id	sid	tid	token	lemma	pos
ejpam-1773	1	1	9_yildiz.dvi	9_yildiz.dvi	NUM
ejpam-1773	1	2	european	european	PROPN
ejpam-1773	1	3	journal	journal	PROPN
ejpam-1773	1	4	of	of	ADP
ejpam-1773	1	5	pure	pure	ADJ
ejpam-1773	1	6	and	and	CCONJ
ejpam-1773	1	7	applied	apply	VERB
ejpam-1773	1	8	mathematics	mathematic	NOUN
ejpam-1773	1	9	vol	vol	NOUN
ejpam-1773	1	10	.	.	PROPN
ejpam-1773	2	1	6	6	NUM
ejpam-1773	2	2	,	,	PUNCT
ejpam-1773	2	3	no	no	INTJ
ejpam-1773	2	4	.	.	NOUN
ejpam-1773	2	5	1	1	NUM
ejpam-1773	2	6	,	,	PUNCT
ejpam-1773	2	7	2013	2013	NUM
ejpam-1773	2	8	,	,	PUNCT
ejpam-1773	2	9	89	89	NUM
ejpam-1773	2	10	-	-	SYM
ejpam-1773	2	11	106	106	NUM
ejpam-1773	2	12	issn	issn	PROPN
ejpam-1773	2	13	1307	1307	NUM
ejpam-1773	2	14	-	-	SYM
ejpam-1773	2	15	5543	5543	NUM
ejpam-1773	2	16	–	–	PUNCT
ejpam-1773	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-1773	2	18	self	self	NOUN
ejpam-1773	2	19	-	-	PUNCT
ejpam-1773	2	20	dual	dual	ADJ
ejpam-1773	2	21	codes	code	NOUN
ejpam-1773	2	22	over	over	ADP
ejpam-1773	2	23	rk	rk	NOUN
ejpam-1773	2	24	and	and	CCONJ
ejpam-1773	2	25	binary	binary	NOUN
ejpam-1773	2	26	self	self	NOUN
ejpam-1773	2	27	-	-	PUNCT
ejpam-1773	2	28	dual	dual	ADJ
ejpam-1773	2	29	codes	code	NOUN
ejpam-1773	2	30	steven	steven	PROPN
ejpam-1773	2	31	dougherty1	dougherty1	PROPN
ejpam-1773	2	32	,	,	PUNCT
ejpam-1773	2	33	bahattin	bahattin	PROPN
ejpam-1773	2	34	yıldız2,∗	yıldız2,∗	PROPN
ejpam-1773	2	35	,	,	PUNCT
ejpam-1773	2	36	suat	suat	NOUN
ejpam-1773	2	37	karadeniz2	karadeniz2	X
ejpam-1773	2	38	1	1	NUM
ejpam-1773	2	39	department	department	NOUN
ejpam-1773	2	40	of	of	ADP
ejpam-1773	2	41	mathematics	mathematics	PROPN
ejpam-1773	2	42	,	,	PUNCT
ejpam-1773	2	43	scranton	scranton	PROPN
ejpam-1773	2	44	university	university	PROPN
ejpam-1773	2	45	,	,	PUNCT
ejpam-1773	2	46	scranton	scranton	PROPN
ejpam-1773	2	47	,	,	PUNCT
ejpam-1773	2	48	pa	pa	PROPN
ejpam-1773	2	49	,	,	PUNCT
ejpam-1773	2	50	usa	usa	PROPN
ejpam-1773	2	51	2	2	NUM
ejpam-1773	2	52	department	department	NOUN
ejpam-1773	2	53	of	of	ADP
ejpam-1773	2	54	mathematics	mathematics	PROPN
ejpam-1773	2	55	,	,	PUNCT
ejpam-1773	2	56	fatih	fatih	PROPN
ejpam-1773	2	57	university	university	PROPN
ejpam-1773	2	58	,	,	PUNCT
ejpam-1773	2	59	istanbul	istanbul	PROPN
ejpam-1773	2	60	,	,	PUNCT
ejpam-1773	2	61	turkey	turkey	PROPN
ejpam-1773	2	62	abstract	abstract	NOUN
ejpam-1773	2	63	.	.	PUNCT
ejpam-1773	3	1	we	we	PRON
ejpam-1773	3	2	study	study	VERB
ejpam-1773	3	3	self	self	NOUN
ejpam-1773	3	4	-	-	PUNCT
ejpam-1773	3	5	dual	dual	ADJ
ejpam-1773	3	6	codes	code	NOUN
ejpam-1773	3	7	over	over	ADP
ejpam-1773	3	8	an	an	DET
ejpam-1773	3	9	infinite	infinite	ADJ
ejpam-1773	3	10	family	family	NOUN
ejpam-1773	3	11	of	of	ADP
ejpam-1773	3	12	rings	ring	NOUN
ejpam-1773	3	13	,	,	PUNCT
ejpam-1773	3	14	denoted	denote	VERB
ejpam-1773	3	15	rk	rk	NOUN
ejpam-1773	3	16	,	,	PUNCT
ejpam-1773	3	17	which	which	PRON
ejpam-1773	3	18	has	have	AUX
ejpam-1773	3	19	been	be	AUX
ejpam-1773	3	20	recently	recently	ADV
ejpam-1773	3	21	introduced	introduce	VERB
ejpam-1773	3	22	to	to	ADP
ejpam-1773	3	23	the	the	DET
ejpam-1773	3	24	literature	literature	NOUN
ejpam-1773	3	25	.	.	PUNCT
ejpam-1773	4	1	we	we	PRON
ejpam-1773	4	2	prove	prove	VERB
ejpam-1773	4	3	that	that	SCONJ
ejpam-1773	4	4	for	for	ADP
ejpam-1773	4	5	each	each	DET
ejpam-1773	4	6	self	self	NOUN
ejpam-1773	4	7	-	-	PUNCT
ejpam-1773	4	8	dual	dual	ADJ
ejpam-1773	4	9	code	code	NOUN
ejpam-1773	4	10	over	over	ADP
ejpam-1773	4	11	rk	rk	PROPN
ejpam-1773	4	12	,	,	PUNCT
ejpam-1773	4	13	k	k	PROPN
ejpam-1773	4	14	≥	≥	NUM
ejpam-1773	4	15	2	2	NUM
ejpam-1773	4	16	,	,	PUNCT
ejpam-1773	4	17	there	there	PRON
ejpam-1773	4	18	exist	exist	VERB
ejpam-1773	4	19	a	a	DET
ejpam-1773	4	20	corresponding	correspond	VERB
ejpam-1773	4	21	binary	binary	ADJ
ejpam-1773	4	22	self	self	NOUN
ejpam-1773	4	23	-	-	PUNCT
ejpam-1773	4	24	dual	dual	ADJ
ejpam-1773	4	25	code	code	NOUN
ejpam-1773	4	26	,	,	PUNCT
ejpam-1773	4	27	a	a	DET
ejpam-1773	4	28	real	real	ADJ
ejpam-1773	4	29	unimodular	unimodular	ADJ
ejpam-1773	4	30	lattice	lattice	NOUN
ejpam-1773	4	31	,	,	PUNCT
ejpam-1773	4	32	a	a	DET
ejpam-1773	4	33	complex	complex	ADJ
ejpam-1773	4	34	unimodular	unimodular	ADJ
ejpam-1773	4	35	lattice	lattice	NOUN
ejpam-1773	4	36	,	,	PUNCT
ejpam-1773	4	37	a	a	DET
ejpam-1773	4	38	quaternionic	quaternionic	ADJ
ejpam-1773	4	39	lattice	lattice	NOUN
ejpam-1773	4	40	and	and	CCONJ
ejpam-1773	4	41	an	an	DET
ejpam-1773	4	42	infinite	infinite	ADJ
ejpam-1773	4	43	family	family	NOUN
ejpam-1773	4	44	of	of	ADP
ejpam-1773	4	45	self	self	NOUN
ejpam-1773	4	46	-	-	PUNCT
ejpam-1773	4	47	dual	dual	ADJ
ejpam-1773	4	48	codes	code	NOUN
ejpam-1773	4	49	.	.	PUNCT
ejpam-1773	5	1	we	we	PRON
ejpam-1773	5	2	prove	prove	VERB
ejpam-1773	5	3	the	the	DET
ejpam-1773	5	4	existence	existence	NOUN
ejpam-1773	5	5	of	of	ADP
ejpam-1773	5	6	type	type	NOUN
ejpam-1773	5	7	ii	ii	PROPN
ejpam-1773	5	8	codes	code	NOUN
ejpam-1773	5	9	of	of	ADP
ejpam-1773	5	10	all	all	DET
ejpam-1773	5	11	lengths	length	NOUN
ejpam-1773	5	12	over	over	ADP
ejpam-1773	5	13	rk	rk	NOUN
ejpam-1773	5	14	,	,	PUNCT
ejpam-1773	5	15	for	for	ADP
ejpam-1773	5	16	k	k	PROPN
ejpam-1773	5	17	≥	≥	NUM
ejpam-1773	5	18	3	3	NUM
ejpam-1773	5	19	,	,	PUNCT
ejpam-1773	5	20	and	and	CCONJ
ejpam-1773	5	21	we	we	PRON
ejpam-1773	5	22	obtain	obtain	VERB
ejpam-1773	5	23	some	some	DET
ejpam-1773	5	24	extremal	extremal	ADJ
ejpam-1773	5	25	binary	binary	ADJ
ejpam-1773	5	26	self	self	NOUN
ejpam-1773	5	27	-	-	PUNCT
ejpam-1773	5	28	dual	dual	ADJ
ejpam-1773	5	29	codes	code	NOUN
ejpam-1773	5	30	including	include	VERB
ejpam-1773	5	31	the	the	DET
ejpam-1773	5	32	extended	extended	ADJ
ejpam-1773	5	33	binary	binary	NOUN
ejpam-1773	5	34	golay	golay	NOUN
ejpam-1773	5	35	code	code	NOUN
ejpam-1773	5	36	as	as	ADP
ejpam-1773	5	37	the	the	DET
ejpam-1773	5	38	gray	gray	ADJ
ejpam-1773	5	39	images	image	NOUN
ejpam-1773	5	40	of	of	ADP
ejpam-1773	5	41	self	self	NOUN
ejpam-1773	5	42	-	-	PUNCT
ejpam-1773	5	43	dual	dual	ADJ
ejpam-1773	5	44	codes	code	NOUN
ejpam-1773	5	45	over	over	ADP
ejpam-1773	5	46	rk	rk	NOUN
ejpam-1773	5	47	for	for	ADP
ejpam-1773	5	48	some	some	DET
ejpam-1773	5	49	suitable	suitable	ADJ
ejpam-1773	5	50	k.	k.	PROPN
ejpam-1773	5	51	the	the	DET
ejpam-1773	5	52	binary	binary	PROPN
ejpam-1773	5	53	self	self	NOUN
ejpam-1773	5	54	-	-	PUNCT
ejpam-1773	5	55	dual	dual	ADJ
ejpam-1773	5	56	codes	code	NOUN
ejpam-1773	5	57	obtained	obtain	VERB
ejpam-1773	5	58	from	from	ADP
ejpam-1773	5	59	rk	rk	PRON
ejpam-1773	5	60	all	all	PRON
ejpam-1773	5	61	have	have	VERB
ejpam-1773	5	62	automorphism	automorphism	NOUN
ejpam-1773	5	63	groups	group	NOUN
ejpam-1773	5	64	whose	whose	DET
ejpam-1773	5	65	orders	order	NOUN
ejpam-1773	5	66	are	be	AUX
ejpam-1773	5	67	a	a	DET
ejpam-1773	5	68	multiple	multiple	NOUN
ejpam-1773	5	69	of	of	ADP
ejpam-1773	5	70	2k	2k	NUM
ejpam-1773	5	71	.	.	PUNCT
ejpam-1773	6	1	2010	2010	NUM
ejpam-1773	6	2	mathematics	mathematic	NOUN
ejpam-1773	6	3	subject	subject	NOUN
ejpam-1773	6	4	classifications	classification	NOUN
ejpam-1773	6	5	:	:	PUNCT
ejpam-1773	6	6	94b05	94b05	NUM
ejpam-1773	6	7	key	key	ADJ
ejpam-1773	6	8	words	word	NOUN
ejpam-1773	6	9	and	and	CCONJ
ejpam-1773	6	10	phrases	phrase	NOUN
ejpam-1773	6	11	:	:	PUNCT
ejpam-1773	6	12	self	self	NOUN
ejpam-1773	6	13	-	-	PUNCT
ejpam-1773	6	14	dual	dual	ADJ
ejpam-1773	6	15	codes	code	NOUN
ejpam-1773	6	16	,	,	PUNCT
ejpam-1773	6	17	codes	code	NOUN
ejpam-1773	6	18	over	over	ADP
ejpam-1773	6	19	rings	ring	NOUN
ejpam-1773	6	20	,	,	PUNCT
ejpam-1773	6	21	extremal	extremal	ADJ
ejpam-1773	6	22	codes	code	NOUN
ejpam-1773	6	23	,	,	PUNCT
ejpam-1773	6	24	lattices	lattice	NOUN
ejpam-1773	6	25	1	1	NUM
ejpam-1773	6	26	.	.	PUNCT
ejpam-1773	7	1	introduction	introduction	NOUN
ejpam-1773	7	2	self	self	NOUN
ejpam-1773	7	3	-	-	PUNCT
ejpam-1773	7	4	dual	dual	ADJ
ejpam-1773	7	5	codes	code	NOUN
ejpam-1773	7	6	are	be	AUX
ejpam-1773	7	7	an	an	DET
ejpam-1773	7	8	important	important	ADJ
ejpam-1773	7	9	class	class	NOUN
ejpam-1773	7	10	of	of	ADP
ejpam-1773	7	11	codes	code	NOUN
ejpam-1773	7	12	and	and	CCONJ
ejpam-1773	7	13	an	an	DET
ejpam-1773	7	14	extensive	extensive	ADJ
ejpam-1773	7	15	literature	literature	NOUN
ejpam-1773	7	16	exists	exist	VERB
ejpam-1773	7	17	on	on	ADP
ejpam-1773	7	18	selfdual	selfdual	ADJ
ejpam-1773	7	19	codes	code	NOUN
ejpam-1773	7	20	over	over	ADP
ejpam-1773	7	21	finite	finite	ADJ
ejpam-1773	7	22	fields	field	NOUN
ejpam-1773	7	23	.	.	PUNCT
ejpam-1773	8	1	self	self	NOUN
ejpam-1773	8	2	-	-	PUNCT
ejpam-1773	8	3	dual	dual	ADJ
ejpam-1773	8	4	codes	code	NOUN
ejpam-1773	8	5	over	over	ADP
ejpam-1773	8	6	rings	ring	NOUN
ejpam-1773	8	7	have	have	AUX
ejpam-1773	8	8	received	receive	VERB
ejpam-1773	8	9	attention	attention	NOUN
ejpam-1773	8	10	especially	especially	ADV
ejpam-1773	8	11	with	with	ADP
ejpam-1773	8	12	respect	respect	NOUN
ejpam-1773	8	13	to	to	ADP
ejpam-1773	8	14	their	their	PRON
ejpam-1773	8	15	connection	connection	NOUN
ejpam-1773	8	16	to	to	ADP
ejpam-1773	8	17	unimodular	unimodular	ADJ
ejpam-1773	8	18	lattices	lattice	NOUN
ejpam-1773	8	19	and	and	CCONJ
ejpam-1773	8	20	invariant	invariant	ADJ
ejpam-1773	8	21	theory	theory	NOUN
ejpam-1773	8	22	;	;	PUNCT
ejpam-1773	8	23	see	see	VERB
ejpam-1773	8	24	[	[	X
ejpam-1773	8	25	4	4	NUM
ejpam-1773	8	26	]	]	PUNCT
ejpam-1773	8	27	,	,	PUNCT
ejpam-1773	8	28	[	[	X
ejpam-1773	8	29	9	9	NUM
ejpam-1773	8	30	]	]	PUNCT
ejpam-1773	8	31	and	and	CCONJ
ejpam-1773	8	32	[	[	X
ejpam-1773	8	33	5	5	NUM
ejpam-1773	8	34	]	]	PUNCT
ejpam-1773	8	35	for	for	ADP
ejpam-1773	8	36	a	a	DET
ejpam-1773	8	37	description	description	NOUN
ejpam-1773	8	38	and	and	CCONJ
ejpam-1773	8	39	extensive	extensive	ADJ
ejpam-1773	8	40	bibliographies	bibliography	NOUN
ejpam-1773	8	41	.	.	PUNCT
ejpam-1773	9	1	they	they	PRON
ejpam-1773	9	2	can	can	AUX
ejpam-1773	9	3	also	also	ADV
ejpam-1773	9	4	be	be	AUX
ejpam-1773	9	5	used	use	VERB
ejpam-1773	9	6	to	to	PART
ejpam-1773	9	7	construct	construct	VERB
ejpam-1773	9	8	designs	design	NOUN
ejpam-1773	9	9	by	by	ADP
ejpam-1773	9	10	using	use	VERB
ejpam-1773	9	11	the	the	DET
ejpam-1773	9	12	assmus	assmus	NOUN
ejpam-1773	9	13	-	-	PUNCT
ejpam-1773	9	14	mattson	mattson	NOUN
ejpam-1773	9	15	theorem	theorem	NOUN
ejpam-1773	9	16	.	.	PUNCT
ejpam-1773	10	1	in	in	ADP
ejpam-1773	10	2	[	[	X
ejpam-1773	10	3	6	6	NUM
ejpam-1773	10	4	]	]	PUNCT
ejpam-1773	10	5	,	,	PUNCT
ejpam-1773	10	6	self	self	NOUN
ejpam-1773	10	7	-	-	PUNCT
ejpam-1773	10	8	dual	dual	ADJ
ejpam-1773	10	9	codes	code	NOUN
ejpam-1773	10	10	were	be	AUX
ejpam-1773	10	11	studied	study	VERB
ejpam-1773	10	12	over	over	ADP
ejpam-1773	10	13	the	the	DET
ejpam-1773	10	14	ring	ring	NOUN
ejpam-1773	10	15	f2	f2	PROPN
ejpam-1773	10	16	+	+	CCONJ
ejpam-1773	10	17	uf2	uf2	NOUN
ejpam-1773	11	1	and	and	CCONJ
ejpam-1773	11	2	they	they	PRON
ejpam-1773	11	3	were	be	AUX
ejpam-1773	11	4	connected	connect	VERB
ejpam-1773	11	5	to	to	ADP
ejpam-1773	11	6	complex	complex	ADJ
ejpam-1773	11	7	unimodular	unimodular	ADJ
ejpam-1773	11	8	lattices	lattice	NOUN
ejpam-1773	11	9	.	.	PUNCT
ejpam-1773	12	1	in	in	ADP
ejpam-1773	12	2	[	[	X
ejpam-1773	12	3	2	2	NUM
ejpam-1773	12	4	]	]	PUNCT
ejpam-1773	12	5	,	,	PUNCT
ejpam-1773	12	6	the	the	DET
ejpam-1773	12	7	ring	ring	NOUN
ejpam-1773	12	8	f2	f2	PROPN
ejpam-1773	12	9	+	+	CCONJ
ejpam-1773	12	10	uf2	uf2	NOUN
ejpam-1773	12	11	was	be	AUX
ejpam-1773	12	12	generalized	generalize	VERB
ejpam-1773	12	13	to	to	ADP
ejpam-1773	12	14	σ2	σ2	PROPN
ejpam-1773	12	15	m	m	PROPN
ejpam-1773	12	16	and	and	CCONJ
ejpam-1773	12	17	self	self	NOUN
ejpam-1773	12	18	-	-	PUNCT
ejpam-1773	12	19	dual	dual	ADJ
ejpam-1773	12	20	codes	code	NOUN
ejpam-1773	12	21	over	over	ADP
ejpam-1773	12	22	this	this	DET
ejpam-1773	12	23	ring	ring	NOUN
ejpam-1773	12	24	were	be	AUX
ejpam-1773	12	25	used	use	VERB
ejpam-1773	12	26	to	to	PART
ejpam-1773	12	27	construct	construct	VERB
ejpam-1773	12	28	quaternionic	quaternionic	ADJ
ejpam-1773	12	29	unimodular	unimodular	ADJ
ejpam-1773	12	30	lattices	lattice	NOUN
ejpam-1773	12	31	and	and	CCONJ
ejpam-1773	12	32	associated	associate	VERB
ejpam-1773	12	33	jacobi	jacobi	PROPN
ejpam-1773	12	34	forms	form	NOUN
ejpam-1773	12	35	.	.	PUNCT
ejpam-1773	13	1	we	we	PRON
ejpam-1773	13	2	shall	shall	AUX
ejpam-1773	13	3	generalize	generalize	VERB
ejpam-1773	13	4	these	these	DET
ejpam-1773	13	5	rings	ring	NOUN
ejpam-1773	13	6	to	to	ADP
ejpam-1773	13	7	an	an	DET
ejpam-1773	13	8	infinite	infinite	ADJ
ejpam-1773	13	9	family	family	NOUN
ejpam-1773	13	10	of	of	ADP
ejpam-1773	13	11	rings	ring	NOUN
ejpam-1773	13	12	denoted	denote	VERB
ejpam-1773	13	13	by	by	ADP
ejpam-1773	13	14	rk	rk	PRON
ejpam-1773	13	15	and	and	CCONJ
ejpam-1773	13	16	use	use	VERB
ejpam-1773	13	17	these	these	DET
ejpam-1773	13	18	rings	ring	NOUN
ejpam-1773	13	19	to	to	PART
ejpam-1773	13	20	construct	construct	VERB
ejpam-1773	13	21	binary	binary	ADJ
ejpam-1773	13	22	self	self	NOUN
ejpam-1773	13	23	-	-	PUNCT
ejpam-1773	13	24	dual	dual	ADJ
ejpam-1773	13	25	codes	code	NOUN
ejpam-1773	13	26	and	and	CCONJ
ejpam-1773	13	27	real	real	ADJ
ejpam-1773	13	28	,	,	PUNCT
ejpam-1773	13	29	complex	complex	ADJ
ejpam-1773	13	30	and	and	CCONJ
ejpam-1773	13	31	quaternionic	quaternionic	ADJ
ejpam-1773	13	32	unimodular	unimodular	ADJ
ejpam-1773	13	33	lattices	lattice	NOUN
ejpam-1773	13	34	.	.	PUNCT
ejpam-1773	14	1	codes	code	NOUN
ejpam-1773	14	2	over	over	ADP
ejpam-1773	14	3	the	the	DET
ejpam-1773	14	4	ring	ring	NOUN
ejpam-1773	14	5	rk	rk	NOUN
ejpam-1773	14	6	were	be	AUX
ejpam-1773	14	7	first	first	ADV
ejpam-1773	14	8	studied	study	VERB
ejpam-1773	14	9	in	in	ADP
ejpam-1773	14	10	[	[	X
ejpam-1773	14	11	7	7	NUM
ejpam-1773	14	12	]	]	PUNCT
ejpam-1773	14	13	.	.	PUNCT
ejpam-1773	15	1	in	in	ADP
ejpam-1773	15	2	the	the	DET
ejpam-1773	15	3	literature	literature	NOUN
ejpam-1773	15	4	there	there	PRON
ejpam-1773	15	5	are	be	VERB
ejpam-1773	15	6	constructions	construction	NOUN
ejpam-1773	15	7	for	for	ADP
ejpam-1773	15	8	extremal	extremal	ADJ
ejpam-1773	15	9	binary	binary	ADJ
ejpam-1773	15	10	self	self	NOUN
ejpam-1773	15	11	-	-	PUNCT
ejpam-1773	15	12	dual	dual	ADJ
ejpam-1773	15	13	codes	code	NOUN
ejpam-1773	15	14	with	with	ADP
ejpam-1773	15	15	automorphism	automorphism	NOUN
ejpam-1773	15	16	groups	group	NOUN
ejpam-1773	15	17	of	of	ADP
ejpam-1773	15	18	order	order	NOUN
ejpam-1773	15	19	2	2	NUM
ejpam-1773	15	20	,	,	PUNCT
ejpam-1773	15	21	p	p	X
ejpam-1773	15	22	(	(	PUNCT
ejpam-1773	15	23	an	an	DET
ejpam-1773	15	24	odd	odd	ADJ
ejpam-1773	15	25	prime	prime	NOUN
ejpam-1773	15	26	)	)	PUNCT
ejpam-1773	15	27	,	,	PUNCT
ejpam-1773	15	28	p2	p2	PROPN
ejpam-1773	15	29	and	and	CCONJ
ejpam-1773	15	30	pq	pq	INTJ
ejpam-1773	15	31	.	.	PROPN
ejpam-1773	15	32	as	as	SCONJ
ejpam-1773	15	33	was	be	AUX
ejpam-1773	15	34	shown	show	VERB
ejpam-1773	15	35	in	in	ADP
ejpam-1773	15	36	[	[	X
ejpam-1773	15	37	7	7	NUM
ejpam-1773	15	38	]	]	PUNCT
ejpam-1773	15	39	,	,	PUNCT
ejpam-1773	15	40	codes	code	NOUN
ejpam-1773	15	41	over	over	ADP
ejpam-1773	15	42	rk	rk	NOUN
ejpam-1773	15	43	are	be	AUX
ejpam-1773	15	44	all	all	ADV
ejpam-1773	15	45	invariant	invariant	ADJ
ejpam-1773	15	46	under	under	ADP
ejpam-1773	15	47	a	a	DET
ejpam-1773	15	48	group	group	NOUN
ejpam-1773	15	49	of	of	ADP
ejpam-1773	15	50	automorphisms	automorphism	NOUN
ejpam-1773	15	51	of	of	ADP
ejpam-1773	15	52	size	size	NOUN
ejpam-1773	15	53	2k	2k	NUM
ejpam-1773	15	54	.	.	PUNCT
ejpam-1773	16	1	this	this	PRON
ejpam-1773	16	2	means	mean	VERB
ejpam-1773	16	3	that	that	SCONJ
ejpam-1773	16	4	self	self	NOUN
ejpam-1773	16	5	-	-	PUNCT
ejpam-1773	16	6	dual	dual	ADJ
ejpam-1773	16	7	codes	code	NOUN
ejpam-1773	16	8	∗corresponding	∗corresponde	VERB
ejpam-1773	16	9	author	author	NOUN
ejpam-1773	16	10	.	.	PUNCT
ejpam-1773	17	1	email	email	NOUN
ejpam-1773	17	2	addresses	address	NOUN
ejpam-1773	17	3	:	:	PUNCT
ejpam-1773	17	4	doughertys1	doughertys1	NUM
ejpam-1773	17	5	�	�	PROPN
ejpam-1773	17	6	s	s	PART
ejpam-1773	17	7	ranton.edu	ranton.edu	NOUN
ejpam-1773	17	8	(	(	PUNCT
ejpam-1773	17	9	s.	s.	PROPN
ejpam-1773	17	10	dougherty	dougherty	PROPN
ejpam-1773	17	11	)	)	PUNCT
ejpam-1773	17	12	,	,	PUNCT
ejpam-1773	17	13	byildiz�fatih.edu.tr	byildiz�fatih.edu.tr	PROPN
ejpam-1773	17	14	(	(	PUNCT
ejpam-1773	17	15	b.yildiz	b.yildiz	NOUN
ejpam-1773	17	16	)	)	PUNCT
ejpam-1773	17	17	,	,	PUNCT
ejpam-1773	17	18	skaradeniz�fatih.edu.tr	skaradeniz�fatih.edu.tr	PROPN
ejpam-1773	17	19	(	(	PUNCT
ejpam-1773	17	20	s.karadeniz	s.karadeniz	NOUN
ejpam-1773	17	21	)	)	PUNCT
ejpam-1773	17	22	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-1773	18	1	89	89	NUM
ejpam-1773	18	2	c	c	X
ejpam-1773	18	3	©	©	PROPN
ejpam-1773	18	4	2013	2013	NUM
ejpam-1773	18	5	ejpam	ejpam	NOUN
ejpam-1773	18	6	all	all	DET
ejpam-1773	18	7	rights	right	NOUN
ejpam-1773	18	8	reserved	reserve	VERB
ejpam-1773	18	9	.	.	PUNCT
ejpam-1773	19	1	s.	s.	PROPN
ejpam-1773	19	2	dougherty	dougherty	PROPN
ejpam-1773	19	3	,	,	PUNCT
ejpam-1773	19	4	b.yıldız	b.yıldız	NOUN
ejpam-1773	19	5	,	,	PUNCT
ejpam-1773	19	6	s.karadeniz	s.karadeniz	NOUN
ejpam-1773	19	7	/	/	SYM
ejpam-1773	19	8	eur	eur	NOUN
ejpam-1773	19	9	.	.	PUNCT
ejpam-1773	20	1	j.	j.	PROPN
ejpam-1773	20	2	pure	pure	PROPN
ejpam-1773	20	3	appl	appl	PROPN
ejpam-1773	20	4	.	.	PROPN
ejpam-1773	20	5	math	math	PROPN
ejpam-1773	20	6	,	,	PUNCT
ejpam-1773	20	7	6	6	NUM
ejpam-1773	20	8	(	(	PUNCT
ejpam-1773	20	9	2013	2013	NUM
ejpam-1773	20	10	)	)	PUNCT
ejpam-1773	20	11	,	,	PUNCT
ejpam-1773	20	12	89	89	NUM
ejpam-1773	20	13	-	-	SYM
ejpam-1773	20	14	106	106	NUM
ejpam-1773	20	15	90	90	NUM
ejpam-1773	20	16	constructed	construct	VERB
ejpam-1773	20	17	from	from	ADP
ejpam-1773	20	18	rk	rk	PRON
ejpam-1773	20	19	will	will	AUX
ejpam-1773	20	20	all	all	ADV
ejpam-1773	20	21	have	have	VERB
ejpam-1773	20	22	automorphism	automorphism	NOUN
ejpam-1773	20	23	groups	group	NOUN
ejpam-1773	20	24	whose	whose	DET
ejpam-1773	20	25	orders	order	NOUN
ejpam-1773	20	26	are	be	AUX
ejpam-1773	20	27	a	a	DET
ejpam-1773	20	28	multiple	multiple	NOUN
ejpam-1773	20	29	of	of	ADP
ejpam-1773	20	30	2k	2k	NUM
ejpam-1773	20	31	.	.	PUNCT
ejpam-1773	21	1	so	so	ADV
ejpam-1773	21	2	,	,	PUNCT
ejpam-1773	21	3	we	we	PRON
ejpam-1773	21	4	believe	believe	VERB
ejpam-1773	21	5	that	that	SCONJ
ejpam-1773	21	6	studying	study	VERB
ejpam-1773	21	7	self	self	NOUN
ejpam-1773	21	8	-	-	PUNCT
ejpam-1773	21	9	dual	dual	ADJ
ejpam-1773	21	10	codes	code	NOUN
ejpam-1773	21	11	over	over	ADP
ejpam-1773	21	12	rk	rk	NOUN
ejpam-1773	21	13	fills	fill	VERB
ejpam-1773	21	14	a	a	DET
ejpam-1773	21	15	gap	gap	NOUN
ejpam-1773	21	16	in	in	ADP
ejpam-1773	21	17	the	the	DET
ejpam-1773	21	18	literature	literature	NOUN
ejpam-1773	21	19	of	of	ADP
ejpam-1773	21	20	binary	binary	PROPN
ejpam-1773	21	21	self	self	NOUN
ejpam-1773	21	22	-	-	PUNCT
ejpam-1773	21	23	dual	dual	ADJ
ejpam-1773	21	24	codes	code	NOUN
ejpam-1773	21	25	.	.	PUNCT
ejpam-1773	22	1	we	we	PRON
ejpam-1773	22	2	have	have	AUX
ejpam-1773	22	3	illustrated	illustrate	VERB
ejpam-1773	22	4	several	several	ADJ
ejpam-1773	22	5	examples	example	NOUN
ejpam-1773	22	6	at	at	ADP
ejpam-1773	22	7	the	the	DET
ejpam-1773	22	8	end	end	NOUN
ejpam-1773	22	9	of	of	ADP
ejpam-1773	22	10	the	the	DET
ejpam-1773	22	11	paper	paper	NOUN
ejpam-1773	22	12	.	.	PUNCT
ejpam-1773	23	1	the	the	DET
ejpam-1773	23	2	rest	rest	NOUN
ejpam-1773	23	3	of	of	ADP
ejpam-1773	23	4	the	the	DET
ejpam-1773	23	5	paper	paper	NOUN
ejpam-1773	23	6	is	be	AUX
ejpam-1773	23	7	organized	organize	VERB
ejpam-1773	23	8	as	as	SCONJ
ejpam-1773	23	9	follows	follow	VERB
ejpam-1773	23	10	:	:	PUNCT
ejpam-1773	23	11	in	in	ADP
ejpam-1773	23	12	section	section	NOUN
ejpam-1773	23	13	2	2	NUM
ejpam-1773	23	14	,	,	PUNCT
ejpam-1773	23	15	we	we	PRON
ejpam-1773	23	16	will	will	AUX
ejpam-1773	23	17	present	present	VERB
ejpam-1773	23	18	some	some	DET
ejpam-1773	23	19	definitions	definition	NOUN
ejpam-1773	23	20	and	and	CCONJ
ejpam-1773	23	21	notations	notation	NOUN
ejpam-1773	23	22	about	about	ADP
ejpam-1773	23	23	the	the	DET
ejpam-1773	23	24	rings	ring	NOUN
ejpam-1773	23	25	rk	rk	NOUN
ejpam-1773	23	26	and	and	CCONJ
ejpam-1773	23	27	about	about	ADP
ejpam-1773	23	28	codes	code	NOUN
ejpam-1773	23	29	over	over	ADP
ejpam-1773	23	30	rk	rk	NOUN
ejpam-1773	23	31	.	.	PUNCT
ejpam-1773	24	1	in	in	ADP
ejpam-1773	24	2	section	section	NOUN
ejpam-1773	24	3	3	3	NUM
ejpam-1773	24	4	,	,	PUNCT
ejpam-1773	24	5	we	we	PRON
ejpam-1773	24	6	will	will	AUX
ejpam-1773	24	7	discuss	discuss	VERB
ejpam-1773	24	8	the	the	DET
ejpam-1773	24	9	projection	projection	NOUN
ejpam-1773	24	10	maps	map	NOUN
ejpam-1773	24	11	and	and	CCONJ
ejpam-1773	24	12	lifts	lift	NOUN
ejpam-1773	24	13	between	between	ADP
ejpam-1773	24	14	rk	rk	NOUN
ejpam-1773	24	15	and	and	CCONJ
ejpam-1773	24	16	rk′	rk′	PROPN
ejpam-1773	24	17	,	,	PUNCT
ejpam-1773	24	18	for	for	ADP
ejpam-1773	24	19	k	k	PROPN
ejpam-1773	24	20	6=	6=	PROPN
ejpam-1773	24	21	k′	k′	PROPN
ejpam-1773	24	22	,	,	PUNCT
ejpam-1773	24	23	in	in	ADP
ejpam-1773	24	24	connection	connection	NOUN
ejpam-1773	24	25	with	with	ADP
ejpam-1773	24	26	self	self	NOUN
ejpam-1773	24	27	-	-	PUNCT
ejpam-1773	24	28	dual	dual	ADJ
ejpam-1773	24	29	codes	code	NOUN
ejpam-1773	24	30	.	.	PUNCT
ejpam-1773	25	1	section	section	NOUN
ejpam-1773	25	2	4	4	NUM
ejpam-1773	25	3	will	will	AUX
ejpam-1773	25	4	consist	consist	VERB
ejpam-1773	25	5	of	of	ADP
ejpam-1773	25	6	the	the	DET
ejpam-1773	25	7	description	description	NOUN
ejpam-1773	25	8	of	of	ADP
ejpam-1773	25	9	the	the	DET
ejpam-1773	25	10	binary	binary	ADJ
ejpam-1773	25	11	images	image	NOUN
ejpam-1773	25	12	of	of	ADP
ejpam-1773	25	13	self	self	NOUN
ejpam-1773	25	14	-	-	PUNCT
ejpam-1773	25	15	dual	dual	ADJ
ejpam-1773	25	16	codes	code	NOUN
ejpam-1773	25	17	over	over	ADP
ejpam-1773	25	18	rk	rk	NOUN
ejpam-1773	25	19	.	.	PUNCT
ejpam-1773	26	1	in	in	ADP
ejpam-1773	26	2	particular	particular	ADJ
ejpam-1773	26	3	,	,	PUNCT
ejpam-1773	26	4	the	the	DET
ejpam-1773	26	5	existence	existence	NOUN
ejpam-1773	26	6	of	of	ADP
ejpam-1773	26	7	type	type	NOUN
ejpam-1773	26	8	ii	ii	PROPN
ejpam-1773	26	9	codes	code	NOUN
ejpam-1773	26	10	of	of	ADP
ejpam-1773	26	11	all	all	DET
ejpam-1773	26	12	lengths	length	NOUN
ejpam-1773	26	13	over	over	ADP
ejpam-1773	26	14	rk	rk	NOUN
ejpam-1773	26	15	,	,	PUNCT
ejpam-1773	26	16	for	for	ADP
ejpam-1773	26	17	k	k	PROPN
ejpam-1773	26	18	≥	≥	NUM
ejpam-1773	26	19	3	3	NUM
ejpam-1773	26	20	,	,	PUNCT
ejpam-1773	26	21	and	and	CCONJ
ejpam-1773	26	22	of	of	ADP
ejpam-1773	26	23	all	all	DET
ejpam-1773	26	24	even	even	ADJ
ejpam-1773	26	25	lengths	length	NOUN
ejpam-1773	26	26	over	over	ADP
ejpam-1773	26	27	r2	r2	PROPN
ejpam-1773	26	28	will	will	AUX
ejpam-1773	26	29	be	be	AUX
ejpam-1773	26	30	established	establish	VERB
ejpam-1773	26	31	.	.	PUNCT
ejpam-1773	27	1	an	an	DET
ejpam-1773	27	2	upper	upper	ADJ
ejpam-1773	27	3	bound	bind	VERB
ejpam-1773	27	4	on	on	ADP
ejpam-1773	27	5	the	the	DET
ejpam-1773	27	6	minimum	minimum	ADJ
ejpam-1773	27	7	lee	lee	PROPN
ejpam-1773	27	8	distance	distance	NOUN
ejpam-1773	27	9	of	of	ADP
ejpam-1773	27	10	self	self	NOUN
ejpam-1773	27	11	-	-	PUNCT
ejpam-1773	27	12	dual	dual	ADJ
ejpam-1773	27	13	codes	code	NOUN
ejpam-1773	27	14	will	will	AUX
ejpam-1773	27	15	also	also	ADV
ejpam-1773	27	16	be	be	AUX
ejpam-1773	27	17	given	give	VERB
ejpam-1773	27	18	.	.	PUNCT
ejpam-1773	28	1	in	in	ADP
ejpam-1773	28	2	section	section	NOUN
ejpam-1773	28	3	5	5	NUM
ejpam-1773	28	4	,	,	PUNCT
ejpam-1773	28	5	we	we	PRON
ejpam-1773	28	6	will	will	AUX
ejpam-1773	28	7	give	give	VERB
ejpam-1773	28	8	a	a	DET
ejpam-1773	28	9	characterization	characterization	NOUN
ejpam-1773	28	10	of	of	ADP
ejpam-1773	28	11	self	self	NOUN
ejpam-1773	28	12	-	-	PUNCT
ejpam-1773	28	13	dual	dual	ADJ
ejpam-1773	28	14	codes	code	NOUN
ejpam-1773	28	15	over	over	ADP
ejpam-1773	28	16	rk	rk	NOUN
ejpam-1773	28	17	of	of	ADP
ejpam-1773	28	18	length	length	NOUN
ejpam-1773	28	19	1	1	NUM
ejpam-1773	28	20	and	and	CCONJ
ejpam-1773	28	21	2	2	NUM
ejpam-1773	28	22	.	.	PUNCT
ejpam-1773	29	1	in	in	ADP
ejpam-1773	29	2	particular	particular	ADJ
ejpam-1773	29	3	,	,	PUNCT
ejpam-1773	29	4	a	a	DET
ejpam-1773	29	5	full	full	ADJ
ejpam-1773	29	6	characterization	characterization	NOUN
ejpam-1773	29	7	of	of	ADP
ejpam-1773	29	8	one	one	NUM
ejpam-1773	29	9	-	-	PUNCT
ejpam-1773	29	10	generator	generator	NOUN
ejpam-1773	29	11	self	self	NOUN
ejpam-1773	29	12	-	-	PUNCT
ejpam-1773	29	13	dual	dual	ADJ
ejpam-1773	29	14	codes	code	NOUN
ejpam-1773	29	15	of	of	ADP
ejpam-1773	29	16	length	length	NOUN
ejpam-1773	29	17	1	1	NUM
ejpam-1773	29	18	and	and	CCONJ
ejpam-1773	29	19	2	2	NUM
ejpam-1773	29	20	will	will	AUX
ejpam-1773	29	21	be	be	AUX
ejpam-1773	29	22	given	give	VERB
ejpam-1773	29	23	.	.	PUNCT
ejpam-1773	30	1	section	section	NOUN
ejpam-1773	30	2	6	6	NUM
ejpam-1773	30	3	will	will	AUX
ejpam-1773	30	4	highlight	highlight	VERB
ejpam-1773	30	5	the	the	DET
ejpam-1773	30	6	connection	connection	NOUN
ejpam-1773	30	7	between	between	ADP
ejpam-1773	30	8	self	self	NOUN
ejpam-1773	30	9	-	-	PUNCT
ejpam-1773	30	10	dual	dual	ADJ
ejpam-1773	30	11	codes	code	NOUN
ejpam-1773	30	12	over	over	ADP
ejpam-1773	30	13	rk	rk	NOUN
ejpam-1773	30	14	and	and	CCONJ
ejpam-1773	30	15	real	real	ADJ
ejpam-1773	30	16	,	,	PUNCT
ejpam-1773	30	17	complex	complex	ADJ
ejpam-1773	30	18	,	,	PUNCT
ejpam-1773	30	19	and	and	CCONJ
ejpam-1773	30	20	quaternionic	quaternionic	ADJ
ejpam-1773	30	21	unimodular	unimodular	ADJ
ejpam-1773	30	22	lattices	lattice	NOUN
ejpam-1773	30	23	.	.	PUNCT
ejpam-1773	31	1	we	we	PRON
ejpam-1773	31	2	will	will	AUX
ejpam-1773	31	3	finish	finish	VERB
ejpam-1773	31	4	the	the	DET
ejpam-1773	31	5	paper	paper	NOUN
ejpam-1773	31	6	with	with	ADP
ejpam-1773	31	7	some	some	DET
ejpam-1773	31	8	examples	example	NOUN
ejpam-1773	31	9	of	of	ADP
ejpam-1773	31	10	extremal	extremal	ADJ
ejpam-1773	31	11	binary	binary	ADJ
ejpam-1773	31	12	self	self	NOUN
ejpam-1773	31	13	-	-	PUNCT
ejpam-1773	31	14	dual	dual	ADJ
ejpam-1773	31	15	codes	code	NOUN
ejpam-1773	31	16	including	include	VERB
ejpam-1773	31	17	the	the	DET
ejpam-1773	31	18	extended	extended	ADJ
ejpam-1773	31	19	binary	binary	NOUN
ejpam-1773	31	20	golay	golay	PROPN
ejpam-1773	31	21	code	code	NOUN
ejpam-1773	31	22	obtained	obtain	VERB
ejpam-1773	31	23	from	from	ADP
ejpam-1773	31	24	the	the	DET
ejpam-1773	31	25	codes	code	NOUN
ejpam-1773	31	26	over	over	ADP
ejpam-1773	31	27	rk	rk	NOUN
ejpam-1773	31	28	for	for	ADP
ejpam-1773	31	29	some	some	DET
ejpam-1773	31	30	suitable	suitable	ADJ
ejpam-1773	31	31	k.	k.	NOUN
ejpam-1773	31	32	2	2	NUM
ejpam-1773	31	33	.	.	PUNCT
ejpam-1773	31	34	definitions	definition	NOUN
ejpam-1773	31	35	and	and	CCONJ
ejpam-1773	31	36	notations	notation	NOUN
ejpam-1773	31	37	for	for	ADP
ejpam-1773	31	38	finite	finite	PROPN
ejpam-1773	31	39	k	k	PROPN
ejpam-1773	31	40	≥	≥	NUM
ejpam-1773	31	41	1	1	NUM
ejpam-1773	31	42	,	,	PUNCT
ejpam-1773	31	43	we	we	PRON
ejpam-1773	31	44	define	define	VERB
ejpam-1773	31	45	a	a	DET
ejpam-1773	31	46	family	family	NOUN
ejpam-1773	31	47	of	of	ADP
ejpam-1773	31	48	rings	ring	NOUN
ejpam-1773	31	49	by	by	ADP
ejpam-1773	31	50	rk	rk	NOUN
ejpam-1773	31	51	=	=	SYM
ejpam-1773	31	52	f2[u1,u2	f2[u1,u2	PROPN
ejpam-1773	31	53	,	,	PUNCT
ejpam-1773	31	54	.	.	PUNCT
ejpam-1773	31	55	.	.	PUNCT
ejpam-1773	32	1	.	.	PUNCT
ejpam-1773	33	1	,	,	PUNCT
ejpam-1773	33	2	uk]/〈u2	uk]/〈u2	PROPN
ejpam-1773	33	3	i	i	PRON
ejpam-1773	33	4	=	=	NOUN
ejpam-1773	33	5	0,uiu	0,uiu	NUM
ejpam-1773	34	1	j	j	NOUN
ejpam-1773	34	2	=	=	SYM
ejpam-1773	34	3	u	u	NOUN
ejpam-1773	34	4	jui	jui	PROPN
ejpam-1773	34	5	〉	〉	PROPN
ejpam-1773	34	6	.	.	PUNCT
ejpam-1773	35	1	(	(	PUNCT
ejpam-1773	35	2	1	1	X
ejpam-1773	35	3	)	)	PUNCT
ejpam-1773	35	4	we	we	PRON
ejpam-1773	35	5	let	let	VERB
ejpam-1773	35	6	r∞	r∞	PROPN
ejpam-1773	35	7	be	be	AUX
ejpam-1773	35	8	the	the	DET
ejpam-1773	35	9	ring	ring	NOUN
ejpam-1773	35	10	r∞	r∞	NOUN
ejpam-1773	35	11	=	=	SYM
ejpam-1773	35	12	f2[u1,u2	f2[u1,u2	PROPN
ejpam-1773	35	13	,	,	PUNCT
ejpam-1773	35	14	.	.	PUNCT
ejpam-1773	35	15	.	.	PUNCT
ejpam-1773	36	1	.]/〈u2	.]/〈u2	PUNCT
ejpam-1773	37	1	i	i	PRON
ejpam-1773	37	2	=	=	NOUN
ejpam-1773	37	3	0,uiu	0,uiu	NUM
ejpam-1773	37	4	j	j	NOUN
ejpam-1773	37	5	=	=	SYM
ejpam-1773	37	6	u	u	NOUN
ejpam-1773	37	7	jui	jui	PROPN
ejpam-1773	37	8	〉	〉	PROPN
ejpam-1773	37	9	,	,	PUNCT
ejpam-1773	37	10	(	(	PUNCT
ejpam-1773	37	11	2	2	NUM
ejpam-1773	37	12	)	)	PUNCT
ejpam-1773	37	13	and	and	CCONJ
ejpam-1773	37	14	r0	r0	NOUN
ejpam-1773	37	15	=	=	SYM
ejpam-1773	37	16	f2	f2	PROPN
ejpam-1773	37	17	.	.	PUNCT
ejpam-1773	38	1	for	for	ADP
ejpam-1773	38	2	all	all	DET
ejpam-1773	38	3	k	k	NOUN
ejpam-1773	38	4	,	,	PUNCT
ejpam-1773	38	5	finite	finite	NOUN
ejpam-1773	38	6	or	or	CCONJ
ejpam-1773	38	7	infinite	infinite	VERB
ejpam-1773	38	8	,	,	PUNCT
ejpam-1773	38	9	rk	rk	NOUN
ejpam-1773	38	10	is	be	AUX
ejpam-1773	38	11	a	a	DET
ejpam-1773	38	12	commutative	commutative	ADJ
ejpam-1773	38	13	ring	ring	NOUN
ejpam-1773	38	14	.	.	PUNCT
ejpam-1773	39	1	note	note	VERB
ejpam-1773	39	2	that	that	SCONJ
ejpam-1773	39	3	the	the	DET
ejpam-1773	39	4	ring	ring	NOUN
ejpam-1773	39	5	r∞	r∞	ADV
ejpam-1773	39	6	is	be	AUX
ejpam-1773	39	7	an	an	DET
ejpam-1773	39	8	infinite	infinite	ADJ
ejpam-1773	39	9	ring	ring	NOUN
ejpam-1773	39	10	while	while	SCONJ
ejpam-1773	39	11	rk	rk	NOUN
ejpam-1773	39	12	is	be	AUX
ejpam-1773	39	13	a	a	DET
ejpam-1773	39	14	finite	finite	ADJ
ejpam-1773	39	15	ring	ring	NOUN
ejpam-1773	39	16	for	for	ADP
ejpam-1773	39	17	finite	finite	ADJ
ejpam-1773	39	18	values	value	NOUN
ejpam-1773	39	19	of	of	ADP
ejpam-1773	39	20	k.	k.	PROPN
ejpam-1773	39	21	to	to	PART
ejpam-1773	39	22	describe	describe	VERB
ejpam-1773	39	23	the	the	DET
ejpam-1773	39	24	elements	element	NOUN
ejpam-1773	39	25	of	of	ADP
ejpam-1773	39	26	rk	rk	PRON
ejpam-1773	39	27	we	we	PRON
ejpam-1773	39	28	let	let	VERB
ejpam-1773	39	29	,	,	PUNCT
ejpam-1773	39	30	for	for	ADP
ejpam-1773	39	31	a⊆	a⊆	PROPN
ejpam-1773	39	32	{	{	PUNCT
ejpam-1773	39	33	1,2	1,2	NUM
ejpam-1773	39	34	,	,	PUNCT
ejpam-1773	39	35	.	.	PUNCT
ejpam-1773	39	36	.	.	PUNCT
ejpam-1773	39	37	.	.	PUNCT
ejpam-1773	40	1	,	,	PUNCT
ejpam-1773	40	2	k	k	X
ejpam-1773	40	3	}	}	PUNCT
ejpam-1773	40	4	ua	ua	NOUN
ejpam-1773	40	5	:	:	PUNCT
ejpam-1773	40	6	=	=	SYM
ejpam-1773	40	7	∏	∏	PROPN
ejpam-1773	40	8	i∈a	i∈a	ADJ
ejpam-1773	40	9	ui	ui	NOUN
ejpam-1773	40	10	(	(	PUNCT
ejpam-1773	40	11	3	3	NUM
ejpam-1773	40	12	)	)	PUNCT
ejpam-1773	40	13	with	with	ADP
ejpam-1773	40	14	u	u	NOUN
ejpam-1773	40	15	;	;	PUNCT
ejpam-1773	40	16	=	=	SYM
ejpam-1773	40	17	1	1	X
ejpam-1773	40	18	.	.	PUNCT
ejpam-1773	41	1	elements	element	NOUN
ejpam-1773	41	2	of	of	ADP
ejpam-1773	41	3	rk	rk	NOUN
ejpam-1773	41	4	,	,	PUNCT
ejpam-1773	41	5	then	then	ADV
ejpam-1773	41	6	can	can	AUX
ejpam-1773	41	7	be	be	AUX
ejpam-1773	41	8	represented	represent	VERB
ejpam-1773	41	9	as	as	ADP
ejpam-1773	41	10	∑	∑	PROPN
ejpam-1773	41	11	a⊆{1,	a⊆{1,	PROPN
ejpam-1773	41	12	...	...	PUNCT
ejpam-1773	41	13	,k	,k	ADJ
ejpam-1773	41	14	}	}	PUNCT
ejpam-1773	41	15	caua	caua	NOUN
ejpam-1773	41	16	,	,	PUNCT
ejpam-1773	41	17	ca	can	AUX
ejpam-1773	41	18	∈	∈	PROPN
ejpam-1773	41	19	f2	f2	PROPN
ejpam-1773	41	20	.	.	PUNCT
ejpam-1773	42	1	(	(	PUNCT
ejpam-1773	42	2	4	4	X
ejpam-1773	42	3	)	)	PUNCT
ejpam-1773	42	4	it	it	PRON
ejpam-1773	42	5	is	be	AUX
ejpam-1773	42	6	easily	easily	ADV
ejpam-1773	42	7	observed	observe	VERB
ejpam-1773	42	8	that	that	SCONJ
ejpam-1773	42	9	the	the	DET
ejpam-1773	42	10	ring	ring	NOUN
ejpam-1773	42	11	rk	rk	NOUN
ejpam-1773	42	12	is	be	AUX
ejpam-1773	42	13	local	local	ADJ
ejpam-1773	42	14	whose	whose	DET
ejpam-1773	42	15	maximal	maximal	ADJ
ejpam-1773	42	16	ideal	ideal	NOUN
ejpam-1773	42	17	is	be	AUX
ejpam-1773	42	18	given	give	VERB
ejpam-1773	42	19	by	by	ADP
ejpam-1773	42	20	〈	〈	PROPN
ejpam-1773	42	21	u1,u2	u1,u2	PROPN
ejpam-1773	42	22	,	,	PUNCT
ejpam-1773	42	23	.	.	PUNCT
ejpam-1773	42	24	.	.	PUNCT
ejpam-1773	43	1	.	.	PUNCT
ejpam-1773	44	1	,	,	PUNCT
ejpam-1773	44	2	uk	uk	PROPN
ejpam-1773	44	3	〉	〉	NOUN
ejpam-1773	44	4	and	and	CCONJ
ejpam-1773	44	5	|rk|	|rk|	NOUN
ejpam-1773	44	6	=	=	SYM
ejpam-1773	44	7	2(2	2(2	NUM
ejpam-1773	45	1	k	k	NOUN
ejpam-1773	45	2	)	)	PUNCT
ejpam-1773	45	3	.	.	PUNCT
ejpam-1773	46	1	rk	rk	NOUN
ejpam-1773	46	2	is	be	AUX
ejpam-1773	46	3	not	not	PART
ejpam-1773	46	4	a	a	DET
ejpam-1773	46	5	principal	principal	ADJ
ejpam-1773	46	6	ideal	ideal	ADJ
ejpam-1773	46	7	ring	ring	NOUN
ejpam-1773	46	8	nor	nor	CCONJ
ejpam-1773	46	9	is	be	AUX
ejpam-1773	46	10	it	it	PRON
ejpam-1773	46	11	a	a	DET
ejpam-1773	46	12	chain	chain	NOUN
ejpam-1773	46	13	ring	ring	NOUN
ejpam-1773	46	14	.	.	PUNCT
ejpam-1773	47	1	but	but	CCONJ
ejpam-1773	47	2	,	,	PUNCT
ejpam-1773	47	3	it	it	PRON
ejpam-1773	47	4	is	be	AUX
ejpam-1773	47	5	a	a	DET
ejpam-1773	47	6	frobenius	frobenius	ADJ
ejpam-1773	47	7	ring	ring	NOUN
ejpam-1773	47	8	.	.	PUNCT
ejpam-1773	48	1	it	it	PRON
ejpam-1773	48	2	is	be	AUX
ejpam-1773	48	3	shown	show	VERB
ejpam-1773	48	4	in	in	ADP
ejpam-1773	48	5	[	[	X
ejpam-1773	48	6	11	11	NUM
ejpam-1773	48	7	]	]	PUNCT
ejpam-1773	48	8	that	that	PRON
ejpam-1773	48	9	codes	code	VERB
ejpam-1773	48	10	over	over	ADP
ejpam-1773	48	11	frobenius	frobenius	ADJ
ejpam-1773	48	12	rings	ring	NOUN
ejpam-1773	48	13	satisfy	satisfy	PROPN
ejpam-1773	48	14	macwilliams	macwilliam	NOUN
ejpam-1773	48	15	theorems	theorem	NOUN
ejpam-1773	48	16	.	.	PUNCT
ejpam-1773	49	1	see	see	VERB
ejpam-1773	49	2	[	[	X
ejpam-1773	49	3	11	11	NUM
ejpam-1773	49	4	]	]	PUNCT
ejpam-1773	49	5	for	for	ADP
ejpam-1773	49	6	other	other	ADJ
ejpam-1773	49	7	foundational	foundational	ADJ
ejpam-1773	49	8	results	result	NOUN
ejpam-1773	49	9	on	on	ADP
ejpam-1773	49	10	codes	code	NOUN
ejpam-1773	49	11	over	over	ADP
ejpam-1773	49	12	frobenius	frobenius	ADJ
ejpam-1773	49	13	rings	ring	NOUN
ejpam-1773	49	14	.	.	PUNCT
ejpam-1773	50	1	s.	s.	PROPN
ejpam-1773	50	2	dougherty	dougherty	PROPN
ejpam-1773	50	3	,	,	PUNCT
ejpam-1773	50	4	b.yıldız	b.yıldız	NOUN
ejpam-1773	50	5	,	,	PUNCT
ejpam-1773	50	6	s.karadeniz	s.karadeniz	NOUN
ejpam-1773	50	7	/	/	SYM
ejpam-1773	50	8	eur	eur	NOUN
ejpam-1773	50	9	.	.	PUNCT
ejpam-1773	51	1	j.	j.	PROPN
ejpam-1773	51	2	pure	pure	PROPN
ejpam-1773	51	3	appl	appl	PROPN
ejpam-1773	51	4	.	.	PROPN
ejpam-1773	51	5	math	math	PROPN
ejpam-1773	51	6	,	,	PUNCT
ejpam-1773	51	7	6	6	NUM
ejpam-1773	51	8	(	(	PUNCT
ejpam-1773	51	9	2013	2013	NUM
ejpam-1773	51	10	)	)	PUNCT
ejpam-1773	51	11	,	,	PUNCT
ejpam-1773	51	12	89	89	NUM
ejpam-1773	51	13	-	-	SYM
ejpam-1773	51	14	106	106	NUM
ejpam-1773	51	15	91	91	NUM
ejpam-1773	51	16	in	in	ADP
ejpam-1773	51	17	[	[	X
ejpam-1773	51	18	7	7	NUM
ejpam-1773	51	19	]	]	PUNCT
ejpam-1773	51	20	,	,	PUNCT
ejpam-1773	51	21	it	it	PRON
ejpam-1773	51	22	is	be	AUX
ejpam-1773	51	23	shown	show	VERB
ejpam-1773	51	24	that	that	SCONJ
ejpam-1773	51	25	an	an	DET
ejpam-1773	51	26	element	element	NOUN
ejpam-1773	51	27	of	of	ADP
ejpam-1773	51	28	rk	rk	NOUN
ejpam-1773	51	29	is	be	AUX
ejpam-1773	51	30	a	a	DET
ejpam-1773	51	31	unit	unit	NOUN
ejpam-1773	51	32	if	if	SCONJ
ejpam-1773	51	33	and	and	CCONJ
ejpam-1773	51	34	only	only	ADV
ejpam-1773	51	35	if	if	SCONJ
ejpam-1773	51	36	the	the	DET
ejpam-1773	51	37	coefficient	coefficient	NOUN
ejpam-1773	51	38	of	of	ADP
