	AUX
ejpam-702	169	11	(	(	PUNCT
ejpam-702	169	12	β5	β5	PROPN
ejpam-702	169	13	à	à	X
ejpam-702	169	14	β5	β5	PROPN
ejpam-702	169	15	)	)	PUNCT
ejpam-702	169	16	=	=	SYM
ejpam-702	169	17	1	1	NUM
ejpam-702	169	18	and	and	CCONJ
ejpam-702	169	19	l2	l2	NOUN
ejpam-702	169	20	=	=	SYM
ejpam-702	169	21	10≡	10≡	NUM
ejpam-702	169	22	4	4	NUM
ejpam-702	169	23	(	(	PUNCT
ejpam-702	169	24	mod	mod	PROPN
ejpam-702	169	25	6	6	NUM
ejpam-702	169	26	)	)	PUNCT
ejpam-702	169	27	and	and	CCONJ
ejpam-702	169	28	its	its	PRON
ejpam-702	169	29	value	value	NOUN
ejpam-702	169	30	is	be	AUX
ejpam-702	169	31	(	(	PUNCT
ejpam-702	169	32	β10	β10	NOUN
ejpam-702	169	33	à	à	X
ejpam-702	169	34	β4	β4	PROPN
ejpam-702	169	35	)	)	PUNCT
ejpam-702	169	36	=	=	SYM
ejpam-702	169	37	−1	−1	NOUN
ejpam-702	169	38	.	.	PUNCT
ejpam-702	170	1	thus	thus	ADV
ejpam-702	170	2	,	,	PUNCT
ejpam-702	170	3	one	one	NUM
ejpam-702	170	4	error	error	NOUN
ejpam-702	170	5	has	have	AUX
ejpam-702	170	6	occurred	occur	VERB
ejpam-702	170	7	in	in	ADP
ejpam-702	170	8	location	location	NOUN
ejpam-702	170	9	6	6	NUM
ejpam-702	170	10	and	and	CCONJ
ejpam-702	170	11	its	its	PRON
ejpam-702	170	12	value	value	NOUN
ejpam-702	170	13	is	be	AUX
ejpam-702	170	14	1	1	NUM
ejpam-702	170	15	,	,	PUNCT
ejpam-702	170	16	and	and	CCONJ
ejpam-702	170	17	another	another	DET
ejpam-702	170	18	one	one	NOUN
ejpam-702	170	19	in	in	ADP
ejpam-702	170	20	location	location	NOUN
ejpam-702	170	21	5	5	NUM
ejpam-702	170	22	and	and	CCONJ
ejpam-702	170	23	its	its	PRON
ejpam-702	170	24	value	value	NOUN
ejpam-702	170	25	is	be	AUX
ejpam-702	170	26	-1	-1	PUNCT
ejpam-702	170	27	.	.	PUNCT
ejpam-702	171	1	theorem	theorem	NOUN
ejpam-702	171	2	3	3	NUM
ejpam-702	171	3	.	.	PUNCT
ejpam-702	172	1	the	the	DET
ejpam-702	172	2	code	code	NOUN
ejpam-702	172	3	defined	define	VERB
ejpam-702	172	4	by	by	ADP
ejpam-702	172	5	the	the	DET
ejpam-702	172	6	parity	parity	NOUN
ejpam-702	172	7	check	check	NOUN
ejpam-702	172	8	matrix	matrix	NOUN
ejpam-702	172	9	(	(	PUNCT
ejpam-702	172	10	3	3	X
ejpam-702	172	11	)	)	PUNCT
ejpam-702	172	12	has	have	VERB
ejpam-702	172	13	the	the	DET
ejpam-702	172	14	minimum	minimum	ADJ
ejpam-702	172	15	distance	distance	NOUN
ejpam-702	172	16	dqm	dqm	X
ejpam-702	172	17	≥	≥	NOUN
ejpam-702	172	18	4	4	NUM
ejpam-702	172	19	if	if	SCONJ
ejpam-702	172	20	π	π	PROPN
ejpam-702	172	21	is	be	AUX
ejpam-702	172	22	a	a	DET