ejpam-1773	51	39	u	u	NOUN
ejpam-1773	51	40	;	;	PUNCT
ejpam-1773	51	41	is	be	AUX
ejpam-1773	51	42	1	1	NUM
ejpam-1773	51	43	and	and	CCONJ
ejpam-1773	51	44	that	that	SCONJ
ejpam-1773	51	45	each	each	DET
ejpam-1773	51	46	unit	unit	NOUN
ejpam-1773	51	47	is	be	AUX
ejpam-1773	51	48	also	also	ADV
ejpam-1773	51	49	its	its	PRON
ejpam-1773	51	50	own	own	ADJ
ejpam-1773	51	51	inverse	inverse	NOUN
ejpam-1773	51	52	.	.	PUNCT
ejpam-1773	52	1	see	see	VERB
ejpam-1773	52	2	[	[	X
ejpam-1773	52	3	7	7	X
ejpam-1773	52	4	]	]	PUNCT
ejpam-1773	52	5	for	for	ADP
ejpam-1773	52	6	proofs	proof	NOUN
ejpam-1773	52	7	of	of	ADP
ejpam-1773	52	8	these	these	DET
ejpam-1773	52	9	and	and	CCONJ
ejpam-1773	52	10	other	other	ADJ
ejpam-1773	52	11	foundational	foundational	ADJ
ejpam-1773	52	12	results	result	NOUN
ejpam-1773	52	13	for	for	ADP
ejpam-1773	52	14	finite	finite	PROPN
ejpam-1773	52	15	k.	k.	PROPN
ejpam-1773	53	1	the	the	DET
ejpam-1773	53	2	proofs	proof	NOUN
ejpam-1773	53	3	are	be	AUX
ejpam-1773	53	4	similar	similar	ADJ
ejpam-1773	53	5	for	for	ADP
ejpam-1773	53	6	r∞.	r∞.	NOUN
ejpam-1773	53	7	throughout	throughout	PROPN
ejpam-1773	53	8	,	,	PUNCT
ejpam-1773	53	9	unless	unless	SCONJ
ejpam-1773	53	10	otherwise	otherwise	ADV
ejpam-1773	53	11	specified	specify	VERB
ejpam-1773	53	12	,	,	PUNCT
ejpam-1773	53	13	k	k	PROPN
ejpam-1773	53	14	can	can	AUX
ejpam-1773	53	15	be	be	AUX
ejpam-1773	53	16	any	any	DET
ejpam-1773	53	17	natural	natural	ADJ
ejpam-1773	53	18	number	number	NOUN
ejpam-1773	53	19	greater	great	ADJ
ejpam-1773	53	20	than	than	ADP
ejpam-1773	53	21	0	0	NUM
ejpam-1773	53	22	or	or	CCONJ
ejpam-1773	53	23	can	can	AUX
ejpam-1773	53	24	be∞.	be∞.	VERB
ejpam-1773	53	25	we	we	PRON
ejpam-1773	53	26	say	say	VERB
ejpam-1773	53	27	that	that	SCONJ
ejpam-1773	53	28	a	a	DET
ejpam-1773	53	29	linear	linear	ADJ
ejpam-1773	53	30	code	code	NOUN
ejpam-1773	53	31	of	of	ADP
ejpam-1773	53	32	length	length	NOUN
ejpam-1773	53	33	n	n	CCONJ
ejpam-1773	53	34	over	over	ADP
ejpam-1773	53	35	rk	rk	NOUN
ejpam-1773	53	36	is	be	AUX
ejpam-1773	53	37	an	an	DET
ejpam-1773	53	38	rk	rk	NOUN
ejpam-1773	53	39	-	-	PUNCT
ejpam-1773	53	40	submodule	submodule	NOUN
ejpam-1773	53	41	of	of	ADP
ejpam-1773	53	42	rn	rn	PROPN
ejpam-1773	53	43	k	k	PROPN
ejpam-1773	53	44	.	.	PUNCT
ejpam-1773	54	1	notice	notice	VERB
ejpam-1773	54	2	that	that	SCONJ
ejpam-1773	54	3	a	a	DET
ejpam-1773	54	4	code	code	NOUN
ejpam-1773	54	5	over	over	ADP
ejpam-1773	54	6	r∞	r∞	PROPN
ejpam-1773	54	7	is	be	AUX
ejpam-1773	54	8	an	an	DET
ejpam-1773	54	9	infinite	infinite	ADJ
ejpam-1773	54	10	module	module	NOUN
ejpam-1773	54	11	.	.	PUNCT
ejpam-1773	55	1	we	we	PRON
ejpam-1773	55	2	define	define	VERB
ejpam-1773	55	3	the	the	DET
ejpam-1773	55	4	inner	inner	ADJ
ejpam-1773	55	5	product	product	NOUN
ejpam-1773	55	6	on	on	ADP
ejpam-1773	55	7	rn	rn	PROPN
ejpam-1773	55	8	k	k	PROPN
ejpam-1773	55	9	in	in	ADP
ejpam-1773	55	10	the	the	DET
ejpam-1773	55	11	usual	usual	ADJ
ejpam-1773	55	12	way	way	NOUN
ejpam-1773	55	13	,	,	PUNCT
ejpam-1773	55	14	that	that	PRON
ejpam-1773	55	15	is	be	AUX
ejpam-1773	55	16	[	[	X
ejpam-1773	55	17	v	v	NOUN
ejpam-1773	55	18	,	,	PUNCT
ejpam-1773	55	19	w]k	w]k	NOUN
ejpam-1773	55	20	=	=	SYM
ejpam-1773	55	21	∑	∑	PUNCT
ejpam-1773	55	22	viwi	viwi	NOUN
ejpam-1773	55	23	.	.	PUNCT
ejpam-1773	56	1	the	the	DET
ejpam-1773	56	2	dual	dual	ADJ
ejpam-1773	56	3	c⊥	c⊥	NOUN
ejpam-1773	56	4	is	be	AUX
ejpam-1773	56	5	defined	define	VERB
ejpam-1773	56	6	as	as	ADP
ejpam-1773	56	7	c⊥	c⊥	NOUN
ejpam-1773	56	8	=	=	PUNCT
ejpam-1773	56	9	{	{	PUNCT
ejpam-1773	56	10	v	v	NUM
ejpam-1773	56	11	∈	∈	PROPN
ejpam-1773	56	12	rn	rn	PROPN
ejpam-1773	56	13	k	k	PROPN
ejpam-1773	57	1	|	|	PROPN
ejpam-1773	58	1	[	[	X
ejpam-1773	58	2	v	v	NOUN
ejpam-1773	58	3	,	,	PUNCT
ejpam-1773	58	4	w]k	w]k	NOUN
ejpam-1773	58	5	=	=	SYM
ejpam-1773	58	6	0	0	NUM
ejpam-1773	58	7	for	for	ADP
ejpam-1773	58	8	all	all	DET
ejpam-1773	58	9	w	w	PROPN
ejpam-1773	58	10	∈	∈	PROPN
ejpam-1773	58	11	c	c	NOUN
ejpam-1773	58	12	}	}	PUNCT
ejpam-1773	58	13	.	.	PUNCT
ejpam-1773	59	1	by	by	ADP
ejpam-1773	59	2	[	[	X
ejpam-1773	59	3	11	11	NUM
ejpam-1773	59	4	]	]	PUNCT
ejpam-1773	59	5	,	,	PUNCT
ejpam-1773	59	6	we	we	PRON
ejpam-1773	59	7	know	know	VERB
ejpam-1773	59	8	that	that	SCONJ
ejpam-1773	59	9	for	for	ADP
ejpam-1773	59	10	finite	finite	NOUN
ejpam-1773	59	11	k	k	PROPN
ejpam-1773	59	12	a	a	DET
ejpam-1773	59	13	linear	linear	PROPN
ejpam-1773	59	14	code	code	NOUN
ejpam-1773	59	15	c	c	NOUN
ejpam-1773	59	16	over	over	ADP
ejpam-1773	59	17	rk	rk	NOUN
ejpam-1773	59	18	of	of	ADP
ejpam-1773	59	19	length	length	NOUN
ejpam-1773	60	1	n	n	PROPN
ejpam-1773	60	2	satisfies	satisfie	NOUN
ejpam-1773	60	3	|c	|c	VERB
ejpam-1773	60	4	||c⊥|	||c⊥|	PROPN
ejpam-1773	60	5	=	=	SYM
ejpam-1773	60	6	|rk|n	|rk|n	NOUN
ejpam-1773	60	7	.	.	NOUN
ejpam-1773	61	1	we	we	PRON
ejpam-1773	61	2	say	say	VERB
ejpam-1773	61	3	that	that	SCONJ
ejpam-1773	61	4	a	a	DET
ejpam-1773	61	5	code	code	NOUN
ejpam-1773	61	6	is	be	AUX
ejpam-1773	61	7	self	self	NOUN
ejpam-1773	61	8	-	-	PUNCT
ejpam-1773	61	9	orthogonal	orthogonal	ADJ
ejpam-1773	61	10	if	if	SCONJ
ejpam-1773	61	11	c	c	PROPN
ejpam-1773	61	12	⊆	⊆	NUM
ejpam-1773	61	13	c⊥	c⊥	NOUN
ejpam-1773	61	14	and	and	CCONJ
ejpam-1773	61	15	self	self	NOUN
ejpam-1773	61	16	-	-	PUNCT
ejpam-1773	61	17	dual	dual	ADJ
ejpam-1773	61	18	if	if	SCONJ
ejpam-1773	61	19	c	c	NOUN
ejpam-1773	61	20	=	=	PUNCT
ejpam-1773	61	21	c⊥.	c⊥.	NOUN
ejpam-1773	61	22	we	we	PRON
ejpam-1773	61	23	define	define	VERB
ejpam-1773	61	24	the	the	DET
ejpam-1773	61	25	gray	gray	ADJ
ejpam-1773	61	26	map	map	NOUN
ejpam-1773	61	27	inductively	inductively	ADV
ejpam-1773	61	28	,	,	PUNCT
ejpam-1773	61	29	extending	extend	VERB
ejpam-1773	61	30	it	it	PRON
ejpam-1773	61	31	naturally	naturally	ADV
ejpam-1773	61	32	from	from	ADP
ejpam-1773	61	33	the	the	DET
ejpam-1773	61	34	gray	gray	ADJ
ejpam-1773	61	35	map	map	NOUN
ejpam-1773	61	36	on	on	ADP
ejpam-1773	61	37	r1	r1	PROPN
ejpam-1773	61	38	from	from	ADP
ejpam-1773	61	39	[	[	X
ejpam-1773	61	40	6	6	NUM
ejpam-1773	61	41	]	]	PUNCT
ejpam-1773	61	42	as	as	SCONJ
ejpam-1773	61	43	follows	follow	VERB
ejpam-1773	61	44	.	.	PUNCT
ejpam-1773	62	1	for	for	ADP
ejpam-1773	62	2	c	c	PROPN
ejpam-1773	62	3	∈	∈	PROPN
ejpam-1773	62	4	rn	rn	PROPN
ejpam-1773	62	5	k	k	PROPN
ejpam-1773	62	6	,	,	PUNCT
ejpam-1773	62	7	we	we	PRON
ejpam-1773	62	8	can	can	AUX
ejpam-1773	62	9	write	write	VERB
ejpam-1773	62	10	c=	c=	NOUN
ejpam-1773	62	11	c1	c1	PROPN
ejpam-1773	62	12	+	+	CCONJ
ejpam-1773	62	13	ukc2	ukc2	PROPN
ejpam-1773	62	14	with	with	ADP
ejpam-1773	62	15	c1,c2	c1,c2	PROPN
ejpam-1773	62	16	∈	∈	PROPN
ejpam-1773	62	17	rn	rn	PROPN
ejpam-1773	62	18	k−1	k−1	PROPN
ejpam-1773	62	19	,	,	PUNCT
ejpam-1773	62	20	then	then	ADV
ejpam-1773	62	21	we	we	PRON
ejpam-1773	62	22	can	can	AUX
ejpam-1773	62	23	define	define	VERB
ejpam-1773	62	24	φk(c	φk(c	PUNCT
ejpam-1773	62	25	)	)	PUNCT
ejpam-1773	62	26	=	=	SYM
ejpam-1773	62	27	�	�	PROPN
ejpam-1773	62	28	φk−1(c2),φk−1(c1	φk−1(c2),φk−1(c1	PUNCT
ejpam-1773	62	29	)	)	PUNCT
ejpam-1773	62	30	+	+	ADJ
ejpam-1773	62	31	φk−1(c2	φk−1(c2	PROPN
ejpam-1773	62	32	)	)	PUNCT
ejpam-1773	62	33	�	�	PROPN
ejpam-1773	62	34	,	,	PUNCT
ejpam-1773	62	35	with	with	ADP
ejpam-1773	62	36	φ0	φ0	PROPN
ejpam-1773	62	37	being	be	AUX
ejpam-1773	62	38	the	the	DET
ejpam-1773	62	39	identity	identity	NOUN
ejpam-1773	62	40	map	map	NOUN
ejpam-1773	62	41	on	on	ADP
ejpam-1773	62	42	f2	f2	PROPN
ejpam-1773	62	43	.	.	PUNCT
ejpam-1773	63	1	the	the	DET
ejpam-1773	63	2	lee	lee	PROPN
ejpam-1773	63	3	weight	weight	NOUN
ejpam-1773	63	4	of	of	ADP
ejpam-1773	63	5	a	a	DET
ejpam-1773	63	6	codeword	codeword	NOUN
ejpam-1773	63	7	is	be	AUX
ejpam-1773	63	8	the	the	DET
ejpam-1773	63	9	hamming	hamming	ADJ
ejpam-1773	63	10	weight	weight	NOUN
ejpam-1773	63	11	of	of	ADP
ejpam-1773	63	12	the	the	DET
ejpam-1773	63	13	image	image	NOUN
ejpam-1773	63	14	of	of	ADP
ejpam-1773	63	15	the	the	DET
ejpam-1773	63	16	codeword	codeword	NOUN
ejpam-1773	63	17	under	under	ADP
ejpam-1773	63	18	φk	φk	ADP
ejpam-1773	63	19	.	.	PUNCT
ejpam-1773	64	1	the	the	DET
ejpam-1773	64	2	lee	lee	PROPN
ejpam-1773	64	3	distance	distance	NOUN
ejpam-1773	64	4	is	be	AUX
ejpam-1773	64	5	defined	define	VERB
ejpam-1773	64	6	similarly	similarly	ADV
ejpam-1773	64	7	.	.	PUNCT
ejpam-1773	65	1	it	it	PRON
ejpam-1773	65	2	’s	’	VERB
ejpam-1773	65	3	clear	clear	ADJ
ejpam-1773	65	4	that	that	SCONJ
ejpam-1773	65	5	the	the	DET
ejpam-1773	65	6	gray	gray	ADJ
ejpam-1773	65	7	map	map	NOUN
ejpam-1773	65	8	φk	φk	ADP
ejpam-1773	65	9	is	be	AUX
ejpam-1773	65	10	a	a	DET
ejpam-1773	65	11	linear	linear	ADJ
ejpam-1773	65	12	weight	weight	NOUN
ejpam-1773	65	13	preserving	preserve	VERB
ejpam-1773	65	14	map	map	NOUN
ejpam-1773	65	15	from	from	ADP
ejpam-1773	65	16	rn	rn	PROPN
ejpam-1773	65	17	k	k	PROPN
ejpam-1773	65	18	to	to	PART
ejpam-1773	65	19	f2kn	f2kn	PROPN
ejpam-1773	65	20	2	2	NUM
ejpam-1773	65	21	as	as	SCONJ
ejpam-1773	65	22	was	be	AUX
ejpam-1773	65	23	shown	show	VERB
ejpam-1773	65	24	in	in	ADP
ejpam-1773	65	25	[	[	X
ejpam-1773	65	26	7	7	NUM
ejpam-1773	65	27	]	]	PUNCT
ejpam-1773	65	28	.	.	PUNCT
ejpam-1773	66	1	if	if	SCONJ
ejpam-1773	66	2	all	all	DET
ejpam-1773	66	3	the	the	DET
ejpam-1773	66	4	codewords	codeword	NOUN
ejpam-1773	66	5	of	of	ADP
ejpam-1773	66	6	a	a	DET
ejpam-1773	66	7	self	self	NOUN
ejpam-1773	66	8	-	-	PUNCT
ejpam-1773	66	9	dual	dual	ADJ
ejpam-1773	66	10	code	code	NOUN
ejpam-1773	66	11	have	have	VERB
ejpam-1773	66	12	doubly	doubly	ADV
ejpam-1773	66	13	-	-	PUNCT
ejpam-1773	66	14	even	even	ADV
ejpam-1773	66	15	lee	lee	PROPN
ejpam-1773	66	16	weight	weight	NOUN
ejpam-1773	66	17	then	then	ADV
ejpam-1773	66	18	the	the	DET
ejpam-1773	66	19	code	code	NOUN
ejpam-1773	66	20	is	be	AUX
ejpam-1773	66	21	said	say	VERB
ejpam-1773	66	22	to	to	PART
ejpam-1773	66	23	be	be	AUX
ejpam-1773	66	24	type	type	NOUN
ejpam-1773	66	25	ii	ii	NOUN
ejpam-1773	66	26	,	,	PUNCT
ejpam-1773	66	27	otherwise	otherwise	ADV
ejpam-1773	66	28	it	it	PRON
ejpam-1773	66	29	is	be	AUX
ejpam-1773	66	30	said	say	VERB
ejpam-1773	66	31	to	to	PART
ejpam-1773	66	32	be	be	AUX
ejpam-1773	66	33	type	type	NOUN
ejpam-1773	66	34	i.	i.	NOUN
ejpam-1773	66	35	it	it	PRON
ejpam-1773	66	36	is	be	AUX
ejpam-1773	66	37	immediate	immediate	ADJ
ejpam-1773	66	38	that	that	SCONJ
ejpam-1773	66	39	φk	φk	SCONJ
ejpam-1773	66	40	is	be	AUX
ejpam-1773	66	41	one	one	NUM
ejpam-1773	66	42	-	-	PUNCT
ejpam-1773	66	43	to	to	ADP
ejpam-1773	66	44	-	-	PUNCT
ejpam-1773	66	45	one	one	NUM
ejpam-1773	66	46	and	and	CCONJ
ejpam-1773	66	47	that	that	DET
ejpam-1773	66	48	wl(ua	wl(ua	NOUN
ejpam-1773	66	49	)	)	PUNCT
ejpam-1773	67	1	=	=	SYM
ejpam-1773	67	2	2|a|	2|a|	NUM
ejpam-1773	67	3	for	for	ADP
ejpam-1773	67	4	each	each	DET
ejpam-1773	67	5	a⊆	a⊆	NOUN
ejpam-1773	67	6	{	{	PUNCT
ejpam-1773	67	7	1,2	1,2	NUM
ejpam-1773	67	8	,	,	PUNCT
ejpam-1773	67	9	.	.	PUNCT
ejpam-1773	67	10	.	.	PUNCT
ejpam-1773	67	11	.	.	PUNCT
ejpam-1773	68	1	,	,	PUNCT
ejpam-1773	68	2	k	k	X
ejpam-1773	68	3	}	}	PUNCT
ejpam-1773	68	4	,	,	PUNCT
ejpam-1773	68	5	see	see	VERB
ejpam-1773	68	6	[	[	X
ejpam-1773	68	7	7	7	X
ejpam-1773	68	8	]	]	PUNCT
ejpam-1773	68	9	for	for	ADP
ejpam-1773	68	10	details	detail	NOUN
ejpam-1773	68	11	.	.	PUNCT
ejpam-1773	69	1	the	the	DET
ejpam-1773	69	2	complete	complete	ADJ
ejpam-1773	69	3	weight	weight	NOUN
ejpam-1773	69	4	enumerator	enumerator	NOUN
ejpam-1773	69	5	of	of	ADP
ejpam-1773	69	6	a	a	DET
ejpam-1773	69	7	code	code	NOUN
ejpam-1773	69	8	c	c	NOUN
ejpam-1773	69	9	over	over	ADP
ejpam-1773	69	10	rn	rn	PROPN
ejpam-1773	69	11	k	k	PROPN
ejpam-1773	69	12	is	be	AUX
ejpam-1773	69	13	defined	define	VERB
ejpam-1773	69	14	as	as	ADP
ejpam-1773	69	15	:	:	PUNCT
ejpam-1773	69	16	cwec	cwec	NOUN
ejpam-1773	69	17	(	(	PUNCT
ejpam-1773	69	18	x	x	X
ejpam-1773	69	19	)	)	PUNCT
ejpam-1773	69	20	=	=	SYM
ejpam-1773	69	21	∑	∑	PUNCT
ejpam-1773	69	22	c∈c	c∈c	NOUN
ejpam-1773	69	23	n	n	CCONJ
ejpam-1773	69	24	∏	∏	PROPN
ejpam-1773	70	1	i=1	i=1	PROPN
ejpam-1773	71	1	xci	xci	PROPN
ejpam-1773	71	2	.	.	PUNCT
ejpam-1773	72	1	(	(	PUNCT
ejpam-1773	72	2	5	5	X
ejpam-1773	72	3	)	)	PUNCT
ejpam-1773	72	4	the	the	DET
ejpam-1773	72	5	hamming	hamming	NOUN
ejpam-1773	72	6	weight	weight	NOUN
ejpam-1773	72	7	of	of	ADP
ejpam-1773	72	8	a	a	DET
ejpam-1773	72	9	vector	vector	NOUN
ejpam-1773	72	10	c	c	NOUN
ejpam-1773	72	11	is	be	AUX
ejpam-1773	72	12	denoted	denote	VERB
ejpam-1773	72	13	by	by	ADP
ejpam-1773	72	14	wt(c	wt(c	NOUN
ejpam-1773	72	15	)	)	PUNCT
ejpam-1773	72	16	and	and	CCONJ
ejpam-1773	72	17	is	be	AUX
ejpam-1773	72	18	the	the	DET
ejpam-1773	72	19	number	number	NOUN
ejpam-1773	72	20	of	of	ADP
ejpam-1773	72	21	non	non	ADJ
ejpam-1773	72	22	-	-	ADJ
ejpam-1773	72	23	zero	zero	NUM
ejpam-1773	72	24	coordinates	coordinate	NOUN
ejpam-1773	72	25	of	of	ADP
ejpam-1773	72	26	the	the	DET
ejpam-1773	72	27	element	element	NOUN
ejpam-1773	72	28	.	.	PUNCT
ejpam-1773	73	1	the	the	DET
ejpam-1773	73	2	minimum	minimum	ADJ
ejpam-1773	73	3	weight	weight	NOUN
ejpam-1773	73	4	is	be	AUX
ejpam-1773	73	5	the	the	DET
ejpam-1773	73	6	minimum	minimum	NOUN
ejpam-1773	73	7	of	of	ADP
ejpam-1773	73	8	all	all	DET
ejpam-1773	73	9	non	non	ADJ
ejpam-1773	73	10	-	-	ADJ
ejpam-1773	73	11	zero	zero	NUM
ejpam-1773	73	12	weights	weight	NOUN
ejpam-1773	73	13	in	in	ADP
ejpam-1773	73	14	the	the	DET
ejpam-1773	73	15	code	code	NOUN
ejpam-1773	73	16	.	.	PUNCT
ejpam-1773	74	1	we	we	PRON
ejpam-1773	74	2	denote	denote	VERB
ejpam-1773	74	3	the	the	DET
ejpam-1773	74	4	minimum	minimum	ADJ
ejpam-1773	74	5	hamming	hamming	NOUN
ejpam-1773	74	6	distance	distance	NOUN
ejpam-1773	74	7	by	by	ADP
ejpam-1773	74	8	dh(c	dh(c	PROPN
ejpam-1773	74	9	)	)	PUNCT
ejpam-1773	74	10	and	and	CCONJ
ejpam-1773	74	11	the	the	DET
ejpam-1773	74	12	minimum	minimum	ADJ
ejpam-1773	74	13	lee	lee	PROPN
ejpam-1773	74	14	distance	distance	NOUN
ejpam-1773	74	15	by	by	ADP
ejpam-1773	74	16	dl(c	dl(c	PROPN
ejpam-1773	74	17	)	)	PUNCT
ejpam-1773	74	18	.	.	PUNCT
ejpam-1773	75	1	the	the	DET
ejpam-1773	75	2	hamming	ham	VERB
ejpam-1773	75	3	weight	weight	NOUN
ejpam-1773	75	4	enumerator	enumerator	NOUN
ejpam-1773	75	5	is	be	AUX
ejpam-1773	75	6	defined	define	VERB
ejpam-1773	75	7	as	as	ADP
ejpam-1773	75	8	:	:	PUNCT
ejpam-1773	75	9	wc	wc	PROPN
ejpam-1773	75	10	(	(	PUNCT
ejpam-1773	75	11	x	x	PROPN
ejpam-1773	75	12	,	,	PUNCT
ejpam-1773	75	13	y	y	PROPN
ejpam-1773	75	14	)	)	PUNCT
ejpam-1773	75	15	=	=	SYM
ejpam-1773	75	16	∑	∑	PUNCT
ejpam-1773	75	17	c∈c	c∈c	NOUN
ejpam-1773	75	18	xn−wt(c	xn−wt(c	PROPN
ejpam-1773	75	19	)	)	PUNCT
ejpam-1773	75	20	ywt(c	ywt(c	NUM
ejpam-1773	75	21	)	)	PUNCT
ejpam-1773	75	22	.	.	PUNCT
ejpam-1773	76	1	(	(	PUNCT
ejpam-1773	76	2	6	6	X
ejpam-1773	76	3	)	)	PUNCT
ejpam-1773	76	4	the	the	DET
ejpam-1773	76	5	lee	lee	PROPN
ejpam-1773	76	6	weight	weight	NOUN
ejpam-1773	76	7	enumerator	enumerator	NOUN
ejpam-1773	76	8	is	be	AUX
ejpam-1773	76	9	defined	define	VERB
ejpam-1773	76	10	to	to	PART
ejpam-1773	76	11	be	be	AUX
ejpam-1773	76	12	lc	lc	PROPN
ejpam-1773	76	13	(	(	PUNCT
ejpam-1773	76	14	z	z	NOUN
ejpam-1773	76	15	)	)	PUNCT
ejpam-1773	76	16	=	=	SYM
ejpam-1773	76	17	∑	∑	PUNCT
ejpam-1773	76	18	c∈c	c∈c	NOUN
ejpam-1773	76	19	zle(c	zle(c	NUM
ejpam-1773	76	20	)	)	PUNCT
ejpam-1773	76	21	,	,	PUNCT
ejpam-1773	76	22	(	(	PUNCT
ejpam-1773	76	23	7	7	X
ejpam-1773	76	24	)	)	PUNCT
ejpam-1773	76	25	where	where	SCONJ
ejpam-1773	76	26	le(c	le(c	NUM
ejpam-1773	76	27	)	)	PUNCT
ejpam-1773	76	28	is	be	AUX
ejpam-1773	76	29	the	the	DET
ejpam-1773	76	30	lee	lee	PROPN
ejpam-1773	76	31	weight	weight	NOUN
ejpam-1773	76	32	of	of	ADP
ejpam-1773	76	33	the	the	DET
ejpam-1773	76	34	codeword	codeword	NOUN
ejpam-1773	76	35	c.	c.	PROPN
ejpam-1773	77	1	the	the	DET
ejpam-1773	77	2	macwilliams	macwilliams	PROPN
ejpam-1773	77	3	relations	relation	NOUN
ejpam-1773	77	4	for	for	ADP
ejpam-1773	77	5	both	both	PRON
ejpam-1773	77	6	of	of	ADP
ejpam-1773	77	7	these	these	DET
ejpam-1773	77	8	weight	weight	NOUN
ejpam-1773	77	9	enumerators	enumerator	NOUN
ejpam-1773	77	10	are	be	AUX
ejpam-1773	77	11	given	give	VERB
ejpam-1773	77	12	in	in	ADP
ejpam-1773	77	13	[	[	X
ejpam-1773	77	14	7	7	NUM
ejpam-1773	77	15	]	]	PUNCT
ejpam-1773	77	16	.	.	PUNCT
ejpam-1773	78	1	s.	s.	PROPN
ejpam-1773	78	2	dougherty	dougherty	PROPN
ejpam-1773	78	3	,	,	PUNCT
ejpam-1773	78	4	b.yıldız	b.yıldız	NOUN
ejpam-1773	78	5	,	,	PUNCT
ejpam-1773	78	6	s.karadeniz	s.karadeniz	NOUN
ejpam-1773	78	7	/	/	SYM
ejpam-1773	78	8	eur	eur	NOUN
ejpam-1773	78	9	.	.	PUNCT
ejpam-1773	79	1	j.	j.	PROPN
ejpam-1773	79	2	pure	pure	PROPN
ejpam-1773	79	3	appl	appl	PROPN
ejpam-1773	79	4	.	.	PROPN
ejpam-1773	79	5	math	math	PROPN
ejpam-1773	79	6	,	,	PUNCT
ejpam-1773	79	7	6	6	NUM
ejpam-1773	79	8	(	(	PUNCT
ejpam-1773	79	9	2013	2013	NUM
ejpam-1773	79	10	)	)	PUNCT
ejpam-1773	79	11	,	,	PUNCT
ejpam-1773	79	12	89	89	NUM
ejpam-1773	79	13	-	-	SYM
ejpam-1773	79	14	106	106	NUM
ejpam-1773	79	15	92	92	NUM
ejpam-1773	79	16	3	3	NUM
ejpam-1773	79	17	.	.	PUNCT
ejpam-1773	79	18	projections	projection	NOUN
ejpam-1773	79	19	and	and	CCONJ
ejpam-1773	79	20	lifts	lift	NOUN
ejpam-1773	79	21	for	for	ADP
ejpam-1773	79	22	j	j	PROPN
ejpam-1773	79	23	≥	≥	PROPN
ejpam-1773	79	24	k	k	PROPN
ejpam-1773	79	25	≥	≥	PROPN
ejpam-1773	79	26	0	0	NUM
ejpam-1773	79	27	,	,	PUNCT
ejpam-1773	79	28	define	define	VERB
ejpam-1773	79	29	π	π	PROPN
ejpam-1773	79	30	j	j	PROPN
ejpam-1773	79	31	,	,	PUNCT
ejpam-1773	79	32	k	k	PROPN
ejpam-1773	79	33	:	:	PUNCT
ejpam-1773	79	34	r	r	NOUN
ejpam-1773	79	35	j	j	PROPN
ejpam-1773	79	36	→	→	SYM
ejpam-1773	79	37	rk	rk	NOUN
ejpam-1773	79	38	by	by	ADP
ejpam-1773	79	39	π	π	PROPN
ejpam-1773	79	40	j	j	PROPN
ejpam-1773	79	41	,	,	PUNCT
ejpam-1773	79	42	k(ui	k(ui	PROPN
ejpam-1773	79	43	)	)	PUNCT
ejpam-1773	80	1	=	=	SYM
ejpam-1773	80	2	0	0	PUNCT
ejpam-1773	81	1	if	if	SCONJ
ejpam-1773	81	2	i	i	PRON
ejpam-1773	81	3	>	>	X
ejpam-1773	81	4	k	k	PROPN
ejpam-1773	81	5	and	and	CCONJ
ejpam-1773	81	6	the	the	DET
ejpam-1773	81	7	identity	identity	NOUN
ejpam-1773	81	8	elsewhere	elsewhere	ADV
ejpam-1773	81	9	.	.	PUNCT
ejpam-1773	82	1	that	that	PRON
ejpam-1773	82	2	is	be	AUX
ejpam-1773	82	3	π	π	PROPN
ejpam-1773	82	4	j	j	PROPN
ejpam-1773	82	5	,	,	PUNCT
ejpam-1773	82	6	k	k	PROPN
ejpam-1773	82	7	is	be	AUX
ejpam-1773	82	8	the	the	DET
ejpam-1773	82	9	projection	projection	NOUN
ejpam-1773	82	10	of	of	ADP
ejpam-1773	82	11	r	r	PROPN
ejpam-1773	82	12	j	j	PROPN
ejpam-1773	82	13	to	to	PART
ejpam-1773	82	14	rk	rk	VERB
ejpam-1773	82	15	.	.	PUNCT
ejpam-1773	82	16	note	note	VERB
ejpam-1773	82	17	that	that	SCONJ
ejpam-1773	82	18	if	if	SCONJ
ejpam-1773	82	19	j	j	PROPN
ejpam-1773	82	20	≤	≤	PROPN
ejpam-1773	82	21	k	k	PROPN
ejpam-1773	82	22	,	,	PUNCT
ejpam-1773	82	23	then	then	ADV
ejpam-1773	82	24	π	π	PROPN
ejpam-1773	82	25	j	j	PROPN
ejpam-1773	82	26	,	,	PUNCT
ejpam-1773	82	27	k	k	PROPN
ejpam-1773	82	28	is	be	AUX
ejpam-1773	82	29	taken	take	VERB
ejpam-1773	82	30	to	to	PART
ejpam-1773	82	31	be	be	AUX
ejpam-1773	82	32	the	the	DET
ejpam-1773	82	33	identity	identity	NOUN
ejpam-1773	82	34	map	map	NOUN
ejpam-1773	82	35	on	on	ADP
ejpam-1773	82	36	r	r	PROPN
ejpam-1773	82	37	j.	j.	PROPN
ejpam-1773	82	38	we	we	PRON
ejpam-1773	82	39	allow	allow	VERB
ejpam-1773	82	40	j	j	PROPN
ejpam-1773	82	41	to	to	ADP
ejpam-1773	82	42	be∞	be∞	PROPN
ejpam-1773	82	43	as	as	ADV
ejpam-1773	82	44	well	well	ADV
ejpam-1773	82	45	and	and	CCONJ
ejpam-1773	82	46	denote	denote	VERB
ejpam-1773	82	47	this	this	DET
ejpam-1773	82	48	map	map	NOUN
ejpam-1773	82	49	by	by	ADP
ejpam-1773	82	50	π∞,k	π∞,k	NOUN
ejpam-1773	82	51	.	.	PUNCT
ejpam-1773	83	1	if	if	SCONJ
ejpam-1773	83	2	c	c	PROPN
ejpam-1773	83	3	=	=	SYM
ejpam-1773	83	4	π	π	PROPN
ejpam-1773	83	5	j	j	PROPN
ejpam-1773	83	6	,	,	PUNCT
ejpam-1773	83	7	k(c	k(c	PROPN
ejpam-1773	83	8	′	′	NUM
ejpam-1773	83	9	)	)	PUNCT
ejpam-1773	83	10	for	for	ADP
ejpam-1773	83	11	some	some	DET
ejpam-1773	83	12	c	c	NOUN
ejpam-1773	83	13	′	′	NOUN
ejpam-1773	83	14	and	and	CCONJ
ejpam-1773	83	15	j	j	PROPN
ejpam-1773	83	16	≥	≥	NUM
ejpam-1773	83	17	k	k	NOUN
ejpam-1773	83	18	,	,	PUNCT
ejpam-1773	83	19	then	then	ADV
ejpam-1773	83	20	c	c	NOUN
ejpam-1773	83	21	′	′	PROPN
ejpam-1773	83	22	is	be	AUX
ejpam-1773	83	23	said	say	VERB
ejpam-1773	83	24	to	to	PART
ejpam-1773	83	25	be	be	AUX
ejpam-1773	83	26	a	a	DET
ejpam-1773	83	27	lift	lift	NOUN
ejpam-1773	83	28	of	of	ADP
ejpam-1773	83	29	c	c	PROPN
ejpam-1773	83	30	.	.	PUNCT
ejpam-1773	84	1	theorem	theorem	NOUN
ejpam-1773	84	2	1	1	X
ejpam-1773	84	3	.	.	PUNCT
ejpam-1773	85	1	let	let	VERB
ejpam-1773	85	2	c	c	PRON
ejpam-1773	85	3	be	be	AUX
ejpam-1773	85	4	a	a	DET
ejpam-1773	85	5	self	self	NOUN
ejpam-1773	85	6	-	-	PUNCT
ejpam-1773	85	7	dual	dual	ADJ
ejpam-1773	85	8	code	code	NOUN
ejpam-1773	85	9	over	over	ADP
ejpam-1773	85	10	r	r	PROPN
ejpam-1773	85	11	j	j	PROPN
ejpam-1773	85	12	then	then	ADV
ejpam-1773	85	13	π	π	PROPN
ejpam-1773	85	14	j	j	PROPN
ejpam-1773	85	15	,	,	PUNCT
ejpam-1773	85	16	k(c	k(c	PROPN
ejpam-1773	85	17	)	)	PUNCT
ejpam-1773	85	18	is	be	AUX
ejpam-1773	85	19	a	a	DET
ejpam-1773	85	20	self	self	NOUN
ejpam-1773	85	21	-	-	PUNCT
ejpam-1773	85	22	orthogonal	orthogonal	ADJ
ejpam-1773	85	23	code	code	NOUN
ejpam-1773	85	24	over	over	ADP
ejpam-1773	85	25	rk	rk	NOUN
ejpam-1773	85	26	.	.	PUNCT
ejpam-1773	85	27	proof	proof	NOUN
ejpam-1773	85	28	.	.	PUNCT
ejpam-1773	86	1	let	let	VERB
ejpam-1773	86	2	v=	v=	NOUN
ejpam-1773	86	3	(	(	PUNCT
ejpam-1773	86	4	v1	v1	NOUN
ejpam-1773	86	5	,	,	PUNCT
ejpam-1773	86	6	.	.	PUNCT
ejpam-1773	86	7	.	.	PUNCT
ejpam-1773	87	1	.	.	PUNCT
ejpam-1773	88	1	,	,	PUNCT
ejpam-1773	88	2	vn	vn	PROPN
ejpam-1773	88	3	)	)	PUNCT
ejpam-1773	88	4	and	and	CCONJ
ejpam-1773	88	5	w=	w=	PROPN
ejpam-1773	88	6	(	(	PUNCT
ejpam-1773	88	7	w1	w1	NOUN
ejpam-1773	88	8	,	,	PUNCT
ejpam-1773	88	9	.	.	PUNCT
ejpam-1773	88	10	.	.	PUNCT
ejpam-1773	89	1	.	.	PUNCT
ejpam-1773	90	1	,	,	PUNCT
ejpam-1773	90	2	wn	wn	AUX
ejpam-1773	90	3	)	)	PUNCT
ejpam-1773	90	4	be	be	VERB
ejpam-1773	90	5	vectors	vector	NOUN
ejpam-1773	90	6	in	in	ADP
ejpam-1773	90	7	c	c	PROPN
ejpam-1773	90	8	.	.	PUNCT
ejpam-1773	91	1	we	we	PRON
ejpam-1773	91	2	have	have	VERB
ejpam-1773	91	3	that	that	PRON
ejpam-1773	91	4	π	π	PROPN
ejpam-1773	91	5	j	j	PROPN
ejpam-1773	91	6	,	,	PUNCT
ejpam-1773	91	7	k	k	PROPN
ejpam-1773	91	8	(	(	PUNCT
ejpam-1773	91	9	∑	∑	ADV
ejpam-1773	91	10	viwi	viwi	NOUN
ejpam-1773	91	11	)	)	PUNCT
ejpam-1773	91	12	=	=	PUNCT
ejpam-1773	92	1	∑	∑	PUNCT
ejpam-1773	92	2	(	(	PUNCT
ejpam-1773	92	3	π	π	PROPN
ejpam-1773	92	4	j	j	PROPN
ejpam-1773	92	5	,	,	PUNCT
ejpam-1773	92	6	k(vi)π	k(vi)π	PROPN
ejpam-1773	92	7	j	j	PROPN
ejpam-1773	92	8	,	,	PUNCT
ejpam-1773	92	9	k(wi	k(wi	NOUN
ejpam-1773	92	10	)	)	PUNCT
ejpam-1773	92	11	)	)	PUNCT
ejpam-1773	92	12	.	.	PUNCT
ejpam-1773	93	1	if	if	SCONJ
ejpam-1773	93	2	∑	∑	PUNCT
ejpam-1773	93	3	viwi	viwi	NOUN
ejpam-1773	93	4	=	=	NOUN
ejpam-1773	93	5	0	0	NUM
ejpam-1773	94	1	in	in	ADP
ejpam-1773	94	2	r	r	PROPN
ejpam-1773	94	3	j	j	PROPN
ejpam-1773	94	4	then	then	ADV
ejpam-1773	94	5	π	π	PROPN
ejpam-1773	94	6	j	j	PROPN
ejpam-1773	94	7	,	,	PUNCT
ejpam-1773	94	8	k(0	k(0	PROPN
ejpam-1773	94	9	)	)	PUNCT
ejpam-1773	94	10	=	=	SYM
ejpam-1773	94	11	0	0	NUM
ejpam-1773	95	1	so	so	ADV
ejpam-1773	95	2	〈	〈	PROPN
ejpam-1773	95	3	π	π	PROPN
ejpam-1773	95	4	j	j	PROPN
ejpam-1773	95	5	,	,	PUNCT
ejpam-1773	95	6	k(v),π	k(v),π	PROPN
ejpam-1773	95	7	j	j	PROPN
ejpam-1773	95	8	,	,	PUNCT
ejpam-1773	95	9	k(w)〉k	k(w)〉k	PROPN
ejpam-1773	95	10	=	=	NOUN
ejpam-1773	95	11	0	0	PROPN
ejpam-1773	95	12	.	.	PUNCT
ejpam-1773	96	1	therefore	therefore	ADV
ejpam-1773	96	2	the	the	DET
ejpam-1773	96	3	code	code	NOUN
ejpam-1773	96	4	is	be	AUX
ejpam-1773	96	5	selforthogonal	selforthogonal	ADJ
ejpam-1773	96	6	.	.	PUNCT
ejpam-1773	97	1	the	the	DET
ejpam-1773	97	2	image	image	NOUN
ejpam-1773	97	3	need	need	AUX
ejpam-1773	97	4	not	not	PART
ejpam-1773	97	5	necessarily	necessarily	ADV
ejpam-1773	97	6	be	be	AUX
ejpam-1773	97	7	self	self	NOUN
ejpam-1773	97	8	-	-	PUNCT
ejpam-1773	97	9	dual	dual	ADJ
ejpam-1773	97	10	.	.	PUNCT
ejpam-1773	98	1	for	for	ADP
ejpam-1773	98	2	example	example	NOUN
ejpam-1773	98	3	,	,	PUNCT
ejpam-1773	98	4	consider	consider	VERB
ejpam-1773	98	5	the	the	DET
ejpam-1773	98	6	code	code	NOUN
ejpam-1773	98	7	〈	〈	NOUN
ejpam-1773	98	8	u2	u2	NOUN
ejpam-1773	98	9	〉	〉	NOUN
ejpam-1773	98	10	in	in	ADP
ejpam-1773	98	11	r2	r2	PROPN
ejpam-1773	98	12	.	.	PUNCT
ejpam-1773	99	1	this	this	DET
ejpam-1773	99	2	code	code	NOUN
ejpam-1773	99	3	is	be	AUX
ejpam-1773	99	4	self	self	NOUN
ejpam-1773	99	5	-	-	PUNCT
ejpam-1773	99	6	dual	dual	ADJ
ejpam-1773	99	7	but	but	CCONJ
ejpam-1773	99	8	its	its	PRON
ejpam-1773	99	9	image	image	NOUN
ejpam-1773	99	10	under	under	ADP
ejpam-1773	99	11	π2,1	π2,1	PROPN
ejpam-1773	99	12	is	be	AUX
ejpam-1773	99	13	the	the	DET
ejpam-1773	99	14	zero	zero	NUM
ejpam-1773	99	15	code	code	NOUN
ejpam-1773	99	16	which	which	PRON
ejpam-1773	99	17	is	be	AUX
ejpam-1773	99	18	not	not	PART
ejpam-1773	99	19	self	self	NOUN
ejpam-1773	99	20	-	-	PUNCT
ejpam-1773	99	21	dual	dual	ADJ
ejpam-1773	99	22	.	.	PUNCT
ejpam-1773	100	1	theorem	theorem	NOUN
ejpam-1773	100	2	2	2	NUM
ejpam-1773	100	3	.	.	PUNCT
ejpam-1773	100	4	let	let	VERB
ejpam-1773	100	5	v1	v1	NOUN
ejpam-1773	100	6	,	,	PUNCT
ejpam-1773	100	7	v2	v2	NOUN
ejpam-1773	100	8	,	,	PUNCT
ejpam-1773	100	9	.	.	PUNCT
ejpam-1773	100	10	.	.	PUNCT
ejpam-1773	101	1	.	.	PUNCT
ejpam-1773	102	1	,	,	PUNCT
ejpam-1773	102	2	vs	vs	ADP
ejpam-1773	102	3	generate	generate	VERB
ejpam-1773	102	4	a	a	DET
ejpam-1773	102	5	self	self	NOUN
ejpam-1773	102	6	-	-	PUNCT
ejpam-1773	102	7	dual	dual	ADJ
ejpam-1773	102	8	code	code	NOUN
ejpam-1773	102	9	over	over	ADP
ejpam-1773	102	10	rk	rk	PROPN
ejpam-1773	102	11	(	(	PUNCT
ejpam-1773	102	12	of	of	ADP
ejpam-1773	102	13	length	length	NOUN
ejpam-1773	102	14	1	1	NUM
ejpam-1773	102	15	)	)	PUNCT
ejpam-1773	102	16	,	,	PUNCT
ejpam-1773	102	17	then	then	ADV
ejpam-1773	102	18	v1	v1	VERB
ejpam-1773	102	19	,	,	PUNCT
ejpam-1773	102	20	v2	v2	PROPN
ejpam-1773	102	21	,	,	PUNCT
ejpam-1773	102	22	.	.	PUNCT
ejpam-1773	102	23	.	.	PUNCT
ejpam-1773	103	1	.	.	PUNCT
ejpam-1773	104	1	,	,	PUNCT
ejpam-1773	104	2	vs	vs	ADP
ejpam-1773	104	3	generate	generate	VERB
ejpam-1773	104	4	a	a	DET
ejpam-1773	104	5	self	self	NOUN
ejpam-1773	104	6	-	-	PUNCT
ejpam-1773	104	7	dual	dual	ADJ
ejpam-1773	104	8	code	code	NOUN
ejpam-1773	104	9	over	over	ADP
ejpam-1773	104	10	r	r	PROPN
ejpam-1773	104	11	j	j	PROPN
ejpam-1773	104	12	for	for	ADP
ejpam-1773	104	13	all	all	DET
ejpam-1773	104	14	j	j	PROPN
ejpam-1773	104	15	>	>	X
ejpam-1773	104	16	k.	k.	PROPN
ejpam-1773	105	1	proof	proof	PROPN
ejpam-1773	105	2	.	.	PUNCT
ejpam-1773	106	1	let	let	VERB
ejpam-1773	106	2	c	c	PRON
ejpam-1773	106	3	j	j	PROPN
ejpam-1773	106	4	be	be	AUX
ejpam-1773	106	5	the	the	DET
ejpam-1773	106	6	code	code	NOUN
ejpam-1773	106	7	generated	generate	VERB
ejpam-1773	106	8	by	by	ADP
ejpam-1773	106	9	v1	v1	PROPN
ejpam-1773	106	10	,	,	PUNCT
ejpam-1773	106	11	v2	v2	PROPN
ejpam-1773	106	12	,	,	PUNCT
ejpam-1773	106	13	.	.	PUNCT
ejpam-1773	106	14	.	.	PUNCT
ejpam-1773	107	1	.	.	PUNCT
ejpam-1773	108	1	,	,	PUNCT
ejpam-1773	108	2	vs	vs	ADP
ejpam-1773	108	3	over	over	ADP
ejpam-1773	108	4	r	r	NOUN
ejpam-1773	108	5	j.	j.	NOUN
ejpam-1773	108	6	we	we	PRON
ejpam-1773	108	7	proceed	proceed	VERB
ejpam-1773	108	8	by	by	ADP
ejpam-1773	108	9	induction	induction	NOUN
ejpam-1773	108	10	.	.	PUNCT
ejpam-1773	109	1	we	we	PRON
ejpam-1773	109	2	know	know	VERB
ejpam-1773	109	3	ck	ck	PROPN
ejpam-1773	109	4	is	be	AUX
ejpam-1773	109	5	a	a	DET
ejpam-1773	109	6	self	self	NOUN
ejpam-1773	109	7	-	-	PUNCT
ejpam-1773	109	8	dual	dual	ADJ
ejpam-1773	109	9	code	code	NOUN
ejpam-1773	109	10	by	by	ADP
ejpam-1773	109	11	assumption	assumption	NOUN
ejpam-1773	109	12	.	.	PUNCT
ejpam-1773	110	1	assume	assume	VERB
ejpam-1773	110	2	c	c	PROPN
ejpam-1773	110	3	j	j	PROPN
ejpam-1773	110	4	is	be	AUX
ejpam-1773	110	5	a	a	DET
ejpam-1773	110	6	self	self	NOUN
ejpam-1773	110	7	-	-	PUNCT
ejpam-1773	110	8	dual	dual	ADJ
ejpam-1773	110	9	code	code	NOUN
ejpam-1773	110	10	.	.	PUNCT
ejpam-1773	111	1	we	we	PRON
ejpam-1773	111	2	have	have	VERB
ejpam-1773	111	3	that	that	PRON
ejpam-1773	111	4	c	c	NOUN
ejpam-1773	112	1	j+1	j+1	NOUN
ejpam-1773	112	2	=	=	PUNCT
ejpam-1773	112	3	c	c	PROPN
ejpam-1773	112	4	j	j	PROPN
ejpam-1773	112	5	⊕	⊕	PROPN
ejpam-1773	112	6	u	u	PROPN
ejpam-1773	112	7	j+1c	j+1c	PROPN
ejpam-1773	112	8	j	j	PROPN
ejpam-1773	112	9	,	,	PUNCT
ejpam-1773	112	10	where	where	SCONJ
ejpam-1773	112	11	c	c	PROPN
ejpam-1773	112	12	j	j	PROPN
ejpam-1773	112	13	∩	∩	NOUN
ejpam-1773	112	14	u	u	PROPN
ejpam-1773	112	15	j+1c	j+1c	PROPN
ejpam-1773	112	16	j	j	PROPN
ejpam-1773	112	17	=	=	X
ejpam-1773	112	18	;	;	PUNCT
ejpam-1773	112	19	.	.	PUNCT
ejpam-1773	113	1	then	then	ADV
ejpam-1773	113	2	we	we	PRON
ejpam-1773	113	3	have	have	VERB
ejpam-1773	113	4	that	that	DET
ejpam-1773	113	5	|c	|c	ADJ
ejpam-1773	113	6	j+1|	j+1|	NOUN
ejpam-1773	113	7	=	=	SYM
ejpam-1773	113	8	|c	|c	ADJ
ejpam-1773	113	9	j||c	j||c	PROPN
ejpam-1773	113	10	j|	j|	NOUN
ejpam-1773	113	11	=	=	SYM
ejpam-1773	114	1	p	p	PROPN
ejpam-1773	114	2	22	22	NUM
ejpam-1773	114	3	j	j	PROPN
ejpam-1773	114	4	p	p	PROPN
ejpam-1773	114	5	22	22	NUM
ejpam-1773	114	6	j	j	NOUN
ejpam-1773	114	7	=	=	PUNCT
ejpam-1773	115	1	p	p	X
ejpam-1773	115	2	22	22	NUM
ejpam-1773	115	3	j+1	j+1	NUM
ejpam-1773	115	4	.	.	PUNCT
ejpam-1773	116	1	then	then	ADV
ejpam-1773	116	2	for	for	ADP
ejpam-1773	116	3	vectors	vector	NOUN
ejpam-1773	116	4	v	v	ADP
ejpam-1773	116	5	,	,	PUNCT
ejpam-1773	116	6	w	w	PROPN
ejpam-1773	116	7	,	,	PUNCT
ejpam-1773	116	8	v′,w′	v′,w′	PROPN
ejpam-1773	116	9	∈	∈	PROPN
ejpam-1773	117	1	c	c	X
ejpam-1773	117	2	j	j	NOUN
ejpam-1773	117	3	we	we	PRON
ejpam-1773	117	4	have	have	VERB
ejpam-1773	117	5	,	,	PUNCT
ejpam-1773	117	6	since	since	SCONJ
ejpam-1773	117	7	c	c	PROPN
ejpam-1773	117	8	j	j	PROPN
ejpam-1773	117	9	is	be	AUX
ejpam-1773	117	10	self	self	NOUN
ejpam-1773	117	11	-	-	PUNCT
ejpam-1773	117	12	dual	dual	ADJ
ejpam-1773	117	13	by	by	ADP
ejpam-1773	117	14	assumption	assumption	NOUN
ejpam-1773	117	15	,	,	PUNCT
ejpam-1773	117	16	[	[	X
ejpam-1773	117	17	v+	v+	ADP
ejpam-1773	117	18	u	u	NOUN
ejpam-1773	117	19	j+1v′,w+	j+1v′,w+	PROPN
ejpam-1773	117	20	u	u	PROPN
ejpam-1773	117	21	j+1w′	j+1w′	PROPN
ejpam-1773	117	22	]	]	PUNCT
ejpam-1773	118	1	j+1	j+1	X
ejpam-1773	118	2	=	=	PUNCT
ejpam-1773	119	1	[	[	X
ejpam-1773	119	2	v	v	NOUN
ejpam-1773	119	3	,	,	PUNCT
ejpam-1773	119	4	w	w	NOUN
ejpam-1773	119	5	]	]	X
ejpam-1773	119	6	j	j	PROPN
ejpam-1773	119	7	+	+	CCONJ
ejpam-1773	119	8	u	u	PROPN
ejpam-1773	119	9	j+1[v	j+1[v	PROPN
ejpam-1773	119	10	,	,	PUNCT
ejpam-1773	119	11	w′	w′	PROPN
ejpam-1773	119	12	]	]	X
ejpam-1773	119	13	j	j	PROPN
ejpam-1773	120	1	+	+	CCONJ
ejpam-1773	120	2	u	u	NOUN
ejpam-1773	120	3	j+1[v	j+1[v	PROPN
ejpam-1773	120	4	′,w	′,w	NOUN
ejpam-1773	120	5	]	]	X
ejpam-1773	120	6	j	j	PROPN
ejpam-1773	120	7	+	+	CCONJ
ejpam-1773	120	8	u2	u2	PROPN
ejpam-1773	120	9	j+1[v	j+1[v	PROPN
ejpam-1773	120	10	′,w′	′,w′	PROPN
ejpam-1773	120	11	]	]	X
ejpam-1773	120	12	j	j	PROPN
ejpam-1773	120	13	=	=	PUNCT
ejpam-1773	120	14	0	0	PROPN
ejpam-1773	120	15	.	.	PUNCT
ejpam-1773	121	1	hence	hence	ADV
ejpam-1773	121	2	c	c	PROPN
ejpam-1773	121	3	j+1	j+1	PROPN
ejpam-1773	121	4	is	be	AUX
ejpam-1773	121	5	self	self	NOUN
ejpam-1773	121	6	-	-	PUNCT
ejpam-1773	121	7	dual	dual	ADJ
ejpam-1773	121	8	since	since	SCONJ
ejpam-1773	121	9	it	it	PRON
ejpam-1773	121	10	is	be	AUX
ejpam-1773	121	11	self	self	NOUN
ejpam-1773	121	12	-	-	PUNCT
ejpam-1773	121	13	orthogonal	orthogonal	ADJ
ejpam-1773	121	14	and	and	CCONJ
ejpam-1773	121	15	has	have	VERB
ejpam-1773	121	16	the	the	DET
ejpam-1773	121	17	proper	proper	ADJ
ejpam-1773	121	18	cardinality	cardinality	NOUN
ejpam-1773	121	19	.	.	PUNCT
ejpam-1773	122	1	therefore	therefore	ADV
ejpam-1773	122	2	by	by	ADP
ejpam-1773	122	3	mathematical	mathematical	ADJ
ejpam-1773	122	4	induction	induction	NOUN
ejpam-1773	122	5	c	c	PROPN
ejpam-1773	122	6	j	j	PROPN
ejpam-1773	122	7	is	be	AUX
ejpam-1773	122	8	a	a	DET
ejpam-1773	122	9	self	self	NOUN
ejpam-1773	122	10	-	-	PUNCT
ejpam-1773	122	11	dual	dual	ADJ
ejpam-1773	122	12	code	code	NOUN
ejpam-1773	122	13	for	for	ADP
ejpam-1773	122	14	all	all	DET
ejpam-1773	122	15	finite	finite	PROPN
ejpam-1773	122	16	j.	j.	PROPN
ejpam-1773	122	17	next	next	ADV
ejpam-1773	122	18	we	we	PRON
ejpam-1773	122	19	shall	shall	AUX
ejpam-1773	122	20	prove	prove	VERB
ejpam-1773	122	21	that	that	SCONJ
ejpam-1773	122	22	c∞	c∞	PROPN
ejpam-1773	122	23	is	be	AUX
ejpam-1773	122	24	self	self	NOUN
ejpam-1773	122	25	-	-	PUNCT
ejpam-1773	122	26	dual	dual	ADJ
ejpam-1773	122	27	.	.	PUNCT
ejpam-1773	123	1	if	if	SCONJ
ejpam-1773	123	2	v	v	NOUN
ejpam-1773	123	3	,	,	PUNCT
ejpam-1773	123	4	w	w	PROPN
ejpam-1773	123	5	∈	∈	PROPN
ejpam-1773	123	6	c∞	c∞	VERB
ejpam-1773	123	7	then	then	ADV
ejpam-1773	123	8	there	there	PRON
ejpam-1773	123	9	exists	exist	VERB
ejpam-1773	123	10	j	j	PROPN
ejpam-1773	123	11	with	with	ADP
ejpam-1773	123	12	v	v	NOUN
ejpam-1773	123	13	,	,	PUNCT
ejpam-1773	123	14	w	w	PROPN
ejpam-1773	123	15	∈	∈	PROPN
ejpam-1773	123	16	c	c	PROPN
ejpam-1773	123	17	j	j	NOUN
ejpam-1773	123	18	and	and	CCONJ
ejpam-1773	123	19	hence	hence	ADV
ejpam-1773	123	20	[	[	X
ejpam-1773	123	21	v	v	NOUN
ejpam-1773	123	22	,	,	PUNCT
ejpam-1773	123	23	w	w	NOUN
ejpam-1773	123	24	]	]	X
ejpam-1773	123	25	j	j	PROPN
ejpam-1773	123	26	=	=	SYM
ejpam-1773	123	27	0	0	NUM
ejpam-1773	123	28	which	which	PRON
ejpam-1773	123	29	implies	imply	VERB
ejpam-1773	123	30	[	[	X
ejpam-1773	123	31	v	v	NOUN
ejpam-1773	123	32	,	,	PUNCT
ejpam-1773	123	33	w]∞	w]∞	PROPN
ejpam-1773	123	34	=	=	PUNCT
ejpam-1773	123	35	0	0	X
ejpam-1773	123	36	.	.	PUNCT
ejpam-1773	124	1	if	if	SCONJ
ejpam-1773	124	2	w	w	PROPN
ejpam-1773	124	3	∈	∈	PROPN
ejpam-1773	124	4	c⊥∞	c⊥∞	X
ejpam-1773	124	5	then	then	ADV
ejpam-1773	124	6	w	w	PROPN
ejpam-1773	124	7	∈	∈	PROPN
ejpam-1773	124	8	c⊥	c⊥	VERB
ejpam-1773	124	9	j	j	NOUN
ejpam-1773	124	10	for	for	ADP
ejpam-1773	124	11	some	some	DET
ejpam-1773	124	12	j	j	NOUN
ejpam-1773	124	13	which	which	PRON
ejpam-1773	124	14	gives	give	VERB
ejpam-1773	124	15	that	that	PRON
ejpam-1773	124	16	w	w	PROPN
ejpam-1773	124	17	∈	∈	PROPN
ejpam-1773	124	18	c	c	PROPN
ejpam-1773	124	19	j	j	NOUN
ejpam-1773	124	20	and	and	CCONJ
ejpam-1773	124	21	hence	hence	ADV
ejpam-1773	124	22	in	in	ADP
ejpam-1773	124	23	c∞.	c∞.	PROPN
ejpam-1773	124	24	therefore	therefore	ADV
ejpam-1773	124	25	c∞	c∞	PROPN
ejpam-1773	124	26	is	be	AUX
ejpam-1773	124	27	self	self	NOUN
ejpam-1773	124	28	-	-	PUNCT
ejpam-1773	124	29	dual	dual	ADJ
ejpam-1773	124	30	.	.	PUNCT
ejpam-1773	125	1	corollary	corollary	ADJ
ejpam-1773	125	2	1	1	NUM
ejpam-1773	125	3	.	.	PUNCT
ejpam-1773	126	1	if	if	SCONJ
ejpam-1773	126	2	c	c	PROPN
ejpam-1773	126	3	is	be	AUX
ejpam-1773	126	4	a	a	DET
ejpam-1773	126	5	self	self	NOUN
ejpam-1773	126	6	-	-	PUNCT
ejpam-1773	126	7	dual	dual	ADJ
ejpam-1773	126	8	code	code	NOUN
ejpam-1773	126	9	over	over	ADP
ejpam-1773	126	10	rk	rk	NOUN
ejpam-1773	126	11	then	then	ADV
ejpam-1773	126	12	there	there	PRON
ejpam-1773	126	13	exists	exist	VERB
ejpam-1773	126	14	a	a	DET
ejpam-1773	126	15	self	self	NOUN