ejpam-702	172	23	prime	prime	NOUN
ejpam-702	172	24	in	in	ADP
ejpam-702	172	25	h(z	h(z	NOUN
ejpam-702	172	26	)	)	PUNCT
ejpam-702	172	27	and	and	CCONJ
ejpam-702	172	28	p	p	NOUN
ejpam-702	172	29	is	be	AUX
ejpam-702	172	30	a	a	DET
ejpam-702	172	31	prime	prime	NOUN
ejpam-702	172	32	in	in	ADP
ejpam-702	172	33	z	z	PROPN
ejpam-702	172	34	,	,	PUNCT
ejpam-702	172	35	where	where	SCONJ
ejpam-702	172	36	p	p	PROPN
ejpam-702	172	37	≥	≥	NOUN
ejpam-702	172	38	13	13	NUM
ejpam-702	172	39	,	,	PUNCT
ejpam-702	172	40	p	p	NOUN
ejpam-702	172	41	=	=	SYM
ejpam-702	172	42	ππ	ππ	NOUN
ejpam-702	172	43	,	,	PUNCT
ejpam-702	172	44	and	and	CCONJ
ejpam-702	172	45	t=1	t=1	PROPN
ejpam-702	172	46	.	.	PUNCT
ejpam-702	172	47	references	reference	VERB
ejpam-702	172	48	676	676	NUM
ejpam-702	172	49	proof	proof	NOUN
ejpam-702	172	50	.	.	PUNCT
ejpam-702	173	1	it	it	PRON
ejpam-702	173	2	is	be	AUX
ejpam-702	173	3	sufficient	sufficient	ADJ
ejpam-702	173	4	to	to	PART
ejpam-702	173	5	show	show	VERB
ejpam-702	173	6	that	that	SCONJ
ejpam-702	173	7	the	the	DET
ejpam-702	173	8	decoder	decoder	NOUN
ejpam-702	173	9	can	can	AUX
ejpam-702	173	10	distinguish	distinguish	VERB
ejpam-702	173	11	single	single	ADJ
ejpam-702	173	12	and	and	CCONJ
ejpam-702	173	13	double	double	ADJ
ejpam-702	173	14	errors	error	NOUN
ejpam-702	173	15	for	for	ADP
ejpam-702	173	16	the	the	DET
ejpam-702	173	17	proof	proof	NOUN
ejpam-702	173	18	.	.	PUNCT
ejpam-702	174	1	suppose	suppose	VERB
ejpam-702	174	2	that	that	SCONJ
ejpam-702	174	3	an	an	DET
ejpam-702	174	4	error	error	NOUN
ejpam-702	174	5	of	of	ADP
ejpam-702	174	6	the	the	DET
ejpam-702	174	7	quaternion	quaternion	NOUN
ejpam-702	174	8	mannheim	mannheim	NOUN
ejpam-702	174	9	weight	weight	NOUN
ejpam-702	174	10	one	one	NOUN
ejpam-702	174	11	did	do	AUX
ejpam-702	174	12	occur	occur	VERB
ejpam-702	174	13	.	.	PUNCT
ejpam-702	175	1	then	then	ADV
ejpam-702	175	2	s3	s3	PROPN
ejpam-702	175	3	1	1	NUM
ejpam-702	175	4	=	=	SYM
ejpam-702	175	5	s3	s3	PROPN
ejpam-702	175	6	6=	6=	ADP
ejpam-702	175	7	0	0	NUM
ejpam-702	175	8	(	(	PUNCT
ejpam-702	175	9	modπ	modπ	NOUN
ejpam-702	175	10	)	)	PUNCT
ejpam-702	175	11	.	.	PUNCT
ejpam-702	176	1	from	from	ADP
ejpam-702	176	2	eq	eq	ADP
ejpam-702	176	3	.	.	PUNCT
ejpam-702	177	1	(	(	PUNCT
ejpam-702	177	2	5	5	NUM
ejpam-702	177	3	)	)	PUNCT
ejpam-702	177	4	,	,	PUNCT
ejpam-702	177	5	we	we	PRON