ejpam-1773	126	16	-	-	PUNCT
ejpam-1773	126	17	dual	dual	ADJ
ejpam-1773	126	18	code	code	NOUN
ejpam-1773	127	1	c	c	NOUN
ejpam-1773	127	2	′	′	NOUN
ejpam-1773	127	3	over	over	ADP
ejpam-1773	127	4	r	r	PROPN
ejpam-1773	127	5	j	j	PROPN
ejpam-1773	127	6	,	,	PUNCT
ejpam-1773	127	7	for	for	ADP
ejpam-1773	127	8	j	j	PROPN
ejpam-1773	127	9	>	>	X
ejpam-1773	127	10	k	k	PROPN
ejpam-1773	127	11	,	,	PUNCT
ejpam-1773	127	12	with	with	ADP
ejpam-1773	127	13	π	π	PROPN
ejpam-1773	127	14	j	j	PROPN
ejpam-1773	127	15	,	,	PUNCT
ejpam-1773	127	16	k(c	k(c	PROPN
ejpam-1773	127	17	′	′	NUM
ejpam-1773	127	18	)	)	PUNCT
ejpam-1773	127	19	=	=	SYM
ejpam-1773	127	20	c.	c.	NOUN
ejpam-1773	127	21	notice	notice	VERB
ejpam-1773	127	22	that	that	SCONJ
ejpam-1773	127	23	the	the	DET
ejpam-1773	127	24	lifts	lift	NOUN
ejpam-1773	127	25	of	of	ADP
ejpam-1773	127	26	a	a	DET
ejpam-1773	127	27	self	self	NOUN
ejpam-1773	127	28	-	-	PUNCT
ejpam-1773	127	29	dual	dual	ADJ
ejpam-1773	127	30	code	code	NOUN
ejpam-1773	127	31	are	be	AUX
ejpam-1773	127	32	also	also	ADV
ejpam-1773	127	33	self	self	NOUN
ejpam-1773	127	34	-	-	PUNCT
ejpam-1773	127	35	dual	dual	ADJ
ejpam-1773	127	36	as	as	SCONJ
ejpam-1773	127	37	we	we	PRON
ejpam-1773	127	38	have	have	AUX
ejpam-1773	127	39	defined	define	VERB
ejpam-1773	127	40	it	it	PRON
ejpam-1773	127	41	,	,	PUNCT
ejpam-1773	127	42	but	but	CCONJ
ejpam-1773	127	43	not	not	PART
ejpam-1773	127	44	all	all	DET
ejpam-1773	127	45	projections	projection	NOUN
ejpam-1773	127	46	are	be	AUX
ejpam-1773	127	47	self	self	NOUN
ejpam-1773	127	48	-	-	PUNCT
ejpam-1773	127	49	dual	dual	ADJ
ejpam-1773	127	50	.	.	PUNCT
ejpam-1773	128	1	for	for	ADP
ejpam-1773	128	2	any	any	DET
ejpam-1773	128	3	ideal	ideal	NOUN
ejpam-1773	128	4	i	i	PRON
ejpam-1773	128	5	of	of	ADP
ejpam-1773	128	6	rk	rk	PRON
ejpam-1773	128	7	we	we	PRON
ejpam-1773	128	8	have	have	VERB
ejpam-1773	128	9	that	that	PRON
ejpam-1773	128	10	ann(i	ann(i	NOUN
ejpam-1773	128	11	)	)	PUNCT
ejpam-1773	128	12	=	=	PUNCT
ejpam-1773	129	1	i⊥.	i⊥.	NOUN
ejpam-1773	129	2	the	the	DET
ejpam-1773	129	3	following	follow	VERB
ejpam-1773	129	4	lemma	lemma	PROPN
ejpam-1773	129	5	appears	appear	VERB
ejpam-1773	129	6	in	in	ADP
ejpam-1773	129	7	[	[	X
ejpam-1773	129	8	7	7	NUM
ejpam-1773	129	9	]	]	PUNCT
ejpam-1773	129	10	.	.	PUNCT
ejpam-1773	130	1	lemma	lemma	PROPN
ejpam-1773	130	2	1	1	NUM
ejpam-1773	130	3	.	.	PUNCT
ejpam-1773	131	1	the	the	DET
ejpam-1773	131	2	code	code	PROPN
ejpam-1773	131	3	〈	〈	PROPN
ejpam-1773	131	4	ui	ui	NOUN
ejpam-1773	131	5	〉	〉	NOUN
ejpam-1773	131	6	of	of	ADP
ejpam-1773	131	7	length	length	NOUN
ejpam-1773	131	8	1	1	NUM
ejpam-1773	131	9	is	be	AUX
ejpam-1773	131	10	a	a	DET
ejpam-1773	131	11	self	self	NOUN
ejpam-1773	131	12	-	-	PUNCT
ejpam-1773	131	13	dual	dual	ADJ
ejpam-1773	131	14	code	code	NOUN
ejpam-1773	131	15	in	in	ADP
ejpam-1773	131	16	rk	rk	NOUN
ejpam-1773	131	17	for	for	ADP
ejpam-1773	131	18	all	all	DET
ejpam-1773	131	19	k	k	PROPN
ejpam-1773	131	20	≥	≥	PROPN
ejpam-1773	131	21	i.	i.	PROPN
ejpam-1773	131	22	s.	s.	PROPN
ejpam-1773	131	23	dougherty	dougherty	PROPN
ejpam-1773	131	24	,	,	PUNCT
ejpam-1773	131	25	b.yıldız	b.yıldız	NOUN
ejpam-1773	131	26	,	,	PUNCT
ejpam-1773	131	27	s.karadeniz	s.karadeniz	NOUN
ejpam-1773	131	28	/	/	SYM
ejpam-1773	131	29	eur	eur	NOUN
ejpam-1773	131	30	.	.	PUNCT
ejpam-1773	132	1	j.	j.	PROPN
ejpam-1773	132	2	pure	pure	PROPN
ejpam-1773	132	3	appl	appl	PROPN
ejpam-1773	132	4	.	.	PROPN
ejpam-1773	132	5	math	math	PROPN
ejpam-1773	132	6	,	,	PUNCT
ejpam-1773	132	7	6	6	NUM
ejpam-1773	132	8	(	(	PUNCT
ejpam-1773	132	9	2013	2013	NUM
ejpam-1773	132	10	)	)	PUNCT
ejpam-1773	132	11	,	,	PUNCT
ejpam-1773	132	12	89	89	NUM
ejpam-1773	132	13	-	-	SYM
ejpam-1773	132	14	106	106	NUM
ejpam-1773	132	15	93	93	NUM
ejpam-1773	132	16	proof	proof	NOUN
ejpam-1773	132	17	.	.	PUNCT
ejpam-1773	133	1	this	this	PRON
ejpam-1773	133	2	was	be	AUX
ejpam-1773	133	3	proved	prove	VERB
ejpam-1773	133	4	for	for	ADP
ejpam-1773	133	5	finite	finite	NOUN
ejpam-1773	133	6	k	k	PROPN
ejpam-1773	133	7	in	in	ADP
ejpam-1773	133	8	[	[	X
ejpam-1773	133	9	7	7	NUM
ejpam-1773	133	10	]	]	PUNCT
ejpam-1773	133	11	.	.	PUNCT
ejpam-1773	134	1	it	it	PRON
ejpam-1773	134	2	is	be	AUX
ejpam-1773	134	3	true	true	ADJ
ejpam-1773	134	4	for	for	ADP
ejpam-1773	134	5	infinite	infinite	ADJ
ejpam-1773	134	6	k	k	PROPN
ejpam-1773	134	7	by	by	ADP
ejpam-1773	134	8	theorem	theorem	NOUN
ejpam-1773	134	9	2	2	NUM
ejpam-1773	134	10	.	.	PUNCT
ejpam-1773	135	1	if	if	SCONJ
ejpam-1773	135	2	c	c	PROPN
ejpam-1773	135	3	and	and	CCONJ
ejpam-1773	135	4	d	d	PROPN
ejpam-1773	135	5	are	be	AUX
ejpam-1773	135	6	self	self	NOUN
ejpam-1773	135	7	-	-	PUNCT
ejpam-1773	135	8	dual	dual	ADJ
ejpam-1773	135	9	codes	code	NOUN
ejpam-1773	135	10	over	over	ADP
ejpam-1773	135	11	rk	rk	NOUN
ejpam-1773	135	12	then	then	ADV
ejpam-1773	135	13	define	define	VERB
ejpam-1773	135	14	c	c	PROPN
ejpam-1773	135	15	×	×	PROPN
ejpam-1773	135	16	d	d	NOUN
ejpam-1773	135	17	as	as	ADP
ejpam-1773	135	18	{	{	PUNCT
ejpam-1773	135	19	(	(	PUNCT
ejpam-1773	135	20	v	v	NOUN
ejpam-1773	135	21	,	,	PUNCT
ejpam-1773	135	22	w	w	NOUN
ejpam-1773	135	23	)	)	PUNCT
ejpam-1773	136	1	|	|	ADV
ejpam-1773	136	2	v	v	ADP
ejpam-1773	136	3	∈	∈	NOUN
ejpam-1773	136	4	c	c	NOUN
ejpam-1773	136	5	,	,	PUNCT
ejpam-1773	136	6	w	w	PROPN
ejpam-1773	136	7	∈	∈	PROPN
ejpam-1773	136	8	d	d	NOUN
ejpam-1773	136	9	}	}	PUNCT
ejpam-1773	136	10	.	.	PUNCT
ejpam-1773	137	1	it	it	PRON
ejpam-1773	137	2	is	be	AUX
ejpam-1773	137	3	easy	easy	ADJ
ejpam-1773	137	4	to	to	PART
ejpam-1773	137	5	see	see	VERB
ejpam-1773	137	6	that	that	SCONJ
ejpam-1773	137	7	this	this	DET
ejpam-1773	137	8	code	code	NOUN
ejpam-1773	137	9	is	be	AUX
ejpam-1773	137	10	self	self	NOUN
ejpam-1773	137	11	-	-	PUNCT
ejpam-1773	137	12	orthogonal	orthogonal	ADJ
ejpam-1773	137	13	and	and	CCONJ
ejpam-1773	137	14	of	of	ADP
ejpam-1773	137	15	the	the	DET
ejpam-1773	137	16	proper	proper	ADJ
ejpam-1773	137	17	cardinality	cardinality	NOUN
ejpam-1773	137	18	.	.	PUNCT
ejpam-1773	138	1	therefore	therefore	ADV
ejpam-1773	138	2	the	the	DET
ejpam-1773	138	3	code	code	NOUN
ejpam-1773	138	4	is	be	AUX
ejpam-1773	138	5	self	self	NOUN
ejpam-1773	138	6	-	-	PUNCT
ejpam-1773	138	7	dual	dual	ADJ
ejpam-1773	138	8	.	.	PUNCT
ejpam-1773	139	1	theorem	theorem	VERB
ejpam-1773	139	2	3	3	NUM
ejpam-1773	139	3	.	.	PUNCT
ejpam-1773	139	4	self	self	NOUN
ejpam-1773	139	5	-	-	PUNCT
ejpam-1773	139	6	dual	dual	ADJ
ejpam-1773	139	7	codes	code	NOUN
ejpam-1773	139	8	over	over	ADP
ejpam-1773	139	9	rk	rk	NOUN
ejpam-1773	139	10	exist	exist	VERB
ejpam-1773	139	11	for	for	ADP
ejpam-1773	139	12	all	all	DET
ejpam-1773	139	13	lengths	length	NOUN
ejpam-1773	139	14	and	and	CCONJ
ejpam-1773	139	15	for	for	ADP
ejpam-1773	139	16	all	all	PRON
ejpam-1773	139	17	k	k	PROPN
ejpam-1773	139	18	≥	≥	NUM
ejpam-1773	139	19	1	1	NUM
ejpam-1773	139	20	.	.	PUNCT
ejpam-1773	140	1	proof	proof	NOUN
ejpam-1773	140	2	.	.	PUNCT
ejpam-1773	141	1	the	the	DET
ejpam-1773	141	2	ideal	ideal	ADJ
ejpam-1773	141	3	iui	iui	NOUN
ejpam-1773	141	4	=	=	SYM
ejpam-1773	141	5	〈	〈	PROPN
ejpam-1773	141	6	ui	ui	NOUN
ejpam-1773	141	7	〉	〉	NOUN
ejpam-1773	141	8	is	be	AUX
ejpam-1773	141	9	a	a	DET
ejpam-1773	141	10	self	self	NOUN
ejpam-1773	141	11	-	-	PUNCT
ejpam-1773	141	12	dual	dual	ADJ
ejpam-1773	141	13	code	code	NOUN
ejpam-1773	141	14	of	of	ADP
ejpam-1773	141	15	length	length	NOUN
ejpam-1773	141	16	1	1	NUM
ejpam-1773	141	17	for	for	ADP
ejpam-1773	141	18	all	all	DET
ejpam-1773	141	19	i	i	PROPN
ejpam-1773	141	20	,	,	PUNCT
ejpam-1773	141	21	by	by	ADP
ejpam-1773	141	22	lemma	lemma	PROPN
ejpam-1773	141	23	1	1	NUM
ejpam-1773	141	24	.	.	PUNCT
ejpam-1773	141	25	by	by	ADP
ejpam-1773	141	26	taking	take	VERB
ejpam-1773	141	27	direct	direct	ADJ
ejpam-1773	141	28	products	product	NOUN
ejpam-1773	141	29	,	,	PUNCT
ejpam-1773	141	30	we	we	PRON
ejpam-1773	141	31	conclude	conclude	VERB
ejpam-1773	141	32	that	that	SCONJ
ejpam-1773	141	33	self	self	NOUN
ejpam-1773	141	34	-	-	PUNCT
ejpam-1773	141	35	dual	dual	ADJ
ejpam-1773	141	36	codes	code	NOUN
ejpam-1773	141	37	exists	exist	VERB
ejpam-1773	141	38	for	for	ADP
ejpam-1773	141	39	all	all	DET
ejpam-1773	141	40	lengths	length	NOUN
ejpam-1773	141	41	,	,	PUNCT
ejpam-1773	141	42	for	for	ADP
ejpam-1773	141	43	all	all	DET
ejpam-1773	141	44	k	k	PROPN
ejpam-1773	141	45	≥	≥	NUM
ejpam-1773	141	46	1	1	NUM
ejpam-1773	141	47	.	.	NOUN
ejpam-1773	141	48	4	4	NUM
ejpam-1773	141	49	.	.	PUNCT
ejpam-1773	141	50	binary	binary	ADJ
ejpam-1773	141	51	images	image	NOUN
ejpam-1773	141	52	the	the	DET
ejpam-1773	141	53	following	follow	VERB
ejpam-1773	141	54	is	be	AUX
ejpam-1773	141	55	defined	define	VERB
ejpam-1773	141	56	in	in	ADP
ejpam-1773	141	57	[	[	X
ejpam-1773	141	58	7	7	NUM
ejpam-1773	141	59	]	]	PUNCT
ejpam-1773	141	60	.	.	PUNCT
ejpam-1773	142	1	view	view	VERB
ejpam-1773	142	2	rk	rk	PROPN
ejpam-1773	142	3	as	as	ADP
ejpam-1773	142	4	a	a	DET
ejpam-1773	142	5	vector	vector	NOUN
ejpam-1773	142	6	space	space	NOUN
ejpam-1773	142	7	over	over	ADP
ejpam-1773	142	8	f2	f2	PROPN
ejpam-1773	142	9	with	with	ADP
ejpam-1773	142	10	basis	basis	NOUN
ejpam-1773	142	11	{	{	PUNCT
ejpam-1773	142	12	ua	ua	NOUN
ejpam-1773	142	13	:	:	PUNCT
ejpam-1773	142	14	a⊆	a⊆	PROPN
ejpam-1773	142	15	{	{	PUNCT
ejpam-1773	142	16	1,2	1,2	NUM
ejpam-1773	142	17	,	,	PUNCT
ejpam-1773	142	18	.	.	PUNCT
ejpam-1773	142	19	.	.	PUNCT
ejpam-1773	143	1	.	.	PUNCT
ejpam-1773	144	1	,	,	PUNCT
ejpam-1773	144	2	k	k	X
ejpam-1773	144	3	}	}	PUNCT
ejpam-1773	144	4	}	}	PUNCT
ejpam-1773	144	5	,	,	PUNCT
ejpam-1773	144	6	and	and	CCONJ
ejpam-1773	144	7	define	define	VERB
ejpam-1773	144	8	the	the	DET
ejpam-1773	144	9	gray	gray	ADJ
ejpam-1773	144	10	map	map	NOUN
ejpam-1773	144	11	of	of	ADP
ejpam-1773	144	12	each	each	DET
ejpam-1773	144	13	ua	ua	NOUN
ejpam-1773	144	14	and	and	CCONJ
ejpam-1773	144	15	then	then	ADV
ejpam-1773	144	16	extend	extend	VERB
ejpam-1773	144	17	it	it	PRON
ejpam-1773	144	18	linearly	linearly	ADV
ejpam-1773	144	19	to	to	ADP
ejpam-1773	144	20	all	all	PRON
ejpam-1773	144	21	of	of	ADP
ejpam-1773	144	22	rk	rk	PRON
ejpam-1773	144	23	.	.	PUNCT
ejpam-1773	145	1	fix	fix	VERB
ejpam-1773	145	2	an	an	DET
ejpam-1773	145	3	ordering	ordering	NOUN
ejpam-1773	145	4	on	on	ADP
ejpam-1773	145	5	the	the	DET
ejpam-1773	145	6	subsets	subset	NOUN
ejpam-1773	145	7	of	of	ADP
ejpam-1773	145	8	{	{	PUNCT
ejpam-1773	145	9	1,2	1,2	NUM
ejpam-1773	145	10	,	,	PUNCT
ejpam-1773	145	11	.	.	PUNCT
ejpam-1773	145	12	.	.	PUNCT
ejpam-1773	146	1	.	.	PUNCT
ejpam-1773	147	1	,	,	PUNCT
ejpam-1773	147	2	k	k	X
ejpam-1773	147	3	}	}	PUNCT
ejpam-1773	147	4	,	,	PUNCT
ejpam-1773	147	5	that	that	PRON
ejpam-1773	147	6	will	will	AUX
ejpam-1773	147	7	be	be	AUX
ejpam-1773	147	8	defined	define	VERB
ejpam-1773	147	9	recursively	recursively	ADV
ejpam-1773	147	10	as	as	SCONJ
ejpam-1773	147	11	follows	follow	VERB
ejpam-1773	147	12	:	:	PUNCT
ejpam-1773	147	13	{	{	PUNCT
ejpam-1773	147	14	1,2	1,2	NUM
ejpam-1773	147	15	,	,	PUNCT
ejpam-1773	147	16	.	.	PUNCT
ejpam-1773	147	17	.	.	PUNCT
ejpam-1773	148	1	.	.	PUNCT
ejpam-1773	149	1	,	,	PUNCT
ejpam-1773	149	2	k	k	X
ejpam-1773	149	3	}	}	PUNCT
ejpam-1773	149	4	=	=	SYM
ejpam-1773	149	5	{	{	PUNCT
ejpam-1773	149	6	1,2	1,2	NUM
ejpam-1773	149	7	,	,	PUNCT
ejpam-1773	149	8	.	.	PUNCT
ejpam-1773	149	9	.	.	PUNCT
ejpam-1773	150	1	.	.	PUNCT
ejpam-1773	151	1	,	,	PUNCT
ejpam-1773	151	2	k−	k−	PROPN
ejpam-1773	151	3	1	1	NUM
ejpam-1773	151	4	}	}	PUNCT
ejpam-1773	151	5	∪	∪	X
ejpam-1773	151	6	{	{	PUNCT
ejpam-1773	151	7	k	k	NOUN
ejpam-1773	151	8	}	}	PUNCT
ejpam-1773	151	9	.	.	PUNCT
ejpam-1773	152	1	we	we	PRON
ejpam-1773	152	2	can	can	AUX
ejpam-1773	152	3	now	now	ADV
ejpam-1773	152	4	define	define	VERB
ejpam-1773	152	5	the	the	DET
ejpam-1773	152	6	coordinate	coordinate	NOUN
ejpam-1773	152	7	-	-	PUNCT
ejpam-1773	152	8	wise	wise	ADJ
ejpam-1773	152	9	gray	gray	ADJ
ejpam-1773	152	10	map	map	NOUN
ejpam-1773	152	11	.	.	PUNCT
ejpam-1773	153	1	we	we	PRON
ejpam-1773	153	2	denote	denote	VERB
ejpam-1773	153	3	this	this	DET
ejpam-1773	153	4	map	map	NOUN
ejpam-1773	153	5	by	by	ADP
ejpam-1773	153	6	ψk	ψk	NOUN
ejpam-1773	153	7	:	:	PUNCT
ejpam-1773	153	8	rk→	rk→	PROPN
ejpam-1773	153	9	f2k	f2k	VERB
ejpam-1773	153	10	2	2	NUM
ejpam-1773	153	11	and	and	CCONJ
ejpam-1773	153	12	define	define	VERB
ejpam-1773	153	13	it	it	PRON
ejpam-1773	153	14	as	as	SCONJ
ejpam-1773	153	15	follows	follow	VERB
ejpam-1773	153	16	:	:	PUNCT
ejpam-1773	153	17	ψk(ua	ψk(ua	X
ejpam-1773	153	18	)	)	PUNCT
ejpam-1773	154	1	=	=	PRON
ejpam-1773	154	2	(	(	PUNCT
ejpam-1773	154	3	cb)b⊆{1,2,	cb)b⊆{1,2,	NOUN
ejpam-1773	154	4	...	...	PUNCT
ejpam-1773	154	5	,k	,k	PUNCT
ejpam-1773	154	6	}	}	PUNCT
ejpam-1773	154	7	,	,	PUNCT
ejpam-1773	154	8	where	where	SCONJ
ejpam-1773	154	9	cb	cb	PROPN
ejpam-1773	154	10	=	=	PRON
ejpam-1773	154	11	(	(	PUNCT
ejpam-1773	154	12	1	1	NUM
ejpam-1773	154	13	if	if	SCONJ
ejpam-1773	154	14	b	b	PROPN
ejpam-1773	154	15	⊆	⊆	SYM
ejpam-1773	154	16	a	a	DET
ejpam-1773	154	17	0	0	NUM
ejpam-1773	154	18	otherwise	otherwise	ADV
ejpam-1773	154	19	.	.	PUNCT
ejpam-1773	154	20	.	.	PUNCT
ejpam-1773	155	1	we	we	PRON
ejpam-1773	155	2	then	then	ADV
ejpam-1773	155	3	extend	extend	VERB
ejpam-1773	155	4	ψk	ψk	ADV
ejpam-1773	155	5	linearly	linearly	ADV
ejpam-1773	155	6	to	to	ADP
ejpam-1773	155	7	all	all	PRON
ejpam-1773	155	8	of	of	ADP
ejpam-1773	155	9	rk	rk	PRON
ejpam-1773	155	10	and	and	CCONJ
ejpam-1773	155	11	define	define	VERB
ejpam-1773	155	12	the	the	DET
ejpam-1773	155	13	lee	lee	PROPN
ejpam-1773	155	14	weight	weight	NOUN
ejpam-1773	155	15	of	of	ADP
ejpam-1773	155	16	an	an	DET
ejpam-1773	155	17	element	element	NOUN
ejpam-1773	155	18	in	in	ADP
ejpam-1773	155	19	rk	rk	NOUN
ejpam-1773	155	20	to	to	PART
ejpam-1773	155	21	be	be	AUX
ejpam-1773	155	22	the	the	DET
ejpam-1773	155	23	hamming	hamming	ADJ
ejpam-1773	155	24	weight	weight	NOUN
ejpam-1773	155	25	of	of	ADP
ejpam-1773	155	26	its	its	PRON
ejpam-1773	155	27	image	image	NOUN
ejpam-1773	155	28	.	.	PUNCT
ejpam-1773	156	1	we	we	PRON
ejpam-1773	156	2	get	get	VERB
ejpam-1773	156	3	a	a	DET
ejpam-1773	156	4	linear	linear	ADJ
ejpam-1773	156	5	distance	distance	NOUN
ejpam-1773	156	6	preserving	preserve	VERB
ejpam-1773	156	7	map	map	NOUN
ejpam-1773	156	8	from	from	ADP
ejpam-1773	156	9	rn	rn	PROPN
ejpam-1773	156	10	k	k	PROPN
ejpam-1773	156	11	to	to	PART
ejpam-1773	156	12	f2k	f2k	PROPN
ejpam-1773	156	13	n	n	ADV
ejpam-1773	156	14	2	2	NUM
ejpam-1773	156	15	.	.	PUNCT
ejpam-1773	157	1	it	it	PRON
ejpam-1773	157	2	follows	follow	VERB
ejpam-1773	157	3	immediately	immediately	ADV
ejpam-1773	157	4	that	that	SCONJ
ejpam-1773	157	5	wl(ua	wl(ua	NOUN
ejpam-1773	157	6	)	)	PUNCT
ejpam-1773	158	1	=	=	SYM
ejpam-1773	158	2	2|a|	2|a|	NUM
ejpam-1773	158	3	.	.	PUNCT
ejpam-1773	159	1	(	(	PUNCT
ejpam-1773	159	2	8)	8)	NUM
ejpam-1773	159	3	the	the	DET
ejpam-1773	159	4	map	map	NOUN
ejpam-1773	159	5	ψk	ψk	INTJ
ejpam-1773	159	6	was	be	AUX
ejpam-1773	159	7	shown	show	VERB
ejpam-1773	159	8	to	to	PART
ejpam-1773	159	9	be	be	AUX
ejpam-1773	159	10	equivalent	equivalent	ADJ
ejpam-1773	159	11	to	to	ADP
ejpam-1773	159	12	φk	φk	ADP
ejpam-1773	159	13	in	in	ADP
ejpam-1773	159	14	[	[	X
ejpam-1773	159	15	7	7	NUM
ejpam-1773	159	16	]	]	PUNCT
ejpam-1773	159	17	.	.	PUNCT
ejpam-1773	160	1	the	the	DET
ejpam-1773	160	2	following	follow	VERB
ejpam-1773	160	3	lemma	lemma	PROPN
ejpam-1773	160	4	also	also	ADV
ejpam-1773	160	5	appears	appear	VERB
ejpam-1773	160	6	in	in	ADP
ejpam-1773	160	7	[	[	X
ejpam-1773	160	8	7	7	NUM
ejpam-1773	160	9	]	]	PUNCT
ejpam-1773	160	10	.	.	PUNCT
ejpam-1773	161	1	lemma	lemma	PROPN
ejpam-1773	161	2	2	2	X
ejpam-1773	161	3	.	.	PUNCT
ejpam-1773	162	1	let	let	VERB
ejpam-1773	162	2	c	c	PRON
ejpam-1773	162	3	be	be	AUX
ejpam-1773	162	4	a	a	DET
ejpam-1773	162	5	linear	linear	ADJ
ejpam-1773	162	6	code	code	NOUN
ejpam-1773	162	7	over	over	ADP
ejpam-1773	162	8	rk	rk	PROPN
ejpam-1773	162	9	of	of	ADP
ejpam-1773	162	10	length	length	NOUN
ejpam-1773	162	11	n.	n.	PROPN
ejpam-1773	162	12	then	then	ADV
ejpam-1773	162	13	ψk(c	ψk(c	NUM
ejpam-1773	162	14	⊥	⊥	NOUN
ejpam-1773	162	15	)	)	PUNCT
ejpam-1773	162	16	=	=	SYM
ejpam-1773	162	17	(	(	PUNCT
ejpam-1773	162	18	ψk(c	ψk(c	NUM
ejpam-1773	162	19	)	)	PUNCT
ejpam-1773	162	20	)	)	PUNCT
ejpam-1773	163	1	⊥	⊥	NOUN
ejpam-1773	163	2	where	where	SCONJ
ejpam-1773	163	3	(	(	PUNCT
ejpam-1773	163	4	ψk(c	ψk(c	NUM
ejpam-1773	163	5	)	)	PUNCT
ejpam-1773	163	6	)	)	PUNCT
ejpam-1773	164	1	⊥	⊥	NOUN
ejpam-1773	164	2	denotes	denote	VERB
ejpam-1773	164	3	the	the	DET
ejpam-1773	164	4	ordinary	ordinary	ADJ
ejpam-1773	164	5	dual	dual	NOUN
ejpam-1773	164	6	of	of	ADP
ejpam-1773	164	7	ψk(c	ψk(c	NUM
ejpam-1773	164	8	)	)	PUNCT
ejpam-1773	164	9	as	as	ADP
ejpam-1773	164	10	a	a	DET
ejpam-1773	164	11	binary	binary	PROPN
ejpam-1773	164	12	code	code	NOUN
ejpam-1773	164	13	.	.	PUNCT
ejpam-1773	165	1	theorem	theorem	ADJ
ejpam-1773	165	2	4	4	NUM
ejpam-1773	165	3	.	.	PUNCT
ejpam-1773	166	1	let	let	VERB
ejpam-1773	166	2	c	c	PRON
ejpam-1773	166	3	be	be	AUX
ejpam-1773	166	4	a	a	DET
ejpam-1773	166	5	self	self	NOUN
ejpam-1773	166	6	-	-	PUNCT
ejpam-1773	166	7	dual	dual	ADJ
ejpam-1773	166	8	code	code	NOUN
ejpam-1773	166	9	over	over	ADP
ejpam-1773	166	10	rk	rk	PROPN
ejpam-1773	166	11	of	of	ADP
ejpam-1773	166	12	length	length	NOUN
ejpam-1773	166	13	n	n	CCONJ
ejpam-1773	166	14	,	,	PUNCT
ejpam-1773	166	15	then	then	ADV
ejpam-1773	166	16	ψk(c	ψk(c	NUM
ejpam-1773	166	17	)	)	PUNCT
ejpam-1773	166	18	is	be	AUX
ejpam-1773	166	19	a	a	DET
ejpam-1773	166	20	binary	binary	ADJ
ejpam-1773	166	21	self	self	NOUN
ejpam-1773	166	22	-	-	PUNCT
ejpam-1773	166	23	dual	dual	ADJ
ejpam-1773	166	24	code	code	NOUN
ejpam-1773	166	25	of	of	ADP
ejpam-1773	166	26	length	length	NOUN
ejpam-1773	166	27	2kn	2kn	ADV
ejpam-1773	166	28	.	.	PUNCT
ejpam-1773	167	1	if	if	SCONJ
ejpam-1773	167	2	c	c	PROPN
ejpam-1773	167	3	is	be	AUX
ejpam-1773	167	4	a	a	DET
ejpam-1773	167	5	type	type	NOUN
ejpam-1773	167	6	ii	ii	PROPN
ejpam-1773	167	7	code	code	NOUN
ejpam-1773	167	8	then	then	ADV
ejpam-1773	167	9	ψk(c	ψk(c	NUM
ejpam-1773	167	10	)	)	PUNCT
ejpam-1773	167	11	is	be	AUX
ejpam-1773	167	12	type	type	NOUN
ejpam-1773	167	13	ii	ii	NOUN
ejpam-1773	167	14	and	and	CCONJ
ejpam-1773	167	15	if	if	SCONJ
ejpam-1773	167	16	c	c	PROPN
ejpam-1773	167	17	is	be	AUX
ejpam-1773	167	18	type	type	NOUN
ejpam-1773	167	19	i	i	PRON
ejpam-1773	167	20	then	then	ADV
ejpam-1773	167	21	ψk(c	ψk(c	NUM
ejpam-1773	167	22	)	)	PUNCT
ejpam-1773	167	23	is	be	AUX
ejpam-1773	167	24	type	type	NOUN
ejpam-1773	167	25	i.	i.	NOUN
ejpam-1773	167	26	proof	proof	NOUN
ejpam-1773	167	27	.	.	PUNCT
ejpam-1773	168	1	if	if	SCONJ
ejpam-1773	168	2	c	c	NOUN
ejpam-1773	168	3	=	=	PUNCT
ejpam-1773	168	4	c⊥	c⊥	X
ejpam-1773	168	5	then	then	ADV
ejpam-1773	168	6	by	by	ADP
ejpam-1773	168	7	lemma	lemma	PROPN
ejpam-1773	168	8	2	2	NUM
ejpam-1773	168	9	,	,	PUNCT
ejpam-1773	168	10	ψk(c	ψk(c	NUM
ejpam-1773	168	11	)	)	PUNCT
ejpam-1773	169	1	=	=	NOUN
ejpam-1773	169	2	ψk(c	ψk(c	NUM
ejpam-1773	169	3	⊥	⊥	NOUN
ejpam-1773	169	4	)	)	PUNCT
ejpam-1773	169	5	=	=	NOUN
ejpam-1773	169	6	ψk(c	ψk(c	NUM
ejpam-1773	169	7	)	)	PUNCT
ejpam-1773	169	8	⊥.	⊥.	NUM
ejpam-1773	169	9	since	since	SCONJ
ejpam-1773	169	10	ψk	ψk	NOUN
ejpam-1773	169	11	is	be	AUX
ejpam-1773	169	12	distance	distance	NOUN
ejpam-1773	169	13	preserving	preserve	VERB
ejpam-1773	169	14	,	,	PUNCT
ejpam-1773	169	15	the	the	DET
ejpam-1773	169	16	following	follow	VERB
ejpam-1773	169	17	corollary	corollary	NOUN
ejpam-1773	169	18	immediately	immediately	ADV
ejpam-1773	169	19	follows	follow	VERB
ejpam-1773	169	20	from	from	ADP
ejpam-1773	169	21	the	the	DET
ejpam-1773	169	22	bounds	bound	NOUN
ejpam-1773	169	23	given	give	VERB
ejpam-1773	169	24	in	in	ADP
ejpam-1773	169	25	[	[	X
ejpam-1773	169	26	10	10	NUM
ejpam-1773	169	27	]	]	PUNCT
ejpam-1773	169	28	.	.	PUNCT
ejpam-1773	170	1	note	note	VERB
ejpam-1773	170	2	that	that	SCONJ
ejpam-1773	170	3	for	for	ADP
ejpam-1773	170	4	k	k	PROPN
ejpam-1773	170	5	≥	≥	NUM
ejpam-1773	170	6	2	2	NUM
ejpam-1773	170	7	,	,	PUNCT
ejpam-1773	170	8	the	the	DET
ejpam-1773	170	9	length	length	NOUN
ejpam-1773	170	10	of	of	ADP
ejpam-1773	170	11	the	the	DET
ejpam-1773	170	12	binary	binary	ADJ
ejpam-1773	170	13	image	image	NOUN
ejpam-1773	170	14	of	of	ADP
ejpam-1773	170	15	a	a	DET
ejpam-1773	170	16	code	code	NOUN
ejpam-1773	170	17	over	over	ADP
ejpam-1773	170	18	rk	rk	NOUN
ejpam-1773	170	19	will	will	AUX
ejpam-1773	170	20	always	always	ADV
ejpam-1773	170	21	be	be	AUX
ejpam-1773	170	22	divisible	divisible	ADJ
ejpam-1773	170	23	by	by	ADP
ejpam-1773	170	24	4	4	NUM
ejpam-1773	170	25	,	,	PUNCT
ejpam-1773	170	26	hence	hence	ADV
ejpam-1773	170	27	the	the	DET
ejpam-1773	170	28	case	case	NOUN
ejpam-1773	170	29	n	n	NUM
ejpam-1773	170	30	≡	≡	PROPN
ejpam-1773	170	31	22	22	NUM
ejpam-1773	170	32	(	(	PUNCT
ejpam-1773	170	33	mod	mod	NOUN
ejpam-1773	170	34	24	24	NUM
ejpam-1773	170	35	)	)	PUNCT
ejpam-1773	170	36	is	be	AUX
ejpam-1773	170	37	not	not	PART
ejpam-1773	170	38	possible	possible	ADJ
ejpam-1773	170	39	for	for	ADP
ejpam-1773	170	40	the	the	DET
ejpam-1773	170	41	image	image	NOUN
ejpam-1773	170	42	of	of	ADP
ejpam-1773	170	43	an	an	DET
ejpam-1773	170	44	rk	rk	NOUN
ejpam-1773	170	45	code	code	NOUN
ejpam-1773	170	46	.	.	PUNCT
ejpam-1773	171	1	hence	hence	ADV
ejpam-1773	171	2	we	we	PRON
ejpam-1773	171	3	need	need	AUX
ejpam-1773	171	4	not	not	PART
ejpam-1773	171	5	consider	consider	VERB
ejpam-1773	171	6	that	that	DET
ejpam-1773	171	7	special	special	ADJ
ejpam-1773	171	8	case	case	NOUN
ejpam-1773	171	9	for	for	ADP
ejpam-1773	171	10	binary	binary	ADJ
ejpam-1773	171	11	codes	code	NOUN
ejpam-1773	171	12	.	.	PUNCT
ejpam-1773	172	1	s.	s.	PROPN
ejpam-1773	172	2	dougherty	dougherty	PROPN
ejpam-1773	172	3	,	,	PUNCT
ejpam-1773	172	4	b.yıldız	b.yıldız	NOUN
ejpam-1773	172	5	,	,	PUNCT
ejpam-1773	172	6	s.karadeniz	s.karadeniz	NOUN
ejpam-1773	172	7	/	/	SYM
ejpam-1773	172	8	eur	eur	NOUN
ejpam-1773	172	9	.	.	PUNCT
ejpam-1773	173	1	j.	j.	PROPN
ejpam-1773	173	2	pure	pure	PROPN
ejpam-1773	173	3	appl	appl	PROPN
ejpam-1773	173	4	.	.	PROPN
ejpam-1773	173	5	math	math	PROPN
ejpam-1773	173	6	,	,	PUNCT
ejpam-1773	173	7	6	6	NUM
ejpam-1773	173	8	(	(	PUNCT
ejpam-1773	173	9	2013	2013	NUM
ejpam-1773	173	10	)	)	PUNCT
ejpam-1773	173	11	,	,	PUNCT
ejpam-1773	173	12	89	89	NUM
ejpam-1773	173	13	-	-	SYM
ejpam-1773	173	14	106	106	NUM
ejpam-1773	173	15	94	94	NUM
ejpam-1773	173	16	corollary	corollary	ADJ
ejpam-1773	173	17	2	2	NUM
ejpam-1773	173	18	.	.	PUNCT
ejpam-1773	174	1	let	let	VERB
ejpam-1773	174	2	dl(n	dl(n	NOUN
ejpam-1773	174	3	,	,	PUNCT
ejpam-1773	174	4	i	i	NOUN
ejpam-1773	174	5	)	)	PUNCT
ejpam-1773	174	6	and	and	CCONJ
ejpam-1773	174	7	dl(n	dl(n	ADJ
ejpam-1773	174	8	,	,	PUNCT
ejpam-1773	174	9	i	i	PRON
ejpam-1773	174	10	i	i	PROPN
ejpam-1773	174	11	)	)	PUNCT
ejpam-1773	174	12	denote	denote	VERB
ejpam-1773	174	13	the	the	DET
ejpam-1773	174	14	minimum	minimum	ADJ
ejpam-1773	174	15	distance	distance	NOUN
ejpam-1773	174	16	of	of	ADP
ejpam-1773	174	17	a	a	DET
ejpam-1773	174	18	type	type	NOUN
ejpam-1773	174	19	i	i	PRON
ejpam-1773	174	20	and	and	CCONJ
ejpam-1773	174	21	type	type	PROPN
ejpam-1773	174	22	ii	ii	PROPN
ejpam-1773	174	23	code	code	NOUN
ejpam-1773	174	24	over	over	ADP
ejpam-1773	174	25	rk	rk	PROPN
ejpam-1773	174	26	of	of	ADP
ejpam-1773	174	27	length	length	NOUN
ejpam-1773	174	28	n	n	CCONJ
ejpam-1773	174	29	,	,	PUNCT
ejpam-1773	174	30	respectively	respectively	ADV
ejpam-1773	174	31	.	.	PUNCT
ejpam-1773	175	1	then	then	ADV
ejpam-1773	175	2	for	for	SCONJ
ejpam-1773	175	3	k	k	PROPN
ejpam-1773	175	4	≥	≥	NUM
ejpam-1773	175	5	2	2	NUM
ejpam-1773	175	6	we	we	PRON
ejpam-1773	175	7	have	have	VERB
ejpam-1773	175	8	dl(n	dl(n	NOUN
ejpam-1773	175	9	,	,	PUNCT
ejpam-1773	175	10	i	i	NOUN
ejpam-1773	175	11	)	)	PUNCT
ejpam-1773	175	12	,	,	PUNCT
ejpam-1773	175	13	dl(n	dl(n	NOUN
ejpam-1773	175	14	,	,	PUNCT
ejpam-1773	176	1	i	i	PRON
ejpam-1773	176	2	i	i	PROPN
ejpam-1773	176	3	)	)	PUNCT
ejpam-1773	176	4	≤	≤	NUM
ejpam-1773	176	5	4	4	NUM
ejpam-1773	176	6	�	�	PROPN
ejpam-1773	176	7	2k−2n	2k−2n	PROPN
ejpam-1773	176	8	6	6	NUM
ejpam-1773	176	9	�	�	PROPN
ejpam-1773	176	10	+	+	CCONJ
ejpam-1773	176	11	4	4	NUM
ejpam-1773	176	12	.	.	X
ejpam-1773	177	1	another	another	DET
ejpam-1773	177	2	corollary	corollary	NOUN
ejpam-1773	177	3	follows	follow	VERB
ejpam-1773	177	4	from	from	ADP
ejpam-1773	177	5	the	the	DET
ejpam-1773	177	6	fact	fact	NOUN
ejpam-1773	177	7	that	that	SCONJ
ejpam-1773	177	8	a	a	DET
ejpam-1773	177	9	self	self	NOUN
ejpam-1773	177	10	-	-	PUNCT
ejpam-1773	177	11	dual	dual	ADJ
ejpam-1773	177	12	binary	binary	PROPN
ejpam-1773	177	13	code	code	NOUN
ejpam-1773	177	14	must	must	AUX
ejpam-1773	177	15	contain	contain	VERB
ejpam-1773	177	16	the	the	DET
ejpam-1773	177	17	all	all	DET
ejpam-1773	177	18	1	1	NUM
ejpam-1773	177	19	-	-	PUNCT
ejpam-1773	177	20	vector	vector	NOUN
ejpam-1773	177	21	,	,	PUNCT
ejpam-1773	177	22	and	and	CCONJ
ejpam-1773	177	23	as	as	ADP
ejpam-1773	177	24	the	the	DET
ejpam-1773	177	25	pre	pre	NOUN
ejpam-1773	177	26	-	-	NOUN
ejpam-1773	177	27	image	image	NOUN
ejpam-1773	177	28	under	under	ADP
ejpam-1773	177	29	ψk	ψk	NOUN
ejpam-1773	177	30	of	of	ADP
ejpam-1773	177	31	the	the	DET
ejpam-1773	177	32	all	all	DET
ejpam-1773	177	33	1	1	NUM
ejpam-1773	177	34	-	-	PUNCT
ejpam-1773	177	35	vector	vector	NOUN
ejpam-1773	177	36	corresponds	correspond	NOUN
ejpam-1773	177	37	to	to	ADP
ejpam-1773	177	38	the	the	DET
ejpam-1773	177	39	all	all	PRON
ejpam-1773	177	40	u1u2	u1u2	NOUN
ejpam-1773	177	41	.	.	PUNCT
ejpam-1773	177	42	.	.	PUNCT
ejpam-1773	177	43	.	.	PUNCT
ejpam-1773	178	1	ukvector	ukvector	NOUN
ejpam-1773	178	2	in	in	ADP
ejpam-1773	178	3	rk	rk	NOUN
ejpam-1773	178	4	we	we	PRON
ejpam-1773	178	5	get	get	VERB
ejpam-1773	178	6	the	the	DET
ejpam-1773	178	7	following	follow	VERB
ejpam-1773	178	8	corollary	corollary	NOUN
ejpam-1773	178	9	.	.	PUNCT
ejpam-1773	179	1	corollary	corollary	ADJ
ejpam-1773	179	2	3	3	NUM
ejpam-1773	179	3	.	.	PUNCT
ejpam-1773	180	1	any	any	DET
ejpam-1773	180	2	self	self	NOUN
ejpam-1773	180	3	-	-	PUNCT
ejpam-1773	180	4	dual	dual	ADJ
ejpam-1773	180	5	code	code	NOUN
ejpam-1773	180	6	over	over	ADP
ejpam-1773	180	7	rk	rk	PROPN
ejpam-1773	180	8	must	must	AUX
ejpam-1773	180	9	contain	contain	VERB
ejpam-1773	180	10	the	the	DET
ejpam-1773	180	11	all	all	DET
ejpam-1773	180	12	u1u2	u1u2	NOUN
ejpam-1773	180	13	.	.	PUNCT
ejpam-1773	180	14	.	.	PUNCT
ejpam-1773	180	15	.	.	PUNCT
ejpam-1773	181	1	uk	uk	PROPN
ejpam-1773	181	2	-	-	NOUN
ejpam-1773	181	3	vector	vector	NOUN
ejpam-1773	181	4	.	.	PUNCT
ejpam-1773	181	5	example	example	NOUN
ejpam-1773	182	1	1	1	NUM
ejpam-1773	182	2	.	.	X
ejpam-1773	183	1	we	we	PRON
ejpam-1773	183	2	have	have	AUX
ejpam-1773	183	3	seen	see	VERB
ejpam-1773	183	4	that	that	SCONJ
ejpam-1773	183	5	〈	〈	PROPN
ejpam-1773	183	6	ui	ui	NOUN
ejpam-1773	183	7	〉	〉	NOUN
ejpam-1773	183	8	is	be	AUX
ejpam-1773	183	9	a	a	DET
ejpam-1773	183	10	self	self	NOUN
ejpam-1773	183	11	-	-	PUNCT
ejpam-1773	183	12	dual	dual	ADJ
ejpam-1773	183	13	code	code	NOUN
ejpam-1773	183	14	of	of	ADP
ejpam-1773	183	15	length	length	NOUN
ejpam-1773	183	16	1	1	NUM
ejpam-1773	183	17	in	in	ADP
ejpam-1773	183	18	rk	rk	NOUN
ejpam-1773	183	19	for	for	ADP
ejpam-1773	183	20	all	all	DET
ejpam-1773	183	21	k	k	NOUN
ejpam-1773	183	22	with	with	ADP
ejpam-1773	183	23	i	i	PROPN
ejpam-1773	183	24	≤	≤	PROPN
ejpam-1773	183	25	k.	k.	PROPN
ejpam-1773	184	1	let	let	VERB
ejpam-1773	184	2	ck	ck	NOUN
ejpam-1773	184	3	=	=	PUNCT
ejpam-1773	184	4	〈	〈	PROPN
ejpam-1773	184	5	ui	ui	NOUN
ejpam-1773	184	6	〉	〉	NOUN
ejpam-1773	184	7	be	be	VERB
ejpam-1773	184	8	the	the	DET
ejpam-1773	184	9	code	code	NOUN
ejpam-1773	184	10	over	over	ADP
ejpam-1773	184	11	rk	rk	PROPN
ejpam-1773	184	12	.	.	PUNCT
ejpam-1773	185	1	then	then	ADV
ejpam-1773	185	2	ψk(ck	ψk(ck	NOUN
ejpam-1773	185	3	)	)	PUNCT
ejpam-1773	185	4	is	be	AUX
ejpam-1773	185	5	a	a	DET
ejpam-1773	185	6	self	self	NOUN
ejpam-1773	185	7	-	-	PUNCT
ejpam-1773	185	8	dual	dual	ADJ
ejpam-1773	185	9	code	code	NOUN
ejpam-1773	185	10	of	of	ADP
ejpam-1773	185	11	length	length	NOUN
ejpam-1773	185	12	2k	2k	NOUN
ejpam-1773	185	13	with	with	ADP
ejpam-1773	185	14	minimum	minimum	ADJ
ejpam-1773	185	15	hamming	hamming	NOUN
ejpam-1773	185	16	distance	distance	NOUN
ejpam-1773	185	17	2	2	NUM
ejpam-1773	185	18	.	.	PUNCT
ejpam-1773	186	1	it	it	PRON
ejpam-1773	186	2	is	be	AUX
ejpam-1773	186	3	well	well	ADV
ejpam-1773	186	4	known	know	VERB
ejpam-1773	186	5	that	that	SCONJ
ejpam-1773	186	6	if	if	SCONJ
ejpam-1773	186	7	a	a	DET
ejpam-1773	186	8	binary	binary	ADJ
ejpam-1773	186	9	type	type	NOUN
ejpam-1773	186	10	ii	ii	PROPN
ejpam-1773	186	11	code	code	NOUN
ejpam-1773	186	12	of	of	ADP
ejpam-1773	186	13	length	length	NOUN
ejpam-1773	186	14	n	n	PROPN
ejpam-1773	186	15	exists	exist	VERB
ejpam-1773	186	16	,	,	PUNCT
ejpam-1773	186	17	then	then	ADV
ejpam-1773	186	18	n	n	PRON
ejpam-1773	186	19	must	must	AUX
ejpam-1773	186	20	be	be	AUX
ejpam-1773	186	21	a	a	DET
ejpam-1773	186	22	multiple	multiple	NOUN
ejpam-1773	186	23	of	of	ADP
ejpam-1773	186	24	8	8	NUM
ejpam-1773	186	25	.	.	PUNCT
ejpam-1773	187	1	we	we	PRON
ejpam-1773	187	2	first	first	ADV
ejpam-1773	187	3	show	show	VERB
ejpam-1773	187	4	that	that	DET
ejpam-1773	187	5	type	type	NOUN
ejpam-1773	187	6	ii	ii	PROPN
ejpam-1773	187	7	codes	code	NOUN
ejpam-1773	187	8	over	over	ADP
ejpam-1773	187	9	rk	rk	NOUN
ejpam-1773	187	10	of	of	ADP
ejpam-1773	187	11	any	any	DET
ejpam-1773	187	12	length	length	NOUN
ejpam-1773	187	13	exist	exist	VERB
ejpam-1773	187	14	for	for	ADP
ejpam-1773	187	15	all	all	DET
ejpam-1773	187	16	k	k	PROPN
ejpam-1773	187	17	≥	≥	NUM
ejpam-1773	187	18	3	3	NUM
ejpam-1773	187	19	.	.	X
ejpam-1773	187	20	note	note	VERB
ejpam-1773	187	21	that	that	SCONJ
ejpam-1773	187	22	,	,	PUNCT
ejpam-1773	187	23	by	by	ADP
ejpam-1773	187	24	taking	take	VERB
ejpam-1773	187	25	direct	direct	ADJ
ejpam-1773	187	26	sums	sum	NOUN
ejpam-1773	187	27	,	,	PUNCT
ejpam-1773	187	28	it	it	PRON
ejpam-1773	187	29	is	be	AUX
ejpam-1773	187	30	enough	enough	ADJ
ejpam-1773	187	31	to	to	PART
ejpam-1773	187	32	show	show	VERB
ejpam-1773	187	33	that	that	DET
ejpam-1773	187	34	type	type	NOUN
ejpam-1773	187	35	ii	ii	PROPN
ejpam-1773	187	36	codes	code	NOUN
ejpam-1773	187	37	of	of	ADP
ejpam-1773	187	38	length	length	NOUN
ejpam-1773	187	39	1	1	NUM
ejpam-1773	187	40	exist	exist	VERB
ejpam-1773	187	41	over	over	ADP
ejpam-1773	187	42	rk	rk	NOUN
ejpam-1773	187	43	for	for	ADP
ejpam-1773	187	44	any	any	DET
ejpam-1773	187	45	k	k	PROPN
ejpam-1773	187	46	≥	≥	NUM
ejpam-1773	187	47	3	3	NUM
ejpam-1773	187	48	.	.	PUNCT
ejpam-1773	188	1	let	let	VERB
ejpam-1773	188	2	k	k	PROPN
ejpam-1773	188	3	≥	≥	NUM
ejpam-1773	188	4	3	3	NUM
ejpam-1773	188	5	,	,	PUNCT
ejpam-1773	188	6	take	take	VERB
ejpam-1773	188	7	the	the	DET
ejpam-1773	188	8	code	code	NOUN
ejpam-1773	188	9	c	c	NOUN
ejpam-1773	188	10	over	over	ADP
ejpam-1773	188	11	rk	rk	NOUN
ejpam-1773	188	12	of	of	ADP
ejpam-1773	188	13	length	length	NOUN
ejpam-1773	188	14	1	1	NUM
ejpam-1773	188	15	generated	generate	VERB
ejpam-1773	188	16	by	by	ADP
ejpam-1773	188	17	{	{	PUNCT
ejpam-1773	188	18	ua	ua	NOUN
ejpam-1773	188	19	:	:	PUNCT
ejpam-1773	188	20	1	1	NUM
ejpam-1773	188	21	∈	∈	PROPN
ejpam-1773	188	22	a	a	PRON
ejpam-1773	188	23	,	,	PUNCT
ejpam-1773	188	24	a	a	PRON
ejpam-1773	188	25	6=	6=	NUM
ejpam-1773	188	26	{	{	PUNCT
ejpam-1773	188	27	1	1	NUM
ejpam-1773	188	28	}	}	PUNCT
ejpam-1773	188	29	}	}	PUNCT
ejpam-1773	188	30	∪	∪	X
ejpam-1773	188	31	{	{	PUNCT
ejpam-1773	188	32	u2u3	u2u3	X
ejpam-1773	188	33	.	.	PUNCT
ejpam-1773	188	34	.	.	PUNCT
ejpam-1773	188	35	.	.	PUNCT
ejpam-1773	189	1	uk	uk	PROPN
ejpam-1773	189	2	}	}	PUNCT
ejpam-1773	189	3	.	.	PUNCT
ejpam-1773	190	1	note	note	VERB
ejpam-1773	190	2	that	that	SCONJ
ejpam-1773	190	3	c	c	PROPN
ejpam-1773	190	4	can	can	AUX
ejpam-1773	190	5	be	be	AUX
ejpam-1773	190	6	viewed	view	VERB
ejpam-1773	190	7	as	as	ADP
ejpam-1773	190	8	an	an	DET
ejpam-1773	190	9	f2	f2	ADJ
ejpam-1773	190	10	-	-	PUNCT
ejpam-1773	190	11	vector	vector	NOUN
ejpam-1773	190	12	space	space	NOUN
ejpam-1773	190	13	with	with	ADP
ejpam-1773	190	14	basis	basis	NOUN
ejpam-1773	190	15	{	{	PUNCT
ejpam-1773	190	16	u1u2,u1u3	u1u2,u1u3	NOUN
ejpam-1773	190	17	,	,	PUNCT
ejpam-1773	190	18	.	.	PUNCT
ejpam-1773	190	19	.	.	PUNCT
ejpam-1773	190	20	.	.	PUNCT
ejpam-1773	191	1	,	,	PUNCT
ejpam-1773	191	2	u1u2	u1u2	NOUN
ejpam-1773	191	3	.	.	PUNCT
ejpam-1773	191	4	.	.	PUNCT
ejpam-1773	191	5	.	.	PUNCT
ejpam-1773	192	1	uk	uk	PROPN
ejpam-1773	192	2	,	,	PUNCT
ejpam-1773	192	3	u2u3	u2u3	PROPN
ejpam-1773	192	4	.	.	PUNCT
ejpam-1773	192	5	.	.	PUNCT
ejpam-1773	192	6	.	.	PUNCT
ejpam-1773	193	1	uk	uk	PROPN
ejpam-1773	193	2	}	}	PUNCT
ejpam-1773	193	3	.	.	PUNCT
ejpam-1773	194	1	since	since	SCONJ
ejpam-1773	194	2	every	every	DET
ejpam-1773	194	3	basis	basis	NOUN
ejpam-1773	194	4	element	element	NOUN
ejpam-1773	194	5	is	be	AUX
ejpam-1773	194	6	orthogonal	orthogonal	ADJ
ejpam-1773	194	7	to	to	ADP
ejpam-1773	194	8	every	every	DET
ejpam-1773	194	9	other	other	ADJ
ejpam-1773	194	10	basis	basis	NOUN
ejpam-1773	194	11	element	element	NOUN
ejpam-1773	194	12	,	,	PUNCT
ejpam-1773	194	13	c	c	PROPN
ejpam-1773	194	14	is	be	AUX
ejpam-1773	194	15	self	self	NOUN
ejpam-1773	194	16	-	-	PUNCT
ejpam-1773	194	17	orthogonal	orthogonal	ADJ
ejpam-1773	194	18	.	.	PUNCT
ejpam-1773	195	1	to	to	PART
ejpam-1773	195	2	prove	prove	VERB
ejpam-1773	195	3	self	self	NOUN
ejpam-1773	195	4	-	-	PUNCT
ejpam-1773	195	5	duality	duality	NOUN
ejpam-1773	195	6	of	of	ADP
ejpam-1773	195	7	c	c	NOUN
ejpam-1773	195	8	we	we	PRON
ejpam-1773	195	9	just	just	ADV
ejpam-1773	195	10	have	have	VERB
ejpam-1773	195	11	to	to	PART
ejpam-1773	195	12	look	look	VERB
ejpam-1773	195	13	at	at	ADP
ejpam-1773	195	14	the	the	DET
ejpam-1773	195	15	size	size	NOUN
ejpam-1773	195	16	.	.	PUNCT
ejpam-1773	196	1	the	the	DET
ejpam-1773	196	2	number	number	NOUN
ejpam-1773	196	3	of	of	ADP
ejpam-1773	196	4	subsets	subset	NOUN
ejpam-1773	196	5	of	of	ADP
ejpam-1773	196	6	{	{	PUNCT
ejpam-1773	196	7	1,2	1,2	NUM
ejpam-1773	196	8	,	,	PUNCT
ejpam-1773	196	9	.	.	PUNCT
ejpam-1773	196	10	.	.	PUNCT
ejpam-1773	196	11	.	.	PUNCT
ejpam-1773	197	1	,	,	PUNCT
ejpam-1773	197	2	k	k	X
ejpam-1773	197	3	}	}	PUNCT
ejpam-1773	197	4	that	that	PRON
ejpam-1773	197	5	contain	contain	VERB
ejpam-1773	197	6	1	1	NUM
ejpam-1773	197	7	properly	properly	ADV
ejpam-1773	197	8	is	be	AUX
ejpam-1773	197	9	2k−1−	2k−1−	NUM
ejpam-1773	197	10	1	1	NUM
ejpam-1773	197	11	.	.	PUNCT
ejpam-1773	197	12	adding	add	VERB
ejpam-1773	197	13	the	the	DET
ejpam-1773	197	14	vector	vector	NOUN
ejpam-1773	197	15	u2u3	u2u3	X
ejpam-1773	197	16	.	.	PUNCT
ejpam-1773	197	17	.	.	PUNCT
ejpam-1773	197	18	.	.	PUNCT
ejpam-1773	198	1	uk	uk	PROPN
ejpam-1773	198	2	,	,	PUNCT
ejpam-1773	198	3	we	we	PRON
ejpam-1773	198	4	see	see	VERB
ejpam-1773	198	5	that	that	SCONJ
ejpam-1773	198	6	|c	|c	ADJ
ejpam-1773	198	7	|=	|=	NOUN
ejpam-1773	198	8	22k−1	22k−1	NUM
ejpam-1773	198	9	=	=	SYM
ejpam-1773	198	10	p	p	NOUN
ejpam-1773	198	11	22k	22k	NOUN
ejpam-1773	198	12	.	.	PUNCT
ejpam-1773	199	1	so	so	ADV
ejpam-1773	199	2	c	c	PROPN
ejpam-1773	199	3	is	be	AUX
ejpam-1773	199	4	self	self	NOUN
ejpam-1773	199	5	-	-	PUNCT
ejpam-1773	199	6	dual	dual	ADJ
ejpam-1773	199	7	.	.	PUNCT
ejpam-1773	200	1	note	note	VERB
ejpam-1773	200	2	that	that	SCONJ
ejpam-1773	200	3	every	every	DET
ejpam-1773	200	4	element	element	NOUN
ejpam-1773	200	5	of	of	ADP
ejpam-1773	200	6	c	c	PROPN
ejpam-1773	200	7	is	be	AUX
ejpam-1773	200	8	an	an	DET
ejpam-1773	200	9	f2	f2	ADJ
ejpam-1773	200	10	-	-	PUNCT
ejpam-1773	200	11	linear	linear	NOUN
ejpam-1773	200	12	combination	combination	NOUN
ejpam-1773	200	13	of	of	ADP
ejpam-1773	200	14	the	the	DET
ejpam-1773	200	15	ua	ua	PROPN
ejpam-1773	200	16	where	where	SCONJ
ejpam-1773	200	17	|a|	|a|	PROPN
ejpam-1773	200	18	≥	≥	NOUN
ejpam-1773	200	19	2	2	NUM
ejpam-1773	200	20	,	,	PUNCT
ejpam-1773	200	21	so	so	ADV
ejpam-1773	200	22	the	the	DET
ejpam-1773	200	23	lee	lee	PROPN
ejpam-1773	200	24	weight	weight	NOUN
ejpam-1773	200	25	of	of	ADP
ejpam-1773	200	26	every	every	DET
ejpam-1773	200	27	codeword	codeword	NOUN
ejpam-1773	200	28	is	be	AUX
ejpam-1773	200	29	divisible	divisible	ADJ
ejpam-1773	200	30	by	by	ADP
ejpam-1773	200	31	4	4	NUM
ejpam-1773	200	32	and	and	CCONJ
ejpam-1773	200	33	the	the	DET
ejpam-1773	200	34	minimum	minimum	ADJ
ejpam-1773	200	35	lee	lee	PROPN
ejpam-1773	200	36	weight	weight	NOUN
ejpam-1773	200	37	of	of	ADP
ejpam-1773	200	38	c	c	PROPN
ejpam-1773	200	39	is	be	AUX
ejpam-1773	200	40	4	4	NUM
ejpam-1773	200	41	.	.	PUNCT
ejpam-1773	201	1	thus	thus	ADV
ejpam-1773	201	2	we	we	PRON
ejpam-1773	201	3	have	have	AUX
ejpam-1773	201	4	proved	prove	VERB
ejpam-1773	201	5	the	the	DET
ejpam-1773	201	6	following	follow	VERB
ejpam-1773	201	7	theorem	theorem	NOUN
ejpam-1773	201	8	.	.	PUNCT
ejpam-1773	201	9	theorem	theorem	ADJ
ejpam-1773	201	10	5	5	NUM
ejpam-1773	201	11	.	.	PUNCT
ejpam-1773	201	12	type	type	NOUN
ejpam-1773	201	13	ii	ii	PROPN
ejpam-1773	201	14	codes	code	NOUN
ejpam-1773	201	15	over	over	ADP
ejpam-1773	201	16	rk	rk	NOUN
ejpam-1773	201	17	of	of	ADP
ejpam-1773	201	18	all	all	DET
ejpam-1773	201	19	lengths	length	NOUN
ejpam-1773	201	20	exist	exist	VERB
ejpam-1773	201	21	for	for	ADP
ejpam-1773	201	22	any	any	DET
ejpam-1773	201	23	k	k	PROPN
ejpam-1773	201	24	≥	≥	NUM
ejpam-1773	201	25	3	3	NUM
ejpam-1773	201	26	.	.	PUNCT
ejpam-1773	202	1	the	the	DET
ejpam-1773	202	2	case	case	NOUN
ejpam-1773	202	3	when	when	SCONJ
ejpam-1773	202	4	k	k	PROPN
ejpam-1773	202	5	=	=	SYM
ejpam-1773	202	6	1	1	NUM
ejpam-1773	202	7	was	be	AUX
ejpam-1773	202	8	resolved	resolve	VERB
ejpam-1773	202	9	in	in	ADP
ejpam-1773	202	10	[	[	X
ejpam-1773	202	11	6	6	NUM
ejpam-1773	202	12	]	]	PUNCT
ejpam-1773	202	13	.	.	PUNCT
ejpam-1773	203	1	so	so	ADV
ejpam-1773	203	2	we	we	PRON
ejpam-1773	203	3	only	only	ADV
ejpam-1773	203	4	need	need	VERB
ejpam-1773	203	5	to	to	PART
ejpam-1773	203	6	look	look	VERB
ejpam-1773	203	7	at	at	ADP
ejpam-1773	203	8	the	the	DET
ejpam-1773	203	9	case	case	NOUN
ejpam-1773	203	10	when	when	SCONJ
ejpam-1773	203	11	k	k	PROPN
ejpam-1773	203	12	=	=	SYM
ejpam-1773	203	13	2	2	X
ejpam-1773	203	14	.	.	X
ejpam-1773	203	15	note	note	VERB
ejpam-1773	203	16	that	that	SCONJ
ejpam-1773	203	17	if	if	SCONJ
ejpam-1773	203	18	c	c	NOUN
ejpam-1773	203	19	is	be	AUX
ejpam-1773	203	20	any	any	DET
ejpam-1773	203	21	linear	linear	ADJ
ejpam-1773	203	22	code	code	NOUN
ejpam-1773	203	23	over	over	ADP
ejpam-1773	203	24	r2	r2	PROPN
ejpam-1773	203	25	of	of	ADP
ejpam-1773	203	26	length	length	NOUN
ejpam-1773	203	27	n	n	CCONJ
ejpam-1773	203	28	,	,	PUNCT
ejpam-1773	203	29	then	then	ADV
ejpam-1773	203	30	ψ2(c	ψ2(c	PROPN
ejpam-1773	203	31	)	)	PUNCT
ejpam-1773	203	32	is	be	AUX
ejpam-1773	203	33	a	a	DET
ejpam-1773	203	34	binary	binary	ADJ
ejpam-1773	203	35	linear	linear	PROPN
ejpam-1773	203	36	code	code	NOUN
ejpam-1773	203	37	of	of	ADP
ejpam-1773	203	38	length	length	NOUN
ejpam-1773	203	39	4n	4n	NOUN
ejpam-1773	203	40	.	.	PUNCT
ejpam-1773	204	1	by	by	ADP
ejpam-1773	204	2	the	the	DET
ejpam-1773	204	3	observation	observation	NOUN
ejpam-1773	204	4	about	about	ADP
ejpam-1773	204	5	the	the	DET
ejpam-1773	204	6	lengths	length	NOUN
ejpam-1773	204	7	of	of	ADP
ejpam-1773	204	8	binary	binary	ADJ
ejpam-1773	204	9	type	type	PROPN
ejpam-1773	204	10	ii	ii	PROPN
ejpam-1773	204	11	codes	code	NOUN
ejpam-1773	204	12	,	,	PUNCT
ejpam-1773	204	13	we	we	PRON
ejpam-1773	204	14	know	know	VERB
ejpam-1773	204	15	that	that	SCONJ
ejpam-1773	204	16	we	we	PRON
ejpam-1773	204	17	should	should	AUX
ejpam-1773	204	18	only	only	ADV
ejpam-1773	204	19	look	look	VERB
ejpam-1773	204	20	for	for	ADP
ejpam-1773	204	21	type	type	NOUN
ejpam-1773	204	22	ii	ii	NOUN
ejpam-1773	204	23	codes	code	NOUN
ejpam-1773	204	24	of	of	ADP
ejpam-1773	204	25	even	even	ADV
ejpam-1773	204	26	lengths	length	NOUN
ejpam-1773	204	27	over	over	ADP
ejpam-1773	204	28	r2	r2	PROPN
ejpam-1773	204	29	.	.	PUNCT
ejpam-1773	205	1	again	again	ADV
ejpam-1773	205	2	,	,	PUNCT
ejpam-1773	205	3	by	by	ADP
ejpam-1773	205	4	taking	take	VERB
ejpam-1773	205	5	direct	direct	ADJ
ejpam-1773	205	6	sums	sum	NOUN
ejpam-1773	205	7	if	if	SCONJ
ejpam-1773	205	8	necessary	necessary	ADJ
ejpam-1773	205	9	,	,	PUNCT
ejpam-1773	205	10	we	we	PRON
ejpam-1773	205	11	only	only	ADV
ejpam-1773	205	12	need	need	VERB
ejpam-1773	205	13	to	to	PART
ejpam-1773	205	14	look	look	VERB
ejpam-1773	205	15	for	for	ADP
ejpam-1773	205	16	a	a	DET
ejpam-1773	205	17	type	type	NOUN
ejpam-1773	205	18	ii	ii	NOUN
ejpam-1773	205	19	code	code	NOUN
ejpam-1773	205	20	of	of	ADP
ejpam-1773	205	21	length	length	NOUN
ejpam-1773	205	22	2	2	NUM
ejpam-1773	205	23	over	over	ADP
ejpam-1773	205	24	r2	r2	PROPN
ejpam-1773	205	25	.	.	PUNCT
ejpam-1773	206	1	indeed	indeed	ADV
ejpam-1773	206	2	,	,	PUNCT
ejpam-1773	206	3	let	let	VERB
ejpam-1773	206	4	c	c	PRON
ejpam-1773	206	5	be	be	AUX
ejpam-1773	206	6	the	the	DET
ejpam-1773	206	7	linear	linear	PROPN
ejpam-1773	206	8	code	code	NOUN
ejpam-1773	206	9	over	over	ADP
ejpam-1773	206	10	r2	r2	PROPN
ejpam-1773	206	11	of	of	ADP
ejpam-1773	206	12	length	length	NOUN
ejpam-1773	206	13	2	2	NUM
ejpam-1773	206	14	,	,	PUNCT
ejpam-1773	206	15	generated	generate	VERB
ejpam-1773	206	16	by	by	ADP
ejpam-1773	206	17	the	the	DET
ejpam-1773	206	18	vector	vector	NOUN
ejpam-1773	206	19	(	(	PUNCT
ejpam-1773	206	20	1,1	1,1	NUM
ejpam-1773	206	21	+	+	SYM
ejpam-1773	206	22	u1u2	u1u2	NOUN
ejpam-1773	206	23	)	)	PUNCT
ejpam-1773	206	24	.	.	PUNCT
ejpam-1773	207	1	it	it	PRON
ejpam-1773	207	2	turns	turn	VERB
ejpam-1773	207	3	out	out	ADP
ejpam-1773	207	4	that	that	SCONJ
ejpam-1773	207	5	c	c	PROPN
ejpam-1773	207	6	is	be	AUX
ejpam-1773	207	7	a	a	DET
ejpam-1773	207	8	self	self	NOUN
ejpam-1773	207	9	-	-	PUNCT
ejpam-1773	207	10	dual	dual	ADJ
ejpam-1773	207	11	code	code	NOUN
ejpam-1773	207	12	with	with	ADP
ejpam-1773	207	13	lee	lee	PROPN
ejpam-1773	207	14	weight	weight	NOUN
ejpam-1773	207	15	enumerator	enumerator	NOUN
ejpam-1773	207	16	1	1	NUM
ejpam-1773	207	17	+	+	SYM
ejpam-1773	207	18	14z4	14z4	NUM
ejpam-1773	207	19	+	+	CCONJ
ejpam-1773	207	20	z8	z8	NOUN
ejpam-1773	207	21	,	,	PUNCT
ejpam-1773	207	22	so	so	SCONJ
ejpam-1773	207	23	it	it	PRON
ejpam-1773	207	24	is	be	AUX
ejpam-1773	207	25	type	type	NOUN
ejpam-1773	207	26	ii	ii	NOUN
ejpam-1773	207	27	.	.	PUNCT
ejpam-1773	208	1	in	in	ADP
ejpam-1773	208	2	fact	fact	NOUN
ejpam-1773	208	3	the	the	DET
ejpam-1773	208	4	binary	binary	ADJ
ejpam-1773	208	5	image	image	NOUN
ejpam-1773	208	6	of	of	ADP
ejpam-1773	208	7	c	c	PROPN
ejpam-1773	208	8	is	be	AUX
ejpam-1773	208	9	an	an	DET
ejpam-1773	208	10	[	[	X
ejpam-1773	208	11	8,4,4	8,4,4	X
ejpam-1773	208	12	]	]	X
ejpam-1773	208	13	code	code	NOUN
ejpam-1773	208	14	which	which	PRON
ejpam-1773	208	15	is	be	AUX
ejpam-1773	208	16	the	the	DET
ejpam-1773	208	17	extended	extended	ADJ
ejpam-1773	208	18	hamming	hamming	NOUN
ejpam-1773	208	19	code	code	NOUN
ejpam-1773	208	20	.	.	PUNCT
ejpam-1773	209	1	thus	thus	ADV
ejpam-1773	209	2	we	we	PRON
ejpam-1773	209	3	have	have	AUX
ejpam-1773	209	4	proved	prove	VERB
ejpam-1773	209	5	that	that	SCONJ
ejpam-1773	209	6	following	follow	VERB
ejpam-1773	209	7	result	result	NOUN
ejpam-1773	209	8	.	.	PUNCT
ejpam-1773	210	1	theorem	theorem	ADJ
ejpam-1773	210	2	6	6	NUM
ejpam-1773	210	3	.	.	PUNCT
ejpam-1773	210	4	type	type	NOUN
ejpam-1773	210	5	ii	ii	PROPN
ejpam-1773	210	6	codes	code	NOUN
ejpam-1773	210	7	exist	exist	VERB
ejpam-1773	210	8	over	over	ADP
ejpam-1773	210	9	r2	r2	NOUN
ejpam-1773	210	10	for	for	ADP
ejpam-1773	210	11	all	all	DET
ejpam-1773	210	12	even	even	ADJ
ejpam-1773	210	13	lengths	length	NOUN
ejpam-1773	210	14	.	.	PUNCT
ejpam-1773	211	1	s.	s.	PROPN
ejpam-1773	211	2	dougherty	dougherty	PROPN
ejpam-1773	211	3	,	,	PUNCT
ejpam-1773	211	4	b.yıldız	b.yıldız	NOUN
ejpam-1773	211	5	,	,	PUNCT
ejpam-1773	211	6	s.karadeniz	s.karadeniz	NOUN
ejpam-1773	211	7	/	/	SYM
ejpam-1773	211	8	eur	eur	NOUN
ejpam-1773	211	9	.	.	PUNCT
ejpam-1773	212	1	j.	j.	PROPN
ejpam-1773	212	2	pure	pure	PROPN
ejpam-1773	212	3	appl	appl	PROPN
ejpam-1773	212	4	.	.	PROPN
ejpam-1773	212	5	math	math	PROPN
ejpam-1773	212	6	,	,	PUNCT
ejpam-1773	212	7	6	6	NUM
ejpam-1773	212	8	(	(	PUNCT
ejpam-1773	212	9	2013	2013	NUM
ejpam-1773	212	10	)	)	PUNCT
ejpam-1773	212	11	,	,	PUNCT
ejpam-1773	212	12	89	89	NUM
ejpam-1773	212	13	-	-	SYM
ejpam-1773	212	14	106	106	NUM
ejpam-1773	212	15	95	95	NUM
ejpam-1773	212	16	consider	consider	VERB
ejpam-1773	212	17	the	the	DET
ejpam-1773	212	18	complete	complete	ADJ
ejpam-1773	212	19	weight	weight	NOUN
ejpam-1773	212	20	enumerator	enumerator	NOUN
ejpam-1773	212	21	of	of	ADP
ejpam-1773	212	22	a	a	DET
ejpam-1773	212	23	self	self	NOUN
ejpam-1773	212	24	-	-	PUNCT
ejpam-1773	212	25	dual	dual	ADJ
ejpam-1773	212	26	code	code	NOUN
ejpam-1773	212	27	c	c	NOUN
ejpam-1773	212	28	.	.	PUNCT
ejpam-1773	213	1	it	it	PRON
ejpam-1773	213	2	is	be	AUX
ejpam-1773	213	3	held	hold	VERB
ejpam-1773	213	4	invariant	invariant	ADJ
ejpam-1773	213	5	by	by	ADP
ejpam-1773	213	6	the	the	DET
ejpam-1773	213	7	action	action	NOUN
ejpam-1773	213	8	of	of	ADP
ejpam-1773	213	9	the	the	DET
ejpam-1773	213	10	macwilliams	macwilliam	NOUN
ejpam-1773	213	11	relations	relation	NOUN
ejpam-1773	213	12	.	.	PUNCT
ejpam-1773	214	1	that	that	PRON
ejpam-1773	214	2	is	be	AUX
ejpam-1773	214	3	the	the	DET
ejpam-1773	214	4	complete	complete	ADJ
ejpam-1773	214	5	weight	weight	NOUN
ejpam-1773	214	6	enumerator	enumerator	NOUN
ejpam-1773	214	7	is	be	AUX
ejpam-1773	214	8	held	hold	VERB
ejpam-1773	214	9	invariant	invariant	ADJ
ejpam-1773	214	10	by	by	ADP
ejpam-1773	214	11	the	the	DET
ejpam-1773	214	12	matrix	matrix	NOUN
ejpam-1773	214	13	mk	mk	NOUN
ejpam-1773	214	14	,	,	PUNCT
ejpam-1773	214	15	where	where	SCONJ
ejpam-1773	214	16	mk	mk	NOUN
ejpam-1773	214	17	=	=	NOUN
ejpam-1773	214	18	1	1	PROPN
ejpam-1773	214	19	p	p	NOUN
ejpam-1773	214	20	22k	22k	NOUN
ejpam-1773	214	21	tk	tk	PROPN
ejpam-1773	214	22	.	.	PUNCT
ejpam-1773	215	1	the	the	DET
ejpam-1773	215	2	matrix	matrix	NOUN
ejpam-1773	215	3	tk	tk	PROPN
ejpam-1773	215	4	is	be	AUX
ejpam-1773	215	5	defined	define	VERB
ejpam-1773	215	6	as	as	SCONJ
ejpam-1773	215	7	follows	follow	VERB
ejpam-1773	215	8	.	.	PUNCT
ejpam-1773	216	1	let	let	VERB
ejpam-1773	216	2	∑	∑	PART
ejpam-1773	216	3	a⊆{1,2,	a⊆{1,2,	VERB
ejpam-1773	216	4	...	...	PUNCT
ejpam-1773	216	5	,k	,k	PUNCT
ejpam-1773	216	6	}	}	PUNCT
ejpam-1773	216	7	caua	caua	NOUN
ejpam-1773	216	8	∈	∈	PROPN
ejpam-1773	216	9	rk	rk	NOUN
ejpam-1773	216	10	.	.	PUNCT
ejpam-1773	217	1	then	then	ADV
ejpam-1773	217	2	(	(	PUNCT
ejpam-1773	217	3	ca	ca	NOUN
ejpam-1773	217	4	)	)	PUNCT
ejpam-1773	217	5	can	can	AUX
ejpam-1773	217	6	be	be	AUX
ejpam-1773	217	7	thought	think	VERB
ejpam-1773	217	8	of	of	ADP
ejpam-1773	217	9	as	as	ADP
ejpam-1773	217	10	a	a	DET
ejpam-1773	217	11	binary	binary	ADJ
ejpam-1773	217	12	vector	vector	NOUN
ejpam-1773	217	13	of	of	ADP
ejpam-1773	217	14	length	length	NOUN
ejpam-1773	217	15	2k	2k	NUM
ejpam-1773	217	16	.	.	PUNCT
ejpam-1773	218	1	let	let	VERB
ejpam-1773	218	2	wt(ca	wt(ca	PROPN
ejpam-1773	218	3	)	)	PUNCT
ejpam-1773	218	4	be	be	AUX
ejpam-1773	218	5	the	the	DET
ejpam-1773	218	6	hamming	hamming	ADJ
ejpam-1773	218	7	weight	weight	NOUN
ejpam-1773	218	8	of	of	ADP
ejpam-1773	218	9	this	this	DET
ejpam-1773	218	10	vector	vector	NOUN
ejpam-1773	218	11	.	.	PUNCT
ejpam-1773	219	1	then	then	ADV
ejpam-1773	219	2	χ1	χ1	PROPN
ejpam-1773	219	3	(	(	PUNCT
ejpam-1773	219	4	∑	∑	ADV
ejpam-1773	219	5	a⊆{1,2,	a⊆{1,2,	VERB
ejpam-1773	219	6	...	...	PUNCT
ejpam-1773	219	7	,k	,k	PUNCT
ejpam-1773	219	8	}	}	PUNCT
ejpam-1773	219	9	caua	caua	NOUN
ejpam-1773	219	10	)	)	PUNCT
ejpam-1773	219	11	=	=	SYM
ejpam-1773	219	12	(	(	PUNCT
ejpam-1773	219	13	−1)wt(ca	−1)wt(ca	PROPN
ejpam-1773	219	14	)	)	PUNCT
ejpam-1773	219	15	.	.	PUNCT
ejpam-1773	220	1	(	(	PUNCT
ejpam-1773	220	2	9	9	X
ejpam-1773	220	3	)	)	PUNCT
ejpam-1773	220	4	let	let	VERB
ejpam-1773	220	5	t	t	NOUN
ejpam-1773	220	6	be	be	AUX
ejpam-1773	220	7	a	a	DET
ejpam-1773	220	8	square	square	ADJ
ejpam-1773	220	9	22k	22k	NOUN
ejpam-1773	220	10	by	by	ADP
ejpam-1773	220	11	22k	22k	NOUN
ejpam-1773	220	12	matrix	matrix	NOUN
ejpam-1773	220	13	indexed	index	VERB
ejpam-1773	220	14	by	by	ADP
ejpam-1773	220	15	the	the	DET
ejpam-1773	220	16	elements	element	NOUN
ejpam-1773	220	17	of	of	ADP
ejpam-1773	220	18	rk	rk	PRON
ejpam-1773	220	19	and	and	CCONJ
ejpam-1773	220	20	define	define	VERB
ejpam-1773	220	21	ta	ta	ADP
ejpam-1773	220	22	,	,	PUNCT
ejpam-1773	220	23	b	b	NOUN
ejpam-1773	220	24	=	=	SYM
ejpam-1773	220	25	χa(b	χa(b	ADJ
ejpam-1773	220	26	)	)	PUNCT
ejpam-1773	220	27	=	=	SYM
ejpam-1773	220	28	χ1(ab	χ1(ab	PROPN
ejpam-1773	220	29	)	)	PUNCT
ejpam-1773	220	30	.	.	PUNCT
ejpam-1773	221	1	(	(	PUNCT
ejpam-1773	221	2	10	10	NUM
ejpam-1773	221	3	)	)	PUNCT
ejpam-1773	221	4	the	the	DET
ejpam-1773	221	5	complete	complete	ADJ
ejpam-1773	221	6	weight	weight	NOUN
ejpam-1773	221	7	enumerator	enumerator	NOUN
ejpam-1773	221	8	is	be	AUX
ejpam-1773	221	9	also	also	ADV
ejpam-1773	221	10	held	hold	VERB
ejpam-1773	221	11	invariant	invariant	ADJ
ejpam-1773	221	12	by	by	ADP
ejpam-1773	221	13	the	the	DET
ejpam-1773	221	14	action	action	NOUN
ejpam-1773	221	15	of	of	ADP
ejpam-1773	221	16	multiplication	multiplication	NOUN
ejpam-1773	221	17	by	by	ADP
ejpam-1773	221	18	a	a	DET
ejpam-1773	221	19	unit	unit	NOUN
ejpam-1773	221	20	.	.	PUNCT
ejpam-1773	222	1	it	it	PRON
ejpam-1773	222	2	is	be	AUX
ejpam-1773	222	3	shown	show	VERB
ejpam-1773	222	4	in	in	ADP
ejpam-1773	222	5	[	[	X
ejpam-1773	222	6	7	7	NUM
ejpam-1773	222	7	]	]	PUNCT
ejpam-1773	222	8	that	that	SCONJ
ejpam-1773	222	9	these	these	DET
ejpam-1773	222	10	actions	action	NOUN
ejpam-1773	222	11	are	be	AUX
ejpam-1773	222	12	all	all	PRON
ejpam-1773	222	13	generated	generate	VERB
ejpam-1773	222	14	by	by	ADP
ejpam-1773	222	15	multiplication	multiplication	NOUN
ejpam-1773	222	16	by	by	ADP
ejpam-1773	222	17	the	the	DET
ejpam-1773	222	18	unit	unit	NOUN
ejpam-1773	222	19	1+us	1+us	PROPN
ejpam-1773	222	20	for	for	ADP
ejpam-1773	222	21	1≤	1≤	PROPN
ejpam-1773	222	22	s	s	PART
ejpam-1773	222	23	≤	≤	PROPN
ejpam-1773	222	24	k.	k.	PROPN
ejpam-1773	222	25	let	let	VERB
ejpam-1773	222	26	as	as	SCONJ
ejpam-1773	222	27	be	be	AUX
ejpam-1773	222	28	the	the	DET
ejpam-1773	222	29	permutation	permutation	NOUN
ejpam-1773	222	30	matrix	matrix	NOUN
ejpam-1773	222	31	that	that	PRON
ejpam-1773	222	32	gives	give	VERB
ejpam-1773	222	33	the	the	DET
ejpam-1773	222	34	permutation	permutation	NOUN
ejpam-1773	222	35	α→	α→	X
ejpam-1773	222	36	(	(	PUNCT
ejpam-1773	222	37	1	1	NUM
ejpam-1773	222	38	+	+	CCONJ
ejpam-1773	222	39	us)α	us)α	ADJ
ejpam-1773	222	40	.	.	PUNCT
ejpam-1773	223	1	then	then	ADV
ejpam-1773	223	2	the	the	DET
ejpam-1773	223	3	group	group	NOUN
ejpam-1773	223	4	of	of	ADP
ejpam-1773	223	5	invariants	invariant	NOUN
ejpam-1773	223	6	of	of	ADP
ejpam-1773	223	7	a	a	DET
ejpam-1773	223	8	type	type	NOUN
ejpam-1773	223	9	i	i	PRON
ejpam-1773	223	10	code	code	VERB
ejpam-1773	223	11	over	over	ADP
ejpam-1773	223	12	rk	rk	PROPN
ejpam-1773	223	13	is	be	AUX
ejpam-1773	223	14	gi	gi	INTJ
ejpam-1773	223	15	=	=	PUNCT
ejpam-1773	223	16	〈	〈	PROPN
ejpam-1773	223	17	mk	mk	PROPN
ejpam-1773	223	18	,	,	PUNCT
ejpam-1773	223	19	a1	a1	PROPN
ejpam-1773	223	20	,	,	PUNCT
ejpam-1773	223	21	.	.	PUNCT
ejpam-1773	223	22	.	.	PUNCT
ejpam-1773	223	23	.	.	PUNCT
ejpam-1773	224	1	,	,	PUNCT
ejpam-1773	224	2	as	as	ADP
ejpam-1773	224	3	〉	〉	X
ejpam-1773	224	4	.	.	PUNCT
ejpam-1773	225	1	(	(	PUNCT
ejpam-1773	225	2	11	11	NUM
ejpam-1773	225	3	)	)	PUNCT
ejpam-1773	225	4	let	let	VERB
ejpam-1773	225	5	bk	bk	PRON
ejpam-1773	225	6	be	be	AUX
ejpam-1773	225	7	the	the	DET
ejpam-1773	225	8	diagonal	diagonal	ADJ
ejpam-1773	225	9	matrix	matrix	NOUN
ejpam-1773	225	10	indexed	index	VERB
ejpam-1773	225	11	by	by	ADP
ejpam-1773	225	12	the	the	DET
ejpam-1773	225	13	elements	element	NOUN
ejpam-1773	225	14	of	of	ADP
ejpam-1773	225	15	rk	rk	PRON
ejpam-1773	225	16	with	with	ADP
ejpam-1773	225	17	(	(	PUNCT
ejpam-1773	225	18	bk)α	bk)α	PROPN
ejpam-1773	225	19	=	=	PUNCT
ejpam-1773	225	20	ile(α	ile(α	PROPN
ejpam-1773	225	21	)	)	PUNCT
ejpam-1773	225	22	,	,	PUNCT
ejpam-1773	225	23	where	where	SCONJ
ejpam-1773	225	24	i2	i2	PROPN
ejpam-1773	225	25	=	=	PROPN
ejpam-1773	225	26	−1	−1	NOUN
ejpam-1773	225	27	.	.	PUNCT
ejpam-1773	226	1	then	then	ADV
ejpam-1773	226	2	the	the	DET
ejpam-1773	226	3	weight	weight	NOUN
ejpam-1773	226	4	enumerator	enumerator	NOUN
ejpam-1773	226	5	of	of	ADP
ejpam-1773	226	6	a	a	DET
ejpam-1773	226	7	type	type	NOUN
ejpam-1773	226	8	ii	ii	PROPN
ejpam-1773	226	9	code	code	NOUN
ejpam-1773	226	10	is	be	AUX
ejpam-1773	226	11	also	also	ADV
ejpam-1773	226	12	held	hold	VERB
ejpam-1773	226	13	invariant	invariant	ADJ
ejpam-1773	226	14	by	by	ADP
ejpam-1773	226	15	bk	bk	NOUN
ejpam-1773	226	16	.	.	PUNCT
ejpam-1773	227	1	then	then	ADV
ejpam-1773	227	2	the	the	DET
ejpam-1773	227	3	group	group	NOUN
ejpam-1773	227	4	of	of	ADP
ejpam-1773	227	5	invariants	invariant	NOUN
ejpam-1773	227	6	of	of	ADP
ejpam-1773	227	7	a	a	DET
ejpam-1773	227	8	type	type	NOUN
ejpam-1773	227	9	ii	ii	PROPN
ejpam-1773	227	10	code	code	NOUN
ejpam-1773	227	11	over	over	ADP
ejpam-1773	227	12	rk	rk	PROPN
ejpam-1773	227	13	is	be	AUX
ejpam-1773	227	14	gi	gi	INTJ
ejpam-1773	227	15	i	i	NOUN
ejpam-1773	227	16	=	=	PUNCT
ejpam-1773	228	1	〈	〈	PROPN
ejpam-1773	228	2	mk	mk	PROPN
ejpam-1773	228	3	,	,	PUNCT
ejpam-1773	228	4	bk	bk	PROPN
ejpam-1773	228	5	,	,	PUNCT
ejpam-1773	228	6	a1	a1	NOUN
ejpam-1773	228	7	,	,	PUNCT
ejpam-1773	228	8	.	.	PUNCT
ejpam-1773	228	9	.	.	PUNCT
ejpam-1773	229	1	.	.	PUNCT
ejpam-1773	230	1	,	,	PUNCT
ejpam-1773	230	2	as	as	ADP
ejpam-1773	230	3	〉	〉	X
ejpam-1773	230	4	.	.	PUNCT
ejpam-1773	231	1	(	(	PUNCT
ejpam-1773	231	2	12	12	NUM
ejpam-1773	231	3	)	)	PUNCT
ejpam-1773	231	4	the	the	DET
ejpam-1773	231	5	invariants	invariant	NOUN
ejpam-1773	231	6	for	for	ADP
ejpam-1773	231	7	the	the	DET
ejpam-1773	231	8	hamming	hamming	ADJ
ejpam-1773	231	9	weight	weight	NOUN
ejpam-1773	231	10	enumerator	enumerator	NOUN
ejpam-1773	231	11	is	be	AUX
ejpam-1773	231	12	the	the	DET
ejpam-1773	231	13	same	same	ADJ
ejpam-1773	231	14	for	for	ADP
ejpam-1773	231	15	any	any	DET
ejpam-1773	231	16	ring	ring	NOUN
ejpam-1773	231	17	of	of	ADP
ejpam-1773	231	18	order	order	NOUN
ejpam-1773	231	19	22k	22k	NOUN
ejpam-1773	231	20	.	.	PUNCT
ejpam-1773	232	1	that	that	ADV
ejpam-1773	232	2	is	be	AUX
ejpam-1773	232	3	,	,	PUNCT
ejpam-1773	232	4	it	it	PRON
ejpam-1773	232	5	is	be	AUX
ejpam-1773	232	6	held	hold	VERB
ejpam-1773	232	7	invariant	invariant	ADJ
ejpam-1773	232	8	by	by	ADP
ejpam-1773	232	9	the	the	DET
ejpam-1773	232	10	matrix	matrix	NOUN
ejpam-1773	232	11	1p	1p	NUM
ejpam-1773	232	12	22k	22k	NOUN
ejpam-1773	232	13	�	�	PROPN
ejpam-1773	232	14	1	1	NUM
ejpam-1773	232	15	(	(	PUNCT
ejpam-1773	232	16	22k	22k	NOUN
ejpam-1773	232	17	−	−	NOUN
ejpam-1773	232	18	1	1	NUM
ejpam-1773	232	19	)	)	SYM
ejpam-1773	232	20	1	1	NUM
ejpam-1773	232	21	−1	−1	NOUN
ejpam-1773	232	22	�	�	PROPN
ejpam-1773	232	23	.	.	PUNCT
ejpam-1773	233	1	the	the	DET
ejpam-1773	233	2	hamming	hamming	NOUN
ejpam-1773	233	3	weight	weight	NOUN
ejpam-1773	233	4	enumerator	enumerator	NOUN
ejpam-1773	233	5	does	do	AUX
ejpam-1773	233	6	not	not	PART
ejpam-1773	233	7	change	change	VERB
ejpam-1773	233	8	for	for	ADP
ejpam-1773	233	9	type	type	NOUN
ejpam-1773	233	10	ii	ii	PROPN
ejpam-1773	233	11	codes	code	NOUN
ejpam-1773	233	12	.	.	PUNCT
ejpam-1773	234	1	it	it	PRON
ejpam-1773	234	2	follows	follow	VERB
ejpam-1773	234	3	that	that	SCONJ
ejpam-1773	234	4	weight	weight	NOUN
ejpam-1773	234	5	enumerator	enumerator	NOUN
ejpam-1773	234	6	is	be	AUX
ejpam-1773	234	7	a	a	DET
ejpam-1773	234	8	polynomial	polynomial	NOUN
ejpam-1773	234	9	in	in	ADP
ejpam-1773	234	10	x	x	PROPN
ejpam-1773	234	11	+	+	CCONJ
ejpam-1773	234	12	(	(	PUNCT
ejpam-1773	234	13	22k	22k	NOUN
ejpam-1773	234	14	−	−	NOUN
ejpam-1773	234	15	1)y	1)y	NUM
ejpam-1773	234	16	and	and	CCONJ
ejpam-1773	234	17	y(x	y(x	PROPN
ejpam-1773	234	18	−	−	PROPN
ejpam-1773	234	19	y	y	PROPN
ejpam-1773	234	20	)	)	PUNCT
ejpam-1773	234	21	.	.	PUNCT
ejpam-1773	235	1	see	see	VERB
ejpam-1773	235	2	[	[	X
ejpam-1773	235	3	8	8	NUM
ejpam-1773	235	4	]	]	PUNCT
ejpam-1773	235	5	for	for	ADP
ejpam-1773	235	6	details	detail	NOUN
ejpam-1773	235	7	.	.	PUNCT
ejpam-1773	236	1	the	the	DET
ejpam-1773	236	2	lee	lee	PROPN
ejpam-1773	236	3	weight	weight	NOUN
ejpam-1773	236	4	enumerator	enumerator	NOUN
ejpam-1773	236	5	for	for	ADP
ejpam-1773	236	6	a	a	DET
ejpam-1773	236	7	code	code	NOUN
ejpam-1773	236	8	over	over	ADP
ejpam-1773	236	9	rk	rk	NOUN
ejpam-1773	236	10	is	be	AUX
ejpam-1773	236	11	indistinguishable	indistinguishable	ADJ
ejpam-1773	236	12	from	from	ADP
ejpam-1773	236	13	the	the	DET
ejpam-1773	236	14	hamming	hamming	ADJ
ejpam-1773	236	15	weight	weight	NOUN
ejpam-1773	236	16	enumerator	enumerator	NOUN
ejpam-1773	236	17	for	for	ADP
ejpam-1773	236	18	binary	binary	ADJ
ejpam-1773	236	19	self	self	NOUN
ejpam-1773	236	20	-	-	PUNCT
ejpam-1773	236	21	dual	dual	ADJ
ejpam-1773	236	22	codes	code	NOUN
ejpam-1773	236	23	.	.	PUNCT
ejpam-1773	237	1	therefore	therefore	ADV
ejpam-1773	237	2	,	,	PUNCT
ejpam-1773	237	3	the	the	DET
ejpam-1773	237	4	lee	lee	PROPN
ejpam-1773	237	5	weight	weight	NOUN
ejpam-1773	237	6	of	of	ADP
ejpam-1773	237	7	a	a	DET
ejpam-1773	237	8	type	type	NOUN
ejpam-1773	237	9	ii	ii	PROPN
ejpam-1773	237	10	code	code	NOUN
ejpam-1773	237	11	is	be	AUX
ejpam-1773	237	12	a	a	DET
ejpam-1773	237	13	polynomial	polynomial	NOUN
ejpam-1773	237	14	in	in	ADP
ejpam-1773	237	15	the	the	DET
ejpam-1773	237	16	weight	weight	NOUN
ejpam-1773	237	17	enumerator	enumerator	NOUN
ejpam-1773	237	18	of	of	ADP
ejpam-1773	237	19	the	the	DET
ejpam-1773	237	20	extended	extended	ADJ
ejpam-1773	237	21	length	length	NOUN
ejpam-1773	237	22	8	8	NUM
ejpam-1773	237	23	hamming	hamming	NOUN
ejpam-1773	237	24	code	code	NOUN
ejpam-1773	237	25	and	and	CCONJ
ejpam-1773	237	26	the	the	DET
ejpam-1773	237	27	extended	extended	ADJ
ejpam-1773	237	28	binary	binary	NOUN
ejpam-1773	237	29	golay	golay	NOUN
ejpam-1773	237	30	code	code	NOUN
ejpam-1773	237	31	of	of	ADP
ejpam-1773	237	32	length	length	NOUN
ejpam-1773	237	33	24	24	NUM
ejpam-1773	237	34	.	.	PUNCT
ejpam-1773	238	1	the	the	DET
ejpam-1773	238	2	lee	lee	PROPN
ejpam-1773	238	3	weight	weight	NOUN
ejpam-1773	238	4	enumerator	enumerator	NOUN
ejpam-1773	238	5	of	of	ADP
ejpam-1773	238	6	a	a	DET
ejpam-1773	238	7	type	type	NOUN
ejpam-1773	238	8	i	i	PRON
ejpam-1773	238	9	code	code	VERB
ejpam-1773	238	10	is	be	AUX
ejpam-1773	238	11	a	a	DET
ejpam-1773	238	12	polynomial	polynomial	NOUN
ejpam-1773	238	13	in	in	ADP
ejpam-1773	238	14	1	1	NUM
ejpam-1773	238	15	+	+	SYM
ejpam-1773	238	16	z2	z2	NOUN
ejpam-1773	238	17	and	and	CCONJ
ejpam-1773	238	18	the	the	DET
ejpam-1773	238	19	weight	weight	NOUN
ejpam-1773	238	20	enumerator	enumerator	NOUN
ejpam-1773	238	21	of	of	ADP
ejpam-1773	238	22	the	the	DET
ejpam-1773	238	23	extended	extended	ADJ
ejpam-1773	238	24	length	length	NOUN
ejpam-1773	238	25	8	8	NUM
ejpam-1773	238	26	hamming	hamming	NOUN
ejpam-1773	238	27	code	code	NOUN
ejpam-1773	238	28	.	.	PUNCT
ejpam-1773	239	1	s.	s.	PROPN
ejpam-1773	239	2	dougherty	dougherty	PROPN
ejpam-1773	239	3	,	,	PUNCT
ejpam-1773	239	4	b.yıldız	b.yıldız	NOUN
ejpam-1773	239	5	,	,	PUNCT
ejpam-1773	239	6	s.karadeniz	s.karadeniz	NOUN
ejpam-1773	239	7	/	/	SYM
ejpam-1773	239	8	eur	eur	NOUN
ejpam-1773	239	9	.	.	PUNCT
ejpam-1773	240	1	j.	j.	PROPN
ejpam-1773	240	2	pure	pure	PROPN
ejpam-1773	240	3	appl	appl	PROPN
ejpam-1773	240	4	.	.	PROPN
ejpam-1773	240	5	math	math	PROPN
ejpam-1773	240	6	,	,	PUNCT
ejpam-1773	240	7	6	6	NUM
ejpam-1773	240	8	(	(	PUNCT
ejpam-1773	240	9	2013	2013	NUM
ejpam-1773	240	10	)	)	PUNCT
ejpam-1773	240	11	,	,	PUNCT
ejpam-1773	240	12	89	89	NUM
ejpam-1773	240	13	-	-	SYM
ejpam-1773	240	14	106	106	NUM
ejpam-1773	240	15	96	96	NUM
ejpam-1773	240	16	5	5	NUM
ejpam-1773	240	17	.	.	PUNCT
ejpam-1773	240	18	self	self	NOUN
ejpam-1773	240	19	-	-	PUNCT
ejpam-1773	240	20	dual	dual	ADJ
ejpam-1773	240	21	codes	code	NOUN
ejpam-1773	240	22	of	of	ADP
ejpam-1773	240	23	length	length	NOUN
ejpam-1773	240	24	1	1	NUM
ejpam-1773	240	25	and	and	CCONJ
ejpam-1773	240	26	2	2	NUM
ejpam-1773	240	27	5.1	5.1	NUM
ejpam-1773	240	28	.	.	PUNCT
ejpam-1773	241	1	length	length	NOUN
ejpam-1773	241	2	1	1	NUM
ejpam-1773	241	3	self	self	NOUN
ejpam-1773	241	4	-	-	PUNCT
ejpam-1773	241	5	dual	dual	ADJ
ejpam-1773	241	6	codes	code	NOUN
ejpam-1773	241	7	over	over	ADP
ejpam-1773	241	8	rk	rk	NOUN
ejpam-1773	241	9	we	we	PRON
ejpam-1773	241	10	first	first	ADV
ejpam-1773	241	11	note	note	VERB
ejpam-1773	241	12	that	that	SCONJ
ejpam-1773	241	13	if	if	SCONJ
ejpam-1773	241	14	a	a	DET
ejpam-1773	241	15	length	length	NOUN
ejpam-1773	241	16	1	1	NUM
ejpam-1773	241	17	code	code	NOUN
ejpam-1773	241	18	c	c	NOUN
ejpam-1773	241	19	,	,	PUNCT
ejpam-1773	241	20	generated	generate	VERB
ejpam-1773	241	21	by	by	ADP
ejpam-1773	241	22	a	a	DET
ejpam-1773	241	23	+	+	NUM
ejpam-1773	241	24	uk	uk	PROPN
ejpam-1773	241	25	b	b	PROPN
ejpam-1773	241	26	,	,	PUNCT
ejpam-1773	241	27	with	with	ADP
ejpam-1773	241	28	a	a	PRON
ejpam-1773	241	29	,	,	PUNCT
ejpam-1773	241	30	b	b	X
ejpam-1773	241	31	∈	∈	PROPN
ejpam-1773	241	32	rk−1	rk−1	NOUN
ejpam-1773	241	33	is	be	AUX
ejpam-1773	241	34	selforthogonal	selforthogonal	ADJ
ejpam-1773	241	35	,	,	PUNCT
ejpam-1773	241	36	then	then	ADV
ejpam-1773	241	37	we	we	PRON
ejpam-1773	241	38	must	must	AUX
ejpam-1773	241	39	have	have	VERB
ejpam-1773	241	40	that	that	SCONJ
ejpam-1773	241	41	a	a	PRON
ejpam-1773	241	42	is	be	AUX
ejpam-1773	241	43	a	a	DET
ejpam-1773	241	44	non	non	ADJ
ejpam-1773	241	45	-	-	NOUN
ejpam-1773	241	46	unit	unit	NOUN
ejpam-1773	241	47	in	in	ADP
ejpam-1773	241	48	rk−1	rk−1	PROPN
ejpam-1773	241	49	,	,	PUNCT
ejpam-1773	241	50	because	because	SCONJ
ejpam-1773	241	51	if	if	SCONJ
ejpam-1773	241	52	a	a	PRON
ejpam-1773	241	53	were	be	AUX
ejpam-1773	241	54	a	a	DET
ejpam-1773	241	55	unit	unit	NOUN
ejpam-1773	241	56	,	,	PUNCT
ejpam-1773	241	57	then	then	ADV
ejpam-1773	241	58	we	we	PRON
ejpam-1773	241	59	would	would	AUX
ejpam-1773	241	60	have	have	VERB
ejpam-1773	241	61	(	(	PUNCT
ejpam-1773	241	62	a+	a+	PUNCT
ejpam-1773	241	63	uk	uk	PROPN
ejpam-1773	241	64	b)2	b)2	PROPN
ejpam-1773	241	65	=	=	SYM
ejpam-1773	241	66	a2	a2	PROPN
ejpam-1773	241	67	=	=	SYM
ejpam-1773	241	68	1	1	NUM
ejpam-1773	241	69	6=	6=	NUM
ejpam-1773	241	70	0	0	NUM
ejpam-1773	241	71	.	.	PUNCT
ejpam-1773	242	1	we	we	PRON
ejpam-1773	242	2	will	will	AUX
ejpam-1773	242	3	prove	prove	VERB
ejpam-1773	242	4	that	that	SCONJ
ejpam-1773	242	5	if	if	SCONJ
ejpam-1773	242	6	a	a	PRON
ejpam-1773	242	7	is	be	AUX
ejpam-1773	242	8	a	a	DET
ejpam-1773	242	9	non	non	ADJ
ejpam-1773	242	10	-	-	NOUN
ejpam-1773	242	11	unit	unit	NOUN
ejpam-1773	242	12	and	and	CCONJ
ejpam-1773	242	13	b	b	NOUN
ejpam-1773	242	14	is	be	AUX
ejpam-1773	242	15	a	a	DET
ejpam-1773	242	16	unit	unit	NOUN
ejpam-1773	242	17	,	,	PUNCT
ejpam-1773	242	18	then	then	ADV
ejpam-1773	242	19	〈	〈	PROPN
ejpam-1773	242	20	a+	a+	PROPN
ejpam-1773	242	21	uk	uk	PROPN
ejpam-1773	242	22	b	b	PROPN
ejpam-1773	242	23	〉	〉	NOUN
ejpam-1773	242	24	is	be	AUX
ejpam-1773	242	25	a	a	DET
ejpam-1773	242	26	self	self	NOUN
ejpam-1773	242	27	-	-	PUNCT
ejpam-1773	242	28	dual	dual	ADJ
ejpam-1773	242	29	code	code	NOUN
ejpam-1773	242	30	.	.	PUNCT
ejpam-1773	243	1	for	for	ADP
ejpam-1773	243	2	this	this	PRON
ejpam-1773	243	3	we	we	PRON
ejpam-1773	243	4	will	will	AUX
ejpam-1773	243	5	first	first	ADV
ejpam-1773	243	6	introduce	introduce	VERB
ejpam-1773	243	7	the	the	DET
ejpam-1773	243	8	following	follow	VERB
ejpam-1773	243	9	map	map	NOUN
ejpam-1773	243	10	:	:	PUNCT
ejpam-1773	243	11	ψk	ψk	NUM
ejpam-1773	243	12	:	:	PUNCT
ejpam-1773	243	13	rk→	rk→	NOUN
ejpam-1773	243	14	r2	r2	PROPN
ejpam-1773	243	15	k−1	k−1	PROPN
ejpam-1773	243	16	defined	define	VERB
ejpam-1773	243	17	by	by	ADP
ejpam-1773	243	18	ψk(a+	ψk(a+	PROPN
ejpam-1773	243	19	uk	uk	PROPN
ejpam-1773	243	20	b	b	PROPN
ejpam-1773	243	21	)	)	PUNCT
ejpam-1773	243	22	=	=	PUNCT
ejpam-1773	243	23	(	(	PUNCT
ejpam-1773	243	24	b	b	NOUN
ejpam-1773	243	25	,	,	PUNCT
ejpam-1773	243	26	a+	a+	PRON
ejpam-1773	243	27	b	b	NOUN
ejpam-1773	243	28	)	)	PUNCT
ejpam-1773	243	29	.	.	PUNCT
ejpam-1773	244	1	(	(	PUNCT
ejpam-1773	244	2	13	13	NUM
ejpam-1773	244	3	)	)	PUNCT
ejpam-1773	244	4	it	it	PRON
ejpam-1773	244	5	is	be	AUX
ejpam-1773	244	6	easy	easy	ADJ
ejpam-1773	244	7	to	to	PART
ejpam-1773	244	8	verify	verify	VERB
ejpam-1773	244	9	that	that	DET
ejpam-1773	244	10	ψk	ψk	NOUN
ejpam-1773	244	11	is	be	AUX
ejpam-1773	244	12	a	a	DET
ejpam-1773	244	13	linear	linear	ADJ
ejpam-1773	244	14	bijection	bijection	NOUN
ejpam-1773	244	15	from	from	ADP
ejpam-1773	244	16	rn	rn	PROPN
ejpam-1773	244	17	k	k	PROPN
ejpam-1773	244	18	to	to	ADP
ejpam-1773	244	19	r2n	r2n	VERB
ejpam-1773	244	20	k−1	k−1	PROPN
ejpam-1773	244	21	and	and	CCONJ
ejpam-1773	244	22	furthermore	furthermore	ADV
ejpam-1773	244	23	it	it	PRON
ejpam-1773	244	24	is	be	AUX
ejpam-1773	244	25	distance	distance	NOUN
ejpam-1773	244	26	preserving	preserve	VERB
ejpam-1773	244	27	.	.	PUNCT
ejpam-1773	245	1	the	the	DET
ejpam-1773	245	2	following	follow	VERB
ejpam-1773	245	3	lemma	lemma	PROPN
ejpam-1773	245	4	will	will	AUX
ejpam-1773	245	5	help	help	VERB
ejpam-1773	245	6	us	we	PRON
ejpam-1773	245	7	resolve	resolve	VERB
ejpam-1773	245	8	the	the	DET
ejpam-1773	245	9	previous	previous	ADJ
ejpam-1773	245	10	question	question	NOUN
ejpam-1773	245	11	.	.	PUNCT
ejpam-1773	246	1	lemma	lemma	PROPN
ejpam-1773	247	1	3	3	X
ejpam-1773	247	2	.	.	PUNCT
ejpam-1773	248	1	if	if	SCONJ
ejpam-1773	248	2	c	c	PROPN
ejpam-1773	248	3	is	be	AUX
ejpam-1773	248	4	a	a	DET
ejpam-1773	248	5	length	length	NOUN
ejpam-1773	248	6	1	1	NUM
ejpam-1773	248	7	code	code	NOUN
ejpam-1773	248	8	over	over	ADP
ejpam-1773	248	9	rk	rk	NOUN
ejpam-1773	248	10	generated	generate	VERB
ejpam-1773	248	11	by	by	ADP
ejpam-1773	248	12	a+	a+	PUNCT
ejpam-1773	248	13	uk	uk	PROPN
ejpam-1773	248	14	b	b	PROPN
ejpam-1773	248	15	with	with	ADP
ejpam-1773	248	16	a	a	DET
ejpam-1773	248	17	,	,	PUNCT
ejpam-1773	248	18	b	b	PROPN
ejpam-1773	248	19	∈	∈	PROPN
ejpam-1773	248	20	rk−1	rk−1	PROPN
ejpam-1773	248	21	,	,	PUNCT
ejpam-1773	248	22	then	then	ADV
ejpam-1773	248	23	ψk(c	ψk(c	NUM
ejpam-1773	248	24	)	)	PUNCT
ejpam-1773	248	25	is	be	AUX
ejpam-1773	248	26	a	a	DET
ejpam-1773	248	27	length	length	NOUN
ejpam-1773	248	28	2	2	NUM
ejpam-1773	248	29	code	code	NOUN
ejpam-1773	248	30	over	over	ADP
ejpam-1773	248	31	rk−1	rk−1	PROPN
ejpam-1773	248	32	generated	generate	VERB
ejpam-1773	248	33	by	by	ADP
ejpam-1773	248	34	(	(	PUNCT
ejpam-1773	248	35	b	b	NOUN
ejpam-1773	248	36	,	,	PUNCT
ejpam-1773	248	37	a+	a+	PRON
ejpam-1773	248	38	b	b	X
ejpam-1773	248	39	)	)	PUNCT
ejpam-1773	248	40	and	and	CCONJ
ejpam-1773	248	41	(	(	PUNCT
ejpam-1773	248	42	a	a	PRON
ejpam-1773	248	43	,	,	PUNCT
ejpam-1773	248	44	a	a	NOUN
ejpam-1773	248	45	)	)	PUNCT
ejpam-1773	248	46	.	.	PUNCT
ejpam-1773	249	1	proof	proof	NOUN
ejpam-1773	249	2	.	.	PUNCT
ejpam-1773	250	1	we	we	PRON
ejpam-1773	250	2	note	note	VERB
ejpam-1773	250	3	that	that	SCONJ
ejpam-1773	250	4	(	(	PUNCT
ejpam-1773	250	5	x	x	X
ejpam-1773	250	6	+	+	PUNCT
ejpam-1773	250	7	uk	uk	PROPN
ejpam-1773	250	8	y)(a+	y)(a+	PROPN
ejpam-1773	250	9	uk	uk	PROPN
ejpam-1773	250	10	b	b	PROPN
ejpam-1773	250	11	)	)	PUNCT
ejpam-1773	250	12	=	=	NOUN
ejpam-1773	250	13	ax	ax	NOUN
ejpam-1773	251	1	+	+	CCONJ
ejpam-1773	251	2	(	(	PUNCT
ejpam-1773	251	3	x	x	X
ejpam-1773	251	4	b+	b+	X
ejpam-1773	251	5	a	a	DET
ejpam-1773	251	6	y)uk	y)uk	PROPN
ejpam-1773	251	7	for	for	ADP
ejpam-1773	251	8	all	all	DET
ejpam-1773	251	9	x	x	SYM
ejpam-1773	251	10	,	,	PUNCT
ejpam-1773	251	11	y	y	PROPN
ejpam-1773	251	12	∈	∈	PROPN
ejpam-1773	251	13	rk−1	rk−1	NOUN
ejpam-1773	251	14	and	and	CCONJ
ejpam-1773	251	15	hence	hence	ADV
ejpam-1773	251	16	ψk((x	ψk((x	X
ejpam-1773	252	1	+	+	CCONJ
ejpam-1773	252	2	uk	uk	PROPN
ejpam-1773	252	3	y)(a+	y)(a+	PROPN
ejpam-1773	252	4	uk	uk	PROPN
ejpam-1773	252	5	b	b	PROPN
ejpam-1773	252	6	)	)	PUNCT
ejpam-1773	252	7	)	)	PUNCT
ejpam-1773	253	1	=	=	SYM
ejpam-1773	253	2	(	(	PUNCT
ejpam-1773	253	3	x	x	X
ejpam-1773	253	4	b+	b+	X
ejpam-1773	253	5	a	a	DET
ejpam-1773	253	6	y	y	PROPN
ejpam-1773	253	7	,	,	PUNCT
ejpam-1773	253	8	x	x	X
ejpam-1773	253	9	b+	b+	X
ejpam-1773	253	10	a	a	DET
ejpam-1773	253	11	y	y	NOUN
ejpam-1773	253	12	+	+	NOUN
ejpam-1773	253	13	ax	ax	NOUN
ejpam-1773	253	14	)	)	PUNCT
ejpam-1773	253	15	=	=	SYM
ejpam-1773	253	16	x(b	x(b	PROPN
ejpam-1773	253	17	,	,	PUNCT
ejpam-1773	253	18	a+	a+	PUNCT
ejpam-1773	253	19	b	b	X
ejpam-1773	253	20	)	)	PUNCT
ejpam-1773	254	1	+	+	CCONJ
ejpam-1773	254	2	y(a	y(a	PROPN
ejpam-1773	254	3	,	,	PUNCT
ejpam-1773	254	4	a	a	PRON
ejpam-1773	254	5	)	)	PUNCT
ejpam-1773	254	6	.	.	PUNCT
ejpam-1773	255	1	since	since	SCONJ
ejpam-1773	255	2	x	x	PROPN
ejpam-1773	255	3	and	and	CCONJ
ejpam-1773	255	4	y	y	PROPN
ejpam-1773	255	5	are	be	AUX
ejpam-1773	255	6	arbitrary	arbitrary	ADJ
ejpam-1773	255	7	elements	element	NOUN
ejpam-1773	255	8	in	in	ADP
ejpam-1773	255	9	rk−1	rk−1	PROPN
ejpam-1773	255	10	,	,	PUNCT
ejpam-1773	255	11	we	we	PRON
ejpam-1773	255	12	see	see	VERB
ejpam-1773	255	13	that	that	PRON
ejpam-1773	255	14	ψk(c	ψk(c	NUM
ejpam-1773	255	15	)	)	PUNCT
ejpam-1773	255	16	must	must	AUX
ejpam-1773	255	17	be	be	AUX
ejpam-1773	255	18	generated	generate	VERB
ejpam-1773	255	19	by	by	ADP
ejpam-1773	255	20	(	(	PUNCT
ejpam-1773	255	21	b	b	NOUN
ejpam-1773	255	22	,	,	PUNCT
ejpam-1773	255	23	a+	a+	PRON
ejpam-1773	255	24	b	b	X
ejpam-1773	255	25	)	)	PUNCT
ejpam-1773	255	26	and	and	CCONJ
ejpam-1773	255	27	(	(	PUNCT
ejpam-1773	255	28	a	a	PRON
ejpam-1773	255	29	,	,	PUNCT
ejpam-1773	255	30	a	a	NOUN
ejpam-1773	255	31	)	)	PUNCT
ejpam-1773	255	32	.	.	PUNCT
ejpam-1773	256	1	theorem	theorem	ADJ
ejpam-1773	256	2	7	7	NUM
ejpam-1773	256	3	.	.	PUNCT
ejpam-1773	257	1	let	let	VERB
ejpam-1773	257	2	c	c	NOUN
ejpam-1773	257	3	be	be	AUX
ejpam-1773	257	4	the	the	DET
ejpam-1773	257	5	length	length	NOUN
ejpam-1773	257	6	1	1	NUM
ejpam-1773	257	7	code	code	NOUN
ejpam-1773	257	8	over	over	ADP
ejpam-1773	257	9	rk	rk	NOUN
ejpam-1773	257	10	generated	generate	VERB
ejpam-1773	257	11	by	by	ADP
ejpam-1773	257	12	a+	a+	PUNCT
ejpam-1773	257	13	uk	uk	PROPN