ejpam-702	177	6	get	get	VERB
ejpam-702	177	7	z1,2	z1,2	ADJ
ejpam-702	177	8	=	=	SYM
ejpam-702	177	9	s1	s1	PROPN
ejpam-702	177	10	±	±	PROPN
ejpam-702	177	11	q	q	PROPN
ejpam-702	177	12	s3	s3	PROPN
ejpam-702	177	13	s1	s1	NOUN
ejpam-702	177	14	2	2	NUM
ejpam-702	177	15	=	=	SYM
ejpam-702	177	16	s1	s1	PROPN
ejpam-702	177	17	±	±	NUM
ejpam-702	177	18	s1	s1	NOUN
ejpam-702	177	19	2	2	NUM
ejpam-702	177	20	.	.	PUNCT
ejpam-702	178	1	in	in	ADP
ejpam-702	178	2	view	view	NOUN
ejpam-702	178	3	of	of	ADP
ejpam-702	178	4	theorem	theorem	NOUN
ejpam-702	178	5	2	2	NUM
ejpam-702	178	6	,	,	PUNCT
ejpam-702	178	7	the	the	DET
ejpam-702	178	8	decoder	decoder	NOUN
ejpam-702	178	9	can	can	AUX
ejpam-702	178	10	distinguish	distinguish	VERB
ejpam-702	178	11	between	between	ADP
ejpam-702	178	12	single	single	ADJ
ejpam-702	178	13	and	and	CCONJ
ejpam-702	178	14	double	double	ADJ
ejpam-702	178	15	errors	error	NOUN
ejpam-702	178	16	.	.	PUNCT
ejpam-702	179	1	5	5	X
ejpam-702	179	2	.	.	X
ejpam-702	179	3	conclusion	conclusion	NOUN
ejpam-702	179	4	in	in	ADP
ejpam-702	179	5	this	this	DET
ejpam-702	179	6	paper	paper	NOUN
ejpam-702	179	7	,	,	PUNCT
ejpam-702	179	8	codes	code	NOUN
ejpam-702	179	9	over	over	ADP
ejpam-702	179	10	the	the	DET
ejpam-702	179	11	subring	subre	VERB
ejpam-702	179	12	r	r	NOUN
ejpam-702	179	13	of	of	ADP
ejpam-702	179	14	the	the	DET
ejpam-702	179	15	quaternion	quaternion	NOUN
ejpam-702	179	16	integer	integer	NOUN
ejpam-702	179	17	ring	ring	NOUN
ejpam-702	179	18	h(z	h(z	PROPN
ejpam-702	179	19	)	)	PUNCT
ejpam-702	179	20	are	be	AUX
ejpam-702	179	21	constructed	construct	VERB
ejpam-702	179	22	and	and	CCONJ
ejpam-702	179	23	decoding	decode	VERB
ejpam-702	179	24	algorithms	algorithm	NOUN
ejpam-702	179	25	of	of	ADP
ejpam-702	179	26	these	these	DET
ejpam-702	179	27	codes	code	NOUN
ejpam-702	179	28	are	be	AUX
ejpam-702	179	29	given	give	VERB
ejpam-702	179	30	.	.	PUNCT
ejpam-702	180	1	moreover	moreover	ADV
ejpam-702	180	2	,	,	PUNCT
ejpam-702	180	3	it	it	PRON
ejpam-702	180	4	is	be	AUX
ejpam-702	180	5	seen	see	VERB
ejpam-702	180	6	that	that	SCONJ
ejpam-702	180	7	all	all	PRON
ejpam-702	180	8	of	of	ADP
ejpam-702	180	9	the	the	DET
ejpam-702	180	10	one	one	NUM
ejpam-702	180	11	quaternion	quaternion	NOUN
ejpam-702	180	12	mannheim	mannheim	NOUN
ejpam-702	180	13	error	error	NOUN
ejpam-702	180	14	correcting	correct	VERB