ejpam-1773	257	14	b	b	PROPN
ejpam-1773	257	15	where	where	SCONJ
ejpam-1773	257	16	a	a	PRON
ejpam-1773	257	17	is	be	AUX
ejpam-1773	257	18	a	a	DET
ejpam-1773	257	19	non	non	ADJ
ejpam-1773	257	20	-	-	NOUN
ejpam-1773	257	21	unit	unit	NOUN
ejpam-1773	257	22	and	and	CCONJ
ejpam-1773	257	23	b	b	NOUN
ejpam-1773	257	24	is	be	AUX
ejpam-1773	257	25	a	a	DET
ejpam-1773	257	26	unit	unit	NOUN
ejpam-1773	257	27	in	in	ADP
ejpam-1773	257	28	rk−1	rk−1	PROPN
ejpam-1773	257	29	.	.	PUNCT
ejpam-1773	258	1	then	then	ADV
ejpam-1773	258	2	c	c	PROPN
ejpam-1773	258	3	is	be	AUX
ejpam-1773	258	4	self	self	NOUN
ejpam-1773	258	5	-	-	PUNCT
ejpam-1773	258	6	dual	dual	ADJ
ejpam-1773	258	7	.	.	PUNCT
ejpam-1773	259	1	proof	proof	NOUN
ejpam-1773	259	2	.	.	PUNCT
ejpam-1773	260	1	we	we	PRON
ejpam-1773	260	2	first	first	ADV
ejpam-1773	260	3	note	note	VERB
ejpam-1773	260	4	that	that	SCONJ
ejpam-1773	260	5	(	(	PUNCT
ejpam-1773	260	6	a+	a+	PUNCT
ejpam-1773	260	7	uk	uk	PROPN
ejpam-1773	260	8	b)(a+	b)(a+	PROPN
ejpam-1773	260	9	uk	uk	PROPN
ejpam-1773	260	10	b	b	PROPN
ejpam-1773	260	11	)	)	PUNCT
ejpam-1773	260	12	=	=	SYM
ejpam-1773	260	13	a2	a2	PROPN
ejpam-1773	260	14	+	+	CCONJ
ejpam-1773	260	15	uk(ab+	uk(ab+	PROPN
ejpam-1773	260	16	ab	ab	PROPN
ejpam-1773	260	17	)	)	PUNCT
ejpam-1773	260	18	=	=	SYM
ejpam-1773	260	19	0	0	PUNCT
ejpam-1773	260	20	since	since	SCONJ
ejpam-1773	260	21	a	a	PRON
ejpam-1773	260	22	is	be	AUX
ejpam-1773	260	23	a	a	DET
ejpam-1773	260	24	non	non	ADJ
ejpam-1773	260	25	-	-	NOUN
ejpam-1773	260	26	unit	unit	NOUN
ejpam-1773	260	27	.	.	PUNCT
ejpam-1773	261	1	therefore	therefore	ADV
ejpam-1773	261	2	,	,	PUNCT
ejpam-1773	261	3	c	c	PROPN
ejpam-1773	261	4	is	be	AUX
ejpam-1773	261	5	a	a	DET
ejpam-1773	261	6	self	self	NOUN
ejpam-1773	261	7	-	-	PUNCT
ejpam-1773	261	8	orthogonal	orthogonal	ADJ
ejpam-1773	261	9	code	code	NOUN
ejpam-1773	261	10	.	.	PUNCT
ejpam-1773	262	1	by	by	ADP
ejpam-1773	262	2	multiplying	multiply	VERB
ejpam-1773	262	3	by	by	ADP
ejpam-1773	262	4	b	b	PROPN
ejpam-1773	262	5	,	,	PUNCT
ejpam-1773	262	6	which	which	PRON
ejpam-1773	262	7	is	be	AUX
ejpam-1773	262	8	a	a	DET
ejpam-1773	262	9	unit	unit	NOUN
ejpam-1773	262	10	,	,	PUNCT
ejpam-1773	262	11	we	we	PRON
ejpam-1773	262	12	might	might	AUX
ejpam-1773	262	13	assume	assume	VERB
ejpam-1773	262	14	that	that	SCONJ
ejpam-1773	262	15	c	c	PROPN
ejpam-1773	262	16	is	be	AUX
ejpam-1773	262	17	generated	generate	VERB
ejpam-1773	262	18	by	by	ADP
ejpam-1773	262	19	a′	a′	PROPN
ejpam-1773	263	1	+	+	PROPN
ejpam-1773	263	2	uk	uk	PROPN
ejpam-1773	263	3	where	where	SCONJ
ejpam-1773	263	4	a′	a′	PROPN
ejpam-1773	263	5	is	be	AUX
ejpam-1773	263	6	a	a	DET
ejpam-1773	263	7	non	non	ADJ
ejpam-1773	263	8	-	-	NOUN
ejpam-1773	263	9	unit	unit	NOUN
ejpam-1773	263	10	in	in	ADP
ejpam-1773	263	11	rk−1	rk−1	PROPN
ejpam-1773	263	12	.	.	PUNCT
ejpam-1773	264	1	since	since	SCONJ
ejpam-1773	264	2	c	c	PROPN
ejpam-1773	264	3	is	be	AUX
ejpam-1773	264	4	self	self	NOUN
ejpam-1773	264	5	-	-	PUNCT
ejpam-1773	264	6	orthogonal	orthogonal	ADJ
ejpam-1773	264	7	we	we	PRON
ejpam-1773	264	8	only	only	ADV
ejpam-1773	264	9	need	need	VERB
ejpam-1773	264	10	to	to	PART
ejpam-1773	264	11	prove	prove	VERB
ejpam-1773	264	12	that	that	SCONJ
ejpam-1773	264	13	it	it	PRON
ejpam-1773	264	14	has	have	VERB
ejpam-1773	264	15	the	the	DET
ejpam-1773	264	16	right	right	ADJ
ejpam-1773	264	17	cardinality	cardinality	NOUN
ejpam-1773	264	18	.	.	PUNCT
ejpam-1773	265	1	but	but	CCONJ
ejpam-1773	265	2	now	now	ADV
ejpam-1773	265	3	looking	look	VERB
ejpam-1773	265	4	at	at	ADP
ejpam-1773	265	5	ψk(c	ψk(c	NUM
ejpam-1773	265	6	)	)	PUNCT
ejpam-1773	265	7	,	,	PUNCT
ejpam-1773	265	8	we	we	PRON
ejpam-1773	265	9	see	see	VERB
ejpam-1773	265	10	that	that	SCONJ
ejpam-1773	265	11	by	by	ADP
ejpam-1773	265	12	lemma	lemma	PROPN
ejpam-1773	265	13	3	3	NUM
ejpam-1773	265	14	,	,	PUNCT
ejpam-1773	265	15	it	it	PRON
ejpam-1773	265	16	is	be	AUX
ejpam-1773	265	17	generated	generate	VERB
ejpam-1773	265	18	by	by	ADP
ejpam-1773	265	19	(	(	PUNCT
ejpam-1773	265	20	1,1	1,1	NUM
ejpam-1773	265	21	+	+	CCONJ
ejpam-1773	265	22	a′	a′	NUM
ejpam-1773	265	23	)	)	PUNCT
ejpam-1773	265	24	and	and	CCONJ
ejpam-1773	265	25	(	(	PUNCT
ejpam-1773	265	26	a′	a′	PROPN
ejpam-1773	265	27	,	,	PUNCT
ejpam-1773	265	28	a′	a′	PROPN
ejpam-1773	265	29	)	)	PUNCT
ejpam-1773	265	30	.	.	PUNCT
ejpam-1773	266	1	since	since	SCONJ
ejpam-1773	266	2	a′(1,1	a′(1,1	NOUN
ejpam-1773	266	3	+	+	NOUN
ejpam-1773	266	4	a′	a′	NOUN
ejpam-1773	266	5	)	)	PUNCT
ejpam-1773	266	6	=	=	NOUN
ejpam-1773	266	7	(	(	PUNCT
ejpam-1773	266	8	a′	a′	PROPN
ejpam-1773	266	9	,	,	PUNCT
ejpam-1773	266	10	a′	a′	PROPN
ejpam-1773	266	11	)	)	PUNCT
ejpam-1773	266	12	,	,	PUNCT
ejpam-1773	266	13	we	we	PRON
ejpam-1773	266	14	see	see	VERB
ejpam-1773	266	15	that	that	PRON
ejpam-1773	266	16	ψk(c	ψk(c	X
ejpam-1773	266	17	)	)	PUNCT
ejpam-1773	266	18	is	be	AUX
ejpam-1773	266	19	actually	actually	ADV
ejpam-1773	266	20	generated	generate	VERB
ejpam-1773	266	21	over	over	ADP
ejpam-1773	266	22	rk−1	rk−1	NOUN
ejpam-1773	266	23	by	by	ADP
ejpam-1773	266	24	(	(	PUNCT
ejpam-1773	266	25	1,1	1,1	NUM
ejpam-1773	266	26	+	+	CCONJ
ejpam-1773	266	27	a′	a′	NUM
ejpam-1773	266	28	)	)	PUNCT
ejpam-1773	266	29	,	,	PUNCT
ejpam-1773	266	30	and	and	CCONJ
ejpam-1773	266	31	so	so	ADV
ejpam-1773	266	32	it	it	PRON
ejpam-1773	266	33	has	have	VERB
ejpam-1773	266	34	size	size	NOUN
ejpam-1773	266	35	22k−1	22k−1	PROPN
ejpam-1773	266	36	.	.	PUNCT
ejpam-1773	267	1	but	but	CCONJ
ejpam-1773	267	2	since	since	SCONJ
ejpam-1773	267	3	ψk	ψk	NOUN
ejpam-1773	267	4	is	be	AUX
ejpam-1773	267	5	bijective	bijective	ADJ
ejpam-1773	267	6	,	,	PUNCT
ejpam-1773	267	7	we	we	PRON
ejpam-1773	267	8	see	see	VERB
ejpam-1773	267	9	that	that	SCONJ
ejpam-1773	267	10	c	c	PROPN
ejpam-1773	267	11	is	be	AUX
ejpam-1773	267	12	a	a	DET
ejpam-1773	267	13	length	length	NOUN
ejpam-1773	267	14	1	1	NUM
ejpam-1773	267	15	code	code	NOUN
ejpam-1773	267	16	over	over	ADP
ejpam-1773	267	17	rk	rk	NOUN
ejpam-1773	267	18	of	of	ADP
ejpam-1773	267	19	size	size	NOUN
ejpam-1773	267	20	22k−1	22k−1	NUM
ejpam-1773	268	1	and	and	CCONJ
ejpam-1773	268	2	so	so	ADV
ejpam-1773	268	3	it	it	PRON
ejpam-1773	268	4	must	must	AUX
ejpam-1773	268	5	be	be	AUX
ejpam-1773	268	6	self	self	NOUN
ejpam-1773	268	7	-	-	PUNCT
ejpam-1773	268	8	dual	dual	ADJ
ejpam-1773	268	9	.	.	PUNCT
ejpam-1773	269	1	note	note	VERB
ejpam-1773	269	2	that	that	SCONJ
ejpam-1773	269	3	by	by	ADP
ejpam-1773	269	4	changing	change	VERB
ejpam-1773	269	5	the	the	DET
ejpam-1773	269	6	indices	index	NOUN
ejpam-1773	269	7	of	of	ADP
ejpam-1773	269	8	the	the	DET
ejpam-1773	269	9	ui	ui	PROPN
ejpam-1773	269	10	if	if	SCONJ
ejpam-1773	269	11	necessary	necessary	ADJ
ejpam-1773	269	12	,	,	PUNCT
ejpam-1773	269	13	we	we	PRON
ejpam-1773	269	14	can	can	AUX
ejpam-1773	269	15	generalize	generalize	VERB
ejpam-1773	269	16	the	the	DET
ejpam-1773	269	17	previous	previous	ADJ
ejpam-1773	269	18	theorem	theorem	NOUN
ejpam-1773	269	19	as	as	SCONJ
ejpam-1773	269	20	follows	follow	VERB
ejpam-1773	269	21	:	:	PUNCT
ejpam-1773	269	22	corollary	corollary	ADJ
ejpam-1773	269	23	4	4	X
ejpam-1773	269	24	.	.	PUNCT
ejpam-1773	270	1	let	let	VERB
ejpam-1773	270	2	c	c	PRON
ejpam-1773	270	3	be	be	AUX
ejpam-1773	270	4	a	a	DET
ejpam-1773	270	5	length	length	NOUN
ejpam-1773	270	6	1	1	NUM
ejpam-1773	270	7	code	code	NOUN
ejpam-1773	270	8	over	over	ADP
ejpam-1773	270	9	rk	rk	NOUN
ejpam-1773	270	10	generated	generate	VERB
ejpam-1773	270	11	by	by	ADP
ejpam-1773	270	12	a+	a+	PRON
ejpam-1773	270	13	ui	ui	PROPN
ejpam-1773	270	14	b	b	PROPN
ejpam-1773	270	15	for	for	ADP
ejpam-1773	270	16	some	some	DET
ejpam-1773	270	17	i	i	PRON
ejpam-1773	270	18	with	with	ADP
ejpam-1773	270	19	1	1	NUM
ejpam-1773	270	20	≤	≤	NUM
ejpam-1773	271	1	i	i	NOUN
ejpam-1773	271	2	≤	≤	PUNCT
ejpam-1773	272	1	k	k	NOUN
ejpam-1773	272	2	,	,	PUNCT
ejpam-1773	272	3	where	where	SCONJ
ejpam-1773	272	4	a	a	PRON
ejpam-1773	272	5	is	be	AUX
ejpam-1773	272	6	a	a	DET
ejpam-1773	272	7	non	non	ADJ
ejpam-1773	272	8	-	-	NOUN
ejpam-1773	272	9	unit	unit	NOUN
ejpam-1773	272	10	and	and	CCONJ
ejpam-1773	272	11	b	b	NOUN
ejpam-1773	272	12	is	be	AUX
ejpam-1773	272	13	a	a	DET
ejpam-1773	272	14	unit	unit	NOUN
ejpam-1773	272	15	in	in	ADP
ejpam-1773	272	16	rk	rk	NOUN
ejpam-1773	272	17	,	,	PUNCT
ejpam-1773	272	18	such	such	ADJ
ejpam-1773	272	19	that	that	SCONJ
ejpam-1773	272	20	a	a	PRON
ejpam-1773	272	21	and	and	CCONJ
ejpam-1773	272	22	b	b	NOUN
ejpam-1773	272	23	are	be	AUX
ejpam-1773	272	24	not	not	PART
ejpam-1773	272	25	aui	aui	NOUN
ejpam-1773	272	26	,	,	PUNCT
ejpam-1773	272	27	nor	nor	CCONJ
ejpam-1773	272	28	is	be	AUX
ejpam-1773	272	29	bui	bui	NOUN
ejpam-1773	272	30	equal	equal	ADJ
ejpam-1773	272	31	to	to	ADP
ejpam-1773	272	32	0	0	NUM
ejpam-1773	272	33	,	,	PUNCT
ejpam-1773	272	34	that	that	ADV
ejpam-1773	272	35	is	is	ADV
ejpam-1773	272	36	,	,	PUNCT
ejpam-1773	272	37	ui	ui	PROPN
ejpam-1773	272	38	is	be	AUX
ejpam-1773	272	39	not	not	PART
ejpam-1773	272	40	a	a	DET
ejpam-1773	272	41	part	part	NOUN
ejpam-1773	272	42	of	of	ADP
ejpam-1773	272	43	either	either	DET
ejpam-1773	272	44	expression	expression	NOUN
ejpam-1773	272	45	.	.	PUNCT
ejpam-1773	273	1	then	then	ADV
ejpam-1773	273	2	c	c	PROPN
ejpam-1773	273	3	is	be	AUX
ejpam-1773	273	4	a	a	DET
ejpam-1773	273	5	self	self	NOUN
ejpam-1773	273	6	-	-	PUNCT
ejpam-1773	273	7	dual	dual	ADJ
ejpam-1773	273	8	code	code	NOUN
ejpam-1773	273	9	.	.	PUNCT
ejpam-1773	274	1	this	this	PRON
ejpam-1773	274	2	gives	give	VERB
ejpam-1773	274	3	us	we	PRON
ejpam-1773	274	4	a	a	DET
ejpam-1773	274	5	large	large	ADJ
ejpam-1773	274	6	class	class	NOUN
ejpam-1773	274	7	of	of	ADP
ejpam-1773	274	8	length	length	NOUN
ejpam-1773	274	9	1	1	NUM
ejpam-1773	274	10	self	self	NOUN
ejpam-1773	274	11	-	-	PUNCT
ejpam-1773	274	12	dual	dual	ADJ
ejpam-1773	274	13	codes	code	NOUN
ejpam-1773	274	14	.	.	PUNCT
ejpam-1773	275	1	namely	namely	ADV
ejpam-1773	275	2	,	,	PUNCT
ejpam-1773	275	3	if	if	SCONJ
ejpam-1773	275	4	the	the	DET
ejpam-1773	275	5	generator	generator	NOUN
ejpam-1773	275	6	is	be	AUX
ejpam-1773	275	7	a	a	DET
ejpam-1773	275	8	non	non	ADJ
ejpam-1773	275	9	-	-	NOUN
ejpam-1773	275	10	unit	unit	NOUN
ejpam-1773	275	11	of	of	ADP
ejpam-1773	275	12	the	the	DET
ejpam-1773	275	13	form	form	NOUN
ejpam-1773	275	14	ui	ui	PROPN
ejpam-1773	276	1	+	+	CCONJ
ejpam-1773	276	2	c	c	NOUN
ejpam-1773	276	3	for	for	ADP
ejpam-1773	276	4	some	some	DET
ejpam-1773	276	5	i	i	PROPN
ejpam-1773	276	6	,	,	PUNCT
ejpam-1773	276	7	then	then	ADV
ejpam-1773	276	8	the	the	DET
ejpam-1773	276	9	code	code	NOUN
ejpam-1773	276	10	it	it	PRON
ejpam-1773	276	11	generates	generate	VERB
ejpam-1773	276	12	turns	turn	VERB
ejpam-1773	276	13	out	out	ADP
ejpam-1773	276	14	to	to	PART
ejpam-1773	276	15	be	be	AUX
ejpam-1773	276	16	self	self	NOUN
ejpam-1773	276	17	-	-	PUNCT
ejpam-1773	276	18	dual	dual	ADJ
ejpam-1773	276	19	.	.	PUNCT
ejpam-1773	277	1	s.	s.	PROPN
ejpam-1773	277	2	dougherty	dougherty	PROPN
ejpam-1773	277	3	,	,	PUNCT
ejpam-1773	277	4	b.yıldız	b.yıldız	NOUN
ejpam-1773	277	5	,	,	PUNCT
ejpam-1773	277	6	s.karadeniz	s.karadeniz	NOUN
ejpam-1773	277	7	/	/	SYM
ejpam-1773	277	8	eur	eur	NOUN
ejpam-1773	277	9	.	.	PUNCT
ejpam-1773	278	1	j.	j.	PROPN
ejpam-1773	278	2	pure	pure	PROPN
ejpam-1773	278	3	appl	appl	PROPN
ejpam-1773	278	4	.	.	PROPN
ejpam-1773	278	5	math	math	PROPN
ejpam-1773	278	6	,	,	PUNCT
ejpam-1773	278	7	6	6	NUM
ejpam-1773	278	8	(	(	PUNCT
ejpam-1773	278	9	2013	2013	NUM
ejpam-1773	278	10	)	)	PUNCT
ejpam-1773	278	11	,	,	PUNCT
ejpam-1773	278	12	89	89	NUM
ejpam-1773	278	13	-	-	SYM
ejpam-1773	278	14	106	106	NUM
ejpam-1773	278	15	97	97	NUM
ejpam-1773	278	16	theorem	theorem	NOUN
ejpam-1773	278	17	8	8	NUM
ejpam-1773	278	18	.	.	PUNCT
ejpam-1773	279	1	every	every	DET
ejpam-1773	279	2	self	self	NOUN
ejpam-1773	279	3	-	-	PUNCT
ejpam-1773	279	4	dual	dual	ADJ
ejpam-1773	279	5	code	code	NOUN
ejpam-1773	279	6	generated	generate	VERB
ejpam-1773	279	7	by	by	ADP
ejpam-1773	279	8	a	a	DET
ejpam-1773	279	9	single	single	ADJ
ejpam-1773	279	10	element	element	NOUN
ejpam-1773	279	11	is	be	AUX
ejpam-1773	279	12	generated	generate	VERB
ejpam-1773	279	13	by	by	ADP
ejpam-1773	279	14	an	an	DET
ejpam-1773	279	15	element	element	NOUN
ejpam-1773	279	16	of	of	ADP
ejpam-1773	279	17	the	the	DET
ejpam-1773	279	18	form	form	NOUN
ejpam-1773	279	19	given	give	VERB
ejpam-1773	279	20	in	in	ADP
ejpam-1773	279	21	theorem	theorem	ADJ
ejpam-1773	279	22	7	7	NUM
ejpam-1773	279	23	.	.	PUNCT
ejpam-1773	279	24	proof	proof	NOUN
ejpam-1773	279	25	.	.	PUNCT
ejpam-1773	280	1	we	we	PRON
ejpam-1773	280	2	shall	shall	AUX
ejpam-1773	280	3	prove	prove	VERB
ejpam-1773	280	4	that	that	SCONJ
ejpam-1773	280	5	if	if	SCONJ
ejpam-1773	280	6	a	a	DET
ejpam-1773	280	7	one	one	NUM
ejpam-1773	280	8	-	-	PUNCT
ejpam-1773	280	9	generator	generator	NOUN
ejpam-1773	280	10	code	code	NOUN
ejpam-1773	280	11	is	be	AUX
ejpam-1773	280	12	not	not	PART
ejpam-1773	280	13	of	of	ADP
ejpam-1773	280	14	the	the	DET
ejpam-1773	280	15	form	form	NOUN
ejpam-1773	280	16	described	describe	VERB
ejpam-1773	280	17	above	above	ADV
ejpam-1773	280	18	,	,	PUNCT
ejpam-1773	280	19	i.e.	i.e.	X
ejpam-1773	280	20	,	,	PUNCT
ejpam-1773	280	21	if	if	SCONJ
ejpam-1773	280	22	every	every	DET
ejpam-1773	280	23	set	set	NOUN
ejpam-1773	280	24	in	in	ADP
ejpam-1773	280	25	the	the	DET
ejpam-1773	280	26	support	support	NOUN
ejpam-1773	280	27	of	of	ADP
ejpam-1773	280	28	the	the	DET
ejpam-1773	280	29	generator	generator	NOUN
ejpam-1773	280	30	contains	contain	VERB
ejpam-1773	280	31	at	at	ADV
ejpam-1773	280	32	least	least	ADV
ejpam-1773	280	33	two	two	NUM
ejpam-1773	280	34	elements	element	NOUN
ejpam-1773	280	35	,	,	PUNCT
ejpam-1773	280	36	then	then	ADV
ejpam-1773	280	37	it	it	PRON
ejpam-1773	280	38	can	can	AUX
ejpam-1773	280	39	not	not	PART
ejpam-1773	280	40	generate	generate	VERB
ejpam-1773	280	41	a	a	DET
ejpam-1773	280	42	self	self	NOUN
ejpam-1773	280	43	-	-	PUNCT
ejpam-1773	280	44	dual	dual	ADJ
ejpam-1773	280	45	code	code	NOUN
ejpam-1773	280	46	over	over	ADP
ejpam-1773	280	47	rk	rk	NOUN
ejpam-1773	280	48	.	.	PUNCT
ejpam-1773	280	49	to	to	PART
ejpam-1773	280	50	prove	prove	VERB
ejpam-1773	280	51	this	this	PRON
ejpam-1773	280	52	,	,	PUNCT
ejpam-1773	280	53	we	we	PRON
ejpam-1773	280	54	let	let	VERB
ejpam-1773	280	55	a	a	PRON
ejpam-1773	280	56	be	be	AUX
ejpam-1773	280	57	a	a	DET
ejpam-1773	280	58	non	non	ADJ
ejpam-1773	280	59	-	-	NOUN
ejpam-1773	280	60	unit	unit	NOUN
ejpam-1773	280	61	in	in	ADP
ejpam-1773	280	62	rk	rk	NOUN
ejpam-1773	280	63	with	with	ADP
ejpam-1773	280	64	k	k	PROPN
ejpam-1773	280	65	≥	≥	NUM
ejpam-1773	280	66	2	2	NUM
ejpam-1773	280	67	and	and	CCONJ
ejpam-1773	280	68	every	every	DET
ejpam-1773	280	69	component	component	NOUN
ejpam-1773	280	70	in	in	ADP
ejpam-1773	280	71	a	a	DET
ejpam-1773	280	72	is	is	NOUN
ejpam-1773	280	73	of	of	ADP
ejpam-1773	280	74	the	the	DET
ejpam-1773	280	75	form	form	NOUN
ejpam-1773	280	76	ua	ua	PROPN
ejpam-1773	280	77	with	with	ADP
ejpam-1773	280	78	|a|	|a|	PROPN
ejpam-1773	280	79	≥	≥	NOUN
ejpam-1773	280	80	2	2	NUM
ejpam-1773	280	81	.	.	PUNCT
ejpam-1773	281	1	we	we	PRON
ejpam-1773	281	2	will	will	AUX
ejpam-1773	281	3	prove	prove	VERB
ejpam-1773	281	4	that	that	SCONJ
ejpam-1773	281	5	c	c	NOUN
ejpam-1773	281	6	=	=	PUNCT
ejpam-1773	281	7	〈	〈	PROPN
ejpam-1773	281	8	a	a	DET
ejpam-1773	281	9	〉	〉	NOUN
ejpam-1773	281	10	can	can	AUX
ejpam-1773	281	11	not	not	PART
ejpam-1773	281	12	be	be	AUX
ejpam-1773	281	13	self	self	NOUN
ejpam-1773	281	14	-	-	PUNCT
ejpam-1773	281	15	dual	dual	ADJ
ejpam-1773	281	16	.	.	PUNCT
ejpam-1773	282	1	of	of	ADP
ejpam-1773	282	2	course	course	NOUN
ejpam-1773	282	3	,	,	PUNCT
ejpam-1773	282	4	c	c	PROPN
ejpam-1773	282	5	is	be	AUX
ejpam-1773	282	6	self	self	NOUN
ejpam-1773	282	7	-	-	PUNCT
ejpam-1773	282	8	orthogonal	orthogonal	ADJ
ejpam-1773	282	9	,	,	PUNCT
ejpam-1773	282	10	giving	give	VERB
ejpam-1773	282	11	that	that	SCONJ
ejpam-1773	282	12	c	c	PROPN
ejpam-1773	282	13	⊆	⊆	NUM
ejpam-1773	282	14	c⊥.	c⊥.	NOUN
ejpam-1773	282	15	to	to	PART
ejpam-1773	282	16	prove	prove	VERB
ejpam-1773	282	17	that	that	SCONJ
ejpam-1773	282	18	c	c	PROPN
ejpam-1773	282	19	is	be	AUX
ejpam-1773	282	20	not	not	PART
ejpam-1773	282	21	self	self	NOUN
ejpam-1773	282	22	-	-	PUNCT
ejpam-1773	282	23	dual	dual	ADJ
ejpam-1773	282	24	,	,	PUNCT
ejpam-1773	282	25	we	we	PRON
ejpam-1773	282	26	will	will	AUX
ejpam-1773	282	27	exhibit	exhibit	VERB
ejpam-1773	282	28	an	an	DET
ejpam-1773	282	29	element	element	NOUN
ejpam-1773	282	30	in	in	ADP
ejpam-1773	282	31	c⊥	c⊥	PROPN
ejpam-1773	282	32	that	that	PRON
ejpam-1773	282	33	is	be	AUX
ejpam-1773	282	34	not	not	PART
ejpam-1773	282	35	in	in	ADP
ejpam-1773	282	36	c	c	NOUN
ejpam-1773	282	37	.	.	PUNCT
ejpam-1773	283	1	we	we	PRON
ejpam-1773	283	2	let	let	VERB
ejpam-1773	283	3	ub	ub	INTJ
ejpam-1773	283	4	be	be	AUX
ejpam-1773	283	5	an	an	DET
ejpam-1773	283	6	element	element	NOUN
ejpam-1773	283	7	with	with	ADP
ejpam-1773	283	8	minimal	minimal	PROPN
ejpam-1773	283	9	b	b	PROPN
ejpam-1773	283	10	6=	6=	NUM
ejpam-1773	283	11	;	;	PUNCT
ejpam-1773	283	12	such	such	ADJ
ejpam-1773	283	13	that	that	SCONJ
ejpam-1773	283	14	a	a	DET
ejpam-1773	283	15	·	·	PUNCT
ejpam-1773	283	16	ub	ub	ADJ
ejpam-1773	283	17	=	=	SYM
ejpam-1773	283	18	0	0	PROPN
ejpam-1773	283	19	.	.	PUNCT
ejpam-1773	284	1	for	for	ADP
ejpam-1773	284	2	example	example	NOUN
ejpam-1773	284	3	,	,	PUNCT
ejpam-1773	284	4	if	if	SCONJ
ejpam-1773	284	5	a	a	DET
ejpam-1773	284	6	=	=	SYM
ejpam-1773	284	7	u1u2	u1u2	NOUN
ejpam-1773	284	8	,	,	PUNCT
ejpam-1773	284	9	we	we	PRON
ejpam-1773	284	10	can	can	AUX
ejpam-1773	284	11	choose	choose	VERB
ejpam-1773	284	12	ub	ub	NOUN
ejpam-1773	284	13	to	to	PART
ejpam-1773	284	14	be	be	AUX
ejpam-1773	284	15	u1	u1	NOUN
ejpam-1773	284	16	or	or	CCONJ
ejpam-1773	284	17	u2	u2	NOUN
ejpam-1773	284	18	.	.	PUNCT
ejpam-1773	285	1	if	if	SCONJ
ejpam-1773	285	2	a	a	DET
ejpam-1773	285	3	=	=	X
ejpam-1773	285	4	u1u2	u1u2	NOUN
ejpam-1773	285	5	+	+	X
ejpam-1773	285	6	u3u4	u3u4	PROPN
ejpam-1773	285	7	,	,	PUNCT
ejpam-1773	285	8	then	then	ADV
ejpam-1773	285	9	we	we	PRON
ejpam-1773	285	10	can	can	AUX
ejpam-1773	285	11	choose	choose	VERB
ejpam-1773	285	12	ub	ub	NOUN
ejpam-1773	285	13	to	to	PART
ejpam-1773	285	14	be	be	AUX
ejpam-1773	285	15	u1u3	u1u3	X
ejpam-1773	285	16	or	or	CCONJ
ejpam-1773	285	17	u1u4	u1u4	PROPN
ejpam-1773	285	18	or	or	CCONJ
ejpam-1773	285	19	u2u3	u2u3	X
ejpam-1773	285	20	or	or	CCONJ
ejpam-1773	285	21	u2u4	u2u4	NOUN
ejpam-1773	285	22	.	.	PUNCT
ejpam-1773	286	1	note	note	VERB
ejpam-1773	286	2	that	that	SCONJ
ejpam-1773	286	3	ub	ub	ADV
ejpam-1773	286	4	=	=	SYM
ejpam-1773	286	5	u1	u1	NOUN
ejpam-1773	286	6	.	.	PUNCT
ejpam-1773	286	7	.	.	PUNCT
ejpam-1773	286	8	.	.	PUNCT
ejpam-1773	287	1	uk	uk	PROPN
ejpam-1773	288	1	if	if	SCONJ
ejpam-1773	288	2	and	and	CCONJ
ejpam-1773	288	3	only	only	ADV
ejpam-1773	288	4	if	if	SCONJ
ejpam-1773	288	5	a	a	PRON
ejpam-1773	288	6	is	be	AUX
ejpam-1773	288	7	of	of	ADP
ejpam-1773	288	8	the	the	DET
ejpam-1773	288	9	form	form	NOUN
ejpam-1773	288	10	u1	u1	NOUN
ejpam-1773	288	11	+	+	CCONJ
ejpam-1773	288	12	u2	u2	PROPN
ejpam-1773	288	13	+	+	X
ejpam-1773	288	14	.	.	PUNCT
ejpam-1773	288	15	.	.	PUNCT
ejpam-1773	289	1	.+	.+	NOUN
ejpam-1773	289	2	uk	uk	PROPN
ejpam-1773	289	3	,	,	PUNCT
ejpam-1773	289	4	so	so	SCONJ
ejpam-1773	289	5	in	in	ADP
ejpam-1773	289	6	our	our	PRON
ejpam-1773	289	7	case	case	NOUN
ejpam-1773	289	8	we	we	PRON
ejpam-1773	289	9	know	know	VERB
ejpam-1773	289	10	that	that	SCONJ
ejpam-1773	289	11	|b|	|b|	PROPN
ejpam-1773	289	12	<	<	X
ejpam-1773	289	13	k.	k.	X
ejpam-1773	289	14	after	after	ADP
ejpam-1773	289	15	rearranging	rearrange	VERB
ejpam-1773	289	16	the	the	DET
ejpam-1773	289	17	indices	index	NOUN
ejpam-1773	289	18	if	if	SCONJ
ejpam-1773	289	19	necessary	necessary	ADJ
ejpam-1773	289	20	,	,	PUNCT
ejpam-1773	289	21	we	we	PRON
ejpam-1773	289	22	might	might	AUX
ejpam-1773	289	23	assume	assume	VERB
ejpam-1773	289	24	,	,	PUNCT
ejpam-1773	289	25	without	without	ADP
ejpam-1773	289	26	loss	loss	NOUN
ejpam-1773	289	27	of	of	ADP
ejpam-1773	289	28	generality	generality	NOUN
ejpam-1773	289	29	,	,	PUNCT
ejpam-1773	289	30	that	that	SCONJ
ejpam-1773	289	31	ub	ub	VERB
ejpam-1773	289	32	=	=	X
ejpam-1773	289	33	u1u2	u1u2	PROPN
ejpam-1773	289	34	.	.	PUNCT
ejpam-1773	289	35	.	.	PUNCT
ejpam-1773	289	36	.	.	PUNCT
ejpam-1773	290	1	us	we	PRON
ejpam-1773	290	2	,	,	PUNCT
ejpam-1773	290	3	with	with	ADP
ejpam-1773	290	4	s	s	PRON
ejpam-1773	290	5	<	<	X
ejpam-1773	290	6	k.	k.	X
ejpam-1773	290	7	now	now	ADV
ejpam-1773	290	8	,	,	PUNCT
ejpam-1773	290	9	by	by	ADP
ejpam-1773	290	10	definition	definition	NOUN
ejpam-1773	290	11	,	,	PUNCT
ejpam-1773	290	12	ub	ub	PROPN
ejpam-1773	290	13	∈	∈	PROPN
ejpam-1773	290	14	c⊥.	c⊥.	NOUN
ejpam-1773	290	15	therefore	therefore	ADV
ejpam-1773	290	16	,	,	PUNCT
ejpam-1773	290	17	it	it	PRON
ejpam-1773	290	18	is	be	AUX
ejpam-1773	290	19	enough	enough	ADJ
ejpam-1773	290	20	to	to	PART
ejpam-1773	290	21	show	show	VERB
ejpam-1773	290	22	that	that	SCONJ
ejpam-1773	290	23	ub	ub	INTJ
ejpam-1773	290	24	6∈	6∈	PROPN
ejpam-1773	290	25	c	c	PROPN
ejpam-1773	290	26	.	.	PUNCT
ejpam-1773	291	1	assume	assume	VERB
ejpam-1773	291	2	that	that	SCONJ
ejpam-1773	291	3	r	r	NOUN
ejpam-1773	291	4	·	·	PUNCT
ejpam-1773	291	5	a	a	DET
ejpam-1773	291	6	=	=	X
ejpam-1773	291	7	u1u2	u1u2	NOUN
ejpam-1773	291	8	.	.	PUNCT
ejpam-1773	291	9	.	.	PUNCT
ejpam-1773	291	10	.	.	PUNCT
ejpam-1773	292	1	us	we	PRON
ejpam-1773	292	2	.	.	PUNCT
ejpam-1773	293	1	if	if	SCONJ
ejpam-1773	293	2	a	a	PRON
ejpam-1773	293	3	contains	contain	VERB
ejpam-1773	293	4	just	just	ADV
ejpam-1773	293	5	one	one	NUM
ejpam-1773	293	6	component	component	NOUN
ejpam-1773	293	7	,	,	PUNCT
ejpam-1773	293	8	then	then	ADV
ejpam-1773	293	9	we	we	PRON
ejpam-1773	293	10	would	would	AUX
ejpam-1773	293	11	have	have	VERB
ejpam-1773	293	12	s	s	NOUN
ejpam-1773	293	13	=	=	SYM
ejpam-1773	293	14	1	1	NUM
ejpam-1773	293	15	and	and	CCONJ
ejpam-1773	293	16	we	we	PRON
ejpam-1773	293	17	know	know	VERB
ejpam-1773	293	18	in	in	ADP
ejpam-1773	293	19	that	that	DET
ejpam-1773	293	20	case	case	NOUN
ejpam-1773	293	21	u1	u1	NOUN
ejpam-1773	293	22	6∈	6∈	PROPN
ejpam-1773	293	23	c	c	NOUN
ejpam-1773	293	24	.	.	PUNCT
ejpam-1773	294	1	because	because	SCONJ
ejpam-1773	294	2	ub	ub	ADJ
ejpam-1773	294	3	=	=	SYM
ejpam-1773	294	4	u1	u1	NOUN
ejpam-1773	294	5	.	.	PUNCT
ejpam-1773	294	6	.	.	PUNCT
ejpam-1773	294	7	.	.	PUNCT
ejpam-1773	295	1	us	we	PRON
ejpam-1773	295	2	,	,	PUNCT
ejpam-1773	295	3	we	we	PRON
ejpam-1773	295	4	must	must	AUX
ejpam-1773	295	5	have	have	VERB
ejpam-1773	295	6	a	a	DET
ejpam-1773	295	7	=	=	SYM
ejpam-1773	295	8	u1a1	u1a1	PROPN
ejpam-1773	295	9	+	+	NOUN
ejpam-1773	295	10	u2a2	u2a2	PROPN
ejpam-1773	295	11	+	+	NUM
ejpam-1773	295	12	.	.	PUNCT
ejpam-1773	295	13	.	.	PUNCT
ejpam-1773	296	1	.+	.+	NOUN
ejpam-1773	296	2	usas	usas	PROPN
ejpam-1773	296	3	,	,	PUNCT
ejpam-1773	296	4	where	where	SCONJ
ejpam-1773	296	5	a1	a1	NOUN
ejpam-1773	296	6	,	,	PUNCT
ejpam-1773	296	7	a2	a2	PROPN
ejpam-1773	296	8	,	,	PUNCT
ejpam-1773	296	9	.	.	PUNCT
ejpam-1773	296	10	.	.	PUNCT
ejpam-1773	296	11	.	.	PUNCT
ejpam-1773	297	1	,	,	PUNCT
ejpam-1773	297	2	as	as	SCONJ
ejpam-1773	297	3	are	be	AUX
ejpam-1773	297	4	non	non	ADJ
ejpam-1773	297	5	-	-	ADJ
ejpam-1773	297	6	zero	zero	NUM
ejpam-1773	297	7	non	non	ADJ
ejpam-1773	297	8	-	-	NOUN
ejpam-1773	297	9	units	unit	NOUN
ejpam-1773	297	10	.	.	PUNCT
ejpam-1773	298	1	additionally	additionally	ADV
ejpam-1773	298	2	,	,	PUNCT
ejpam-1773	298	3	ai	ai	INTJ
ejpam-1773	298	4	does	do	AUX
ejpam-1773	298	5	not	not	PART
ejpam-1773	298	6	contain	contain	VERB
ejpam-1773	298	7	any	any	PRON
ejpam-1773	298	8	of	of	ADP
ejpam-1773	298	9	the	the	DET
ejpam-1773	298	10	u1,u2	u1,u2	PROPN
ejpam-1773	298	11	,	,	PUNCT
ejpam-1773	298	12	.	.	PUNCT
ejpam-1773	298	13	.	.	PUNCT
ejpam-1773	299	1	.	.	PUNCT
ejpam-1773	300	1	,	,	PUNCT
ejpam-1773	300	2	ui−1	ui−1	PROPN
ejpam-1773	300	3	for	for	ADP
ejpam-1773	300	4	i	i	PRON
ejpam-1773	300	5	=	=	NOUN
ejpam-1773	300	6	2,3	2,3	NUM
ejpam-1773	300	7	.	.	PUNCT
ejpam-1773	300	8	.	.	PUNCT
ejpam-1773	300	9	.	.	PUNCT
ejpam-1773	301	1	,	,	PUNCT
ejpam-1773	301	2	s.	s.	PROPN
ejpam-1773	301	3	since	since	SCONJ
ejpam-1773	301	4	as	as	SCONJ
ejpam-1773	301	5	contains	contain	VERB
ejpam-1773	301	6	some	some	PRON
ejpam-1773	301	7	of	of	ADP
ejpam-1773	301	8	us+1	us+1	NOUN
ejpam-1773	301	9	,	,	PUNCT
ejpam-1773	301	10	.	.	PUNCT
ejpam-1773	301	11	.	.	PUNCT
ejpam-1773	302	1	.	.	PUNCT
ejpam-1773	303	1	,	,	PUNCT
ejpam-1773	303	2	uk	uk	PROPN
ejpam-1773	303	3	,	,	PUNCT
ejpam-1773	303	4	in	in	ADP
ejpam-1773	303	5	order	order	NOUN
ejpam-1773	303	6	for	for	SCONJ
ejpam-1773	303	7	ra	ra	PROPN
ejpam-1773	303	8	to	to	PART
ejpam-1773	303	9	be	be	AUX
ejpam-1773	303	10	u1u2	u1u2	X
ejpam-1773	303	11	.	.	PUNCT
ejpam-1773	303	12	.	.	PUNCT
ejpam-1773	303	13	.	.	PUNCT
ejpam-1773	304	1	us	we	PRON
ejpam-1773	304	2	,	,	PUNCT
ejpam-1773	304	3	r	r	NOUN
ejpam-1773	304	4	must	must	AUX
ejpam-1773	304	5	contain	contain	VERB
ejpam-1773	304	6	us	we	PRON
ejpam-1773	304	7	.	.	PUNCT
ejpam-1773	305	1	therefore	therefore	ADV
ejpam-1773	305	2	we	we	PRON
ejpam-1773	305	3	can	can	AUX
ejpam-1773	305	4	write	write	VERB
ejpam-1773	305	5	r	r	NOUN
ejpam-1773	305	6	=	=	SYM
ejpam-1773	305	7	r1us	r1us	NOUN
ejpam-1773	305	8	.	.	PUNCT
ejpam-1773	306	1	now	now	ADV
ejpam-1773	306	2	,	,	PUNCT
ejpam-1773	306	3	aus	aus	PROPN
ejpam-1773	306	4	=	=	PUNCT
ejpam-1773	306	5	u1usa1	u1usa1	X
ejpam-1773	306	6	+	+	NOUN
ejpam-1773	306	7	u2usa2	u2usa2	NOUN
ejpam-1773	306	8	+	+	X
ejpam-1773	306	9	.	.	PUNCT
ejpam-1773	306	10	.	.	PUNCT
ejpam-1773	306	11	.	.	PUNCT
ejpam-1773	307	1	us−1usas−1	us−1usas−1	PROPN
ejpam-1773	307	2	.	.	PUNCT
ejpam-1773	308	1	we	we	PRON
ejpam-1773	308	2	know	know	VERB
ejpam-1773	308	3	that	that	DET
ejpam-1773	308	4	us−1usas−1	us−1usas−1	ADJ
ejpam-1773	308	5	6=	6=	NUM
ejpam-1773	308	6	0	0	NUM
ejpam-1773	308	7	,	,	PUNCT
ejpam-1773	308	8	because	because	SCONJ
ejpam-1773	308	9	if	if	SCONJ
ejpam-1773	308	10	it	it	PRON
ejpam-1773	308	11	were	be	AUX
ejpam-1773	308	12	,	,	PUNCT
ejpam-1773	308	13	ub\{s−1	ub\{s−1	NOUN
ejpam-1773	308	14	}	}	PUNCT
ejpam-1773	308	15	would	would	AUX
ejpam-1773	308	16	annihilate	annihilate	VERB
ejpam-1773	308	17	a	a	PRON
ejpam-1773	308	18	,	,	PUNCT
ejpam-1773	308	19	contradicting	contradict	VERB
ejpam-1773	308	20	the	the	DET
ejpam-1773	308	21	minimality	minimality	NOUN
ejpam-1773	308	22	of	of	ADP
ejpam-1773	308	23	b.	b.	PROPN
ejpam-1773	308	24	again	again	ADV
ejpam-1773	308	25	since	since	SCONJ
ejpam-1773	308	26	usas−1	usas−1	PROPN
ejpam-1773	308	27	contains	contain	VERB
ejpam-1773	308	28	elements	element	NOUN
ejpam-1773	308	29	from	from	ADP
ejpam-1773	308	30	us+1	us+1	PROPN
ejpam-1773	308	31	,	,	PUNCT
ejpam-1773	308	32	.	.	PUNCT
ejpam-1773	308	33	.	.	PUNCT
ejpam-1773	308	34	.	.	PUNCT
ejpam-1773	309	1	,	,	PUNCT
ejpam-1773	309	2	uk	uk	PROPN
ejpam-1773	309	3	,	,	PUNCT
ejpam-1773	309	4	we	we	PRON
ejpam-1773	309	5	must	must	AUX
ejpam-1773	309	6	have	have	VERB
ejpam-1773	309	7	that	that	DET
ejpam-1773	309	8	r1	r1	PROPN
ejpam-1773	309	9	contains	contain	VERB
ejpam-1773	309	10	us−1	us−1	PROPN
ejpam-1773	309	11	.	.	PUNCT
ejpam-1773	310	1	continuing	continue	VERB
ejpam-1773	310	2	this	this	DET
ejpam-1773	310	3	way	way	NOUN
ejpam-1773	310	4	,	,	PUNCT
ejpam-1773	310	5	we	we	PRON
ejpam-1773	310	6	see	see	VERB
ejpam-1773	310	7	that	that	SCONJ
ejpam-1773	310	8	r	r	NOUN
ejpam-1773	310	9	=	=	SYM
ejpam-1773	310	10	u1u2	u1u2	NOUN
ejpam-1773	310	11	.	.	PUNCT
ejpam-1773	310	12	.	.	PUNCT
ejpam-1773	310	13	.	.	PUNCT
ejpam-1773	311	1	us	we	PRON
ejpam-1773	311	2	.	.	PUNCT
ejpam-1773	312	1	but	but	CCONJ
ejpam-1773	312	2	in	in	ADP
ejpam-1773	312	3	that	that	DET
ejpam-1773	312	4	case	case	NOUN
ejpam-1773	312	5	it	it	PRON
ejpam-1773	312	6	is	be	AUX
ejpam-1773	312	7	impossible	impossible	ADJ
ejpam-1773	312	8	to	to	PART
ejpam-1773	312	9	have	have	VERB
ejpam-1773	312	10	ra	ra	NOUN
ejpam-1773	312	11	=	=	NOUN
ejpam-1773	312	12	u1u2	u1u2	PROPN
ejpam-1773	312	13	.	.	PUNCT
ejpam-1773	312	14	.	.	PUNCT
ejpam-1773	312	15	.	.	PUNCT
ejpam-1773	313	1	us	we	PRON
ejpam-1773	313	2	.	.	PUNCT
ejpam-1773	314	1	thus	thus	ADV
ejpam-1773	314	2	,	,	PUNCT
ejpam-1773	314	3	we	we	PRON
ejpam-1773	314	4	have	have	AUX
ejpam-1773	314	5	classified	classify	VERB
ejpam-1773	314	6	all	all	DET
ejpam-1773	314	7	one	one	NUM
ejpam-1773	314	8	-	-	PUNCT
ejpam-1773	314	9	generator	generator	NOUN
ejpam-1773	314	10	length	length	NOUN
ejpam-1773	314	11	1	1	NUM
ejpam-1773	314	12	self	self	NOUN
ejpam-1773	314	13	-	-	PUNCT
ejpam-1773	314	14	dual	dual	ADJ
ejpam-1773	314	15	codes	code	NOUN
ejpam-1773	314	16	over	over	ADP
ejpam-1773	314	17	rk	rk	PROPN
ejpam-1773	314	18	for	for	ADP
ejpam-1773	314	19	k	k	PROPN
ejpam-1773	314	20	≥	≥	PROPN
ejpam-1773	314	21	2	2	NUM
ejpam-1773	314	22	.	.	PUNCT
ejpam-1773	314	23	not	not	PART
ejpam-1773	314	24	all	all	PRON
ejpam-1773	314	25	length	length	NOUN
ejpam-1773	314	26	one	one	NUM
ejpam-1773	314	27	self	self	NOUN
ejpam-1773	314	28	-	-	PUNCT
ejpam-1773	314	29	dual	dual	ADJ
ejpam-1773	314	30	codes	code	NOUN
ejpam-1773	314	31	are	be	AUX
ejpam-1773	314	32	principal	principal	ADJ
ejpam-1773	314	33	ideals	ideal	NOUN
ejpam-1773	314	34	.	.	PUNCT
ejpam-1773	315	1	for	for	ADP
ejpam-1773	315	2	example	example	NOUN
ejpam-1773	315	3	,	,	PUNCT
ejpam-1773	315	4	the	the	DET
ejpam-1773	315	5	code	code	NOUN
ejpam-1773	315	6	〈	〈	NOUN
ejpam-1773	315	7	u1u2,u1u3,u2u3	u1u2,u1u3,u2u3	NOUN
ejpam-1773	315	8	〉	〉	NOUN
ejpam-1773	315	9	over	over	ADP
ejpam-1773	315	10	r3	r3	PROPN
ejpam-1773	315	11	is	be	AUX
ejpam-1773	315	12	self	self	NOUN
ejpam-1773	315	13	-	-	PUNCT
ejpam-1773	315	14	dual	dual	ADJ
ejpam-1773	315	15	but	but	CCONJ
ejpam-1773	315	16	not	not	PART
ejpam-1773	315	17	principal	principal	ADJ
ejpam-1773	315	18	.	.	PUNCT
ejpam-1773	316	1	we	we	PRON
ejpam-1773	316	2	generalize	generalize	VERB
ejpam-1773	316	3	this	this	PRON
ejpam-1773	316	4	to	to	ADP
ejpam-1773	316	5	the	the	DET
ejpam-1773	316	6	following	follow	VERB
ejpam-1773	316	7	theorem	theorem	PROPN
ejpam-1773	316	8	.	.	PUNCT
ejpam-1773	316	9	theorem	theorem	NOUN
ejpam-1773	316	10	9	9	NUM
ejpam-1773	316	11	.	.	PUNCT
ejpam-1773	317	1	let	let	VERB
ejpam-1773	317	2	k	k	PROPN
ejpam-1773	317	3	be	be	AUX
ejpam-1773	317	4	odd	odd	ADJ
ejpam-1773	317	5	,	,	PUNCT
ejpam-1773	317	6	with	with	ADP
ejpam-1773	317	7	d1	d1	PROPN
ejpam-1773	317	8	,	,	PUNCT
ejpam-1773	317	9	d2	d2	PROPN
ejpam-1773	317	10	,	,	PUNCT
ejpam-1773	317	11	.	.	PUNCT
ejpam-1773	317	12	.	.	PUNCT
ejpam-1773	318	1	.	.	PUNCT
ejpam-1773	319	1	,	,	PUNCT
ejpam-1773	319	2	ds	ds	VERB
ejpam-1773	319	3	the	the	DET
ejpam-1773	319	4	subsets	subset	NOUN
ejpam-1773	319	5	of	of	ADP
ejpam-1773	319	6	{	{	PUNCT
ejpam-1773	319	7	1,2	1,2	NUM
ejpam-1773	319	8	,	,	PUNCT
ejpam-1773	319	9	.	.	PUNCT
ejpam-1773	319	10	.	.	PUNCT
ejpam-1773	319	11	.	.	PUNCT
ejpam-1773	320	1	,	,	PUNCT
ejpam-1773	320	2	k	k	X
ejpam-1773	320	3	}	}	PUNCT
ejpam-1773	320	4	of	of	ADP
ejpam-1773	320	5	size	size	NOUN
ejpam-1773	320	6	⌈	⌈	X
ejpam-1773	320	7	k	k	PROPN
ejpam-1773	320	8	2	2	NUM
ejpam-1773	320	9	⌉	⌉	NOUN
ejpam-1773	320	10	where	where	SCONJ
ejpam-1773	320	11	s	s	AUX
ejpam-1773	320	12	=	=	SYM
ejpam-1773	320	13	�	�	PROPN
ejpam-1773	320	14	k	k	PROPN
ejpam-1773	320	15	⌈	⌈	PROPN
ejpam-1773	320	16	k	k	PROPN
ejpam-1773	320	17	2	2	NUM
ejpam-1773	320	18	⌉	⌉	NOUN
ejpam-1773	320	19	�	�	PROPN
ejpam-1773	320	20	.	.	PUNCT
ejpam-1773	321	1	then	then	ADV
ejpam-1773	321	2	c	c	X
ejpam-1773	321	3	=	=	PUNCT
ejpam-1773	322	1	〈	〈	PROPN
ejpam-1773	322	2	ud1	ud1	NOUN
ejpam-1773	322	3	,	,	PUNCT
ejpam-1773	322	4	ud2	ud2	NOUN
ejpam-1773	322	5	,	,	PUNCT
ejpam-1773	322	6	.	.	PUNCT
ejpam-1773	322	7	.	.	PUNCT
ejpam-1773	323	1	.	.	PUNCT
ejpam-1773	324	1	,	,	PUNCT
ejpam-1773	324	2	uds	uds	PROPN
ejpam-1773	324	3	〉	〉	PROPN
ejpam-1773	324	4	is	be	AUX
ejpam-1773	324	5	a	a	DET
ejpam-1773	324	6	self	self	NOUN
ejpam-1773	324	7	-	-	PUNCT
ejpam-1773	324	8	dual	dual	ADJ
ejpam-1773	324	9	code	code	NOUN
ejpam-1773	324	10	of	of	ADP
ejpam-1773	324	11	length	length	NOUN
ejpam-1773	324	12	1	1	NUM
ejpam-1773	324	13	.	.	PUNCT
ejpam-1773	325	1	proof	proof	NOUN
ejpam-1773	325	2	.	.	PUNCT
ejpam-1773	326	1	for	for	ADP
ejpam-1773	326	2	any	any	DET
ejpam-1773	326	3	di	di	NOUN
ejpam-1773	326	4	,	,	PUNCT
ejpam-1773	326	5	d	d	PROPN
ejpam-1773	326	6	j	j	PROPN
ejpam-1773	326	7	we	we	PRON
ejpam-1773	326	8	have	have	VERB
ejpam-1773	326	9	|di	|di	VERB
ejpam-1773	326	10	∪	∪	PROPN
ejpam-1773	326	11	d	d	X
ejpam-1773	326	12	j	j	PROPN
ejpam-1773	326	13	|	|	NOUN
ejpam-1773	326	14	=	=	SYM
ejpam-1773	326	15	|di|+	|di|+	NOUN
ejpam-1773	326	16	|d	|d	NOUN
ejpam-1773	326	17	j|	j|	PROPN
ejpam-1773	326	18	−	−	PROPN
ejpam-1773	326	19	|di	|di	NUM
ejpam-1773	326	20	∩	∩	PROPN
ejpam-1773	326	21	d	d	X
ejpam-1773	326	22	j	j	PROPN
ejpam-1773	326	23	|	|	ADV
ejpam-1773	326	24	.	.	PUNCT
ejpam-1773	327	1	since	since	SCONJ
ejpam-1773	327	2	⌈	⌈	PROPN
ejpam-1773	327	3	k	k	PROPN
ejpam-1773	327	4	2	2	NUM
ejpam-1773	327	5	⌉+	⌉+	NOUN
ejpam-1773	327	6	⌈	⌈	NOUN
ejpam-1773	327	7	k	k	PROPN
ejpam-1773	327	8	2	2	NUM
ejpam-1773	327	9	⌉	⌉	X
ejpam-1773	327	10	>	>	X
ejpam-1773	327	11	k	k	PROPN
ejpam-1773	327	12	and	and	CCONJ
ejpam-1773	327	13	the	the	DET
ejpam-1773	327	14	maximum	maximum	NOUN
ejpam-1773	327	15	of	of	ADP
ejpam-1773	327	16	|di	|di	NUM
ejpam-1773	327	17	∪	∪	PROPN
ejpam-1773	327	18	d	d	X
ejpam-1773	327	19	j	j	PROPN
ejpam-1773	328	1	|	|	ADV
ejpam-1773	328	2	is	be	AUX
ejpam-1773	328	3	k	k	INTJ
ejpam-1773	328	4	we	we	PRON
ejpam-1773	328	5	have	have	VERB
ejpam-1773	328	6	|di	|di	NUM
ejpam-1773	328	7	∩	∩	NOUN
ejpam-1773	328	8	d	d	X
ejpam-1773	328	9	j	j	PROPN
ejpam-1773	328	10	|	|	ADV
ejpam-1773	328	11	>	>	X
ejpam-1773	328	12	0	0	X
ejpam-1773	328	13	.	.	PUNCT
ejpam-1773	329	1	this	this	PRON
ejpam-1773	329	2	implies	imply	VERB
ejpam-1773	329	3	that	that	SCONJ
ejpam-1773	329	4	udi	udi	PROPN
ejpam-1773	329	5	ud	ud	INTJ
ejpam-1773	329	6	j	j	PROPN
ejpam-1773	329	7	=	=	PUNCT
ejpam-1773	329	8	0	0	PROPN
ejpam-1773	329	9	for	for	ADP
ejpam-1773	329	10	all	all	DET
ejpam-1773	329	11	i	i	PROPN
ejpam-1773	329	12	,	,	PUNCT
ejpam-1773	329	13	j.	j.	PROPN
ejpam-1773	329	14	hence	hence	PROPN
ejpam-1773	329	15	c	c	PROPN
ejpam-1773	329	16	is	be	AUX
ejpam-1773	329	17	self	self	NOUN
ejpam-1773	329	18	-	-	PUNCT
ejpam-1773	329	19	orthogonal	orthogonal	NOUN
ejpam-1773	329	20	.	.	PUNCT
ejpam-1773	330	1	assume	assume	VERB
ejpam-1773	330	2	that	that	SCONJ
ejpam-1773	330	3	∑	∑	ADP
ejpam-1773	330	4	ub	ub	INTJ
ejpam-1773	330	5	∈	∈	PROPN
ejpam-1773	330	6	c⊥.	c⊥.	NOUN
ejpam-1773	330	7	this	this	PRON
ejpam-1773	330	8	implies	imply	VERB
ejpam-1773	330	9	that	that	SCONJ
ejpam-1773	330	10	(	(	PUNCT
ejpam-1773	330	11	∑	∑	PUNCT
ejpam-1773	330	12	ub)udi	ub)udi	X
ejpam-1773	330	13	=	=	NOUN
ejpam-1773	330	14	0	0	NUM
ejpam-1773	330	15	for	for	ADP
ejpam-1773	330	16	all	all	DET
ejpam-1773	330	17	i.	i.	NOUN
ejpam-1773	330	18	if	if	SCONJ
ejpam-1773	330	19	is	be	AUX
ejpam-1773	330	20	easy	easy	ADJ
ejpam-1773	330	21	to	to	PART
ejpam-1773	330	22	see	see	VERB
ejpam-1773	330	23	that	that	SCONJ
ejpam-1773	330	24	this	this	PRON
ejpam-1773	330	25	implies	imply	VERB
ejpam-1773	330	26	that	that	SCONJ
ejpam-1773	330	27	ubudi	ubudi	ADJ
ejpam-1773	330	28	=	=	NOUN
ejpam-1773	330	29	0	0	NUM
ejpam-1773	330	30	for	for	ADP
ejpam-1773	330	31	all	all	DET
ejpam-1773	330	32	i.	i.	NOUN
ejpam-1773	330	33	thus	thus	ADV
ejpam-1773	330	34	b	b	X
ejpam-1773	330	35	∩	∩	X
ejpam-1773	330	36	di	di	X
ejpam-1773	330	37	6=	6=	PROPN
ejpam-1773	330	38	;	;	PUNCT
ejpam-1773	330	39	for	for	ADP
ejpam-1773	330	40	all	all	DET
ejpam-1773	330	41	i.	i.	NOUN
ejpam-1773	330	42	this	this	PRON
ejpam-1773	330	43	implies	imply	VERB
ejpam-1773	330	44	that	that	SCONJ
ejpam-1773	330	45	each	each	DET
ejpam-1773	330	46	b	b	NOUN
ejpam-1773	330	47	must	must	AUX
ejpam-1773	330	48	have	have	VERB
ejpam-1773	330	49	cardinality	cardinality	NOUN
ejpam-1773	330	50	at	at	ADP
ejpam-1773	330	51	least	least	ADJ
ejpam-1773	331	1	k	k	NOUN
ejpam-1773	331	2	−	−	PROPN
ejpam-1773	331	3	⌈	⌈	PROPN
ejpam-1773	331	4	k	k	PROPN
ejpam-1773	331	5	2	2	NUM
ejpam-1773	331	6	⌉+	⌉+	SYM
ejpam-1773	331	7	1	1	NUM
ejpam-1773	331	8	=	=	SYM
ejpam-1773	331	9	⌈	⌈	NOUN
ejpam-1773	331	10	k	k	X
ejpam-1773	331	11	2	2	NUM
ejpam-1773	331	12	⌉	⌉	NOUN
ejpam-1773	331	13	when	when	SCONJ
ejpam-1773	331	14	k	k	PROPN
ejpam-1773	331	15	is	be	AUX
ejpam-1773	331	16	odd	odd	ADJ
ejpam-1773	331	17	.	.	PUNCT
ejpam-1773	332	1	thus	thus	ADV
ejpam-1773	332	2	b	b	X
ejpam-1773	332	3	is	be	AUX
ejpam-1773	332	4	a	a	DET
ejpam-1773	332	5	subset	subset	NOUN
ejpam-1773	332	6	of	of	ADP
ejpam-1773	332	7	{	{	PUNCT
ejpam-1773	332	8	1,2	1,2	NUM
ejpam-1773	332	9	,	,	PUNCT
ejpam-1773	332	10	.	.	PUNCT
ejpam-1773	332	11	.	.	PUNCT
ejpam-1773	333	1	.	.	PUNCT
ejpam-1773	334	1	,	,	PUNCT
ejpam-1773	334	2	k	k	X
ejpam-1773	334	3	}	}	PUNCT
ejpam-1773	334	4	with	with	ADP
ejpam-1773	334	5	cardinality	cardinality	NOUN
ejpam-1773	334	6	at	at	ADP
ejpam-1773	334	7	least	least	ADJ
ejpam-1773	334	8	⌈	⌈	NOUN
ejpam-1773	334	9	k	k	NOUN
ejpam-1773	334	10	2	2	NUM
ejpam-1773	334	11	⌉.	⌉.	ADV
ejpam-1773	334	12	hence	hence	ADV
ejpam-1773	334	13	ub	ub	ADP
ejpam-1773	334	14	∈	∈	PROPN
ejpam-1773	334	15	〈	〈	PROPN
ejpam-1773	334	16	ud1	ud1	NOUN
ejpam-1773	334	17	,	,	PUNCT
ejpam-1773	334	18	ud2	ud2	NOUN
ejpam-1773	334	19	,	,	PUNCT
ejpam-1773	334	20	.	.	PUNCT
ejpam-1773	334	21	.	.	PUNCT
ejpam-1773	334	22	.	.	PUNCT
ejpam-1773	335	1	,	,	PUNCT
ejpam-1773	335	2	uds	uds	PROPN
ejpam-1773	335	3	〉	〉	PROPN
ejpam-1773	335	4	,	,	PUNCT
ejpam-1773	335	5	that	that	PRON
ejpam-1773	335	6	is	be	AUX
ejpam-1773	335	7	ub	ub	ADP
ejpam-1773	335	8	∈	∈	PROPN
ejpam-1773	335	9	c	c	PROPN
ejpam-1773	335	10	.	.	PUNCT
ejpam-1773	336	1	therefore	therefore	ADV
ejpam-1773	336	2	c	c	X
ejpam-1773	336	3	=	=	SYM
ejpam-1773	336	4	c⊥.	c⊥.	PROPN
ejpam-1773	336	5	s.	s.	PROPN
ejpam-1773	336	6	dougherty	dougherty	PROPN
ejpam-1773	336	7	,	,	PUNCT
ejpam-1773	336	8	b.yıldız	b.yıldız	NOUN
ejpam-1773	336	9	,	,	PUNCT
ejpam-1773	336	10	s.karadeniz	s.karadeniz	NOUN
ejpam-1773	336	11	/	/	SYM
ejpam-1773	336	12	eur	eur	NOUN
ejpam-1773	336	13	.	.	PUNCT
ejpam-1773	337	1	j.	j.	PROPN
ejpam-1773	337	2	pure	pure	PROPN
ejpam-1773	337	3	appl	appl	PROPN
ejpam-1773	337	4	.	.	PROPN
ejpam-1773	337	5	math	math	PROPN
ejpam-1773	337	6	,	,	PUNCT
ejpam-1773	337	7	6	6	NUM
ejpam-1773	337	8	(	(	PUNCT
ejpam-1773	337	9	2013	2013	NUM
ejpam-1773	337	10	)	)	PUNCT
ejpam-1773	337	11	,	,	PUNCT
ejpam-1773	337	12	89	89	NUM
ejpam-1773	337	13	-	-	SYM
ejpam-1773	337	14	106	106	NUM
ejpam-1773	337	15	98	98	NUM
ejpam-1773	337	16	note	note	NOUN
ejpam-1773	337	17	that	that	SCONJ
ejpam-1773	337	18	for	for	ADP
ejpam-1773	337	19	k	k	PROPN
ejpam-1773	337	20	even	even	ADV
ejpam-1773	337	21	one	one	PRON
ejpam-1773	337	22	would	would	AUX
ejpam-1773	337	23	need	need	VERB
ejpam-1773	337	24	sets	set	NOUN
ejpam-1773	337	25	of	of	ADP
ejpam-1773	337	26	size	size	NOUN
ejpam-1773	337	27	k	k	PROPN
ejpam-1773	337	28	2	2	NUM
ejpam-1773	337	29	+	+	CCONJ
ejpam-1773	337	30	1	1	NUM
ejpam-1773	337	31	to	to	PART
ejpam-1773	337	32	be	be	AUX
ejpam-1773	337	33	self	self	NOUN
ejpam-1773	337	34	-	-	PUNCT
ejpam-1773	337	35	orthogonal	orthogonal	ADJ
ejpam-1773	337	36	.	.	PUNCT
ejpam-1773	338	1	but	but	CCONJ
ejpam-1773	338	2	k−	k−	PROPN
ejpam-1773	338	3	(	(	PUNCT
ejpam-1773	338	4	k	k	PROPN
ejpam-1773	338	5	2	2	NUM
ejpam-1773	338	6	+1	+1	NOUN
ejpam-1773	338	7	)	)	PUNCT
ejpam-1773	338	8	=	=	SYM
ejpam-1773	339	1	k	k	PROPN
ejpam-1773	339	2	2	2	NUM
ejpam-1773	339	3	−1	−1	NOUN
ejpam-1773	339	4	.	.	PUNCT
ejpam-1773	340	1	hence	hence	ADV
ejpam-1773	340	2	the	the	DET
ejpam-1773	340	3	code	code	NOUN
ejpam-1773	340	4	is	be	AUX
ejpam-1773	340	5	not	not	PART
ejpam-1773	340	6	self	self	NOUN
ejpam-1773	340	7	-	-	PUNCT
ejpam-1773	340	8	dual	dual	ADJ
ejpam-1773	340	9	since	since	SCONJ
ejpam-1773	340	10	there	there	PRON
ejpam-1773	340	11	exist	exist	VERB
ejpam-1773	340	12	sets	set	NOUN
ejpam-1773	340	13	of	of	ADP
ejpam-1773	340	14	size	size	NOUN
ejpam-1773	340	15	k	k	PROPN
ejpam-1773	340	16	2	2	NUM
ejpam-1773	340	17	−1	−1	NOUN
ejpam-1773	340	18	that	that	PRON
ejpam-1773	340	19	are	be	AUX
ejpam-1773	340	20	not	not	PART
ejpam-1773	340	21	disjoint	disjoint	ADJ
ejpam-1773	340	22	from	from	ADP
ejpam-1773	340	23	all	all	DET
ejpam-1773	340	24	sets	set	NOUN
ejpam-1773	340	25	of	of	ADP
ejpam-1773	340	26	size	size	NOUN
ejpam-1773	340	27	k	k	PROPN
ejpam-1773	340	28	2	2	NUM
ejpam-1773	340	29	+1	+1	PROPN
ejpam-1773	340	30	.	.	PUNCT
ejpam-1773	341	1	for	for	ADP
ejpam-1773	341	2	example	example	NOUN
ejpam-1773	341	3	,	,	PUNCT
ejpam-1773	341	4	if	if	SCONJ
ejpam-1773	341	5	k	k	PROPN
ejpam-1773	341	6	=	=	SYM
ejpam-1773	341	7	4	4	NUM
ejpam-1773	341	8	,	,	PUNCT
ejpam-1773	341	9	the	the	DET
ejpam-1773	341	10	code	code	NOUN
ejpam-1773	341	11	〈	〈	NOUN
ejpam-1773	341	12	u1u2u3,u1u2u4,u1u3u4,u2u3u4	u1u2u3,u1u2u4,u1u3u4,u2u3u4	NOUN
ejpam-1773	341	13	〉	〉	NOUN
ejpam-1773	341	14	would	would	AUX
ejpam-1773	341	15	generate	generate	VERB
ejpam-1773	341	16	a	a	DET
ejpam-1773	341	17	self	self	NOUN
ejpam-1773	341	18	-	-	PUNCT
ejpam-1773	341	19	orthogonal	orthogonal	ADJ
ejpam-1773	341	20	code	code	NOUN
ejpam-1773	341	21	but	but	CCONJ
ejpam-1773	341	22	u1u2	u1u2	ADP
ejpam-1773	341	23	∈	∈	NOUN
ejpam-1773	341	24	c⊥	c⊥	NOUN
ejpam-1773	341	25	and	and	CCONJ
ejpam-1773	341	26	not	not	PART
ejpam-1773	341	27	in	in	ADP
ejpam-1773	341	28	c	c	PROPN
ejpam-1773	341	29	.	.	PUNCT
ejpam-1773	342	1	5.2	5.2	NUM
ejpam-1773	342	2	.	.	PUNCT
ejpam-1773	342	3	length	length	NOUN
ejpam-1773	342	4	2	2	NUM
ejpam-1773	342	5	self	self	NOUN
ejpam-1773	342	6	-	-	PUNCT
ejpam-1773	342	7	dual	dual	ADJ
ejpam-1773	342	8	codes	code	NOUN
ejpam-1773	342	9	over	over	ADP
ejpam-1773	342	10	rk	rk	NOUN
ejpam-1773	342	11	we	we	PRON
ejpam-1773	342	12	first	first	ADV
ejpam-1773	342	13	note	note	VERB
ejpam-1773	342	14	that	that	SCONJ
ejpam-1773	342	15	,	,	PUNCT
ejpam-1773	342	16	for	for	ADP
ejpam-1773	342	17	any	any	DET
ejpam-1773	342	18	a	a	DET
ejpam-1773	342	19	∈	∈	ADJ
ejpam-1773	342	20	rk	rk	NOUN
ejpam-1773	342	21	,	,	PUNCT
ejpam-1773	342	22	with	with	ADP
ejpam-1773	342	23	k	k	PROPN
ejpam-1773	342	24	≥	≥	NUM
ejpam-1773	342	25	1	1	NUM
ejpam-1773	342	26	,	,	PUNCT
ejpam-1773	342	27	a2	a2	PROPN
ejpam-1773	342	28	=	=	NOUN
ejpam-1773	342	29	1	1	NUM
ejpam-1773	342	30	if	if	SCONJ
ejpam-1773	342	31	a	a	PRON
ejpam-1773	342	32	is	be	AUX
ejpam-1773	342	33	a	a	DET
ejpam-1773	342	34	unit	unit	NOUN
ejpam-1773	342	35	and	and	CCONJ
ejpam-1773	342	36	a2	a2	PROPN
ejpam-1773	342	37	=	=	SYM
ejpam-1773	342	38	0	0	PUNCT
ejpam-1773	343	1	otherwise	otherwise	ADV
ejpam-1773	343	2	.	.	PUNCT
ejpam-1773	344	1	this	this	PRON
ejpam-1773	344	2	tells	tell	VERB
ejpam-1773	344	3	us	we	PRON
ejpam-1773	344	4	that	that	SCONJ
ejpam-1773	344	5	every	every	DET
ejpam-1773	344	6	codeword	codeword	NOUN
ejpam-1773	344	7	in	in	ADP
ejpam-1773	344	8	a	a	DET
ejpam-1773	344	9	length	length	NOUN
ejpam-1773	344	10	2	2	NUM
ejpam-1773	344	11	self	self	NOUN
ejpam-1773	344	12	-	-	PUNCT
ejpam-1773	344	13	dual	dual	ADJ
ejpam-1773	344	14	code	code	NOUN
ejpam-1773	344	15	over	over	ADP
ejpam-1773	344	16	rk	rk	PROPN
ejpam-1773	344	17	must	must	AUX
ejpam-1773	344	18	be	be	AUX
ejpam-1773	344	19	of	of	ADP
ejpam-1773	344	20	the	the	DET
ejpam-1773	344	21	form	form	NOUN
ejpam-1773	344	22	(	(	PUNCT
ejpam-1773	344	23	a1	a1	NOUN
ejpam-1773	344	24	,	,	PUNCT
ejpam-1773	344	25	a2	a2	PROPN
ejpam-1773	344	26	)	)	PUNCT
ejpam-1773	344	27	where	where	SCONJ
ejpam-1773	344	28	the	the	DET
ejpam-1773	344	29	ai	ai	NOUN
ejpam-1773	344	30	are	be	AUX
ejpam-1773	344	31	units	unit	NOUN
ejpam-1773	344	32	or	or	CCONJ
ejpam-1773	344	33	of	of	ADP
ejpam-1773	344	34	the	the	DET
ejpam-1773	344	35	form	form	NOUN
ejpam-1773	344	36	(	(	PUNCT
ejpam-1773	344	37	b1	b1	NOUN
ejpam-1773	344	38	,	,	PUNCT
ejpam-1773	344	39	b2	b2	NOUN
ejpam-1773	344	40	)	)	PUNCT
ejpam-1773	344	41	where	where	SCONJ
ejpam-1773	344	42	the	the	DET
ejpam-1773	344	43	bi	bi	NOUN
ejpam-1773	344	44	are	be	AUX
ejpam-1773	344	45	non	non	ADJ
ejpam-1773	344	46	-	-	NOUN
ejpam-1773	344	47	units	unit	NOUN
ejpam-1773	344	48	.	.	PUNCT
ejpam-1773	345	1	we	we	PRON
ejpam-1773	345	2	first	first	ADV
ejpam-1773	345	3	start	start	VERB
ejpam-1773	345	4	with	with	ADP
ejpam-1773	345	5	the	the	DET
ejpam-1773	345	6	following	follow	VERB
ejpam-1773	345	7	proposition	proposition	NOUN
ejpam-1773	345	8	.	.	PUNCT
ejpam-1773	346	1	proposition	proposition	NOUN
ejpam-1773	346	2	1	1	NUM
ejpam-1773	346	3	.	.	PUNCT
ejpam-1773	347	1	let	let	VERB
ejpam-1773	347	2	c	c	PRON
ejpam-1773	347	3	be	be	AUX
ejpam-1773	347	4	a	a	DET
ejpam-1773	347	5	linear	linear	ADJ
ejpam-1773	347	6	code	code	NOUN
ejpam-1773	347	7	over	over	ADP
ejpam-1773	347	8	rk	rk	PROPN
ejpam-1773	347	9	of	of	ADP
ejpam-1773	347	10	length	length	NOUN
ejpam-1773	347	11	2	2	NUM
ejpam-1773	347	12	generated	generate	VERB
ejpam-1773	347	13	by	by	ADP
ejpam-1773	347	14	(	(	PUNCT
ejpam-1773	347	15	1,1	1,1	NUM
ejpam-1773	347	16	+	+	NOUN
ejpam-1773	347	17	u1u2	u1u2	NOUN
ejpam-1773	347	18	.	.	PUNCT
ejpam-1773	347	19	.	.	PUNCT
ejpam-1773	347	20	.	.	PUNCT
ejpam-1773	348	1	uk	uk	PROPN
ejpam-1773	348	2	)	)	PUNCT
ejpam-1773	348	3	with	with	ADP
ejpam-1773	348	4	k	k	PROPN
ejpam-1773	348	5	≥	≥	NUM
ejpam-1773	348	6	2	2	NUM
ejpam-1773	348	7	.	.	PUNCT
ejpam-1773	349	1	then	then	ADV
ejpam-1773	349	2	c	c	PROPN
ejpam-1773	349	3	is	be	AUX
ejpam-1773	349	4	a	a	DET
ejpam-1773	349	5	type	type	NOUN
ejpam-1773	349	6	ii	ii	PROPN
ejpam-1773	349	7	code	code	NOUN
ejpam-1773	349	8	with	with	ADP
ejpam-1773	349	9	minimum	minimum	ADJ
ejpam-1773	349	10	distance	distance	NOUN
ejpam-1773	349	11	4	4	NUM
ejpam-1773	349	12	.	.	PUNCT
ejpam-1773	350	1	proof	proof	NOUN
ejpam-1773	350	2	.	.	PUNCT
ejpam-1773	351	1	we	we	PRON
ejpam-1773	351	2	have	have	VERB
ejpam-1773	351	3	that	that	DET
ejpam-1773	351	4	〈	〈	PROPN
ejpam-1773	351	5	(	(	PUNCT
ejpam-1773	351	6	1,1+u1u2	1,1+u1u2	NUM
ejpam-1773	351	7	.	.	PUNCT
ejpam-1773	351	8	.	.	PUNCT
ejpam-1773	351	9	.	.	PUNCT
ejpam-1773	352	1	uk	uk	PROPN
ejpam-1773	352	2	)	)	PUNCT
ejpam-1773	352	3	,	,	PUNCT
ejpam-1773	352	4	(	(	PUNCT
ejpam-1773	352	5	1,1+u1u2	1,1+u1u2	NUM
ejpam-1773	352	6	.	.	PUNCT
ejpam-1773	352	7	.	.	PUNCT
ejpam-1773	352	8	.	.	PUNCT
ejpam-1773	353	1	uk)〉k	uk)〉k	NOUN
ejpam-1773	354	1	=	=	PUNCT
ejpam-1773	354	2	0	0	NUM
ejpam-1773	355	1	in	in	ADP
ejpam-1773	355	2	rk	rk	NOUN
ejpam-1773	355	3	giving	give	VERB
ejpam-1773	355	4	that	that	SCONJ
ejpam-1773	355	5	c	c	PROPN
ejpam-1773	355	6	is	be	AUX
ejpam-1773	355	7	selforthogonal	selforthogonal	ADJ
ejpam-1773	355	8	.	.	PUNCT
ejpam-1773	356	1	because	because	SCONJ
ejpam-1773	356	2	there	there	PRON
ejpam-1773	356	3	is	be	VERB
ejpam-1773	356	4	a	a	DET
ejpam-1773	356	5	1	1	NUM
ejpam-1773	356	6	in	in	ADP
ejpam-1773	356	7	the	the	DET
ejpam-1773	356	8	first	first	ADJ
ejpam-1773	356	9	coordinate	coordinate	NOUN
ejpam-1773	356	10	,	,	PUNCT
ejpam-1773	356	11	every	every	DET
ejpam-1773	356	12	rk	rk	NOUN
ejpam-1773	356	13	-	-	PUNCT
ejpam-1773	356	14	multiple	multiple	NOUN
ejpam-1773	356	15	of	of	ADP
ejpam-1773	356	16	(	(	PUNCT
ejpam-1773	356	17	1,1	1,1	NUM
ejpam-1773	356	18	+	+	NUM
ejpam-1773	356	19	u1	u1	NOUN
ejpam-1773	356	20	.	.	PUNCT
ejpam-1773	356	21	.	.	PUNCT
ejpam-1773	357	1	.	.	PUNCT
ejpam-1773	358	1	uk	uk	PROPN
ejpam-1773	358	2	)	)	PUNCT
ejpam-1773	358	3	is	be	AUX
ejpam-1773	358	4	distinct	distinct	ADJ
ejpam-1773	358	5	,	,	PUNCT
ejpam-1773	358	6	and	and	CCONJ
ejpam-1773	358	7	so	so	ADV
ejpam-1773	358	8	we	we	PRON
ejpam-1773	358	9	have	have	VERB
ejpam-1773	358	10	|c	|c	ADJ
ejpam-1773	358	11	|	|	NOUN
ejpam-1773	358	12	=	=	SYM
ejpam-1773	358	13	|rk|	|rk|	NOUN
ejpam-1773	358	14	=	=	NOUN
ejpam-1773	358	15	22k	22k	NOUN
ejpam-1773	358	16	.	.	PUNCT
ejpam-1773	359	1	since	since	SCONJ
ejpam-1773	359	2	|r2	|r2	NOUN
ejpam-1773	359	3	k	k	PROPN
ejpam-1773	359	4	|	|	ADV
ejpam-1773	359	5	=	=	SYM
ejpam-1773	359	6	22k+1	22k+1	NUM
ejpam-1773	359	7	=	=	SYM
ejpam-1773	359	8	|c	|c	PUNCT
ejpam-1773	359	9	|	|	ADV
ejpam-1773	359	10	·	·	PUNCT
ejpam-1773	359	11	|c⊥|	|c⊥|	NOUN
ejpam-1773	359	12	we	we	PRON
ejpam-1773	359	13	see	see	VERB
ejpam-1773	359	14	that	that	SCONJ
ejpam-1773	359	15	|c⊥|	|c⊥|	PROPN
ejpam-1773	359	16	=	=	ADJ
ejpam-1773	359	17	|c	|c	ADJ
ejpam-1773	359	18	|=	|=	NOUN
ejpam-1773	359	19	22k	22k	NOUN
ejpam-1773	359	20	.	.	PUNCT
ejpam-1773	360	1	we	we	PRON
ejpam-1773	360	2	know	know	VERB
ejpam-1773	360	3	that	that	SCONJ
ejpam-1773	360	4	c	c	PROPN
ejpam-1773	360	5	is	be	AUX
ejpam-1773	360	6	self	self	NOUN
ejpam-1773	360	7	-	-	PUNCT
ejpam-1773	360	8	orthogonal	orthogonal	NOUN
ejpam-1773	360	9	which	which	PRON
ejpam-1773	360	10	implies	imply	VERB
ejpam-1773	360	11	that	that	SCONJ
ejpam-1773	360	12	c	c	PROPN
ejpam-1773	360	13	is	be	AUX
ejpam-1773	360	14	self	self	NOUN
ejpam-1773	360	15	-	-	PUNCT
ejpam-1773	360	16	dual	dual	ADJ
ejpam-1773	360	17	.	.	PUNCT
ejpam-1773	361	1	to	to	PART
ejpam-1773	361	2	prove	prove	VERB
ejpam-1773	361	3	that	that	SCONJ
ejpam-1773	361	4	the	the	DET
ejpam-1773	361	5	weight	weight	NOUN
ejpam-1773	361	6	of	of	ADP
ejpam-1773	361	7	every	every	DET
ejpam-1773	361	8	element	element	NOUN
ejpam-1773	361	9	is	be	AUX
ejpam-1773	361	10	divisible	divisible	ADJ
ejpam-1773	361	11	by	by	ADP
ejpam-1773	361	12	4	4	NUM
ejpam-1773	361	13	,	,	PUNCT
ejpam-1773	361	14	we	we	PRON
ejpam-1773	361	15	first	first	ADV
ejpam-1773	361	16	observe	observe	VERB
ejpam-1773	361	17	that	that	SCONJ
ejpam-1773	361	18	a	a	DET
ejpam-1773	361	19	·	·	PUNCT
ejpam-1773	361	20	(	(	PUNCT
ejpam-1773	361	21	u1u2	u1u2	NOUN
ejpam-1773	361	22	.	.	PUNCT
ejpam-1773	361	23	.	.	PUNCT
ejpam-1773	361	24	.	.	PUNCT
ejpam-1773	362	1	uk	uk	PROPN
ejpam-1773	362	2	)	)	PUNCT
ejpam-1773	362	3	=	=	PRON
ejpam-1773	362	4	(	(	PUNCT
ejpam-1773	362	5	u1u2	u1u2	NOUN
ejpam-1773	362	6	.	.	PUNCT
ejpam-1773	362	7	.	.	PUNCT
ejpam-1773	362	8	.	.	PUNCT
ejpam-1773	363	1	uk	uk	PROPN
ejpam-1773	363	2	if	if	SCONJ
ejpam-1773	363	3	a	a	PRON
ejpam-1773	363	4	is	be	AUX
ejpam-1773	363	5	a	a	DET
ejpam-1773	363	6	unit	unit	NOUN
ejpam-1773	363	7	0	0	NUM
ejpam-1773	363	8	if	if	SCONJ
ejpam-1773	363	9	a	a	PRON
ejpam-1773	363	10	is	be	AUX
ejpam-1773	363	11	a	a	DET
ejpam-1773	363	12	non	non	ADJ
ejpam-1773	363	13	-	-	NOUN
ejpam-1773	363	14	unit	unit	NOUN
ejpam-1773	363	15	.	.	PUNCT
ejpam-1773	364	1	this	this	PRON
ejpam-1773	364	2	means	mean	VERB
ejpam-1773	364	3	we	we	PRON
ejpam-1773	364	4	have	have	VERB
ejpam-1773	364	5	a	a	DET
ejpam-1773	364	6	·	·	PUNCT
ejpam-1773	364	7	(	(	PUNCT
ejpam-1773	364	8	1,1	1,1	NUM
ejpam-1773	364	9	+	+	SYM
ejpam-1773	364	10	u1u2	u1u2	NOUN
ejpam-1773	364	11	.	.	PUNCT
ejpam-1773	364	12	.	.	PUNCT
ejpam-1773	364	13	.	.	PUNCT
ejpam-1773	365	1	uk	uk	PROPN
ejpam-1773	365	2	)	)	PUNCT
ejpam-1773	365	3	=	=	PRON
ejpam-1773	366	1	(	(	PUNCT
ejpam-1773	366	2	(	(	PUNCT
ejpam-1773	366	3	a	a	X
ejpam-1773	366	4	,	,	PUNCT
ejpam-1773	366	5	a+	a+	PRON
ejpam-1773	366	6	u1u2	u1u2	NOUN
ejpam-1773	366	7	.	.	PUNCT
ejpam-1773	366	8	.	.	PUNCT
ejpam-1773	366	9	.	.	PUNCT
ejpam-1773	367	1	uk	uk	PROPN
ejpam-1773	367	2	)	)	PUNCT
ejpam-1773	367	3	if	if	SCONJ
ejpam-1773	367	4	a	a	PRON
ejpam-1773	367	5	is	be	AUX
ejpam-1773	367	6	a	a	DET
ejpam-1773	367	7	unit	unit	NOUN
ejpam-1773	367	8	(	(	PUNCT
ejpam-1773	367	9	a	a	DET
ejpam-1773	367	10	,	,	PUNCT
ejpam-1773	367	11	a	a	X
ejpam-1773	367	12	)	)	PUNCT
ejpam-1773	367	13	if	if	SCONJ
ejpam-1773	367	14	a	a	PRON
ejpam-1773	367	15	is	be	AUX
ejpam-1773	367	16	a	a	DET
ejpam-1773	367	17	non	non	ADJ
ejpam-1773	367	18	-	-	NOUN
ejpam-1773	367	19	unit	unit	NOUN
ejpam-1773	367	20	.	.	PUNCT
ejpam-1773	368	1	if	if	SCONJ
ejpam-1773	368	2	a	a	PRON
ejpam-1773	368	3	is	be	AUX
ejpam-1773	368	4	a	a	DET
ejpam-1773	368	5	non	non	ADJ
ejpam-1773	368	6	-	-	NOUN
ejpam-1773	368	7	unit	unit	NOUN
ejpam-1773	368	8	,	,	PUNCT
ejpam-1773	368	9	then	then	ADV
ejpam-1773	368	10	wl(a(1,1	wl(a(1,1	ADJ
ejpam-1773	368	11	+	+	X
ejpam-1773	368	12	u1	u1	NOUN
ejpam-1773	368	13	.	.	PUNCT
ejpam-1773	368	14	.	.	PUNCT
ejpam-1773	369	1	.	.	PUNCT
ejpam-1773	370	1	uk	uk	PROPN
ejpam-1773	370	2	)	)	PUNCT
ejpam-1773	370	3	)	)	PUNCT
ejpam-1773	371	1	=	=	PUNCT
ejpam-1773	371	2	wl(a	wl(a	X
ejpam-1773	371	3	,	,	PUNCT
ejpam-1773	371	4	a	a	PRON
ejpam-1773	371	5	)	)	PUNCT
ejpam-1773	371	6	=	=	SYM
ejpam-1773	371	7	2wl(a	2wl(a	NUM
ejpam-1773	371	8	)	)	PUNCT
ejpam-1773	371	9	.	.	PUNCT
ejpam-1773	372	1	therefore	therefore	ADV
ejpam-1773	372	2	the	the	DET
ejpam-1773	372	3	lee	lee	PROPN
ejpam-1773	372	4	weight	weight	PROPN
ejpam-1773	372	5	is	be	AUX
ejpam-1773	372	6	a	a	DET
ejpam-1773	372	7	multiple	multiple	NOUN
ejpam-1773	372	8	of	of	ADP
ejpam-1773	372	9	4	4	NUM
ejpam-1773	372	10	since	since	ADV
ejpam-1773	372	11	,	,	PUNCT
ejpam-1773	372	12	by	by	ADP
ejpam-1773	372	13	[	[	PUNCT
ejpam-1773	372	14	7	7	NUM
ejpam-1773	372	15	]	]	PUNCT
ejpam-1773	372	16	,	,	PUNCT
ejpam-1773	372	17	we	we	PRON
ejpam-1773	372	18	know	know	VERB
ejpam-1773	372	19	that	that	SCONJ
ejpam-1773	372	20	the	the	DET
ejpam-1773	372	21	lee	lee	PROPN
ejpam-1773	372	22	weight	weight	NOUN
ejpam-1773	372	23	of	of	ADP
ejpam-1773	372	24	every	every	DET
ejpam-1773	372	25	non	non	ADJ
ejpam-1773	372	26	-	-	ADJ
ejpam-1773	372	27	zero	zero	ADJ
ejpam-1773	372	28	non	non	ADJ
ejpam-1773	372	29	-	-	NOUN
ejpam-1773	372	30	unit	unit	NOUN
ejpam-1773	372	31	is	be	AUX
ejpam-1773	372	32	even	even	ADV
ejpam-1773	372	33	.	.	PUNCT
ejpam-1773	373	1	if	if	SCONJ
ejpam-1773	373	2	a	a	PRON
ejpam-1773	373	3	is	be	AUX
ejpam-1773	373	4	a	a	DET
ejpam-1773	373	5	unit	unit	NOUN
ejpam-1773	373	6	,	,	PUNCT
ejpam-1773	373	7	then	then	ADV
ejpam-1773	373	8	wl(a(1,1	wl(a(1,1	ADJ
ejpam-1773	373	9	+	+	X
ejpam-1773	373	10	u1	u1	NOUN
ejpam-1773	373	11	.	.	PUNCT
ejpam-1773	373	12	.	.	PUNCT
ejpam-1773	373	13	.	.	PUNCT
ejpam-1773	374	1	uk	uk	PROPN
ejpam-1773	374	2	)	)	PUNCT
ejpam-1773	374	3	)	)	PUNCT
ejpam-1773	375	1	=	=	PUNCT
ejpam-1773	375	2	wl(a	wl(a	X
ejpam-1773	375	3	,	,	PUNCT
ejpam-1773	375	4	a+	a+	X
ejpam-1773	375	5	u1u2	u1u2	NOUN
ejpam-1773	375	6	.	.	PUNCT
ejpam-1773	375	7	.	.	PUNCT
ejpam-1773	375	8	.	.	PUNCT
ejpam-1773	376	1	uk	uk	PROPN
ejpam-1773	376	2	)	)	PUNCT
ejpam-1773	376	3	=	=	PUNCT
ejpam-1773	376	4	wl(a	wl(a	X
ejpam-1773	376	5	)	)	PUNCT
ejpam-1773	377	1	+	+	ADJ
ejpam-1773	377	2	wl(a+	wl(a+	PROPN
ejpam-1773	377	3	u1	u1	NOUN
ejpam-1773	377	4	.	.	PUNCT
ejpam-1773	377	5	.	.	PUNCT
ejpam-1773	377	6	.	.	PUNCT
ejpam-1773	378	1	uk	uk	PROPN
ejpam-1773	378	2	)	)	PUNCT
ejpam-1773	378	3	=	=	SYM
ejpam-1773	378	4	2k	2k	NUM
ejpam-1773	378	5	by	by	ADP
ejpam-1773	378	6	[	[	X
ejpam-1773	378	7	7	7	NUM
ejpam-1773	378	8	]	]	PUNCT
ejpam-1773	378	9	.	.	PUNCT
ejpam-1773	379	1	when	when	SCONJ
ejpam-1773	379	2	k	k	PROPN
ejpam-1773	379	3	≥	≥	VERB
ejpam-1773	379	4	2	2	NUM
ejpam-1773	379	5	this	this	PRON
ejpam-1773	379	6	is	be	AUX
ejpam-1773	379	7	divisible	divisible	ADJ
ejpam-1773	379	8	by	by	ADP
ejpam-1773	379	9	4	4	NUM
ejpam-1773	379	10	.	.	PUNCT
ejpam-1773	380	1	we	we	PRON
ejpam-1773	380	2	have	have	AUX
ejpam-1773	380	3	found	find	VERB
ejpam-1773	380	4	a	a	DET
ejpam-1773	380	5	class	class	NOUN
ejpam-1773	380	6	of	of	ADP
ejpam-1773	380	7	type	type	NOUN
ejpam-1773	380	8	ii	ii	PROPN
ejpam-1773	380	9	codes	code	NOUN
ejpam-1773	380	10	over	over	ADP
ejpam-1773	380	11	rk	rk	NOUN
ejpam-1773	380	12	of	of	ADP
ejpam-1773	380	13	length	length	NOUN
ejpam-1773	380	14	2	2	NUM
ejpam-1773	380	15	for	for	ADP
ejpam-1773	380	16	all	all	DET
ejpam-1773	380	17	k	k	PROPN
ejpam-1773	380	18	≥	≥	NUM
ejpam-1773	380	19	2	2	NUM
ejpam-1773	380	20	.	.	PUNCT
ejpam-1773	381	1	the	the	DET
ejpam-1773	381	2	binary	binary	ADJ
ejpam-1773	381	3	images	image	NOUN
ejpam-1773	381	4	of	of	ADP
ejpam-1773	381	5	these	these	DET
ejpam-1773	381	6	codes	code	NOUN
ejpam-1773	381	7	are	be	AUX
ejpam-1773	381	8	type	type	NOUN
ejpam-1773	381	9	ii	ii	NOUN
ejpam-1773	381	10	codes	code	NOUN
ejpam-1773	381	11	with	with	ADP
ejpam-1773	381	12	parameters	parameter	NOUN
ejpam-1773	381	13	[	[	X
ejpam-1773	381	14	2k+1	2k+1	NOUN
ejpam-1773	381	15	,	,	PUNCT
ejpam-1773	381	16	2k	2k	NUM
ejpam-1773	381	17	,	,	PUNCT
ejpam-1773	381	18	4	4	NUM
ejpam-1773	381	19	]	]	PUNCT
ejpam-1773	381	20	,	,	PUNCT
ejpam-1773	381	21	which	which	PRON
ejpam-1773	381	22	are	be	AUX
ejpam-1773	381	23	extremal	extremal	ADJ
ejpam-1773	382	1	when	when	SCONJ
ejpam-1773	382	2	k	k	PROPN
ejpam-1773	382	3	=	=	SYM
ejpam-1773	382	4	2	2	NUM
ejpam-1773	382	5	and	and	CCONJ
ejpam-1773	382	6	k	k	NOUN
ejpam-1773	382	7	=	=	NOUN
ejpam-1773	382	8	3	3	X
ejpam-1773	382	9	.	.	PUNCT
ejpam-1773	383	1	the	the	DET
ejpam-1773	383	2	following	follow	VERB
ejpam-1773	383	3	proposition	proposition	NOUN
ejpam-1773	383	4	can	can	AUX
ejpam-1773	383	5	be	be	AUX
ejpam-1773	383	6	proven	prove	VERB
ejpam-1773	383	7	in	in	ADP
ejpam-1773	383	8	exactly	exactly	ADV
ejpam-1773	383	9	the	the	DET
ejpam-1773	383	10	same	same	ADJ
ejpam-1773	383	11	way	way	NOUN
ejpam-1773	383	12	as	as	ADP
ejpam-1773	383	13	the	the	DET
ejpam-1773	383	14	previous	previous	ADJ
ejpam-1773	383	15	proposition	proposition	NOUN
ejpam-1773	383	16	.	.	PUNCT
ejpam-1773	384	1	s.	s.	PROPN
ejpam-1773	384	2	dougherty	dougherty	PROPN
ejpam-1773	384	3	,	,	PUNCT
ejpam-1773	384	4	b.yıldız	b.yıldız	NOUN
ejpam-1773	384	5	,	,	PUNCT
ejpam-1773	384	6	s.karadeniz	s.karadeniz	NOUN
ejpam-1773	384	7	/	/	SYM
ejpam-1773	384	8	eur	eur	NOUN
ejpam-1773	384	9	.	.	PUNCT
ejpam-1773	385	1	j.	j.	PROPN
ejpam-1773	385	2	pure	pure	PROPN
ejpam-1773	385	3	appl	appl	PROPN
ejpam-1773	385	4	.	.	PROPN
ejpam-1773	385	5	math	math	PROPN
ejpam-1773	385	6	,	,	PUNCT
ejpam-1773	385	7	6	6	NUM
ejpam-1773	385	8	(	(	PUNCT
ejpam-1773	385	9	2013	2013	NUM
ejpam-1773	385	10	)	)	PUNCT
ejpam-1773	385	11	,	,	PUNCT
ejpam-1773	385	12	89	89	NUM
ejpam-1773	385	13	-	-	SYM
ejpam-1773	385	14	106	106	NUM
ejpam-1773	385	15	99	99	NUM
ejpam-1773	385	16	proposition	proposition	NOUN
ejpam-1773	385	17	2	2	NUM
ejpam-1773	385	18	.	.	PUNCT
ejpam-1773	386	1	let	let	VERB
ejpam-1773	386	2	c	c	NOUN
ejpam-1773	386	3	be	be	AUX
ejpam-1773	386	4	the	the	DET
ejpam-1773	386	5	length	length	NOUN
ejpam-1773	386	6	2	2	NUM
ejpam-1773	386	7	code	code	NOUN
ejpam-1773	386	8	over	over	ADP
ejpam-1773	386	9	rk	rk	NOUN
ejpam-1773	386	10	,	,	PUNCT
ejpam-1773	386	11	for	for	ADP
ejpam-1773	386	12	k	k	PROPN
ejpam-1773	386	13	≥	≥	PROPN
ejpam-1773	386	14	4	4	NUM
ejpam-1773	386	15	,	,	PUNCT
ejpam-1773	386	16	generated	generate	VERB
ejpam-1773	386	17	by	by	ADP
ejpam-1773	386	18	c=	c=	NOUN
ejpam-1773	386	19	(	(	PUNCT
ejpam-1773	386	20	1,1+u1u2+u3	1,1+u1u2+u3	NUM
ejpam-1773	386	21	.	.	PUNCT
ejpam-1773	386	22	.	.	PUNCT
ejpam-1773	386	23	.	.	PUNCT
ejpam-1773	387	1	uk	uk	PROPN
ejpam-1773	387	2	)	)	PUNCT
ejpam-1773	387	3	.	.	PUNCT
ejpam-1773	388	1	then	then	ADV
ejpam-1773	388	2	c	c	PROPN
ejpam-1773	388	3	is	be	AUX
ejpam-1773	388	4	a	a	DET
ejpam-1773	388	5	type	type	NOUN
ejpam-1773	388	6	ii	ii	NOUN
ejpam-1773	388	7	self	self	NOUN
ejpam-1773	388	8	-	-	PUNCT
ejpam-1773	388	9	dual	dual	ADJ
ejpam-1773	388	10	code	code	NOUN
ejpam-1773	388	11	over	over	ADP
ejpam-1773	388	12	rk	rk	NOUN
ejpam-1773	388	13	with	with	ADP
ejpam-1773	388	14	minimum	minimum	NOUN
ejpam-1773	388	15	lee	lee	PROPN
ejpam-1773	388	16	distance	distance	PROPN
ejpam-1773	388	17	8	8	NUM
ejpam-1773	388	18	.	.	PUNCT
ejpam-1773	389	1	hence	hence	ADV
ejpam-1773	389	2	the	the	DET
ejpam-1773	389	3	binary	binary	ADJ
ejpam-1773	389	4	image	image	NOUN
ejpam-1773	389	5	is	be	AUX
ejpam-1773	389	6	an	an	DET
ejpam-1773	389	7	extremal	extremal	ADJ
ejpam-1773	389	8	type	type	NOUN
ejpam-1773	389	9	ii	ii	PROPN
ejpam-1773	389	10	code	code	NOUN
ejpam-1773	389	11	when	when	SCONJ
ejpam-1773	389	12	k	k	PROPN
ejpam-1773	389	13	=	=	SYM
ejpam-1773	389	14	4	4	X
ejpam-1773	389	15	.	.	PUNCT
ejpam-1773	390	1	the	the	DET
ejpam-1773	390	2	following	follow	VERB
ejpam-1773	390	3	is	be	AUX
ejpam-1773	390	4	an	an	DET
ejpam-1773	390	5	easy	easy	ADJ
ejpam-1773	390	6	observation	observation	NOUN
ejpam-1773	390	7	that	that	PRON
ejpam-1773	390	8	can	can	AUX
ejpam-1773	390	9	be	be	AUX
ejpam-1773	390	10	proven	prove	VERB
ejpam-1773	390	11	in	in	ADP
ejpam-1773	390	12	the	the	DET
ejpam-1773	390	13	same	same	ADJ
ejpam-1773	390	14	manner	manner	NOUN
ejpam-1773	390	15	.	.	PUNCT
ejpam-1773	391	1	proposition	proposition	NOUN
ejpam-1773	391	2	3	3	X
ejpam-1773	391	3	.	.	PUNCT
ejpam-1773	392	1	let	let	VERB
ejpam-1773	392	2	c	c	PRON
ejpam-1773	392	3	be	be	AUX
ejpam-1773	392	4	a	a	DET
ejpam-1773	392	5	linear	linear	ADJ
ejpam-1773	392	6	code	code	NOUN
ejpam-1773	392	7	over	over	ADP
ejpam-1773	392	8	rk	rk	PROPN
ejpam-1773	392	9	of	of	ADP
ejpam-1773	392	10	length	length	NOUN
ejpam-1773	392	11	2	2	NUM
ejpam-1773	392	12	generated	generate	VERB
ejpam-1773	392	13	by	by	ADP
ejpam-1773	392	14	(	(	PUNCT
ejpam-1773	392	15	a	a	DET
ejpam-1773	392	16	,	,	PUNCT
ejpam-1773	392	17	b	b	NOUN
ejpam-1773	392	18	)	)	PUNCT
ejpam-1773	392	19	where	where	SCONJ
ejpam-1773	392	20	a	a	PRON
ejpam-1773	392	21	and	and	CCONJ
ejpam-1773	392	22	b	b	NOUN
ejpam-1773	392	23	are	be	AUX
ejpam-1773	392	24	units	unit	NOUN
ejpam-1773	392	25	in	in	ADP
ejpam-1773	392	26	rk	rk	NOUN
ejpam-1773	392	27	.	.	PUNCT
ejpam-1773	393	1	then	then	ADV
ejpam-1773	393	2	c	c	PROPN
ejpam-1773	393	3	is	be	AUX
ejpam-1773	393	4	a	a	DET
ejpam-1773	393	5	self	self	NOUN
ejpam-1773	393	6	dual	dual	ADJ
ejpam-1773	393	7	code	code	NOUN
ejpam-1773	393	8	.	.	PUNCT
ejpam-1773	394	1	conversely	conversely	ADV
ejpam-1773	394	2	,	,	PUNCT
ejpam-1773	394	3	any	any	DET
ejpam-1773	394	4	self	self	NOUN
ejpam-1773	394	5	-	-	PUNCT
ejpam-1773	394	6	dual	dual	ADJ
ejpam-1773	394	7	code	code	NOUN
ejpam-1773	394	8	of	of	ADP
ejpam-1773	394	9	length	length	NOUN
ejpam-1773	394	10	2	2	NUM
ejpam-1773	394	11	over	over	ADP
ejpam-1773	394	12	rk	rk	NOUN
ejpam-1773	394	13	that	that	PRON
ejpam-1773	394	14	contains	contain	VERB
ejpam-1773	394	15	a	a	DET
ejpam-1773	394	16	vector	vector	NOUN
ejpam-1773	394	17	of	of	ADP
ejpam-1773	394	18	the	the	DET
ejpam-1773	394	19	form	form	NOUN
ejpam-1773	394	20	(	(	PUNCT
ejpam-1773	394	21	a1	a1	NOUN
ejpam-1773	394	22	,	,	PUNCT
ejpam-1773	394	23	a2	a2	PROPN
ejpam-1773	394	24	)	)	PUNCT
ejpam-1773	394	25	,	,	PUNCT
ejpam-1773	394	26	where	where	SCONJ
ejpam-1773	394	27	the	the	DET
ejpam-1773	394	28	ai	ai	NOUN
ejpam-1773	394	29	are	be	AUX
ejpam-1773	394	30	units	unit	NOUN
ejpam-1773	394	31	,	,	PUNCT
ejpam-1773	394	32	must	must	AUX
ejpam-1773	394	33	be	be	AUX
ejpam-1773	394	34	generated	generate	VERB
ejpam-1773	394	35	by	by	ADP
ejpam-1773	394	36	that	that	DET
ejpam-1773	394	37	vector	vector	NOUN
ejpam-1773	394	38	and	and	CCONJ
ejpam-1773	394	39	hence	hence	ADV
ejpam-1773	394	40	be	be	AUX
ejpam-1773	394	41	a	a	DET
ejpam-1773	394	42	one	one	NUM
ejpam-1773	394	43	-	-	PUNCT
ejpam-1773	394	44	generator	generator	NOUN
ejpam-1773	394	45	code	code	NOUN
ejpam-1773	394	46	.	.	PUNCT
ejpam-1773	395	1	of	of	ADP
ejpam-1773	395	2	course	course	NOUN
ejpam-1773	395	3	,	,	PUNCT
ejpam-1773	395	4	a	a	DET
ejpam-1773	395	5	vector	vector	NOUN
ejpam-1773	395	6	of	of	ADP
ejpam-1773	395	7	the	the	DET
ejpam-1773	395	8	form	form	NOUN
ejpam-1773	395	9	(	(	PUNCT
ejpam-1773	395	10	b1	b1	NOUN
ejpam-1773	395	11	,	,	PUNCT
ejpam-1773	395	12	b2	b2	NOUN
ejpam-1773	395	13	)	)	PUNCT
ejpam-1773	395	14	where	where	SCONJ
ejpam-1773	395	15	the	the	DET
ejpam-1773	395	16	bi	bi	NOUN
ejpam-1773	395	17	are	be	AUX
ejpam-1773	395	18	non	non	ADJ
ejpam-1773	395	19	-	-	NOUN
ejpam-1773	395	20	units	unit	NOUN
ejpam-1773	395	21	,	,	PUNCT
ejpam-1773	395	22	can	can	AUX
ejpam-1773	395	23	not	not	PART
ejpam-1773	395	24	generate	generate	VERB
ejpam-1773	395	25	a	a	DET
ejpam-1773	395	26	self	self	NOUN
ejpam-1773	395	27	-	-	PUNCT
ejpam-1773	395	28	dual	dual	ADJ
ejpam-1773	395	29	code	code	NOUN
ejpam-1773	395	30	by	by	ADP
ejpam-1773	395	31	itself	itself	PRON
ejpam-1773	395	32	,	,	PUNCT
ejpam-1773	395	33	because	because	SCONJ
ejpam-1773	395	34	multiplying	multiply	VERB
ejpam-1773	395	35	it	it	PRON
ejpam-1773	395	36	by	by	ADP
ejpam-1773	395	37	u1	u1	NOUN
ejpam-1773	395	38	.	.	PUNCT
ejpam-1773	395	39	.	.	PUNCT
ejpam-1773	395	40	.	.	PUNCT
ejpam-1773	396	1	uk	uk	PROPN
ejpam-1773	396	2	would	would	AUX
ejpam-1773	396	3	yield	yield	VERB
ejpam-1773	396	4	the	the	DET
ejpam-1773	396	5	zero	zero	NUM
ejpam-1773	396	6	vector	vector	NOUN
ejpam-1773	396	7	,	,	PUNCT
ejpam-1773	396	8	hence	hence	ADV
ejpam-1773	396	9	the	the	DET
ejpam-1773	396	10	size	size	NOUN
ejpam-1773	396	11	of	of	ADP
ejpam-1773	396	12	such	such	DET
ejpam-1773	396	13	a	a	DET
ejpam-1773	396	14	code	code	NOUN
ejpam-1773	396	15	can	can	AUX
ejpam-1773	396	16	be	be	AUX
ejpam-1773	396	17	at	at	ADP
ejpam-1773	396	18	most	most	ADJ
ejpam-1773	396	19	22k−1	22k−1	NUM
ejpam-1773	396	20	.	.	PUNCT
ejpam-1773	397	1	thus	thus	ADV
ejpam-1773	397	2	we	we	PRON
ejpam-1773	397	3	need	need	VERB
ejpam-1773	397	4	a	a	DET
ejpam-1773	397	5	second	second	ADJ
ejpam-1773	397	6	generator	generator	NOUN
ejpam-1773	397	7	in	in	ADP
ejpam-1773	397	8	such	such	DET
ejpam-1773	397	9	a	a	DET
ejpam-1773	397	10	case	case	NOUN
ejpam-1773	397	11	.	.	PUNCT
ejpam-1773	398	1	6	6	X
ejpam-1773	398	2	.	.	X
ejpam-1773	398	3	lattices	lattice	NOUN
ejpam-1773	398	4	there	there	PRON
ejpam-1773	398	5	is	be	VERB
ejpam-1773	398	6	a	a	DET
ejpam-1773	398	7	vast	vast	ADJ
ejpam-1773	398	8	literature	literature	NOUN
ejpam-1773	398	9	connecting	connect	VERB
ejpam-1773	398	10	codes	code	NOUN
ejpam-1773	398	11	and	and	CCONJ
ejpam-1773	398	12	lattices	lattice	NOUN
ejpam-1773	398	13	.	.	PUNCT
ejpam-1773	399	1	see	see	VERB
ejpam-1773	399	2	[	[	X
ejpam-1773	399	3	3	3	X
ejpam-1773	399	4	]	]	PUNCT
ejpam-1773	399	5	for	for	ADP
ejpam-1773	399	6	details	detail	NOUN
ejpam-1773	399	7	and	and	CCONJ
ejpam-1773	399	8	an	an	DET
ejpam-1773	399	9	extensive	extensive	ADJ
ejpam-1773	399	10	literature	literature	NOUN
ejpam-1773	399	11	.	.	PUNCT
ejpam-1773	400	1	let	let	VERB
ejpam-1773	400	2	f	f	PRON
ejpam-1773	400	3	be	be	AUX
ejpam-1773	400	4	either	either	CCONJ
ejpam-1773	400	5	r	r	NOUN
ejpam-1773	400	6	,	,	PUNCT
ejpam-1773	400	7	c	c	NOUN
ejpam-1773	400	8	or	or	CCONJ
ejpam-1773	400	9	h	h	NOUN
ejpam-1773	400	10	and	and	CCONJ
ejpam-1773	400	11	let	let	VERB
ejpam-1773	400	12	o	o	NOUN
ejpam-1773	400	13	be	be	AUX
ejpam-1773	400	14	z	z	PROPN
ejpam-1773	400	15	,	,	PUNCT
ejpam-1773	400	16	z[i	z[i	PROPN
ejpam-1773	400	17	]	]	X
ejpam-1773	400	18	,	,	PUNCT
ejpam-1773	400	19	or	or	CCONJ
ejpam-1773	400	20	z[i	z[i	NUM
ejpam-1773	400	21	,	,	PUNCT
ejpam-1773	400	22	j	j	PROPN
ejpam-1773	400	23	,	,	PUNCT
ejpam-1773	400	24	k	k	X
ejpam-1773	400	25	]	]	X
ejpam-1773	400	26	,	,	PUNCT
ejpam-1773	400	27	respectively	respectively	ADV
ejpam-1773	400	28	.	.	PUNCT
ejpam-1773	401	1	a	a	DET
ejpam-1773	401	2	lattice	lattice	NOUN
ejpam-1773	401	3	in	in	ADP
ejpam-1773	401	4	f	f	PROPN
ejpam-1773	401	5	n	n	X
ejpam-1773	401	6	is	be	AUX
ejpam-1773	401	7	a	a	DET
ejpam-1773	401	8	free	free	ADJ
ejpam-1773	401	9	o	o	NOUN
ejpam-1773	401	10	-module	-module	NOUN
ejpam-1773	401	11	.	.	PUNCT
ejpam-1773	402	1	the	the	DET
ejpam-1773	402	2	standard	standard	ADJ
ejpam-1773	402	3	inner	inner	ADJ
ejpam-1773	402	4	product	product	NOUN
ejpam-1773	402	5	attached	attach	VERB
ejpam-1773	402	6	to	to	ADP
ejpam-1773	402	7	the	the	DET
ejpam-1773	402	8	ambient	ambient	ADJ
ejpam-1773	402	9	space	space	NOUN
ejpam-1773	402	10	is	be	AUX
ejpam-1773	402	11	defined	define	VERB
ejpam-1773	402	12	as	as	ADP
ejpam-1773	402	13	v	v	NOUN
ejpam-1773	402	14	·	·	PUNCT
ejpam-1773	402	15	u	u	NOUN
ejpam-1773	402	16	=	=	SYM
ejpam-1773	402	17	∑	∑	PUNCT
ejpam-1773	402	18	viui	viui	PROPN
ejpam-1773	402	19	,	,	PUNCT
ejpam-1773	402	20	(	(	PUNCT
ejpam-1773	402	21	14	14	NUM
ejpam-1773	402	22	)	)	PUNCT
ejpam-1773	402	23	where	where	SCONJ
ejpam-1773	402	24	the	the	DET
ejpam-1773	402	25	involution	involution	NOUN
ejpam-1773	402	26	is	be	AUX
ejpam-1773	402	27	the	the	DET
ejpam-1773	402	28	identity	identity	NOUN
ejpam-1773	402	29	for	for	ADP
ejpam-1773	402	30	the	the	DET
ejpam-1773	402	31	real	real	ADJ
ejpam-1773	402	32	numbers	number	NOUN
ejpam-1773	402	33	and	and	CCONJ
ejpam-1773	402	34	the	the	DET
ejpam-1773	402	35	standard	standard	ADJ
ejpam-1773	402	36	involution	involution	NOUN
ejpam-1773	402	37	for	for	ADP
ejpam-1773	402	38	the	the	DET
ejpam-1773	402	39	complex	complex	ADJ
ejpam-1773	402	40	numbers	number	NOUN
ejpam-1773	402	41	and	and	CCONJ
ejpam-1773	402	42	the	the	DET
ejpam-1773	402	43	quaternions	quaternion	NOUN
ejpam-1773	402	44	.	.	PUNCT
ejpam-1773	403	1	we	we	PRON
ejpam-1773	403	2	define	define	VERB
ejpam-1773	403	3	l∗	l∗	PROPN
ejpam-1773	403	4	=	=	SYM
ejpam-1773	403	5	{	{	PUNCT
ejpam-1773	403	6	u	u	NOUN
ejpam-1773	403	7	∈	∈	PROPN
ejpam-1773	403	8	f	f	NOUN
ejpam-1773	403	9	n	n	CCONJ
ejpam-1773	403	10	|	|	ADV
ejpam-1773	403	11	u	u	X
ejpam-1773	403	12	·	·	PUNCT
ejpam-1773	403	13	v	v	ADP
ejpam-1773	403	14	∈	∈	X
ejpam-1773	403	15	o	o	NOUN
ejpam-1773	403	16	for	for	ADP
ejpam-1773	403	17	all	all	PRON
ejpam-1773	403	18	v	v	ADP
ejpam-1773	403	19	∈	∈	NOUN
ejpam-1773	403	20	l	l	NOUN
ejpam-1773	403	21	}	}	PUNCT
ejpam-1773	403	22	.	.	PUNCT
ejpam-1773	404	1	for	for	ADP
ejpam-1773	404	2	the	the	DET
ejpam-1773	404	3	quaternions	quaternion	NOUN
ejpam-1773	404	4	we	we	PRON
ejpam-1773	404	5	only	only	ADV
ejpam-1773	404	6	need	need	VERB
ejpam-1773	404	7	to	to	PART
ejpam-1773	404	8	define	define	VERB
ejpam-1773	404	9	one	one	NUM
ejpam-1773	404	10	orthogonal	orthogonal	NOUN
ejpam-1773	404	11	here	here	ADV
ejpam-1773	404	12	since	since	SCONJ
ejpam-1773	404	13	u·v	u·v	ADJ
ejpam-1773	404	14	∈	∈	PROPN
ejpam-1773	404	15	o	o	NOUN
ejpam-1773	405	1	if	if	SCONJ
ejpam-1773	405	2	and	and	CCONJ
ejpam-1773	405	3	only	only	ADV
ejpam-1773	405	4	if	if	SCONJ
ejpam-1773	405	5	v·u	v·u	NOUN
ejpam-1773	405	6	∈	∈	PROPN
ejpam-1773	405	7	o	o	INTJ
ejpam-1773	405	8	.	.	PUNCT
ejpam-1773	406	1	if	if	SCONJ
ejpam-1773	406	2	the	the	DET
ejpam-1773	406	3	lattice	lattice	PROPN
ejpam-1773	406	4	l	l	PROPN
ejpam-1773	406	5	satisfies	satisfie	NOUN
ejpam-1773	406	6	l	l	NOUN
ejpam-1773	406	7	⊆	⊆	NUM
ejpam-1773	406	8	l∗	l∗	NOUN
ejpam-1773	406	9	it	it	PRON
ejpam-1773	406	10	is	be	AUX
ejpam-1773	406	11	said	say	VERB
ejpam-1773	406	12	to	to	PART
ejpam-1773	406	13	be	be	AUX