ejpam-702	180	15	codes	code	NOUN
ejpam-702	180	16	are	be	AUX
ejpam-702	180	17	perfect	perfect	ADJ
ejpam-702	180	18	.	.	PUNCT
ejpam-702	181	1	furthermore	furthermore	ADV
ejpam-702	181	2	,	,	PUNCT
ejpam-702	181	3	these	these	DET
ejpam-702	181	4	codes	code	NOUN
ejpam-702	181	5	are	be	AUX
ejpam-702	181	6	constructed	construct	VERB
ejpam-702	181	7	using	use	VERB
ejpam-702	181	8	a	a	DET
ejpam-702	181	9	metric	metric	NOUN
ejpam-702	181	10	which	which	PRON
ejpam-702	181	11	is	be	AUX
ejpam-702	181	12	called	call	VERB
ejpam-702	181	13	quaternion	quaternion	PROPN
ejpam-702	181	14	mannheim	mannheim	NOUN
ejpam-702	181	15	metric	metric	NOUN
ejpam-702	181	16	.	.	PUNCT
ejpam-702	182	1	acknowledgements	acknowledgement	NOUN
ejpam-702	182	2	the	the	DET
ejpam-702	182	3	authors	author	NOUN
ejpam-702	182	4	would	would	AUX
ejpam-702	182	5	like	like	VERB
ejpam-702	182	6	to	to	PART
ejpam-702	182	7	thank	thank	VERB
ejpam-702	182	8	the	the	DET
ejpam-702	182	9	anonymous	anonymous	ADJ
ejpam-702	182	10	referees	referee	NOUN
ejpam-702	182	11	for	for	ADP
ejpam-702	182	12	their	their	PRON
ejpam-702	182	13	helpful	helpful	ADJ
ejpam-702	182	14	comments	comment	NOUN
ejpam-702	182	15	and	and	CCONJ
ejpam-702	182	16	suggestions	suggestion	NOUN
ejpam-702	182	17	.	.	PUNCT
ejpam-702	183	1	references	reference	NOUN
ejpam-702	183	2	[	[	X
ejpam-702	183	3	1	1	NUM
ejpam-702	183	4	]	]	X
ejpam-702	183	5	k	k	PROPN
ejpam-702	183	6	huber	huber	PROPN
ejpam-702	183	7	.	.	PUNCT
ejpam-702	184	1	codes	code	NOUN
ejpam-702	184	2	over	over	ADP
ejpam-702	184	3	gaussian	gaussian	ADJ
ejpam-702	184	4	integers	integer	NOUN
ejpam-702	184	5	,	,	PUNCT
ejpam-702	184	6	ieee	ieee	NOUN
ejpam-702	184	7	trans	tran	NOUN
ejpam-702	184	8	.	.	PUNCT
ejpam-702	185	1	inform.theory	inform.theory	ADJ
ejpam-702	185	2	,	,	PUNCT
ejpam-702	185	3	vol	vol	NOUN
ejpam-702	185	4	.	.	PROPN
ejpam-702	185	5	40	40	NUM
ejpam-702	185	6	,	,	PUNCT
ejpam-702	185	7	pp	pp	ADJ
ejpam-702	185	8	.	.	PUNCT
ejpam-702	186	1	207	207	NUM
ejpam-702	186	2	-	-	SYM
ejpam-702	186	3	216	216	NUM
ejpam-702	186	4	,	,	PUNCT
ejpam-702	186	5	jan	jan	PROPN
ejpam-702	186	6	.	.	PROPN
ejpam-702	186	7	1994	1994	NUM
ejpam-702	186	8	.	.	PUNCT
ejpam-702	187	1	[	[	X
ejpam-702	187	2	2	2	NUM
ejpam-702	187	3	]	]	PUNCT
ejpam-702	187	4	e	e	PROPN
ejpam-702	187	5	r	r	NOUN
ejpam-702	187	6	berlekamp	berlekamp	NOUN
ejpam-702	187	7	.	.	PUNCT