ejpam-1773	406	14	integral	integral	ADJ
ejpam-1773	406	15	and	and	CCONJ
ejpam-1773	406	16	if	if	SCONJ
ejpam-1773	406	17	the	the	DET
ejpam-1773	406	18	lattice	lattice	PROPN
ejpam-1773	406	19	l	l	PROPN
ejpam-1773	406	20	satisfies	satisfie	NOUN
ejpam-1773	406	21	l	l	NOUN
ejpam-1773	406	22	=	=	PUNCT
ejpam-1773	406	23	l∗	l∗	PROPN
ejpam-1773	406	24	then	then	ADV
ejpam-1773	406	25	it	it	PRON
ejpam-1773	406	26	is	be	AUX
ejpam-1773	406	27	said	say	VERB
ejpam-1773	406	28	to	to	PART
ejpam-1773	406	29	be	be	AUX
ejpam-1773	406	30	unimodular	unimodular	ADJ
ejpam-1773	406	31	.	.	PUNCT
ejpam-1773	407	1	the	the	DET
ejpam-1773	407	2	norm	norm	NOUN
ejpam-1773	407	3	of	of	ADP
ejpam-1773	407	4	a	a	DET
ejpam-1773	407	5	vector	vector	NOUN
ejpam-1773	407	6	v	v	NOUN
ejpam-1773	407	7	is	be	AUX
ejpam-1773	407	8	n(v	n(v	PROPN
ejpam-1773	407	9	)	)	PUNCT
ejpam-1773	407	10	=	=	SYM
ejpam-1773	407	11	v	v	X
ejpam-1773	407	12	·	·	PUNCT
ejpam-1773	408	1	v.	v.	CCONJ
ejpam-1773	408	2	if	if	SCONJ
ejpam-1773	408	3	the	the	DET
ejpam-1773	408	4	norm	norm	NOUN
ejpam-1773	408	5	of	of	ADP
ejpam-1773	408	6	every	every	DET
ejpam-1773	408	7	vector	vector	NOUN
ejpam-1773	408	8	in	in	ADP
ejpam-1773	408	9	a	a	DET
ejpam-1773	408	10	unimodular	unimodular	ADJ
ejpam-1773	408	11	lattice	lattice	NOUN
ejpam-1773	408	12	is	be	AUX
ejpam-1773	408	13	an	an	DET
ejpam-1773	408	14	even	even	ADV
ejpam-1773	408	15	integer	integer	NOUN
ejpam-1773	408	16	then	then	ADV
ejpam-1773	408	17	we	we	PRON
ejpam-1773	408	18	say	say	VERB
ejpam-1773	408	19	the	the	DET
ejpam-1773	408	20	lattice	lattice	NOUN
ejpam-1773	408	21	is	be	AUX
ejpam-1773	408	22	even	even	ADV
ejpam-1773	408	23	.	.	PUNCT
ejpam-1773	409	1	we	we	PRON
ejpam-1773	409	2	describe	describe	VERB
ejpam-1773	409	3	a	a	DET
ejpam-1773	409	4	family	family	NOUN
ejpam-1773	409	5	of	of	ADP
ejpam-1773	409	6	reduction	reduction	NOUN
ejpam-1773	409	7	maps	map	NOUN
ejpam-1773	409	8	.	.	PUNCT
ejpam-1773	410	1	define	define	VERB
ejpam-1773	410	2	hh	hh	PROPN
ejpam-1773	410	3	:	:	PUNCT
ejpam-1773	410	4	o	o	PROPN
ejpam-1773	410	5	n→	n→	PROPN
ejpam-1773	410	6	rn	rn	PROPN
ejpam-1773	410	7	2	2	NUM
ejpam-1773	410	8	,	,	PUNCT
ejpam-1773	410	9	(	(	PUNCT
ejpam-1773	410	10	15	15	NUM
ejpam-1773	410	11	)	)	PUNCT
ejpam-1773	410	12	to	to	PART
ejpam-1773	410	13	be	be	AUX
ejpam-1773	410	14	the	the	DET
ejpam-1773	410	15	linear	linear	ADJ
ejpam-1773	410	16	map	map	NOUN
ejpam-1773	410	17	where	where	SCONJ
ejpam-1773	410	18	hh(i+	hh(i+	PROPN
ejpam-1773	410	19	1	1	NUM
ejpam-1773	410	20	)	)	PUNCT
ejpam-1773	410	21	=	=	SYM
ejpam-1773	410	22	u1	u1	NOUN
ejpam-1773	410	23	,	,	PUNCT
ejpam-1773	410	24	hh(j+	hh(j+	PROPN
ejpam-1773	410	25	1	1	NUM
ejpam-1773	410	26	)	)	PUNCT
ejpam-1773	410	27	=	=	SYM
ejpam-1773	410	28	u2	u2	NOUN
ejpam-1773	410	29	,	,	PUNCT
ejpam-1773	410	30	and	and	CCONJ
ejpam-1773	410	31	hh(k+	hh(k+	PROPN
ejpam-1773	410	32	1	1	NUM
ejpam-1773	410	33	)	)	PUNCT
ejpam-1773	410	34	=	=	NOUN
ejpam-1773	410	35	u1	u1	NOUN
ejpam-1773	410	36	+	+	CCONJ
ejpam-1773	410	37	u2	u2	PROPN
ejpam-1773	410	38	+	+	CCONJ
ejpam-1773	410	39	u1u2	u1u2	PROPN
ejpam-1773	410	40	.	.	NOUN
ejpam-1773	410	41	define	define	VERB
ejpam-1773	410	42	hc	hc	X
ejpam-1773	410	43	:	:	PUNCT
ejpam-1773	410	44	o	o	PROPN
ejpam-1773	410	45	n→	n→	PROPN
ejpam-1773	410	46	rn	rn	PROPN
ejpam-1773	410	47	1	1	NUM
ejpam-1773	410	48	,	,	PUNCT
ejpam-1773	410	49	(	(	PUNCT
ejpam-1773	410	50	16	16	NUM
ejpam-1773	410	51	)	)	PUNCT
ejpam-1773	410	52	to	to	PART
ejpam-1773	410	53	be	be	AUX
ejpam-1773	410	54	the	the	DET
ejpam-1773	410	55	linear	linear	ADJ
ejpam-1773	410	56	map	map	NOUN
ejpam-1773	410	57	where	where	SCONJ
ejpam-1773	410	58	hc(i+	hc(i+	PROPN
ejpam-1773	410	59	1	1	NUM
ejpam-1773	410	60	)	)	PUNCT
ejpam-1773	410	61	=	=	NOUN
ejpam-1773	410	62	u1	u1	NOUN
ejpam-1773	410	63	.	.	PUNCT
ejpam-1773	411	1	define	define	VERB
ejpam-1773	411	2	hr	hr	NOUN
ejpam-1773	411	3	:	:	PUNCT
ejpam-1773	411	4	o	o	PROPN
ejpam-1773	411	5	n→	n→	PROPN
ejpam-1773	411	6	rn	rn	PROPN
ejpam-1773	411	7	0	0	PROPN
ejpam-1773	411	8	,	,	PUNCT
ejpam-1773	411	9	(	(	PUNCT
ejpam-1773	411	10	17	17	NUM
ejpam-1773	411	11	)	)	PUNCT
ejpam-1773	411	12	where	where	SCONJ
ejpam-1773	411	13	r0	r0	NOUN
ejpam-1773	411	14	=	=	PUNCT
ejpam-1773	411	15	f2	f2	PROPN
ejpam-1773	411	16	and	and	CCONJ
ejpam-1773	411	17	hr(n	hr(n	NOUN
ejpam-1773	411	18	)	)	PUNCT
ejpam-1773	411	19	=	=	SYM
ejpam-1773	412	1	n	n	X
ejpam-1773	412	2	(	(	PUNCT
ejpam-1773	412	3	mod	mod	NOUN
ejpam-1773	412	4	2	2	NUM
ejpam-1773	412	5	)	)	PUNCT
ejpam-1773	412	6	.	.	PUNCT
ejpam-1773	413	1	s.	s.	PROPN
ejpam-1773	413	2	dougherty	dougherty	PROPN
ejpam-1773	413	3	,	,	PUNCT
ejpam-1773	413	4	b.yıldız	b.yıldız	NOUN
ejpam-1773	413	5	,	,	PUNCT
ejpam-1773	413	6	s.karadeniz	s.karadeniz	NOUN
ejpam-1773	413	7	/	/	SYM
ejpam-1773	413	8	eur	eur	NOUN
ejpam-1773	413	9	.	.	PUNCT
ejpam-1773	414	1	j.	j.	PROPN
ejpam-1773	414	2	pure	pure	PROPN
ejpam-1773	414	3	appl	appl	PROPN
ejpam-1773	414	4	.	.	PROPN
ejpam-1773	414	5	math	math	PROPN
ejpam-1773	414	6	,	,	PUNCT
ejpam-1773	414	7	6	6	NUM
ejpam-1773	414	8	(	(	PUNCT
ejpam-1773	414	9	2013	2013	NUM
ejpam-1773	414	10	)	)	PUNCT
ejpam-1773	414	11	,	,	PUNCT
ejpam-1773	414	12	89	89	NUM
ejpam-1773	414	13	-	-	SYM
ejpam-1773	414	14	106	106	NUM
ejpam-1773	414	15	100	100	NUM
ejpam-1773	414	16	each	each	PRON
ejpam-1773	414	17	of	of	ADP
ejpam-1773	414	18	these	these	DET
ejpam-1773	414	19	maps	map	NOUN
ejpam-1773	414	20	is	be	AUX
ejpam-1773	414	21	a	a	DET
ejpam-1773	414	22	ring	ring	NOUN
ejpam-1773	414	23	homomorphism	homomorphism	NOUN
ejpam-1773	414	24	and	and	CCONJ
ejpam-1773	414	25	it	it	PRON
ejpam-1773	414	26	can	can	AUX
ejpam-1773	414	27	be	be	AUX
ejpam-1773	414	28	seen	see	VERB
ejpam-1773	414	29	that	that	SCONJ
ejpam-1773	414	30	h−1(c	h−1(c	NOUN
ejpam-1773	414	31	)	)	PUNCT
ejpam-1773	414	32	is	be	AUX
ejpam-1773	414	33	a	a	DET
ejpam-1773	414	34	free	free	ADJ
ejpam-1773	414	35	o	o	NOUN
ejpam-1773	414	36	-module	-module	NOUN
ejpam-1773	414	37	.	.	PUNCT
ejpam-1773	415	1	the	the	DET
ejpam-1773	415	2	lattices	lattice	NOUN
ejpam-1773	415	3	induced	induce	VERB
ejpam-1773	415	4	from	from	ADP
ejpam-1773	415	5	a	a	DET
ejpam-1773	415	6	code	code	NOUN
ejpam-1773	415	7	c	c	NOUN
ejpam-1773	415	8	are	be	AUX
ejpam-1773	415	9	defined	define	VERB
ejpam-1773	415	10	as	as	SCONJ
ejpam-1773	415	11	follows	follow	VERB
ejpam-1773	415	12	:	:	PUNCT
ejpam-1773	415	13	λh(c	λh(c	NOUN
ejpam-1773	415	14	)	)	PUNCT
ejpam-1773	415	15	:	:	PUNCT
ejpam-1773	416	1	=	=	NOUN
ejpam-1773	416	2	1p	1p	NUM
ejpam-1773	416	3	2	2	NUM
ejpam-1773	416	4	h−1	h−1	PROPN
ejpam-1773	416	5	h	h	NOUN
ejpam-1773	416	6	(	(	PUNCT
ejpam-1773	416	7	c	c	X
ejpam-1773	416	8	)	)	PUNCT
ejpam-1773	416	9	=	=	PRON
ejpam-1773	416	10	{	{	PUNCT
ejpam-1773	416	11	v	v	NUM
ejpam-1773	416	12	∈	∈	NOUN
ejpam-1773	416	13	o	o	NOUN
ejpam-1773	416	14	n	n	CCONJ
ejpam-1773	416	15	|	|	ADV
ejpam-1773	416	16	v	v	X
ejpam-1773	416	17	(	(	PUNCT
ejpam-1773	416	18	mod	mod	ADJ
ejpam-1773	416	19	2o	2o	PROPN
ejpam-1773	416	20	)	)	PUNCT
ejpam-1773	416	21	∈	∈	PROPN
ejpam-1773	416	22	c	c	NOUN
ejpam-1773	416	23	}	}	PUNCT
ejpam-1773	416	24	.	.	PUNCT
ejpam-1773	417	1	(	(	PUNCT
ejpam-1773	417	2	18	18	NUM
ejpam-1773	417	3	)	)	PUNCT
ejpam-1773	417	4	λc(c	λc(c	PUNCT
ejpam-1773	417	5	)	)	PUNCT
ejpam-1773	417	6	:	:	PUNCT
ejpam-1773	418	1	=	=	NOUN
ejpam-1773	418	2	1p	1p	NUM
ejpam-1773	418	3	2	2	NUM
ejpam-1773	419	1	h−1	h−1	PROPN
ejpam-1773	419	2	c	c	NOUN
ejpam-1773	419	3	(	(	PUNCT
ejpam-1773	419	4	c	c	NOUN
ejpam-1773	419	5	)	)	PUNCT
ejpam-1773	419	6	=	=	PRON
ejpam-1773	419	7	{	{	PUNCT
ejpam-1773	419	8	v	v	NUM
ejpam-1773	419	9	∈	∈	NOUN
ejpam-1773	419	10	o	o	NOUN
ejpam-1773	419	11	n	n	CCONJ
ejpam-1773	419	12	|	|	ADV
ejpam-1773	419	13	v	v	X
ejpam-1773	419	14	(	(	PUNCT
ejpam-1773	419	15	mod	mod	ADJ
ejpam-1773	419	16	2o	2o	PROPN
ejpam-1773	419	17	)	)	PUNCT
ejpam-1773	419	18	∈	∈	PROPN
ejpam-1773	419	19	c	c	NOUN
ejpam-1773	419	20	}	}	PUNCT
ejpam-1773	419	21	.	.	PUNCT
ejpam-1773	420	1	(	(	PUNCT
ejpam-1773	420	2	19	19	NUM
ejpam-1773	420	3	)	)	PUNCT
ejpam-1773	420	4	λr(c	λr(c	PUNCT
ejpam-1773	420	5	)	)	PUNCT
ejpam-1773	420	6	:	:	PUNCT
ejpam-1773	421	1	=	=	NOUN
ejpam-1773	421	2	1p	1p	NUM
ejpam-1773	421	3	2	2	NUM
ejpam-1773	421	4	h−1	h−1	NOUN
ejpam-1773	421	5	r	r	NOUN
ejpam-1773	421	6	(	(	PUNCT
ejpam-1773	421	7	c	c	NOUN
ejpam-1773	421	8	)	)	PUNCT
ejpam-1773	421	9	=	=	PRON
ejpam-1773	421	10	{	{	PUNCT
ejpam-1773	421	11	v	v	NUM
ejpam-1773	421	12	∈	∈	NOUN
ejpam-1773	421	13	o	o	NOUN
ejpam-1773	421	14	n	n	CCONJ
ejpam-1773	421	15	|	|	ADV
ejpam-1773	421	16	v	v	X
ejpam-1773	421	17	(	(	PUNCT
ejpam-1773	421	18	mod	mod	ADJ
ejpam-1773	421	19	2o	2o	PROPN
ejpam-1773	421	20	)	)	PUNCT
ejpam-1773	421	21	∈	∈	PROPN
ejpam-1773	421	22	c	c	NOUN
ejpam-1773	421	23	}	}	PUNCT
ejpam-1773	421	24	.	.	PUNCT
ejpam-1773	422	1	(	(	PUNCT
ejpam-1773	422	2	20	20	X
ejpam-1773	422	3	)	)	PUNCT
ejpam-1773	422	4	lemma	lemma	PROPN
ejpam-1773	422	5	4	4	NUM
ejpam-1773	422	6	.	.	PUNCT
ejpam-1773	423	1	if	if	SCONJ
ejpam-1773	423	2	c	c	PROPN
ejpam-1773	423	3	is	be	AUX
ejpam-1773	423	4	a	a	DET
ejpam-1773	423	5	self	self	NOUN
ejpam-1773	423	6	-	-	PUNCT
ejpam-1773	423	7	dual	dual	ADJ
ejpam-1773	423	8	code	code	NOUN
ejpam-1773	423	9	over	over	ADP
ejpam-1773	423	10	r2	r2	PROPN
ejpam-1773	423	11	then	then	ADV
ejpam-1773	423	12	λh(c	λh(c	NOUN
ejpam-1773	423	13	)	)	PUNCT
ejpam-1773	423	14	is	be	AUX
ejpam-1773	423	15	a	a	DET
ejpam-1773	423	16	quaternionic	quaternionic	ADJ
ejpam-1773	423	17	unimodular	unimodular	ADJ
ejpam-1773	423	18	lattice	lattice	NOUN
ejpam-1773	423	19	.	.	PUNCT
ejpam-1773	424	1	if	if	SCONJ
ejpam-1773	424	2	c	c	PROPN
ejpam-1773	424	3	is	be	AUX
ejpam-1773	424	4	a	a	DET
ejpam-1773	424	5	self	self	NOUN
ejpam-1773	424	6	-	-	PUNCT
ejpam-1773	424	7	dual	dual	ADJ
ejpam-1773	424	8	code	code	NOUN
ejpam-1773	424	9	over	over	ADP
ejpam-1773	424	10	r1	r1	PROPN
ejpam-1773	424	11	then	then	ADV
ejpam-1773	424	12	λc(c	λc(c	PUNCT
ejpam-1773	424	13	)	)	PUNCT
ejpam-1773	424	14	is	be	AUX
ejpam-1773	424	15	a	a	DET
ejpam-1773	424	16	complex	complex	ADJ
ejpam-1773	424	17	unimodular	unimodular	ADJ
ejpam-1773	424	18	lattice	lattice	NOUN
ejpam-1773	424	19	.	.	PUNCT
ejpam-1773	425	1	if	if	SCONJ
ejpam-1773	425	2	c	c	PROPN
ejpam-1773	425	3	is	be	AUX
ejpam-1773	425	4	a	a	DET
ejpam-1773	425	5	self	self	NOUN
ejpam-1773	425	6	-	-	PUNCT
ejpam-1773	425	7	dual	dual	ADJ
ejpam-1773	425	8	code	code	NOUN
ejpam-1773	425	9	over	over	ADP
ejpam-1773	425	10	r0	r0	NOUN
ejpam-1773	425	11	=	=	SYM
ejpam-1773	425	12	f2	f2	PROPN
ejpam-1773	425	13	then	then	ADV
ejpam-1773	425	14	λr(c	λr(c	PUNCT
ejpam-1773	425	15	)	)	PUNCT
ejpam-1773	425	16	is	be	AUX
ejpam-1773	425	17	a	a	DET
ejpam-1773	425	18	real	real	ADJ
ejpam-1773	425	19	unimodular	unimodular	ADJ
ejpam-1773	425	20	lattice	lattice	NOUN
ejpam-1773	425	21	.	.	PUNCT
ejpam-1773	426	1	proof	proof	NOUN
ejpam-1773	426	2	.	.	PUNCT
ejpam-1773	427	1	the	the	DET
ejpam-1773	427	2	first	first	ADJ
ejpam-1773	427	3	result	result	NOUN
ejpam-1773	427	4	can	can	AUX
ejpam-1773	427	5	be	be	AUX
ejpam-1773	427	6	found	find	VERB
ejpam-1773	427	7	in	in	ADP
ejpam-1773	427	8	[	[	X
ejpam-1773	427	9	2	2	NUM
ejpam-1773	427	10	]	]	PUNCT
ejpam-1773	427	11	.	.	PUNCT
ejpam-1773	428	1	notice	notice	VERB
ejpam-1773	428	2	that	that	SCONJ
ejpam-1773	428	3	in	in	ADP
ejpam-1773	428	4	the	the	DET
ejpam-1773	428	5	notation	notation	NOUN
ejpam-1773	428	6	of	of	ADP
ejpam-1773	428	7	[	[	X
ejpam-1773	428	8	2	2	NUM
ejpam-1773	428	9	]	]	PUNCT
ejpam-1773	428	10	,	,	PUNCT
ejpam-1773	428	11	α	α	PROPN
ejpam-1773	428	12	corresponds	correspond	VERB
ejpam-1773	428	13	to	to	PART
ejpam-1773	428	14	u1	u1	VERB
ejpam-1773	428	15	,	,	PUNCT
ejpam-1773	428	16	β	β	X
ejpam-1773	428	17	corresponds	correspond	VERB
ejpam-1773	428	18	to	to	ADP
ejpam-1773	428	19	u2	u2	NOUN
ejpam-1773	428	20	and	and	CCONJ
ejpam-1773	428	21	γ	γ	NOUN
ejpam-1773	428	22	corresponds	correspond	NOUN
ejpam-1773	428	23	to	to	ADP
ejpam-1773	428	24	u1+u2+u1u2	u1+u2+u1u2	NUM
ejpam-1773	428	25	.	.	PUNCT
ejpam-1773	429	1	the	the	DET
ejpam-1773	429	2	second	second	ADJ
ejpam-1773	429	3	result	result	NOUN
ejpam-1773	429	4	can	can	AUX
ejpam-1773	429	5	be	be	AUX
ejpam-1773	429	6	found	find	VERB
ejpam-1773	429	7	in	in	ADP
ejpam-1773	429	8	[	[	X
ejpam-1773	429	9	6	6	NUM
ejpam-1773	429	10	]	]	PUNCT
ejpam-1773	429	11	where	where	SCONJ
ejpam-1773	429	12	the	the	DET
ejpam-1773	429	13	rings	ring	NOUN
ejpam-1773	429	14	is	be	AUX
ejpam-1773	429	15	written	write	VERB
ejpam-1773	429	16	as	as	ADP
ejpam-1773	429	17	f2	f2	PROPN
ejpam-1773	429	18	+	+	CCONJ
ejpam-1773	429	19	uf2	uf2	NOUN
ejpam-1773	429	20	and	and	CCONJ
ejpam-1773	429	21	u	u	NOUN
ejpam-1773	429	22	corresponds	correspond	VERB
ejpam-1773	429	23	to	to	PART
ejpam-1773	429	24	u1	u1	VERB
ejpam-1773	429	25	.	.	PUNCT
ejpam-1773	430	1	the	the	DET
ejpam-1773	430	2	third	third	ADJ
ejpam-1773	430	3	result	result	NOUN
ejpam-1773	430	4	can	can	AUX
ejpam-1773	430	5	be	be	AUX
ejpam-1773	430	6	found	find	VERB
ejpam-1773	430	7	in	in	ADP
ejpam-1773	430	8	[	[	X
ejpam-1773	430	9	1	1	NUM
ejpam-1773	430	10	]	]	PUNCT
ejpam-1773	430	11	and	and	CCONJ
ejpam-1773	430	12	numerous	numerous	ADJ
ejpam-1773	430	13	other	other	ADJ
ejpam-1773	430	14	papers	paper	NOUN
ejpam-1773	430	15	,	,	PUNCT
ejpam-1773	430	16	see	see	VERB
ejpam-1773	430	17	[	[	X
ejpam-1773	430	18	3	3	NUM
ejpam-1773	430	19	]	]	PUNCT
ejpam-1773	430	20	.	.	PUNCT
ejpam-1773	431	1	let	let	VERB
ejpam-1773	431	2	k	k	PROPN
ejpam-1773	431	3	≥	≥	NUM
ejpam-1773	431	4	2	2	NUM
ejpam-1773	431	5	.	.	PUNCT
ejpam-1773	432	1	for	for	ADP
ejpam-1773	432	2	α	α	PROPN
ejpam-1773	432	3	∈	∈	PROPN
ejpam-1773	432	4	rk	rk	NOUN
ejpam-1773	432	5	write	write	VERB
ejpam-1773	432	6	α	α	NOUN
ejpam-1773	432	7	=	=	PUNCT
ejpam-1773	433	1	α0	α0	ADJ
ejpam-1773	434	1	+	+	NUM
ejpam-1773	434	2	α1uk−1	α1uk−1	NUM
ejpam-1773	434	3	+	+	CCONJ
ejpam-1773	434	4	α2uk	α2uk	PUNCT
ejpam-1773	435	1	+	+	CCONJ
ejpam-1773	435	2	α3uk−1uk	α3uk−1uk	NOUN
ejpam-1773	435	3	with	with	ADP
ejpam-1773	435	4	αi	αi	PROPN
ejpam-1773	435	5	∈	∈	PROPN
ejpam-1773	435	6	rk−2	rk−2	PROPN
ejpam-1773	435	7	.	.	PUNCT
ejpam-1773	436	1	then	then	ADV
ejpam-1773	436	2	define	define	VERB
ejpam-1773	436	3	φ2	φ2	PROPN
ejpam-1773	436	4	:	:	PUNCT
ejpam-1773	436	5	rk→	rk→	VERB
ejpam-1773	436	6	r2k−2	r2k−2	PROPN
ejpam-1773	436	7	2	2	NUM
ejpam-1773	436	8	by	by	ADP
ejpam-1773	436	9	φ2(α	φ2(α	NOUN
ejpam-1773	436	10	)	)	PUNCT
ejpam-1773	436	11	=	=	SYM
ejpam-1773	436	12	φk−2(α0	φk−2(α0	PROPN
ejpam-1773	436	13	)	)	PUNCT
ejpam-1773	437	1	+	+	PUNCT
ejpam-1773	437	2	φk−2(α1)u1	φk−2(α1)u1	VERB
ejpam-1773	437	3	+	+	ADJ
ejpam-1773	437	4	φk−2(α2)u2	φk−2(α2)u2	NOUN
ejpam-1773	437	5	+	+	ADJ
ejpam-1773	437	6	φk−2(α3)u1u2	φk−2(α3)u1u2	X
ejpam-1773	437	7	.	.	PUNCT
ejpam-1773	438	1	for	for	ADP
ejpam-1773	438	2	α	α	PROPN
ejpam-1773	438	3	∈	∈	PROPN
ejpam-1773	438	4	rk	rk	NOUN
ejpam-1773	438	5	write	write	VERB
ejpam-1773	438	6	α	α	NOUN
ejpam-1773	438	7	=	=	PUNCT
ejpam-1773	439	1	α0	α0	PROPN
ejpam-1773	440	1	+	+	PROPN
ejpam-1773	441	1	α1uk	α1uk	VERB
ejpam-1773	441	2	with	with	ADP
ejpam-1773	441	3	αi	αi	PROPN
ejpam-1773	441	4	∈	∈	PROPN
ejpam-1773	441	5	rk−1	rk−1	PROPN
ejpam-1773	441	6	.	.	PUNCT
ejpam-1773	442	1	then	then	ADV
ejpam-1773	442	2	define	define	VERB
ejpam-1773	442	3	φ1	φ1	PROPN
ejpam-1773	442	4	:	:	PUNCT
ejpam-1773	442	5	rk→	rk→	VERB
ejpam-1773	442	6	r2k−1	r2k−1	PROPN
ejpam-1773	442	7	2	2	NUM
ejpam-1773	442	8	by	by	ADP
ejpam-1773	442	9	φ1(α	φ1(α	NOUN
ejpam-1773	442	10	)	)	PUNCT
ejpam-1773	442	11	=	=	SYM
ejpam-1773	442	12	φk−1(α0	φk−1(α0	PROPN
ejpam-1773	442	13	)	)	PUNCT
ejpam-1773	443	1	+	+	PUNCT
ejpam-1773	443	2	φk−1(α1)u1	φk−1(α1)u1	PROPN
ejpam-1773	443	3	.	.	PUNCT
ejpam-1773	443	4	theorem	theorem	VERB
ejpam-1773	443	5	10	10	NUM
ejpam-1773	443	6	.	.	PUNCT
ejpam-1773	444	1	let	let	VERB
ejpam-1773	444	2	k	k	PROPN
ejpam-1773	444	3	≥	≥	NUM
ejpam-1773	444	4	2	2	NUM
ejpam-1773	444	5	.	.	PUNCT
ejpam-1773	445	1	if	if	SCONJ
ejpam-1773	445	2	c	c	PROPN
ejpam-1773	445	3	is	be	AUX
ejpam-1773	445	4	a	a	DET
ejpam-1773	445	5	self	self	NOUN
ejpam-1773	445	6	-	-	PUNCT
ejpam-1773	445	7	dual	dual	ADJ
ejpam-1773	445	8	code	code	NOUN
ejpam-1773	445	9	over	over	ADP
ejpam-1773	445	10	rk	rk	PROPN
ejpam-1773	445	11	of	of	ADP
ejpam-1773	445	12	length	length	NOUN
ejpam-1773	445	13	n	n	PROPN
ejpam-1773	445	14	then	then	ADV
ejpam-1773	445	15	φ2(c	φ2(c	NUM
ejpam-1773	445	16	)	)	PUNCT
ejpam-1773	445	17	is	be	AUX
ejpam-1773	445	18	a	a	DET
ejpam-1773	445	19	self	self	NOUN
ejpam-1773	445	20	-	-	PUNCT
ejpam-1773	445	21	dual	dual	ADJ
ejpam-1773	445	22	code	code	NOUN
ejpam-1773	445	23	over	over	ADP
ejpam-1773	445	24	r2	r2	PROPN
ejpam-1773	445	25	of	of	ADP
ejpam-1773	445	26	length	length	NOUN
ejpam-1773	445	27	2k−2n	2k−2n	PROPN
ejpam-1773	445	28	.	.	PUNCT
ejpam-1773	446	1	if	if	SCONJ
ejpam-1773	446	2	c	c	PROPN
ejpam-1773	446	3	is	be	AUX
ejpam-1773	446	4	a	a	DET
ejpam-1773	446	5	self	self	NOUN
ejpam-1773	446	6	-	-	PUNCT
ejpam-1773	446	7	dual	dual	ADJ
ejpam-1773	446	8	code	code	NOUN
ejpam-1773	446	9	over	over	ADP
ejpam-1773	446	10	rk	rk	PROPN
ejpam-1773	446	11	of	of	ADP
ejpam-1773	446	12	length	length	NOUN
ejpam-1773	446	13	n	n	PROPN
ejpam-1773	446	14	then	then	ADV
ejpam-1773	446	15	φ1(c	φ1(c	X
ejpam-1773	446	16	)	)	PUNCT
ejpam-1773	446	17	is	be	AUX
ejpam-1773	446	18	a	a	DET
ejpam-1773	446	19	self	self	NOUN
ejpam-1773	446	20	-	-	PUNCT
ejpam-1773	446	21	dual	dual	ADJ
ejpam-1773	446	22	code	code	NOUN
ejpam-1773	446	23	over	over	ADP
ejpam-1773	446	24	r2	r2	PROPN
ejpam-1773	446	25	of	of	ADP
ejpam-1773	446	26	length	length	NOUN
ejpam-1773	446	27	2k−1n	2k−1n	NOUN
ejpam-1773	446	28	.	.	PUNCT
ejpam-1773	447	1	proof	proof	NOUN
ejpam-1773	447	2	.	.	PUNCT
ejpam-1773	448	1	the	the	DET
ejpam-1773	448	2	proof	proof	NOUN
ejpam-1773	448	3	follows	follow	VERB
ejpam-1773	448	4	from	from	ADP
ejpam-1773	448	5	theorem	theorem	ADJ
ejpam-1773	448	6	4	4	NUM
ejpam-1773	448	7	.	.	PUNCT
ejpam-1773	448	8	theorem	theorem	NOUN
ejpam-1773	448	9	11	11	NUM
ejpam-1773	448	10	.	.	PUNCT
ejpam-1773	449	1	let	let	VERB
ejpam-1773	449	2	c	c	PRON
ejpam-1773	449	3	be	be	AUX
ejpam-1773	449	4	a	a	DET
ejpam-1773	449	5	self	self	NOUN
ejpam-1773	449	6	-	-	PUNCT
ejpam-1773	449	7	dual	dual	ADJ
ejpam-1773	449	8	code	code	NOUN
ejpam-1773	449	9	over	over	ADP
ejpam-1773	449	10	rk	rk	PROPN
ejpam-1773	449	11	of	of	ADP
ejpam-1773	449	12	length	length	NOUN
ejpam-1773	449	13	n	n	CCONJ
ejpam-1773	449	14	,	,	PUNCT
ejpam-1773	449	15	k	k	X
ejpam-1773	449	16	≥	≥	NUM
ejpam-1773	449	17	2	2	NUM
ejpam-1773	449	18	with	with	ADP
ejpam-1773	449	19	i	i	PRON
ejpam-1773	449	20	,	,	PUNCT
ejpam-1773	449	21	j	j	PROPN
ejpam-1773	449	22	≤	≤	PROPN
ejpam-1773	449	23	k.	k.	PROPN
ejpam-1773	449	24	then	then	ADV
ejpam-1773	449	25	λh(φ2(c	λh(φ2(c	PROPN
ejpam-1773	449	26	)	)	PUNCT
ejpam-1773	449	27	)	)	PUNCT
ejpam-1773	449	28	is	be	AUX
ejpam-1773	449	29	a	a	DET
ejpam-1773	449	30	quaternionic	quaternionic	ADJ
ejpam-1773	449	31	unimodular	unimodular	ADJ
ejpam-1773	449	32	lattice	lattice	NOUN
ejpam-1773	449	33	of	of	ADP
ejpam-1773	449	34	length	length	NOUN
ejpam-1773	449	35	2k−2n	2k−2n	PROPN
ejpam-1773	449	36	,	,	PUNCT
ejpam-1773	449	37	λc(φ1(c	λc(φ1(c	PROPN
ejpam-1773	449	38	)	)	PUNCT
ejpam-1773	449	39	)	)	PUNCT
ejpam-1773	449	40	is	be	AUX
ejpam-1773	449	41	a	a	DET
ejpam-1773	449	42	complex	complex	ADJ
ejpam-1773	449	43	unimodular	unimodular	ADJ
ejpam-1773	449	44	lattice	lattice	NOUN
ejpam-1773	449	45	of	of	ADP
ejpam-1773	449	46	length	length	NOUN
ejpam-1773	449	47	2k−1n	2k−1n	NOUN
ejpam-1773	449	48	,	,	PUNCT
ejpam-1773	449	49	and	and	CCONJ
ejpam-1773	449	50	λr(φk(c	λr(φk(c	NUM
ejpam-1773	449	51	)	)	PUNCT
ejpam-1773	449	52	)	)	PUNCT
ejpam-1773	450	1	is	be	AUX
ejpam-1773	450	2	a	a	DET
ejpam-1773	450	3	real	real	ADV
ejpam-1773	450	4	unimodular	unimodular	ADJ
ejpam-1773	450	5	lattice	lattice	NOUN
ejpam-1773	450	6	of	of	ADP
ejpam-1773	450	7	length	length	NOUN
ejpam-1773	450	8	2kn	2kn	ADV
ejpam-1773	450	9	.	.	PUNCT
ejpam-1773	451	1	proof	proof	NOUN
ejpam-1773	451	2	.	.	PUNCT
ejpam-1773	452	1	follows	follow	VERB
ejpam-1773	452	2	by	by	ADP
ejpam-1773	452	3	applying	apply	VERB
ejpam-1773	452	4	lemma	lemma	PROPN
ejpam-1773	452	5	4	4	NUM
ejpam-1773	452	6	and	and	CCONJ
ejpam-1773	452	7	theorem	theorem	VERB
ejpam-1773	452	8	10	10	NUM
ejpam-1773	452	9	.	.	PUNCT
ejpam-1773	453	1	7	7	X
ejpam-1773	453	2	.	.	X
ejpam-1773	453	3	extremal	extremal	ADJ
ejpam-1773	453	4	binary	binary	PROPN
ejpam-1773	453	5	self	self	NOUN
ejpam-1773	453	6	-	-	PUNCT
ejpam-1773	453	7	dual	dual	ADJ
ejpam-1773	453	8	codes	code	NOUN
ejpam-1773	453	9	obtained	obtain	VERB
ejpam-1773	453	10	from	from	ADP
ejpam-1773	453	11	codes	code	NOUN
ejpam-1773	453	12	over	over	ADP
ejpam-1773	453	13	rk	rk	NOUN
ejpam-1773	453	14	note	note	VERB
ejpam-1773	453	15	that	that	SCONJ
ejpam-1773	453	16	,	,	PUNCT
ejpam-1773	453	17	in	in	ADP
ejpam-1773	453	18	section	section	NOUN
ejpam-1773	453	19	5	5	NUM
ejpam-1773	453	20	,	,	PUNCT
ejpam-1773	453	21	we	we	PRON
ejpam-1773	453	22	introduced	introduce	VERB
ejpam-1773	453	23	the	the	DET
ejpam-1773	453	24	map	map	NOUN
ejpam-1773	453	25	ψk	ψk	INTJ
ejpam-1773	453	26	:	:	PUNCT
ejpam-1773	453	27	rk→	rk→	NOUN
ejpam-1773	453	28	r2	r2	PROPN
ejpam-1773	453	29	k−1	k−1	PROPN
ejpam-1773	453	30	s.	s.	PROPN
ejpam-1773	453	31	dougherty	dougherty	PROPN
ejpam-1773	453	32	,	,	PUNCT
ejpam-1773	453	33	b.yıldız	b.yıldız	NOUN
ejpam-1773	453	34	,	,	PUNCT
ejpam-1773	453	35	s.karadeniz	s.karadeniz	NOUN
ejpam-1773	453	36	/	/	SYM
ejpam-1773	453	37	eur	eur	NOUN
ejpam-1773	453	38	.	.	PUNCT
ejpam-1773	454	1	j.	j.	PROPN
ejpam-1773	454	2	pure	pure	PROPN
ejpam-1773	454	3	appl	appl	PROPN
ejpam-1773	454	4	.	.	PROPN
ejpam-1773	454	5	math	math	PROPN
ejpam-1773	454	6	,	,	PUNCT
ejpam-1773	454	7	6	6	NUM
ejpam-1773	454	8	(	(	PUNCT
ejpam-1773	454	9	2013	2013	NUM
ejpam-1773	454	10	)	)	PUNCT
ejpam-1773	454	11	,	,	PUNCT
ejpam-1773	454	12	89	89	NUM
ejpam-1773	454	13	-	-	SYM
ejpam-1773	454	14	106	106	NUM
ejpam-1773	454	15	101	101	NUM
ejpam-1773	454	16	given	give	VERB
ejpam-1773	454	17	by	by	ADP
ejpam-1773	454	18	ψk(a+	ψk(a+	PROPN
ejpam-1773	454	19	uk	uk	PROPN
ejpam-1773	454	20	b	b	PROPN
ejpam-1773	454	21	)	)	PUNCT
ejpam-1773	454	22	=	=	PUNCT
ejpam-1773	454	23	(	(	PUNCT
ejpam-1773	454	24	b	b	NOUN
ejpam-1773	454	25	,	,	PUNCT
ejpam-1773	454	26	a+	a+	PRON
ejpam-1773	454	27	b	b	NOUN
ejpam-1773	454	28	)	)	PUNCT
ejpam-1773	454	29	.	.	PUNCT
ejpam-1773	455	1	it	it	PRON
ejpam-1773	455	2	is	be	AUX
ejpam-1773	455	3	easy	easy	ADJ
ejpam-1773	455	4	to	to	PART
ejpam-1773	455	5	verify	verify	VERB
ejpam-1773	455	6	that	that	DET
ejpam-1773	455	7	ψk	ψk	NOUN
ejpam-1773	455	8	is	be	AUX
ejpam-1773	455	9	a	a	DET
ejpam-1773	455	10	linear	linear	ADJ
ejpam-1773	455	11	bijection	bijection	NOUN
ejpam-1773	455	12	from	from	ADP
ejpam-1773	455	13	rn	rn	PROPN
ejpam-1773	455	14	k	k	PROPN
ejpam-1773	455	15	to	to	ADP
ejpam-1773	455	16	r2n	r2n	VERB
ejpam-1773	455	17	k−1	k−1	PROPN
ejpam-1773	455	18	and	and	CCONJ
ejpam-1773	455	19	furthermore	furthermore	ADV
ejpam-1773	455	20	it	it	PRON
ejpam-1773	455	21	is	be	AUX
ejpam-1773	455	22	distance	distance	NOUN
ejpam-1773	455	23	preserving	preserve	VERB
ejpam-1773	455	24	.	.	PUNCT
ejpam-1773	456	1	now	now	ADV
ejpam-1773	456	2	let	let	VERB
ejpam-1773	456	3	c1+ukd1,c2+ukd2	c1+ukd1,c2+ukd2	PROPN
ejpam-1773	456	4	be	be	AUX
ejpam-1773	456	5	two	two	NUM
ejpam-1773	456	6	vectors	vector	NOUN
ejpam-1773	456	7	in	in	ADP
ejpam-1773	456	8	rn	rn	PROPN
ejpam-1773	456	9	k	k	PROPN
ejpam-1773	456	10	such	such	ADJ
ejpam-1773	456	11	that	that	SCONJ
ejpam-1773	456	12	<	<	X
ejpam-1773	456	13	c1	c1	PROPN
ejpam-1773	456	14	+	+	CCONJ
ejpam-1773	456	15	ukd1,c2	ukd1,c2	PROPN
ejpam-1773	456	16	+	+	CCONJ
ejpam-1773	456	17	ukd2	ukd2	NOUN
ejpam-1773	456	18	>	>	X
ejpam-1773	456	19	k=	k=	X
ejpam-1773	456	20	0	0	X
ejpam-1773	456	21	.	.	PUNCT
ejpam-1773	457	1	this	this	PRON
ejpam-1773	457	2	means	mean	VERB
ejpam-1773	457	3	<	<	X
ejpam-1773	457	4	c1,c2	c1,c2	PROPN
ejpam-1773	457	5	>	>	X
ejpam-1773	457	6	k−1=	k−1=	PROPN
ejpam-1773	457	7	0	0	NUM
ejpam-1773	457	8	,	,	PUNCT
ejpam-1773	457	9	<	<	X
ejpam-1773	457	10	c1,d2	c1,d2	PROPN
ejpam-1773	457	11	>	>	PUNCT
ejpam-1773	457	12	k−1	k−1	PROPN
ejpam-1773	458	1	+	+	X
ejpam-1773	458	2	<	<	X
ejpam-1773	458	3	c2,d1	c2,d1	PROPN
ejpam-1773	458	4	>	>	X
ejpam-1773	458	5	k−1=	k−1=	PROPN
ejpam-1773	458	6	0	0	NUM
ejpam-1773	458	7	.	.	PUNCT
ejpam-1773	459	1	(	(	PUNCT
ejpam-1773	459	2	21	21	NUM
ejpam-1773	459	3	)	)	PUNCT
ejpam-1773	459	4	it	it	PRON
ejpam-1773	459	5	follows	follow	VERB
ejpam-1773	459	6	that	that	SCONJ
ejpam-1773	459	7	<	<	X
ejpam-1773	459	8	ψk(c1	ψk(c1	PUNCT
ejpam-1773	459	9	+	+	X
ejpam-1773	459	10	ukd1),ψk(c2	ukd1),ψk(c2	PROPN
ejpam-1773	459	11	+	+	ADJ
ejpam-1773	459	12	ukd2)>k−1=	ukd2)>k−1=	ADJ
ejpam-1773	459	13	<	<	X
ejpam-1773	459	14	(	(	PUNCT
ejpam-1773	459	15	d1,c1	d1,c1	PROPN
ejpam-1773	459	16	+	+	CCONJ
ejpam-1773	459	17	d1	d1	NOUN
ejpam-1773	459	18	)	)	PUNCT
ejpam-1773	459	19	,	,	PUNCT
ejpam-1773	459	20	(	(	PUNCT
ejpam-1773	459	21	d2,c2	d2,c2	X
ejpam-1773	459	22	+	+	CCONJ
ejpam-1773	459	23	d2)>k−1	d2)>k−1	VERB
ejpam-1773	459	24	=	=	X
ejpam-1773	459	25	<	<	X
ejpam-1773	459	26	d1,d2	d1,d2	PROPN
ejpam-1773	459	27	>	>	PUNCT
ejpam-1773	459	28	k−1	k−1	PROPN
ejpam-1773	459	29	+	+	CCONJ
ejpam-1773	459	30	<	<	X
ejpam-1773	459	31	c1	c1	NOUN
ejpam-1773	459	32	+	+	CCONJ
ejpam-1773	459	33	d1,c2	d1,c2	PROPN
ejpam-1773	459	34	+	+	CCONJ
ejpam-1773	459	35	d2	d2	PROPN
ejpam-1773	459	36	>	>	X
ejpam-1773	459	37	k−1	k−1	PROPN
ejpam-1773	459	38	=	=	PUNCT
ejpam-1773	459	39	<	<	X
ejpam-1773	459	40	d1,d2	d1,d2	PROPN
ejpam-1773	459	41	>	>	PUNCT
ejpam-1773	459	42	k−1	k−1	PROPN
ejpam-1773	459	43	+	+	X
ejpam-1773	459	44	<	<	X
ejpam-1773	459	45	c1,c2	c1,c2	PROPN
ejpam-1773	459	46	>	>	PUNCT
ejpam-1773	459	47	k−1	k−1	PROPN
ejpam-1773	460	1	+	+	X
ejpam-1773	460	2	<	<	X
ejpam-1773	460	3	c1,d2	c1,d2	PROPN
ejpam-1773	460	4	>	>	PUNCT
ejpam-1773	460	5	k−1	k−1	PROPN
ejpam-1773	460	6	+	+	X
ejpam-1773	460	7	<	<	X
ejpam-1773	460	8	c2,d1	c2,d1	PROPN
ejpam-1773	460	9	>	>	PUNCT
ejpam-1773	460	10	k−1	k−1	PROPN
ejpam-1773	461	1	+	+	X
ejpam-1773	461	2	<	<	X
ejpam-1773	461	3	d1,d2	d1,d2	PROPN
ejpam-1773	461	4	>	>	PUNCT
ejpam-1773	461	5	k−1	k−1	PROPN
ejpam-1773	461	6	=	=	SYM
ejpam-1773	461	7	0	0	NUM
ejpam-1773	461	8	by	by	ADP
ejpam-1773	461	9	(	(	PUNCT
ejpam-1773	461	10	21	21	NUM
ejpam-1773	461	11	)	)	PUNCT
ejpam-1773	461	12	.	.	PUNCT
ejpam-1773	462	1	this	this	PRON
ejpam-1773	462	2	leads	lead	VERB
ejpam-1773	462	3	to	to	ADP
ejpam-1773	462	4	the	the	DET
ejpam-1773	462	5	following	follow	VERB
ejpam-1773	462	6	lemma	lemma	PROPN
ejpam-1773	462	7	:	:	PUNCT
ejpam-1773	462	8	lemma	lemma	PROPN
ejpam-1773	462	9	5	5	X
ejpam-1773	462	10	.	.	PUNCT
ejpam-1773	463	1	if	if	SCONJ
ejpam-1773	463	2	c	c	PROPN
ejpam-1773	463	3	is	be	AUX
ejpam-1773	463	4	a	a	DET
ejpam-1773	463	5	self	self	NOUN
ejpam-1773	463	6	-	-	PUNCT
ejpam-1773	463	7	dual	dual	ADJ
ejpam-1773	463	8	code	code	NOUN
ejpam-1773	463	9	over	over	ADP
ejpam-1773	463	10	rk	rk	PROPN
ejpam-1773	463	11	of	of	ADP
ejpam-1773	463	12	length	length	NOUN
ejpam-1773	463	13	n	n	CCONJ
ejpam-1773	463	14	,	,	PUNCT
ejpam-1773	463	15	then	then	ADV
ejpam-1773	463	16	ψk(c	ψk(c	NUM
ejpam-1773	463	17	)	)	PUNCT
ejpam-1773	463	18	is	be	AUX
ejpam-1773	463	19	a	a	DET
ejpam-1773	463	20	self	self	NOUN
ejpam-1773	463	21	-	-	PUNCT
ejpam-1773	463	22	dual	dual	ADJ
ejpam-1773	463	23	code	code	NOUN
ejpam-1773	463	24	over	over	ADP
ejpam-1773	463	25	rk−1	rk−1	PROPN
ejpam-1773	463	26	of	of	ADP
ejpam-1773	463	27	length	length	NOUN
ejpam-1773	463	28	2n	2n	NUM
ejpam-1773	463	29	.	.	PUNCT
ejpam-1773	464	1	proof	proof	NOUN
ejpam-1773	464	2	.	.	PUNCT
ejpam-1773	465	1	note	note	VERB
ejpam-1773	465	2	that	that	SCONJ
ejpam-1773	465	3	the	the	DET
ejpam-1773	465	4	above	above	ADJ
ejpam-1773	465	5	observation	observation	NOUN
ejpam-1773	465	6	tells	tell	VERB
ejpam-1773	465	7	us	we	PRON
ejpam-1773	465	8	that	that	SCONJ
ejpam-1773	465	9	ψ	ψ	AUX
ejpam-1773	465	10	preserves	preserve	VERB
ejpam-1773	465	11	self	self	NOUN
ejpam-1773	465	12	-	-	PUNCT
ejpam-1773	465	13	orthogonality	orthogonality	NOUN
ejpam-1773	465	14	.	.	PUNCT
ejpam-1773	466	1	but	but	CCONJ
ejpam-1773	466	2	,	,	PUNCT
ejpam-1773	466	3	since	since	SCONJ
ejpam-1773	466	4	ψk	ψk	NOUN
ejpam-1773	466	5	is	be	AUX
ejpam-1773	466	6	an	an	DET
ejpam-1773	466	7	injective	injective	ADJ
ejpam-1773	466	8	map	map	NOUN
ejpam-1773	466	9	,	,	PUNCT
ejpam-1773	466	10	the	the	DET
ejpam-1773	466	11	sizes	size	NOUN
ejpam-1773	466	12	of	of	ADP
ejpam-1773	466	13	the	the	DET
ejpam-1773	466	14	codes	code	NOUN
ejpam-1773	466	15	are	be	AUX
ejpam-1773	466	16	preserved	preserve	VERB
ejpam-1773	466	17	as	as	ADV
ejpam-1773	466	18	well	well	ADV
ejpam-1773	466	19	,	,	PUNCT
ejpam-1773	466	20	which	which	PRON
ejpam-1773	466	21	implies	imply	VERB
ejpam-1773	466	22	that	that	SCONJ
ejpam-1773	466	23	if	if	SCONJ
ejpam-1773	466	24	c	c	PROPN
ejpam-1773	466	25	is	be	AUX
ejpam-1773	466	26	a	a	DET
ejpam-1773	466	27	self	self	NOUN
ejpam-1773	466	28	-	-	PUNCT
ejpam-1773	466	29	dual	dual	ADJ
ejpam-1773	466	30	code	code	NOUN
ejpam-1773	466	31	of	of	ADP
ejpam-1773	466	32	length	length	NOUN
ejpam-1773	466	33	n	n	CCONJ
ejpam-1773	466	34	over	over	ADP
ejpam-1773	466	35	rk	rk	NOUN
ejpam-1773	466	36	,	,	PUNCT
ejpam-1773	466	37	then	then	ADV
ejpam-1773	466	38	ψk(c	ψk(c	NUM
ejpam-1773	466	39	)	)	PUNCT
ejpam-1773	466	40	is	be	AUX
ejpam-1773	466	41	a	a	DET
ejpam-1773	466	42	self	self	NOUN
ejpam-1773	466	43	-	-	PUNCT
ejpam-1773	466	44	dual	dual	ADJ
ejpam-1773	466	45	code	code	NOUN
ejpam-1773	466	46	of	of	ADP
ejpam-1773	466	47	length	length	NOUN
ejpam-1773	466	48	2n	2n	NUM
ejpam-1773	466	49	over	over	ADP
ejpam-1773	466	50	rk−1	rk−1	PROPN
ejpam-1773	466	51	.	.	PUNCT
ejpam-1773	467	1	combining	combine	VERB
ejpam-1773	467	2	this	this	PRON
ejpam-1773	467	3	with	with	ADP
ejpam-1773	467	4	theorem	theorem	ADJ
ejpam-1773	467	5	4	4	NUM
ejpam-1773	467	6	,	,	PUNCT
ejpam-1773	467	7	we	we	PRON
ejpam-1773	467	8	obtain	obtain	VERB
ejpam-1773	467	9	the	the	DET
ejpam-1773	467	10	following	following	ADJ
ejpam-1773	467	11	result	result	NOUN
ejpam-1773	467	12	:	:	PUNCT
ejpam-1773	467	13	theorem	theorem	NOUN
ejpam-1773	467	14	12	12	NUM
ejpam-1773	467	15	.	.	PUNCT
ejpam-1773	468	1	suppose	suppose	VERB
ejpam-1773	468	2	c	c	NOUN
ejpam-1773	468	3	is	be	AUX
ejpam-1773	468	4	a	a	DET
ejpam-1773	468	5	self	self	NOUN
ejpam-1773	468	6	-	-	PUNCT
ejpam-1773	468	7	dual	dual	ADJ
ejpam-1773	468	8	code	code	NOUN
ejpam-1773	468	9	over	over	ADP
ejpam-1773	468	10	rk	rk	PROPN
ejpam-1773	468	11	of	of	ADP
ejpam-1773	468	12	length	length	NOUN
ejpam-1773	468	13	n	n	CCONJ
ejpam-1773	468	14	,	,	PUNCT
ejpam-1773	468	15	and	and	CCONJ
ejpam-1773	468	16	that	that	SCONJ
ejpam-1773	468	17	its	its	PRON
ejpam-1773	468	18	binary	binary	ADJ
ejpam-1773	468	19	image	image	NOUN
ejpam-1773	468	20	ψk(c	ψk(c	NUM
ejpam-1773	468	21	)	)	PUNCT
ejpam-1773	468	22	is	be	AUX
ejpam-1773	468	23	a	a	DET
ejpam-1773	468	24	binary	binary	ADJ
ejpam-1773	468	25	self	self	NOUN
ejpam-1773	468	26	-	-	PUNCT
ejpam-1773	468	27	dual	dual	ADJ
ejpam-1773	468	28	code	code	NOUN
ejpam-1773	468	29	with	with	ADP
ejpam-1773	468	30	parameters	parameter	NOUN
ejpam-1773	468	31	[	[	X
ejpam-1773	468	32	2kn	2kn	ADJ
ejpam-1773	468	33	,	,	PUNCT
ejpam-1773	468	34	2k−1n	2k−1n	NUM
ejpam-1773	468	35	,	,	PUNCT
ejpam-1773	468	36	d	d	X
ejpam-1773	468	37	]	]	X
ejpam-1773	468	38	.	.	PUNCT
ejpam-1773	469	1	then	then	ADV
ejpam-1773	469	2	there	there	PRON
ejpam-1773	469	3	exists	exist	VERB
ejpam-1773	469	4	a	a	DET
ejpam-1773	469	5	self	self	NOUN
ejpam-1773	469	6	-	-	PUNCT
ejpam-1773	469	7	dual	dual	ADJ
ejpam-1773	469	8	code	code	NOUN
ejpam-1773	469	9	d	d	PROPN
ejpam-1773	469	10	over	over	ADP
ejpam-1773	469	11	rk−1	rk−1	NOUN
ejpam-1773	469	12	of	of	ADP
ejpam-1773	469	13	length	length	NOUN
ejpam-1773	469	14	2n	2n	NUM
ejpam-1773	469	15	such	such	ADJ
ejpam-1773	469	16	that	that	DET
ejpam-1773	469	17	ψk−1(d	ψk−1(d	NOUN
ejpam-1773	469	18	)	)	PUNCT
ejpam-1773	469	19	is	be	AUX
ejpam-1773	469	20	a	a	DET
ejpam-1773	469	21	binary	binary	ADJ
ejpam-1773	469	22	self	self	NOUN
ejpam-1773	469	23	-	-	PUNCT
ejpam-1773	469	24	dual	dual	ADJ
ejpam-1773	469	25	code	code	NOUN
ejpam-1773	469	26	with	with	ADP
ejpam-1773	469	27	the	the	DET
ejpam-1773	469	28	same	same	ADJ
ejpam-1773	469	29	parameters	parameter	NOUN
ejpam-1773	469	30	and	and	CCONJ
ejpam-1773	469	31	moreover	moreover	ADV
ejpam-1773	469	32	is	be	AUX
ejpam-1773	469	33	equivalent	equivalent	ADJ
ejpam-1773	469	34	to	to	ADP
ejpam-1773	469	35	ψk(c	ψk(c	NUM
ejpam-1773	469	36	)	)	PUNCT
ejpam-1773	469	37	.	.	PUNCT
ejpam-1773	470	1	consequently	consequently	ADV
ejpam-1773	470	2	,	,	PUNCT
ejpam-1773	470	3	when	when	SCONJ
ejpam-1773	470	4	we	we	PRON
ejpam-1773	470	5	are	be	AUX
ejpam-1773	470	6	trying	try	VERB
ejpam-1773	470	7	to	to	PART
ejpam-1773	470	8	get	get	VERB
ejpam-1773	470	9	some	some	DET
ejpam-1773	470	10	known	know	VERB
ejpam-1773	470	11	binary	binary	NOUN
ejpam-1773	470	12	codes	code	NOUN
ejpam-1773	470	13	as	as	ADP
ejpam-1773	470	14	the	the	DET
ejpam-1773	470	15	images	image	NOUN
ejpam-1773	470	16	of	of	ADP
ejpam-1773	470	17	linear	linear	NOUN
ejpam-1773	470	18	codes	code	NOUN
ejpam-1773	470	19	over	over	ADP
ejpam-1773	470	20	rk	rk	NOUN
ejpam-1773	470	21	,	,	PUNCT
ejpam-1773	470	22	it	it	PRON
ejpam-1773	470	23	suffices	suffice	VERB
ejpam-1773	470	24	to	to	PART
ejpam-1773	470	25	find	find	VERB
ejpam-1773	470	26	the	the	DET
ejpam-1773	470	27	largest	large	ADJ
ejpam-1773	470	28	k	k	NOUN
ejpam-1773	470	29	for	for	ADP
ejpam-1773	470	30	which	which	PRON
ejpam-1773	470	31	we	we	PRON
ejpam-1773	470	32	can	can	AUX
ejpam-1773	470	33	do	do	VERB
ejpam-1773	470	34	that	that	PRON
ejpam-1773	470	35	.	.	PUNCT
ejpam-1773	471	1	because	because	SCONJ
ejpam-1773	471	2	if	if	SCONJ
ejpam-1773	471	3	it	it	PRON
ejpam-1773	471	4	is	be	AUX
ejpam-1773	471	5	linear	linear	ADJ
ejpam-1773	471	6	over	over	ADP
ejpam-1773	471	7	rk	rk	NOUN
ejpam-1773	471	8	,	,	PUNCT
ejpam-1773	471	9	then	then	ADV
ejpam-1773	471	10	it	it	PRON
ejpam-1773	471	11	will	will	AUX
ejpam-1773	471	12	be	be	AUX
ejpam-1773	471	13	linear	linear	ADJ
ejpam-1773	471	14	over	over	ADP
ejpam-1773	471	15	ri	ri	NOUN
ejpam-1773	471	16	for	for	ADP
ejpam-1773	471	17	all	all	PRON
ejpam-1773	471	18	i	i	PRON
ejpam-1773	471	19	≤	≤	ADJ
ejpam-1773	471	20	k.	k.	PROPN
ejpam-1773	471	21	7.1	7.1	NUM
ejpam-1773	471	22	.	.	PUNCT
ejpam-1773	472	1	examples	example	NOUN
ejpam-1773	472	2	we	we	PRON
ejpam-1773	472	3	are	be	AUX
ejpam-1773	472	4	now	now	ADV
ejpam-1773	472	5	ready	ready	ADJ
ejpam-1773	472	6	to	to	PART
ejpam-1773	472	7	give	give	VERB
ejpam-1773	472	8	some	some	DET
ejpam-1773	472	9	known	know	VERB
ejpam-1773	472	10	binary	binary	NOUN
ejpam-1773	472	11	self	self	NOUN
ejpam-1773	472	12	-	-	PUNCT
ejpam-1773	472	13	dual	dual	ADJ
ejpam-1773	472	14	codes	code	NOUN
ejpam-1773	472	15	as	as	ADP
ejpam-1773	472	16	the	the	DET
ejpam-1773	472	17	images	image	NOUN
ejpam-1773	472	18	of	of	ADP