ejpam-702	188	1	algebraic	algebraic	ADJ
ejpam-702	188	2	coding	coding	PROPN
ejpam-702	188	3	theory	theory	NOUN
ejpam-702	188	4	,	,	PUNCT
ejpam-702	188	5	laguna	laguna	PROPN
ejpam-702	188	6	hills	hills	PROPN
ejpam-702	188	7	,	,	PUNCT
ejpam-702	188	8	ca	can	AUX
ejpam-702	188	9	:	:	PUNCT
ejpam-702	188	10	aegan	aegan	PROPN
ejpam-702	188	11	park	park	NOUN
ejpam-702	188	12	,	,	PUNCT
ejpam-702	188	13	1984	1984	NUM
ejpam-702	188	14	.	.	PUNCT
ejpam-702	189	1	[	[	X
ejpam-702	189	2	3	3	X
ejpam-702	189	3	]	]	X
ejpam-702	189	4	k	k	PROPN
ejpam-702	189	5	huber	huber	PROPN
ejpam-702	189	6	.	.	PUNCT
ejpam-702	190	1	codes	code	NOUN
ejpam-702	190	2	over	over	ADP
ejpam-702	190	3	eisenstein	eisenstein	PROPN
ejpam-702	190	4	-	-	PUNCT
ejpam-702	190	5	jacobi	jacobi	PROPN
ejpam-702	190	6	integers	integer	NOUN
ejpam-702	190	7	,	,	PUNCT
ejpam-702	190	8	ams	am	NOUN
ejpam-702	190	9	.	.	PUNCT
ejpam-702	191	1	contemp	contemp	NOUN
ejpam-702	191	2	.	.	PUNCT
ejpam-702	192	1	math	math	NOUN
ejpam-702	192	2	.	.	PUNCT
ejpam-702	193	1	,	,	PUNCT
ejpam-702	193	2	vol	vol	NOUN
ejpam-702	193	3	.	.	PROPN
ejpam-702	194	1	158	158	NUM
ejpam-702	194	2	,	,	PUNCT
ejpam-702	194	3	pp	pp	ADJ
ejpam-702	194	4	.	.	PUNCT
ejpam-702	195	1	165179	165179	NUM
ejpam-702	195	2	,	,	PUNCT
ejpam-702	195	3	1994	1994	NUM
ejpam-702	195	4	.	.	PUNCT
ejpam-702	196	1	[	[	X
ejpam-702	196	2	4	4	X
ejpam-702	196	3	]	]	PUNCT
ejpam-702	196	4	g	g	PROPN
ejpam-702	196	5	davidoff	davidoff	NOUN
ejpam-702	196	6	.	.	PUNCT
ejpam-702	197	1	p	p	X
ejpam-702	197	2	sarnak	sarnak	PROPN
ejpam-702	197	3	.	.	PUNCT
ejpam-702	198	1	and	and	CCONJ
ejpam-702	198	2	a	a	DET
ejpam-702	198	3	valette	valette	NOUN
ejpam-702	198	4	.	.	PUNCT
ejpam-702	199	1	elementary	elementary	ADJ
ejpam-702	199	2	number	number	NOUN
ejpam-702	199	3	theory	theory	NOUN
ejpam-702	199	4	,	,	PUNCT
ejpam-702	199	5	group	group	NOUN
ejpam-702	199	6	theory	theory	NOUN
ejpam-702	199	7	,	,	PUNCT
ejpam-702	199	8	and	and	CCONJ
ejpam-702	199	9	ramanujan	ramanujan	NOUN
ejpam-702	199	10	graphs	graph	NOUN
ejpam-702	199	11	,	,	PUNCT
ejpam-702	199	12	cambridge	cambridge	PROPN
ejpam-702	199	13	university	university	PROPN
ejpam-702	199	14	pres	pres	PROPN
ejpam-702	199	15	,	,	PUNCT
ejpam-702	199	16	2003	2003	NUM
ejpam-702	199	17	.	.	PUNCT
ejpam-702	200	1	appendix	appendix	NOUN
ejpam-702	200	2	references	reference	NOUN
ejpam-702	200	3	677table	677table	PROPN
ejpam-702	200	4	1	1	NUM
ejpam-702	200	5	:	:	PUNCT
ejpam-702	200	6	table	table	NOUN
ejpam-702	200	7	of	of	ADP
ejpam-702	200	8	p	p	X
ejpam-702	200	9	,	,	PUNCT
ejpam-702	200	10	π	π	PROPN
ejpam-702	200	11	,	,	PUNCT
ejpam-702	200	12	a	a	DET
ejpam-702	200	13	primitive	primitive	ADJ
ejpam-702	200	14	element	element	NOUN
ejpam-702	200	15	α	α	PROPN
ejpam-702	200	16	∈	∈	PROPN
ejpam-702	200	17	rπ	rπ	NOUN
ejpam-702	200	18	,	,	PUNCT
ejpam-702	200	19	u	u	NOUN
ejpam-702	200	20	and	and	CCONJ
ejpam-702	200	21	v	v	NOUN
ejpam-702	200	22	(	(	PUNCT
ejpam-702	200	23	where	where	SCONJ
ejpam-702	200	24	1=	1=	NUM
ejpam-702	200	25	uπ+	uπ+	ADJ
ejpam-702	200	26	vπ	vπ	PROPN
ejpam-702	200	27	)	)	PUNCT
ejpam-702	200	28	,	,	PUNCT
ejpam-702	200	29	for	for	ADP
ejpam-702	200	30	p	p	X
ejpam-702	200	31	<	<	X
ejpam-702	200	32	23	23	NUM
ejpam-702	200	33	.	.	PUNCT
ejpam-702	201	1	p	p	X
ejpam-702	201	2	π	π	PROPN
ejpam-702	201	3	α	α	X
ejpam-702	201	4	u	u	NOUN
ejpam-702	201	5	v	v	ADP
ejpam-702	201	6	7	7	NUM
ejpam-702	201	7	2	2	NUM
ejpam-702	201	8	+	+	NUM
ejpam-702	201	9	i	i	PRON
ejpam-702	201	10	+	+	CCONJ
ejpam-702	201	11	j+	j+	NUM
ejpam-702	201	12	k	k	PROPN
ejpam-702	201	13	1−	1−	NUM
ejpam-702	202	1	i	i	PRON
ejpam-702	202	2	−	−	PROPN
ejpam-702	203	1	j−	j−	PROPN
ejpam-702	203	2	k	k	PROPN
ejpam-702	203	3	i+	i+	NUM
ejpam-702	204	1	j+	j+	NUM
ejpam-702	204	2	k	k	NOUN
ejpam-702	204	3	2	2	NUM
ejpam-702	204	4	11	11	NUM
ejpam-702	204	5	3	3	NUM
ejpam-702	204	6	+	+	NUM
ejpam-702	204	7	i+	i+	NOUN
ejpam-702	204	8	j	j	PROPN
ejpam-702	204	9	−1−	−1−	PROPN
ejpam-702	205	1	i	i	PRON
ejpam-702	205	2	−	−	PROPN
ejpam-702	205	3	j	j	PROPN
ejpam-702	206	1	−1	−1	NOUN
ejpam-702	206	2	+	+	CCONJ
ejpam-702	207	1	i	i	PROPN
ejpam-702	207	2	+	+	NUM
ejpam-702	207	3	j	j	PROPN
ejpam-702	207	4	2	2	NUM
ejpam-702	207	5	13	13	NUM
ejpam-702	207	6	1	1	NUM
ejpam-702	207	7	+	+	NUM
ejpam-702	207	8	2i+	2i+	NUM
ejpam-702	207	9	2	2	NUM
ejpam-702	207	10	j+	j+	NUM
ejpam-702	207	11	2k	2k	NUM
ejpam-702	207	12	2	2	NUM
ejpam-702	207	13	i+	i+	NUM
ejpam-702	207	14	j+	j+	NUM
ejpam-702	207	15	k	k	PROPN
ejpam-702	207	16	1	1	NUM
ejpam-702	207	17	+	+	NUM
ejpam-702	207	18	i	i	PRON
ejpam-702	207	19	+	+	CCONJ
ejpam-702	208	1	j+	j+	NUM
ejpam-702	208	2	k	k	PROPN
ejpam-702	208	3	17	17	NUM
ejpam-702	208	4	3	3	NUM
ejpam-702	208	5	+	+	NUM
ejpam-702	208	6	2i+	2i+	NUM
ejpam-702	208	7	2	2	NUM
ejpam-702	208	8	j	j	NOUN
ejpam-702	209	1	i	i	PRON
ejpam-702	209	2	+	+	NUM
ejpam-702	209	3	j	j	PROPN
ejpam-702	209	4	-31	-31	NUM
ejpam-702	209	5	2	2	NUM
ejpam-702	209	6	+	+	NUM
ejpam-702	209	7	22i+	22i+	NUM
ejpam-702	209	8	22	22	NUM
ejpam-702	209	9	j	j	NOUN
ejpam-702	209	10	19	19	NUM
ejpam-702	209	11	4	4	NUM
ejpam-702	209	12	+	+	NUM
ejpam-702	209	13	i	i	PRON
ejpam-702	209	14	+	+	CCONJ
ejpam-702	210	1	j+	j+	NUM
ejpam-702	210	2	k	k	NOUN
ejpam-702	210	3	2	2	NUM
ejpam-702	210	4	3i+	3i+	NUM
ejpam-702	210	5	3	3	NUM
ejpam-702	210	6	j+	j+	NUM
ejpam-702	210	7	3k	3k	NUM
ejpam-702	211	1	4−	4−	NUM
ejpam-702	212	1	2i−	2i−	NUM
ejpam-702	212	2	2	2	NUM
ejpam-702	212	3	j−	j−	NOUN
ejpam-702	212	4	2k	2k	NUM
ejpam-702	212	5	table	table	NOUN
ejpam-702	212	6	2	2	NUM
ejpam-702	212	7	:	:	PUNCT
ejpam-702	212	8	powers	power	NOUN
ejpam-702	212	9	of	of	ADP
ejpam-702	212	10	the	the	DET
ejpam-702	212	11	element	element	NOUN
ejpam-702	212	12	α=	α=	NOUN
ejpam-702	212	13	1−	1−	NUM
ejpam-702	213	1	i	i	PRON
ejpam-702	213	2	−	−	PROPN
ejpam-702	213	3	j−	j−	VERB
ejpam-702	213	4	k	k	PROPN
ejpam-702	213	5	whi	whi	PROPN
ejpam-702	213	6	h	h	NOUN
ejpam-702	213	7	is	be	AUX
ejpam-702	213	8	a	a	DET
ejpam-702	213	9	root	root	NOUN
ejpam-702	213	10	of	of	ADP
ejpam-702	213	11	x3	x3	ADJ
ejpam-702	213	12	+	+	NOUN
ejpam-702	213	13	1	1	X
ejpam-702	213	14	.	.	X
ejpam-702	213	15	s	s	VERB
ejpam-702	213	16	αs	αs	INTJ
ejpam-702	213	17	s	s	NOUN
ejpam-702	213	18	αs	αs	INTJ
ejpam-702	213	19	0	0	NUM
ejpam-702	213	20	1	1	NUM
ejpam-702	213	21	4	4	NUM
ejpam-702	213	22	−1	−1	NOUN
ejpam-702	213	23	+	+	NUM
ejpam-702	213	24	i+	i+	NUM
ejpam-702	214	1	j+	j+	NUM
ejpam-702	214	2	k	k	NOUN
ejpam-702	214	3	1	1	NUM
ejpam-702	214	4	1−	1−	NUM
ejpam-702	214	5	i−	i−	PROPN
ejpam-702	214	6	j−	j−	PROPN
ejpam-702	214	7	k	k	PROPN
ejpam-702	214	8	5	5	NUM
ejpam-702	215	1	i	i	NOUN
ejpam-702	215	2	+	+	CCONJ
ejpam-702	216	1	j+	j+	NUM
ejpam-702	216	2	k	k	PROPN
ejpam-702	216	3	2	2	NUM
ejpam-702	216	4	−i	−i	NOUN
ejpam-702	216	5	−	−	PROPN
ejpam-702	216	6	j−	j−	VERB
ejpam-702	216	7	k	k	PROPN
ejpam-702	216	8	6	6	NUM
ejpam-702	216	9	1	1	NUM
ejpam-702	216	10	3	3	NUM
ejpam-702	216	11	-1	-1	SYM
ejpam-702	216	12	7	7	NUM
ejpam-702	216	13	1−	1−	NUM
ejpam-702	217	1	i	i	PRON
ejpam-702	217	2	−	−	PROPN
ejpam-702	217	3	j−	j−	VERB
ejpam-702	217	4	k	k	PROPN
ejpam-702	217	5	table	table	NOUN
ejpam-702	217	6	3	3	NUM
ejpam-702	217	7	:	:	PUNCT
ejpam-702	217	8	powers	power	NOUN
ejpam-702	217	9	of	of	ADP
ejpam-702	217	10	the	the	DET
ejpam-702	217	11	element	element	NOUN
ejpam-702	217	12	β	β	NOUN
ejpam-702	217	13	=	=	SYM
ejpam-702	217	14	2	2	NUM
ejpam-702	217	15	whi	whi	NOUN
ejpam-702	217	16	h	h	NOUN
ejpam-702	217	17	is	be	AUX
ejpam-702	217	18	a	a	DET
ejpam-702	217	19	root	root	NOUN
ejpam-702	217	20	of	of	ADP
ejpam-702	217	21	x6	x6	PROPN
ejpam-702	217	22	+	+	CCONJ
ejpam-702	217	23	1	1	NUM
ejpam-702	217	24	.	.	X
ejpam-702	217	25	s	s	PART
ejpam-702	217	26	β	β	X
ejpam-702	217	27	s	s	X
ejpam-702	217	28	s	s	X
ejpam-702	217	29	β	β	X
ejpam-702	217	30	s	s	X
ejpam-702	217	31	0	0	NUM
ejpam-702	217	32	1	1	NUM
ejpam-702	217	33	8	8	NUM
ejpam-702	217	34	2−	2−	NUM
ejpam-702	217	35	i	i	PRON
ejpam-702	217	36	−	−	PROPN
ejpam-702	218	1	j−	j−	VERB
ejpam-702	218	2	k	k	PROPN
ejpam-702	218	3	1	1	NUM
ejpam-702	218	4	2	2	NUM
ejpam-702	218	5	9	9	NUM
ejpam-702	218	6	−1	−1	NOUN
ejpam-702	219	1	+	+	NOUN
ejpam-702	219	2	i	i	PRON
ejpam-702	219	3	+	+	CCONJ
ejpam-702	219	4	j+	j+	NUM
ejpam-702	219	5	k	k	NOUN
ejpam-702	219	6	2	2	NUM
ejpam-702	219	7	−2	−2	NOUN
ejpam-702	220	1	+	+	CCONJ
ejpam-702	220	2	i	i	PRON
ejpam-702	220	3	+	+	CCONJ
ejpam-702	221	1	j+	j+	NUM
ejpam-702	221	2	k	k	PROPN
ejpam-702	221	3	10	10	NUM
ejpam-702	221	4	-3	-3	SYM
ejpam-702	221	5	3	3	NUM
ejpam-702	221	6	1−	1−	NUM
ejpam-702	222	1	i	i	PRON
ejpam-702	222	2	−	−	PROPN
ejpam-702	222	3	j−	j−	VERB
ejpam-702	222	4	k	k	PROPN
ejpam-702	222	5	11	11	NUM
ejpam-702	223	1	−i−	−i−	ADV
ejpam-702	223	2	j−	j−	PROPN
ejpam-702	223	3	k	k	PROPN
ejpam-702	223	4	4	4	NUM
ejpam-702	223	5	3	3	NUM
ejpam-702	223	6	12	12	NUM
ejpam-702	223	7	1	1	NUM
ejpam-702	223	8	5	5	NUM
ejpam-702	223	9	i+	i+	NUM
ejpam-702	223	10	j+	j+	NUM
ejpam-702	223	11	k	k	PROPN
ejpam-702	223	12	13	13	NUM
ejpam-702	223	13	2	2	NUM
ejpam-702	223	14	6	6	NUM
ejpam-702	223	15	-1	-1	SYM
ejpam-702	223	16	14	14	NUM
ejpam-702	223	17	−2	−2	NOUN
ejpam-702	224	1	+	+	CCONJ
ejpam-702	224	2	i	i	PRON
ejpam-702	224	3	+	+	CCONJ
ejpam-702	224	4	j+	j+	NUM
ejpam-702	224	5	k	k	PROPN
ejpam-702	224	6	7	7	NUM
ejpam-702	224	7	-2	-2	NOUN
ejpam-702	224	8	15	15	NUM
ejpam-702	224	9	1−	1−	NUM
ejpam-702	225	1	i	i	PRON
ejpam-702	225	2	−	−	PROPN
ejpam-702	225	3	j−	j−	VERB
ejpam-702	225	4	k	k	PROPN