ejpam-1773	472	19	self	self	NOUN
ejpam-1773	472	20	-	-	PUNCT
ejpam-1773	472	21	dual	dual	ADJ
ejpam-1773	472	22	codes	code	NOUN
ejpam-1773	472	23	over	over	ADP
ejpam-1773	472	24	rk	rk	PROPN
ejpam-1773	472	25	.	.	PUNCT
ejpam-1773	472	26	corollary	corollary	ADJ
ejpam-1773	472	27	4.4	4.4	NUM
ejpam-1773	472	28	in	in	ADP
ejpam-1773	472	29	[	[	X
ejpam-1773	472	30	7	7	NUM
ejpam-1773	472	31	]	]	PUNCT
ejpam-1773	472	32	states	state	NOUN
ejpam-1773	472	33	that	that	SCONJ
ejpam-1773	472	34	if	if	SCONJ
ejpam-1773	472	35	a	a	DET
ejpam-1773	472	36	binary	binary	ADJ
ejpam-1773	472	37	code	code	NOUN
ejpam-1773	472	38	is	be	AUX
ejpam-1773	472	39	the	the	DET
ejpam-1773	472	40	image	image	NOUN
ejpam-1773	472	41	of	of	ADP
ejpam-1773	472	42	a	a	DET
ejpam-1773	472	43	code	code	NOUN
ejpam-1773	472	44	over	over	ADP
ejpam-1773	472	45	rk	rk	NOUN
ejpam-1773	472	46	then	then	ADV
ejpam-1773	472	47	the	the	DET
ejpam-1773	472	48	automorphism	automorphism	NOUN
ejpam-1773	472	49	group	group	NOUN
ejpam-1773	472	50	of	of	ADP
ejpam-1773	472	51	the	the	DET
ejpam-1773	472	52	code	code	NOUN
ejpam-1773	472	53	contains	contain	VERB
ejpam-1773	472	54	k	k	PROPN
ejpam-1773	472	55	distinct	distinct	ADJ
ejpam-1773	472	56	automorphisms	automorphism	NOUN
ejpam-1773	472	57	which	which	PRON
ejpam-1773	472	58	are	be	AUX
ejpam-1773	472	59	involutions	involution	NOUN
ejpam-1773	472	60	corresponding	correspond	VERB
ejpam-1773	472	61	to	to	ADP
ejpam-1773	472	62	multiplication	multiplication	NOUN
ejpam-1773	472	63	in	in	ADP
ejpam-1773	472	64	the	the	DET
ejpam-1773	472	65	ring	ring	NOUN
ejpam-1773	472	66	by	by	ADP
ejpam-1773	472	67	1+ui	1+ui	NUM
ejpam-1773	472	68	for	for	ADP
ejpam-1773	472	69	i	i	PRON
ejpam-1773	472	70	=	=	NOUN
ejpam-1773	472	71	1	1	NUM
ejpam-1773	472	72	.	.	PUNCT
ejpam-1773	472	73	.	.	PUNCT
ejpam-1773	472	74	.	.	PUNCT
ejpam-1773	473	1	k.	k.	PROPN
ejpam-1773	473	2	hence	hence	ADV
ejpam-1773	473	3	,	,	PUNCT
ejpam-1773	473	4	the	the	DET
ejpam-1773	473	5	codes	code	NOUN
ejpam-1773	473	6	described	describe	VERB
ejpam-1773	473	7	below	below	ADV
ejpam-1773	473	8	have	have	VERB
ejpam-1773	473	9	a	a	DET
ejpam-1773	473	10	rich	rich	ADJ
ejpam-1773	473	11	automorphism	automorphism	NOUN
ejpam-1773	473	12	structure	structure	NOUN
ejpam-1773	473	13	containing	contain	VERB
ejpam-1773	473	14	at	at	ADP
ejpam-1773	473	15	least	least	ADJ
ejpam-1773	473	16	the	the	DET
ejpam-1773	473	17	group	group	NOUN
ejpam-1773	473	18	generated	generate	VERB
ejpam-1773	473	19	by	by	ADP
ejpam-1773	473	20	these	these	DET
ejpam-1773	473	21	involutions	involution	NOUN
ejpam-1773	473	22	.	.	PUNCT
ejpam-1773	474	1	in	in	ADP
ejpam-1773	474	2	general	general	ADJ
ejpam-1773	474	3	,	,	PUNCT
ejpam-1773	474	4	it	it	PRON
ejpam-1773	474	5	is	be	AUX
ejpam-1773	474	6	important	important	ADJ
ejpam-1773	474	7	to	to	PART
ejpam-1773	474	8	find	find	VERB
ejpam-1773	474	9	the	the	DET
ejpam-1773	474	10	largest	large	ADJ
ejpam-1773	474	11	k	k	NOUN
ejpam-1773	474	12	such	such	ADJ
ejpam-1773	474	13	that	that	SCONJ
ejpam-1773	474	14	a	a	DET
ejpam-1773	474	15	binary	binary	PROPN
ejpam-1773	474	16	code	code	NOUN
ejpam-1773	474	17	is	be	AUX
ejpam-1773	474	18	the	the	DET
ejpam-1773	474	19	image	image	NOUN
ejpam-1773	474	20	of	of	ADP
ejpam-1773	474	21	a	a	DET
ejpam-1773	474	22	code	code	NOUN
ejpam-1773	474	23	over	over	ADP
ejpam-1773	474	24	rk	rk	NOUN
ejpam-1773	474	25	since	since	SCONJ
ejpam-1773	474	26	this	this	PRON
ejpam-1773	474	27	says	say	VERB
ejpam-1773	474	28	the	the	DET
ejpam-1773	474	29	most	most	ADJ
ejpam-1773	474	30	about	about	ADP
ejpam-1773	474	31	its	its	PRON
ejpam-1773	474	32	automorphism	automorphism	NOUN
ejpam-1773	474	33	group	group	NOUN
ejpam-1773	474	34	.	.	PUNCT
ejpam-1773	475	1	s.	s.	PROPN
ejpam-1773	475	2	dougherty	dougherty	PROPN
ejpam-1773	475	3	,	,	PUNCT
ejpam-1773	475	4	b.yıldız	b.yıldız	NOUN
ejpam-1773	475	5	,	,	PUNCT
ejpam-1773	475	6	s.karadeniz	s.karadeniz	NOUN
ejpam-1773	475	7	/	/	SYM
ejpam-1773	475	8	eur	eur	NOUN
ejpam-1773	475	9	.	.	PUNCT
ejpam-1773	476	1	j.	j.	PROPN
ejpam-1773	476	2	pure	pure	PROPN
ejpam-1773	476	3	appl	appl	PROPN
ejpam-1773	476	4	.	.	PROPN
ejpam-1773	476	5	math	math	PROPN
ejpam-1773	476	6	,	,	PUNCT
ejpam-1773	476	7	6	6	NUM
ejpam-1773	476	8	(	(	PUNCT
ejpam-1773	476	9	2013	2013	NUM
ejpam-1773	476	10	)	)	PUNCT
ejpam-1773	476	11	,	,	PUNCT
ejpam-1773	476	12	89	89	NUM
ejpam-1773	476	13	-	-	SYM
ejpam-1773	476	14	106	106	NUM
ejpam-1773	476	15	102	102	NUM
ejpam-1773	476	16	7.2	7.2	NUM
ejpam-1773	476	17	.	.	PUNCT
ejpam-1773	477	1	[	[	X
ejpam-1773	477	2	8	8	NUM
ejpam-1773	477	3	,	,	PUNCT
ejpam-1773	477	4	4	4	NUM
ejpam-1773	477	5	,	,	PUNCT
ejpam-1773	477	6	4	4	NUM
ejpam-1773	477	7	]	]	X
ejpam-1773	477	8	binary	binary	ADJ
ejpam-1773	477	9	self	self	NOUN
ejpam-1773	477	10	-	-	PUNCT
ejpam-1773	477	11	dual	dual	ADJ
ejpam-1773	477	12	code	code	NOUN
ejpam-1773	477	13	because	because	SCONJ
ejpam-1773	477	14	of	of	ADP
ejpam-1773	477	15	the	the	DET
ejpam-1773	477	16	length	length	NOUN
ejpam-1773	477	17	of	of	ADP
ejpam-1773	477	18	the	the	DET
ejpam-1773	477	19	code	code	NOUN
ejpam-1773	477	20	,	,	PUNCT
ejpam-1773	477	21	the	the	DET
ejpam-1773	477	22	largest	large	ADJ
ejpam-1773	477	23	k	k	NOUN
ejpam-1773	477	24	for	for	ADP
ejpam-1773	477	25	which	which	PRON
ejpam-1773	477	26	the	the	DET
ejpam-1773	477	27	code	code	NOUN
ejpam-1773	477	28	can	can	AUX
ejpam-1773	477	29	be	be	AUX
ejpam-1773	477	30	the	the	DET
ejpam-1773	477	31	image	image	NOUN
ejpam-1773	477	32	of	of	ADP
ejpam-1773	477	33	a	a	DET
ejpam-1773	477	34	code	code	NOUN
ejpam-1773	477	35	over	over	ADP
ejpam-1773	477	36	rk	rk	PROPN
ejpam-1773	477	37	is	be	AUX
ejpam-1773	477	38	3	3	NUM
ejpam-1773	477	39	.	.	PUNCT
ejpam-1773	478	1	if	if	SCONJ
ejpam-1773	478	2	we	we	PRON
ejpam-1773	478	3	take	take	VERB
ejpam-1773	478	4	c1	c1	PROPN
ejpam-1773	478	5	to	to	PART
ejpam-1773	478	6	be	be	AUX
ejpam-1773	478	7	the	the	DET
ejpam-1773	478	8	linear	linear	ADJ
ejpam-1773	478	9	code	code	NOUN
ejpam-1773	478	10	of	of	ADP
ejpam-1773	478	11	length	length	NOUN
ejpam-1773	478	12	1	1	NUM
ejpam-1773	478	13	over	over	ADP
ejpam-1773	478	14	r3	r3	PROPN
ejpam-1773	478	15	generated	generate	VERB
ejpam-1773	478	16	by	by	ADP
ejpam-1773	478	17	u1u2	u1u2	NOUN
ejpam-1773	478	18	,	,	PUNCT
ejpam-1773	478	19	u1u3	u1u3	X
ejpam-1773	478	20	and	and	CCONJ
ejpam-1773	478	21	u2u3	u2u3	X
ejpam-1773	478	22	,	,	PUNCT
ejpam-1773	478	23	then	then	ADV
ejpam-1773	478	24	c1	c1	PROPN
ejpam-1773	478	25	is	be	AUX
ejpam-1773	478	26	a	a	DET
ejpam-1773	478	27	self	self	NOUN
ejpam-1773	478	28	-	-	PUNCT
ejpam-1773	478	29	dual	dual	ADJ
ejpam-1773	478	30	code	code	NOUN
ejpam-1773	478	31	with	with	ADP
ejpam-1773	478	32	weight	weight	NOUN
ejpam-1773	478	33	enumerator	enumerator	NOUN
ejpam-1773	478	34	1	1	NUM
ejpam-1773	478	35	+	+	SYM
ejpam-1773	478	36	14z4	14z4	NUM
ejpam-1773	478	37	+	+	CCONJ
ejpam-1773	478	38	z8	z8	NOUN
ejpam-1773	478	39	.	.	PUNCT
ejpam-1773	479	1	the	the	DET
ejpam-1773	479	2	binary	binary	ADJ
ejpam-1773	479	3	image	image	NOUN
ejpam-1773	479	4	is	be	AUX
ejpam-1773	479	5	an	an	DET
ejpam-1773	479	6	extremal	extremal	ADJ
ejpam-1773	479	7	type	type	NOUN
ejpam-1773	479	8	ii	ii	PROPN
ejpam-1773	479	9	code	code	NOUN
ejpam-1773	479	10	with	with	ADP
ejpam-1773	479	11	parameters	parameter	NOUN
ejpam-1773	479	12	[	[	X
ejpam-1773	479	13	8,4,4	8,4,4	X
ejpam-1773	479	14	]	]	PUNCT
ejpam-1773	479	15	.	.	PUNCT
ejpam-1773	480	1	by	by	ADP
ejpam-1773	480	2	the	the	DET
ejpam-1773	480	3	above	above	ADJ
ejpam-1773	480	4	argument	argument	NOUN
ejpam-1773	480	5	,	,	PUNCT
ejpam-1773	480	6	we	we	PRON
ejpam-1773	480	7	know	know	VERB
ejpam-1773	480	8	that	that	SCONJ
ejpam-1773	480	9	we	we	PRON
ejpam-1773	480	10	can	can	AUX
ejpam-1773	480	11	find	find	VERB
ejpam-1773	480	12	the	the	DET
ejpam-1773	480	13	same	same	ADJ
ejpam-1773	480	14	code	code	NOUN
ejpam-1773	480	15	to	to	PART
ejpam-1773	480	16	be	be	AUX
ejpam-1773	480	17	linear	linear	ADJ
ejpam-1773	480	18	over	over	ADP
ejpam-1773	480	19	r2	r2	PROPN
ejpam-1773	480	20	as	as	ADV
ejpam-1773	480	21	well	well	ADV
ejpam-1773	480	22	.	.	PUNCT
ejpam-1773	481	1	in	in	ADP
ejpam-1773	481	2	that	that	DET
ejpam-1773	481	3	case	case	NOUN
ejpam-1773	481	4	the	the	DET
ejpam-1773	481	5	generator	generator	NOUN
ejpam-1773	481	6	can	can	AUX
ejpam-1773	481	7	be	be	AUX
ejpam-1773	481	8	taken	take	VERB
ejpam-1773	481	9	as	as	ADP
ejpam-1773	481	10	the	the	DET
ejpam-1773	481	11	vector	vector	NOUN
ejpam-1773	481	12	(	(	PUNCT
ejpam-1773	481	13	1,1	1,1	NUM
ejpam-1773	481	14	+	+	SYM
ejpam-1773	481	15	u1u2	u1u2	NOUN
ejpam-1773	481	16	)	)	PUNCT
ejpam-1773	481	17	.	.	PUNCT
ejpam-1773	482	1	7.3	7.3	NUM
ejpam-1773	482	2	.	.	PUNCT
ejpam-1773	483	1	[	[	X
ejpam-1773	483	2	16	16	NUM
ejpam-1773	483	3	,	,	PUNCT
ejpam-1773	483	4	8	8	NUM
ejpam-1773	483	5	,	,	PUNCT
ejpam-1773	483	6	4	4	NUM
ejpam-1773	483	7	]	]	X
ejpam-1773	483	8	binary	binary	ADJ
ejpam-1773	483	9	self	self	NOUN
ejpam-1773	483	10	-	-	PUNCT
ejpam-1773	483	11	dual	dual	ADJ
ejpam-1773	483	12	code	code	NOUN
ejpam-1773	483	13	for	for	ADP
ejpam-1773	483	14	length	length	NOUN
ejpam-1773	483	15	16	16	NUM
ejpam-1773	483	16	,	,	PUNCT
ejpam-1773	483	17	the	the	DET
ejpam-1773	483	18	largest	large	ADJ
ejpam-1773	483	19	k	k	NOUN
ejpam-1773	483	20	for	for	ADP
ejpam-1773	483	21	which	which	PRON
ejpam-1773	483	22	the	the	DET
ejpam-1773	483	23	code	code	NOUN
ejpam-1773	483	24	can	can	AUX
ejpam-1773	483	25	be	be	AUX
ejpam-1773	483	26	the	the	DET
ejpam-1773	483	27	image	image	NOUN
ejpam-1773	483	28	of	of	ADP
ejpam-1773	483	29	a	a	DET
ejpam-1773	483	30	code	code	NOUN
ejpam-1773	483	31	over	over	ADP
ejpam-1773	483	32	rk	rk	NOUN
ejpam-1773	483	33	is	be	AUX
ejpam-1773	483	34	4	4	NUM
ejpam-1773	483	35	.	.	PUNCT
ejpam-1773	484	1	we	we	PRON
ejpam-1773	484	2	take	take	VERB
ejpam-1773	484	3	the	the	DET
ejpam-1773	484	4	code	code	NOUN
ejpam-1773	484	5	c2	c2	PROPN
ejpam-1773	484	6	to	to	PART
ejpam-1773	484	7	be	be	AUX
ejpam-1773	484	8	the	the	DET
ejpam-1773	484	9	length	length	NOUN
ejpam-1773	484	10	1	1	NUM
ejpam-1773	484	11	code	code	NOUN
ejpam-1773	484	12	over	over	ADP
ejpam-1773	484	13	r4	r4	PROPN
ejpam-1773	484	14	generated	generate	VERB
ejpam-1773	484	15	by	by	ADP
ejpam-1773	484	16	u1u2	u1u2	NOUN
ejpam-1773	484	17	,	,	PUNCT
ejpam-1773	484	18	u1u3	u1u3	NOUN
ejpam-1773	484	19	,	,	PUNCT
ejpam-1773	484	20	u1u4	u1u4	ADJ
ejpam-1773	484	21	,	,	PUNCT
ejpam-1773	484	22	u2u3u4	u2u3u4	NOUN
ejpam-1773	484	23	.	.	PUNCT
ejpam-1773	485	1	the	the	DET
ejpam-1773	485	2	code	code	NOUN
ejpam-1773	485	3	c2	c2	PROPN
ejpam-1773	485	4	turns	turn	VERB
ejpam-1773	485	5	out	out	ADP
ejpam-1773	485	6	to	to	PART
ejpam-1773	485	7	be	be	AUX
ejpam-1773	485	8	a	a	DET
ejpam-1773	485	9	self	self	NOUN
ejpam-1773	485	10	-	-	PUNCT
ejpam-1773	485	11	dual	dual	ADJ
ejpam-1773	485	12	code	code	NOUN
ejpam-1773	485	13	with	with	ADP
ejpam-1773	485	14	lee	lee	PROPN
ejpam-1773	485	15	weight	weight	NOUN
ejpam-1773	485	16	enumerator	enumerator	NOUN
ejpam-1773	485	17	lc2	lc2	NOUN
ejpam-1773	485	18	(	(	PUNCT
ejpam-1773	485	19	z	z	NOUN
ejpam-1773	485	20	)	)	PUNCT
ejpam-1773	485	21	=	=	SYM
ejpam-1773	486	1	1	1	NUM
ejpam-1773	486	2	+	+	NUM
ejpam-1773	486	3	28z4	28z4	NUM
ejpam-1773	486	4	+	+	NUM
ejpam-1773	486	5	198z8	198z8	NUM
ejpam-1773	486	6	+	+	NUM
ejpam-1773	486	7	28z12	28z12	NUM
ejpam-1773	486	8	+	+	CCONJ
ejpam-1773	486	9	z16	z16	NOUN
ejpam-1773	486	10	.	.	PUNCT
ejpam-1773	487	1	the	the	DET
ejpam-1773	487	2	code	code	NOUN
ejpam-1773	487	3	ψ4(c2	ψ4(c2	NOUN
ejpam-1773	487	4	)	)	PUNCT
ejpam-1773	487	5	is	be	AUX
ejpam-1773	487	6	a	a	DET
ejpam-1773	487	7	binary	binary	ADJ
ejpam-1773	487	8	type	type	NOUN
ejpam-1773	487	9	ii	ii	PROPN
ejpam-1773	487	10	code	code	NOUN
ejpam-1773	487	11	with	with	ADP
ejpam-1773	487	12	parameters	parameter	NOUN
ejpam-1773	487	13	[	[	X
ejpam-1773	487	14	16,8,4	16,8,4	X
ejpam-1773	487	15	]	]	PUNCT
ejpam-1773	487	16	and	and	CCONJ
ejpam-1773	487	17	is	be	AUX
ejpam-1773	487	18	extremal	extremal	ADJ
ejpam-1773	487	19	.	.	PUNCT
ejpam-1773	488	1	we	we	PRON
ejpam-1773	488	2	know	know	VERB
ejpam-1773	488	3	that	that	SCONJ
ejpam-1773	488	4	we	we	PRON
ejpam-1773	488	5	can	can	AUX
ejpam-1773	488	6	get	get	VERB
ejpam-1773	488	7	the	the	DET
ejpam-1773	488	8	same	same	ADJ
ejpam-1773	488	9	code	code	NOUN
ejpam-1773	488	10	from	from	ADP
ejpam-1773	488	11	r2	r2	PROPN
ejpam-1773	488	12	and	and	CCONJ
ejpam-1773	488	13	r3	r3	PROPN
ejpam-1773	488	14	as	as	ADV
ejpam-1773	488	15	well	well	ADV
ejpam-1773	488	16	.	.	PUNCT
ejpam-1773	489	1	in	in	ADP
ejpam-1773	489	2	particular	particular	ADJ
ejpam-1773	489	3	,	,	PUNCT
ejpam-1773	489	4	ψ3	ψ3	NOUN
ejpam-1773	489	5	(	(	PUNCT
ejpam-1773	489	6	<	<	X
ejpam-1773	489	7	(	(	PUNCT
ejpam-1773	489	8	1,1	1,1	NUM
ejpam-1773	489	9	+	+	CCONJ
ejpam-1773	489	10	u1u2u3	u1u2u3	ADJ
ejpam-1773	489	11	)	)	PUNCT
ejpam-1773	489	12	>	>	PUNCT
ejpam-1773	489	13	)	)	PUNCT
ejpam-1773	489	14	and	and	CCONJ
ejpam-1773	489	15	ψ2	ψ2	NOUN
ejpam-1773	489	16	(	(	PUNCT
ejpam-1773	489	17	<	<	X
ejpam-1773	489	18	(	(	PUNCT
ejpam-1773	489	19	1,1	1,1	NUM
ejpam-1773	489	20	+	+	SYM
ejpam-1773	489	21	u1u2	u1u2	NOUN
ejpam-1773	489	22	,	,	PUNCT
ejpam-1773	489	23	1	1	NUM
ejpam-1773	489	24	+	+	NUM
ejpam-1773	489	25	u1	u1	NOUN
ejpam-1773	489	26	,	,	PUNCT
ejpam-1773	489	27	1	1	NUM
ejpam-1773	489	28	+	+	NUM
ejpam-1773	489	29	u1	u1	NOUN
ejpam-1773	489	30	+	+	CCONJ
ejpam-1773	489	31	u1u2	u1u2	NOUN
ejpam-1773	489	32	)	)	PUNCT
ejpam-1773	489	33	,	,	PUNCT
ejpam-1773	489	34	(	(	PUNCT
ejpam-1773	489	35	0,0,1	0,0,1	NOUN
ejpam-1773	489	36	+	+	NUM
ejpam-1773	489	37	u1	u1	NOUN
ejpam-1773	489	38	,	,	PUNCT
ejpam-1773	489	39	1	1	NUM
ejpam-1773	489	40	+	+	NUM
ejpam-1773	489	41	u1	u1	NOUN
ejpam-1773	489	42	+	+	CCONJ
ejpam-1773	489	43	u1u2	u1u2	NOUN
ejpam-1773	489	44	)	)	PUNCT
ejpam-1773	489	45	>	>	PUNCT
ejpam-1773	489	46	)	)	PUNCT
ejpam-1773	489	47	have	have	VERB
ejpam-1773	489	48	the	the	DET
ejpam-1773	489	49	same	same	ADJ
ejpam-1773	489	50	parameters	parameter	NOUN
ejpam-1773	489	51	and	and	CCONJ
ejpam-1773	489	52	weight	weight	NOUN
ejpam-1773	489	53	enumerators	enumerator	NOUN
ejpam-1773	489	54	.	.	PUNCT
ejpam-1773	490	1	7.4	7.4	NUM
ejpam-1773	490	2	.	.	PUNCT
ejpam-1773	491	1	the	the	DET
ejpam-1773	491	2	extended	extend	VERB
ejpam-1773	491	3	golay	golay	NOUN
ejpam-1773	491	4	code	code	NOUN
ejpam-1773	491	5	let	let	VERB
ejpam-1773	491	6	c3	c3	PROPN
ejpam-1773	491	7	be	be	AUX
ejpam-1773	491	8	the	the	DET
ejpam-1773	491	9	linear	linear	PROPN
ejpam-1773	491	10	code	code	NOUN
ejpam-1773	491	11	over	over	ADP
ejpam-1773	491	12	r3	r3	PROPN
ejpam-1773	491	13	of	of	ADP
ejpam-1773	491	14	length	length	NOUN
ejpam-1773	491	15	3	3	NUM
ejpam-1773	491	16	generated	generate	VERB
ejpam-1773	491	17	by	by	ADP
ejpam-1773	491	18	the	the	DET
ejpam-1773	491	19	following	follow	VERB
ejpam-1773	491	20	vectors	vector	NOUN
ejpam-1773	491	21	(	(	PUNCT
ejpam-1773	491	22	u2,u1	u2,u1	PROPN
ejpam-1773	491	23	+	+	NUM
ejpam-1773	491	24	u3	u3	NOUN
ejpam-1773	491	25	+	+	CCONJ
ejpam-1773	491	26	u1u2,u1	u1u2,u1	ADJ
ejpam-1773	491	27	+	+	X
ejpam-1773	491	28	u1u2	u1u2	NOUN
ejpam-1773	491	29	)	)	PUNCT
ejpam-1773	491	30	,	,	PUNCT
ejpam-1773	491	31	(	(	PUNCT
ejpam-1773	491	32	u1	u1	NOUN
ejpam-1773	491	33	+	+	CCONJ
ejpam-1773	491	34	u2,u1	u2,u1	PROPN
ejpam-1773	491	35	+	+	CCONJ
ejpam-1773	491	36	u1u2,u2	u1u2,u2	NOUN
ejpam-1773	491	37	+	+	CCONJ
ejpam-1773	491	38	u3	u3	NOUN
ejpam-1773	491	39	+	+	CCONJ
ejpam-1773	491	40	u1u2	u1u2	NOUN
ejpam-1773	491	41	)	)	PUNCT
ejpam-1773	491	42	,	,	PUNCT
ejpam-1773	491	43	(	(	PUNCT
ejpam-1773	491	44	u3,u2	u3,u2	PROPN
ejpam-1773	491	45	+	+	CCONJ
ejpam-1773	491	46	u1u3,u1	u1u3,u1	ADJ
ejpam-1773	491	47	+	+	CCONJ
ejpam-1773	491	48	u2	u2	NOUN
ejpam-1773	491	49	)	)	PUNCT
ejpam-1773	491	50	.	.	PUNCT
ejpam-1773	492	1	then	then	ADV
ejpam-1773	492	2	ψ3(c3	ψ3(c3	NOUN
ejpam-1773	492	3	)	)	PUNCT
ejpam-1773	492	4	is	be	AUX
ejpam-1773	492	5	a	a	DET
ejpam-1773	492	6	binary	binary	ADJ
ejpam-1773	492	7	type	type	NOUN
ejpam-1773	492	8	ii	ii	PROPN
ejpam-1773	492	9	self	self	NOUN
ejpam-1773	492	10	-	-	PUNCT
ejpam-1773	492	11	dual	dual	ADJ
ejpam-1773	492	12	code	code	NOUN
ejpam-1773	492	13	with	with	ADP
ejpam-1773	492	14	parameters	parameter	NOUN
ejpam-1773	492	15	[	[	X
ejpam-1773	492	16	24,12,8	24,12,8	X
ejpam-1773	492	17	]	]	PUNCT
ejpam-1773	492	18	and	and	CCONJ
ejpam-1773	492	19	c3	c3	PROPN
ejpam-1773	492	20	has	have	VERB
ejpam-1773	492	21	lee	lee	PROPN
ejpam-1773	492	22	weight	weight	NOUN
ejpam-1773	492	23	enumerator	enumerator	NOUN
ejpam-1773	492	24	lc3	lc3	NOUN
ejpam-1773	492	25	(	(	PUNCT
ejpam-1773	492	26	z	z	NOUN
ejpam-1773	492	27	)	)	PUNCT
ejpam-1773	492	28	=	=	SYM
ejpam-1773	493	1	1	1	NUM
ejpam-1773	493	2	+	+	NUM
ejpam-1773	493	3	759z8	759z8	NUM
ejpam-1773	493	4	+	+	NOUN
ejpam-1773	493	5	2576z12	2576z12	NOUN
ejpam-1773	493	6	+	+	CCONJ
ejpam-1773	493	7	759z16	759z16	NUM
ejpam-1773	493	8	+	+	X
ejpam-1773	493	9	z24	z24	PROPN
ejpam-1773	493	10	,	,	PUNCT
ejpam-1773	493	11	which	which	PRON
ejpam-1773	493	12	is	be	AUX
ejpam-1773	493	13	the	the	DET
ejpam-1773	493	14	weight	weight	NOUN
ejpam-1773	493	15	enumerator	enumerator	NOUN
ejpam-1773	493	16	of	of	ADP
ejpam-1773	493	17	the	the	DET
ejpam-1773	493	18	extended	extended	ADJ
ejpam-1773	493	19	binary	binary	NOUN
ejpam-1773	493	20	golay	golay	NOUN
ejpam-1773	493	21	code	code	NOUN
ejpam-1773	493	22	.	.	PUNCT
ejpam-1773	494	1	we	we	PRON
ejpam-1773	494	2	can	can	AUX
ejpam-1773	494	3	of	of	ADP
ejpam-1773	494	4	course	course	NOUN
ejpam-1773	494	5	get	get	VERB
ejpam-1773	494	6	the	the	DET
ejpam-1773	494	7	same	same	ADJ
ejpam-1773	494	8	code	code	NOUN
ejpam-1773	494	9	from	from	ADP
ejpam-1773	494	10	r2	r2	PROPN
ejpam-1773	494	11	as	as	ADV
ejpam-1773	494	12	well	well	ADV
ejpam-1773	494	13	.	.	PUNCT
ejpam-1773	495	1	in	in	ADP
ejpam-1773	495	2	fact	fact	NOUN
ejpam-1773	495	3	,	,	PUNCT
ejpam-1773	495	4	if	if	SCONJ
ejpam-1773	495	5	d	d	PROPN
ejpam-1773	495	6	is	be	AUX
ejpam-1773	495	7	the	the	DET
ejpam-1773	495	8	linear	linear	PROPN
ejpam-1773	495	9	code	code	NOUN
ejpam-1773	495	10	over	over	ADP
ejpam-1773	495	11	r2	r2	PROPN
ejpam-1773	495	12	of	of	ADP
ejpam-1773	495	13	length	length	NOUN
ejpam-1773	495	14	6	6	NUM
ejpam-1773	495	15	generated	generate	VERB
ejpam-1773	495	16	by	by	ADP
ejpam-1773	495	17	(	(	PUNCT
ejpam-1773	495	18	1,0,0,1	1,0,0,1	NUM
ejpam-1773	495	19	+	+	SYM
ejpam-1773	495	20	u1u2,u2,u1	u1u2,u2,u1	PROPN
ejpam-1773	495	21	+	+	NUM
ejpam-1773	495	22	u2	u2	NOUN
ejpam-1773	495	23	)	)	PUNCT
ejpam-1773	495	24	,	,	PUNCT
ejpam-1773	495	25	(	(	PUNCT
ejpam-1773	495	26	0,1,0,u2	0,1,0,u2	NUM
ejpam-1773	495	27	,	,	PUNCT
ejpam-1773	495	28	1	1	NUM
ejpam-1773	495	29	+	+	NUM
ejpam-1773	495	30	u1	u1	NOUN
ejpam-1773	495	31	+	+	CCONJ
ejpam-1773	495	32	u1u2,u1	u1u2,u1	PROPN
ejpam-1773	495	33	+	+	NUM
ejpam-1773	495	34	u1u2	u1u2	NOUN
ejpam-1773	495	35	)	)	PUNCT
ejpam-1773	495	36	and	and	CCONJ
ejpam-1773	495	37	(	(	PUNCT
ejpam-1773	495	38	0,0,1,u1	0,0,1,u1	NUM
ejpam-1773	496	1	+	+	CCONJ
ejpam-1773	496	2	u2,u1	u2,u1	PROPN
ejpam-1773	496	3	+	+	NUM
ejpam-1773	496	4	u1u2	u1u2	PROPN
ejpam-1773	496	5	,	,	PUNCT
ejpam-1773	496	6	1	1	NUM
ejpam-1773	496	7	+	+	NUM
ejpam-1773	496	8	u2	u2	PROPN
ejpam-1773	496	9	+	+	CCONJ
ejpam-1773	496	10	u1u2	u1u2	NOUN
ejpam-1773	496	11	)	)	PUNCT
ejpam-1773	496	12	,	,	PUNCT
ejpam-1773	496	13	then	then	ADV
ejpam-1773	496	14	ψ2(d	ψ2(d	NUM
ejpam-1773	496	15	)	)	PUNCT
ejpam-1773	496	16	has	have	VERB
ejpam-1773	496	17	the	the	DET
ejpam-1773	496	18	same	same	ADJ
ejpam-1773	496	19	parameters	parameter	NOUN
ejpam-1773	496	20	and	and	CCONJ
ejpam-1773	496	21	the	the	DET
ejpam-1773	496	22	weight	weight	NOUN
ejpam-1773	496	23	enumerator	enumerator	NOUN
ejpam-1773	496	24	.	.	PUNCT
ejpam-1773	497	1	this	this	DET
ejpam-1773	497	2	code	code	NOUN
ejpam-1773	497	3	together	together	ADV
ejpam-1773	497	4	with	with	ADP
ejpam-1773	497	5	the	the	DET
ejpam-1773	497	6	map	map	NOUN
ejpam-1773	498	1	λr	λr	INTJ
ejpam-1773	498	2	produces	produce	VERB
ejpam-1773	498	3	the	the	DET
ejpam-1773	498	4	leech	leech	NOUN
ejpam-1773	498	5	lattice	lattice	NOUN
ejpam-1773	498	6	.	.	PUNCT
ejpam-1773	499	1	s.	s.	PROPN
ejpam-1773	499	2	dougherty	dougherty	PROPN
ejpam-1773	499	3	,	,	PUNCT
ejpam-1773	499	4	b.yıldız	b.yıldız	NOUN
ejpam-1773	499	5	,	,	PUNCT
ejpam-1773	499	6	s.karadeniz	s.karadeniz	NOUN
ejpam-1773	499	7	/	/	SYM
ejpam-1773	499	8	eur	eur	NOUN
ejpam-1773	499	9	.	.	PUNCT
ejpam-1773	500	1	j.	j.	PROPN
ejpam-1773	500	2	pure	pure	PROPN
ejpam-1773	500	3	appl	appl	PROPN
ejpam-1773	500	4	.	.	PROPN
ejpam-1773	500	5	math	math	PROPN
ejpam-1773	500	6	,	,	PUNCT
ejpam-1773	500	7	6	6	NUM
ejpam-1773	500	8	(	(	PUNCT
ejpam-1773	500	9	2013	2013	NUM
ejpam-1773	500	10	)	)	PUNCT
ejpam-1773	500	11	,	,	PUNCT
ejpam-1773	500	12	89	89	NUM
ejpam-1773	500	13	-	-	SYM
ejpam-1773	500	14	106	106	NUM
ejpam-1773	500	15	103	103	NUM
ejpam-1773	500	16	7.5	7.5	NUM
ejpam-1773	500	17	.	.	PUNCT
ejpam-1773	501	1	binary	binary	ADJ
ejpam-1773	501	2	self	self	NOUN
ejpam-1773	501	3	-	-	PUNCT
ejpam-1773	501	4	dual	dual	ADJ
ejpam-1773	501	5	code	code	NOUN
ejpam-1773	501	6	with	with	ADP
ejpam-1773	501	7	parameters	parameter	NOUN
ejpam-1773	501	8	[	[	X
ejpam-1773	501	9	32	32	NUM
ejpam-1773	501	10	,	,	PUNCT
ejpam-1773	501	11	16	16	NUM
ejpam-1773	501	12	,	,	PUNCT
ejpam-1773	501	13	8	8	NUM
ejpam-1773	501	14	]	]	PUNCT
ejpam-1773	501	15	let	let	VERB
ejpam-1773	501	16	c4	c4	NOUN
ejpam-1773	501	17	be	be	AUX
ejpam-1773	501	18	the	the	DET
ejpam-1773	501	19	linear	linear	PROPN
ejpam-1773	501	20	code	code	NOUN
ejpam-1773	501	21	over	over	ADP
ejpam-1773	501	22	r5	r5	PROPN
ejpam-1773	501	23	of	of	ADP
ejpam-1773	501	24	length	length	NOUN
ejpam-1773	501	25	1	1	NUM
ejpam-1773	501	26	generated	generate	VERB
ejpam-1773	501	27	by	by	ADP
ejpam-1773	501	28	{	{	PUNCT
ejpam-1773	501	29	uiu	uiu	PROPN
ejpam-1773	501	30	juk	juk	PROPN
ejpam-1773	502	1	|	|	ADV
ejpam-1773	502	2	1≤	1≤	INTJ
ejpam-1773	503	1	i	i	PRON
ejpam-1773	503	2	<	<	X
ejpam-1773	503	3	j	j	X
ejpam-1773	503	4	<	<	X
ejpam-1773	503	5	k	k	X
ejpam-1773	503	6	≤	≤	NUM
ejpam-1773	503	7	5	5	NUM
ejpam-1773	503	8	}	}	PUNCT
ejpam-1773	503	9	.	.	PUNCT
ejpam-1773	504	1	then	then	ADV
ejpam-1773	504	2	c4	c4	NOUN
ejpam-1773	504	3	is	be	AUX
ejpam-1773	504	4	a	a	DET
ejpam-1773	504	5	self	self	NOUN
ejpam-1773	504	6	-	-	PUNCT
ejpam-1773	504	7	dual	dual	ADJ
ejpam-1773	504	8	type	type	NOUN
ejpam-1773	504	9	ii	ii	PROPN
ejpam-1773	504	10	code	code	NOUN
ejpam-1773	504	11	by	by	ADP
ejpam-1773	504	12	theorem	theorem	NOUN
ejpam-1773	504	13	9	9	NUM
ejpam-1773	504	14	,	,	PUNCT
ejpam-1773	504	15	and	and	CCONJ
ejpam-1773	504	16	has	have	VERB
ejpam-1773	504	17	lee	lee	PROPN
ejpam-1773	504	18	weight	weight	NOUN
ejpam-1773	504	19	enumerator	enumerator	NOUN
ejpam-1773	504	20	lc4	lc4	PROPN
ejpam-1773	504	21	(	(	PUNCT
ejpam-1773	504	22	z	z	NOUN
ejpam-1773	504	23	)	)	PUNCT
ejpam-1773	504	24	=	=	SYM
ejpam-1773	505	1	1	1	NUM
ejpam-1773	505	2	+	+	NUM
ejpam-1773	505	3	620z8	620z8	NUM
ejpam-1773	505	4	+	+	NOUN
ejpam-1773	505	5	13888z12	13888z12	NOUN
ejpam-1773	505	6	+	+	NOUN
ejpam-1773	505	7	36518z16	36518z16	NUM
ejpam-1773	505	8	+	+	NUM
ejpam-1773	505	9	1388z20	1388z20	NUM
ejpam-1773	505	10	+	+	SYM
ejpam-1773	505	11	620z24	620z24	NUM
ejpam-1773	505	12	+	+	NUM
ejpam-1773	505	13	z32	z32	NOUN
ejpam-1773	505	14	.	.	PUNCT
ejpam-1773	506	1	so	so	ADV
ejpam-1773	506	2	we	we	PRON
ejpam-1773	506	3	see	see	VERB
ejpam-1773	506	4	that	that	PRON
ejpam-1773	506	5	ψ5(c4	ψ5(c4	NOUN
ejpam-1773	506	6	)	)	PUNCT
ejpam-1773	506	7	is	be	AUX
ejpam-1773	506	8	an	an	DET
ejpam-1773	506	9	extremal	extremal	ADJ
ejpam-1773	506	10	binary	binary	ADJ
ejpam-1773	506	11	type	type	NOUN
ejpam-1773	506	12	ii	ii	PROPN
ejpam-1773	506	13	code	code	NOUN
ejpam-1773	506	14	of	of	ADP
ejpam-1773	506	15	parameters	parameter	NOUN
ejpam-1773	507	1	[	[	X
ejpam-1773	507	2	32,16,8	32,16,8	NOUN
ejpam-1773	507	3	]	]	X
ejpam-1773	507	4	.	.	PUNCT
ejpam-1773	508	1	of	of	ADV
ejpam-1773	508	2	course	course	NOUN
ejpam-1773	508	3	by	by	ADP
ejpam-1773	508	4	the	the	DET
ejpam-1773	508	5	argument	argument	NOUN
ejpam-1773	508	6	given	give	VERB
ejpam-1773	508	7	at	at	ADP
ejpam-1773	508	8	the	the	DET
ejpam-1773	508	9	beginning	beginning	NOUN
ejpam-1773	508	10	of	of	ADP
ejpam-1773	508	11	the	the	DET
ejpam-1773	508	12	section	section	NOUN
ejpam-1773	508	13	we	we	PRON
ejpam-1773	508	14	know	know	VERB
ejpam-1773	508	15	that	that	SCONJ
ejpam-1773	508	16	we	we	PRON
ejpam-1773	508	17	can	can	AUX
ejpam-1773	508	18	get	get	VERB
ejpam-1773	508	19	the	the	DET
ejpam-1773	508	20	same	same	ADJ
ejpam-1773	508	21	code	code	NOUN
ejpam-1773	508	22	from	from	ADP
ejpam-1773	508	23	r2,r3	r2,r3	PROPN
ejpam-1773	508	24	and	and	CCONJ
ejpam-1773	508	25	r4	r4	VERB
ejpam-1773	508	26	as	as	ADV
ejpam-1773	508	27	well	well	ADV
ejpam-1773	508	28	.	.	PUNCT
ejpam-1773	509	1	for	for	ADP
ejpam-1773	509	2	example	example	NOUN
ejpam-1773	509	3	,	,	PUNCT
ejpam-1773	509	4	if	if	SCONJ
ejpam-1773	509	5	e	e	NOUN
ejpam-1773	509	6	is	be	AUX
ejpam-1773	509	7	the	the	DET
ejpam-1773	509	8	linear	linear	PROPN
ejpam-1773	509	9	code	code	NOUN
ejpam-1773	509	10	over	over	ADP
ejpam-1773	509	11	r4	r4	NOUN
ejpam-1773	509	12	of	of	ADP
ejpam-1773	509	13	length	length	NOUN
ejpam-1773	509	14	2	2	NUM
ejpam-1773	509	15	generated	generate	VERB
ejpam-1773	509	16	by	by	ADP
ejpam-1773	509	17	the	the	DET
ejpam-1773	509	18	vector	vector	NOUN
ejpam-1773	509	19	(	(	PUNCT
ejpam-1773	509	20	1,1	1,1	NUM
ejpam-1773	509	21	+	+	SYM
ejpam-1773	509	22	u1u2	u1u2	X
ejpam-1773	509	23	+	+	NUM
ejpam-1773	509	24	u3u4	u3u4	NOUN
ejpam-1773	509	25	)	)	PUNCT
ejpam-1773	509	26	,	,	PUNCT
ejpam-1773	509	27	then	then	ADV
ejpam-1773	509	28	ψ4(e	ψ4(e	PROPN
ejpam-1773	509	29	)	)	PUNCT
ejpam-1773	509	30	has	have	VERB
ejpam-1773	509	31	the	the	DET
ejpam-1773	509	32	same	same	ADJ
ejpam-1773	509	33	parameters	parameter	NOUN
ejpam-1773	509	34	and	and	CCONJ
ejpam-1773	509	35	the	the	DET
ejpam-1773	509	36	weight	weight	NOUN
ejpam-1773	509	37	enumerator	enumerator	NOUN
ejpam-1773	509	38	as	as	ADP
ejpam-1773	509	39	the	the	DET
ejpam-1773	509	40	above	above	ADP
ejpam-1773	509	41	one	one	NUM
ejpam-1773	509	42	.	.	PUNCT
ejpam-1773	510	1	this	this	DET
ejpam-1773	510	2	code	code	NOUN
ejpam-1773	510	3	is	be	AUX
ejpam-1773	510	4	an	an	DET
ejpam-1773	510	5	example	example	NOUN
ejpam-1773	510	6	of	of	ADP
ejpam-1773	510	7	a	a	DET
ejpam-1773	510	8	code	code	NOUN
ejpam-1773	510	9	constructed	construct	VERB
ejpam-1773	510	10	using	use	VERB
ejpam-1773	510	11	theorem	theorem	NOUN
ejpam-1773	510	12	9	9	NUM
ejpam-1773	510	13	.	.	PUNCT
ejpam-1773	511	1	it	it	PRON
ejpam-1773	511	2	is	be	AUX
ejpam-1773	511	3	easy	easy	ADJ
ejpam-1773	511	4	to	to	PART
ejpam-1773	511	5	see	see	VERB
ejpam-1773	511	6	that	that	SCONJ
ejpam-1773	511	7	any	any	DET
ejpam-1773	511	8	code	code	NOUN
ejpam-1773	511	9	constructed	construct	VERB
ejpam-1773	511	10	with	with	ADP
ejpam-1773	511	11	this	this	DET
ejpam-1773	511	12	theorem	theorem	NOUN
ejpam-1773	511	13	over	over	ADP
ejpam-1773	511	14	rk	rk	NOUN
ejpam-1773	511	15	will	will	AUX
ejpam-1773	511	16	be	be	AUX
ejpam-1773	511	17	a	a	DET
ejpam-1773	511	18	[	[	X
ejpam-1773	511	19	2k	2k	NUM
ejpam-1773	511	20	,	,	PUNCT
ejpam-1773	511	21	2k−1	2k−1	NUM
ejpam-1773	511	22	,	,	PUNCT
ejpam-1773	511	23	2⌈	2⌈	NUM
ejpam-1773	511	24	k	k	NOUN
ejpam-1773	511	25	2	2	NUM
ejpam-1773	511	26	⌉	⌉	NOUN
ejpam-1773	511	27	]	]	X
ejpam-1773	511	28	binary	binary	ADJ
ejpam-1773	511	29	self	self	NOUN
ejpam-1773	511	30	-	-	PUNCT
ejpam-1773	511	31	dual	dual	ADJ
ejpam-1773	511	32	code	code	NOUN
ejpam-1773	511	33	.	.	PUNCT
ejpam-1773	512	1	the	the	DET
ejpam-1773	512	2	next	next	ADJ
ejpam-1773	512	3	in	in	ADP
ejpam-1773	512	4	the	the	DET
ejpam-1773	512	5	family	family	NOUN
ejpam-1773	512	6	would	would	AUX
ejpam-1773	512	7	be	be	AUX
ejpam-1773	512	8	a	a	DET
ejpam-1773	512	9	[	[	X
ejpam-1773	512	10	128,64,16	128,64,16	NUM
ejpam-1773	512	11	]	]	X
ejpam-1773	512	12	code	code	NOUN
ejpam-1773	512	13	.	.	PUNCT
ejpam-1773	513	1	7.6	7.6	NUM
ejpam-1773	513	2	.	.	PUNCT
ejpam-1773	513	3	binary	binary	ADJ
ejpam-1773	513	4	self	self	NOUN
ejpam-1773	513	5	-	-	PUNCT
ejpam-1773	513	6	dual	dual	ADJ
ejpam-1773	513	7	code	code	NOUN
ejpam-1773	513	8	with	with	ADP
ejpam-1773	513	9	parameters	parameter	NOUN
ejpam-1773	513	10	[	[	X
ejpam-1773	513	11	40	40	NUM
ejpam-1773	513	12	,	,	PUNCT
ejpam-1773	513	13	20	20	NUM
ejpam-1773	513	14	,	,	PUNCT
ejpam-1773	513	15	8	8	NUM
ejpam-1773	513	16	]	]	PUNCT
ejpam-1773	513	17	let	let	VERB
ejpam-1773	513	18	c5	c5	PROPN
ejpam-1773	513	19	be	be	AUX
ejpam-1773	513	20	the	the	DET
ejpam-1773	513	21	linear	linear	PROPN
ejpam-1773	513	22	code	code	NOUN
ejpam-1773	513	23	over	over	ADP
ejpam-1773	513	24	r2	r2	PROPN
ejpam-1773	513	25	generated	generate	VERB
ejpam-1773	513	26	by	by	ADP
ejpam-1773	513	27	the	the	DET
ejpam-1773	513	28	matrix	matrix	NOUN
ejpam-1773	513	29	[	[	X
ejpam-1773	513	30	i5|a	i5|a	NOUN
ejpam-1773	513	31	]	]	PUNCT
ejpam-1773	513	32	where	where	SCONJ
ejpam-1773	513	33	a=	a=	PROPN
ejpam-1773	513	34			NOUN
ejpam-1773	513	35			ADJ
ejpam-1773	513	36			ADJ
ejpam-1773	513	37			ADJ
ejpam-1773	513	38			ADJ
ejpam-1773	513	39			ADJ
ejpam-1773	513	40			NUM
ejpam-1773	514	1	1	1	NUM
ejpam-1773	514	2	+	+	NUM
ejpam-1773	514	3	u1u2	u1u2	NOUN
ejpam-1773	514	4	u1	u1	NOUN
ejpam-1773	514	5	u1	u1	NOUN
ejpam-1773	514	6	u1	u1	NOUN
ejpam-1773	514	7	+	+	CCONJ
ejpam-1773	514	8	u2	u2	PROPN
ejpam-1773	514	9	u2	u2	PROPN
ejpam-1773	514	10	u1	u1	NOUN
ejpam-1773	514	11	1	1	NUM
ejpam-1773	514	12	+	+	NUM
ejpam-1773	514	13	u1u2	u1u2	NOUN
ejpam-1773	514	14	u1	u1	NOUN
ejpam-1773	514	15	+	+	CCONJ
ejpam-1773	514	16	u2	u2	PROPN
ejpam-1773	514	17	u1	u1	NOUN
ejpam-1773	514	18	u2	u2	PROPN
ejpam-1773	514	19	u1	u1	NOUN
ejpam-1773	514	20	u1	u1	NOUN
ejpam-1773	514	21	+	+	CCONJ
ejpam-1773	514	22	u2	u2	PROPN
ejpam-1773	514	23	+	+	CCONJ
ejpam-1773	514	24	u1u2	u1u2	PROPN
ejpam-1773	514	25	1	1	NUM
ejpam-1773	514	26	+	+	NUM
ejpam-1773	514	27	u1u2	u1u2	NOUN
ejpam-1773	514	28	u1u2	u1u2	NOUN
ejpam-1773	514	29	u1	u1	NOUN
ejpam-1773	514	30	+	+	CCONJ
ejpam-1773	514	31	u2	u2	PROPN
ejpam-1773	514	32	+	+	CCONJ
ejpam-1773	514	33	u1u2	u1u2	NOUN
ejpam-1773	514	34	u1	u1	NOUN
ejpam-1773	514	35	+	+	CCONJ
ejpam-1773	514	36	u2	u2	NOUN
ejpam-1773	514	37	u1	u1	NOUN
ejpam-1773	514	38	+	+	CCONJ
ejpam-1773	514	39	u1u2	u1u2	X
ejpam-1773	514	40	0	0	NUM
ejpam-1773	514	41	1	1	NUM
ejpam-1773	514	42	+	+	NUM
ejpam-1773	514	43	u1u2	u1u2	PROPN
ejpam-1773	514	44	u2	u2	PROPN
ejpam-1773	514	45	u2	u2	PROPN
ejpam-1773	514	46	+	+	CCONJ
ejpam-1773	514	47	u1u2	u1u2	PROPN
ejpam-1773	514	48	u2	u2	NOUN
ejpam-1773	514	49	u1	u1	NOUN
ejpam-1773	514	50	+	+	CCONJ
ejpam-1773	514	51	u2	u2	PROPN
ejpam-1773	514	52	+	+	CCONJ
ejpam-1773	514	53	u1u2	u1u2	NOUN
ejpam-1773	514	54	u2	u2	PROPN
ejpam-1773	514	55	1	1	NUM
ejpam-1773	514	56	+	+	NUM
ejpam-1773	514	57	u1u2	u1u2	NOUN
ejpam-1773	514	58			PROPN
ejpam-1773	514	59			PROPN
ejpam-1773	514	60			PROPN
ejpam-1773	514	61			PROPN
ejpam-1773	514	62			PROPN
ejpam-1773	514	63			PROPN
ejpam-1773	514	64			PROPN
ejpam-1773	514	65	.	.	PUNCT
ejpam-1773	515	1	then	then	ADV
ejpam-1773	515	2	c5	c5	PROPN
ejpam-1773	515	3	is	be	AUX
ejpam-1773	515	4	a	a	DET
ejpam-1773	515	5	self	self	NOUN
ejpam-1773	515	6	-	-	PUNCT
ejpam-1773	515	7	dual	dual	ADJ
ejpam-1773	515	8	code	code	NOUN
ejpam-1773	515	9	over	over	ADP
ejpam-1773	515	10	r2	r2	PROPN
ejpam-1773	515	11	of	of	ADP
ejpam-1773	515	12	length	length	NOUN
ejpam-1773	515	13	10	10	NUM
ejpam-1773	515	14	with	with	ADP
ejpam-1773	515	15	weight	weight	NOUN
ejpam-1773	515	16	enumerator	enumerator	NOUN
ejpam-1773	515	17	1	1	NUM
ejpam-1773	515	18	+	+	SYM
ejpam-1773	515	19	125z8	125z8	NUM
ejpam-1773	515	20	+	+	NUM
ejpam-1773	515	21	1664z10	1664z10	NUM
ejpam-1773	515	22	+	+	X
ejpam-1773	515	23	10720z12	10720z12	NUM
ejpam-1773	515	24	+	+	NUM
ejpam-1773	515	25	.	.	PUNCT
ejpam-1773	515	26	.	.	PUNCT
ejpam-1773	516	1	..	..	PUNCT
ejpam-1773	517	1	the	the	DET
ejpam-1773	517	2	binary	binary	PROPN
ejpam-1773	517	3	image	image	NOUN
ejpam-1773	517	4	ψ2(c5	ψ2(c5	PROPN
ejpam-1773	517	5	)	)	PUNCT
ejpam-1773	517	6	is	be	AUX
ejpam-1773	517	7	a	a	DET
ejpam-1773	517	8	en	en	X
ejpam-1773	517	9	extremal	extremal	ADJ
ejpam-1773	517	10	singly	singly	ADV
ejpam-1773	517	11	-	-	PUNCT
ejpam-1773	517	12	even	even	ADV
ejpam-1773	517	13	self	self	NOUN
ejpam-1773	517	14	-	-	PUNCT
ejpam-1773	517	15	dual	dual	ADJ
ejpam-1773	517	16	code	code	NOUN
ejpam-1773	517	17	with	with	ADP
ejpam-1773	517	18	parameters	parameter	NOUN
ejpam-1773	517	19	[	[	X
ejpam-1773	517	20	40,20,8	40,20,8	NUM
ejpam-1773	517	21	]	]	PUNCT
ejpam-1773	517	22	and	and	CCONJ
ejpam-1773	517	23	has	have	VERB
ejpam-1773	517	24	an	an	DET
ejpam-1773	517	25	automorphism	automorphism	NOUN
ejpam-1773	517	26	group	group	NOUN
ejpam-1773	517	27	of	of	ADP
ejpam-1773	517	28	order	order	NOUN
ejpam-1773	517	29	27	27	NUM
ejpam-1773	517	30	.	.	PUNCT
ejpam-1773	518	1	7.7	7.7	NUM
ejpam-1773	518	2	.	.	PUNCT
ejpam-1773	518	3	binary	binary	ADJ
ejpam-1773	518	4	self	self	NOUN
ejpam-1773	518	5	-	-	PUNCT
ejpam-1773	518	6	dual	dual	ADJ
ejpam-1773	518	7	code	code	NOUN
ejpam-1773	518	8	with	with	ADP
ejpam-1773	518	9	parameters	parameter	NOUN
ejpam-1773	518	10	[	[	X
ejpam-1773	518	11	44	44	NUM
ejpam-1773	518	12	,	,	PUNCT
ejpam-1773	518	13	22	22	NUM
ejpam-1773	518	14	,	,	PUNCT
ejpam-1773	518	15	8	8	NUM
ejpam-1773	518	16	]	]	PUNCT
ejpam-1773	518	17	let	let	VERB
ejpam-1773	518	18	c6	c6	PROPN
ejpam-1773	518	19	be	be	AUX
ejpam-1773	518	20	the	the	DET
ejpam-1773	518	21	linear	linear	PROPN
ejpam-1773	518	22	code	code	NOUN
ejpam-1773	518	23	over	over	ADP
ejpam-1773	518	24	r2	r2	PROPN
ejpam-1773	518	25	of	of	ADP
ejpam-1773	518	26	length	length	NOUN
ejpam-1773	518	27	11	11	NUM
ejpam-1773	518	28	generated	generate	VERB
ejpam-1773	518	29	by	by	ADP
ejpam-1773	518	30	the	the	DET
ejpam-1773	518	31	matrix	matrix	NOUN
ejpam-1773	518	32	�	�	PROPN
ejpam-1773	518	33	i5	i5	PROPN
ejpam-1773	518	34	|	|	NOUN
ejpam-1773	518	35	0	0	NUM
ejpam-1773	519	1	|	|	ADV
ejpam-1773	519	2	a	a	DET
ejpam-1773	519	3	�	�	NOUN
ejpam-1773	519	4	where	where	SCONJ
ejpam-1773	519	5	a	a	PRON
ejpam-1773	519	6	is	be	AUX
ejpam-1773	519	7	the	the	DET
ejpam-1773	519	8	6×	6×	NUM
ejpam-1773	519	9	6	6	NUM
ejpam-1773	519	10	matrix	matrix	NOUN
ejpam-1773	519	11	given	give	VERB
ejpam-1773	519	12	by	by	ADP
ejpam-1773	519	13	a=	a=	ADJ
ejpam-1773	519	14			NOUN
ejpam-1773	519	15			NOUN
ejpam-1773	519	16			NOUN
ejpam-1773	519	17	1	1	NUM
ejpam-1773	519	18	+	+	NUM
ejpam-1773	519	19	u2	u2	NOUN
ejpam-1773	519	20	+	+	CCONJ
ejpam-1773	519	21	u1u2	u1u2	NOUN
ejpam-1773	519	22	u1u2	u1u2	NOUN
ejpam-1773	519	23	1	1	NUM
ejpam-1773	519	24	+	+	NUM
ejpam-1773	519	25	u2	u2	NOUN
ejpam-1773	519	26	+	+	CCONJ
ejpam-1773	519	27	u1u2	u1u2	PROPN
ejpam-1773	519	28	1	1	NUM
ejpam-1773	519	29	+	+	NUM
ejpam-1773	519	30	u2	u2	NOUN
ejpam-1773	519	31	+	+	CCONJ
ejpam-1773	519	32	u1u2	u1u2	PROPN
ejpam-1773	519	33	1	1	NUM
ejpam-1773	519	34	+	+	NUM
ejpam-1773	519	35	u2	u2	PROPN
ejpam-1773	519	36	1	1	NUM
ejpam-1773	519	37	+	+	NUM
ejpam-1773	519	38	u1	u1	NOUN
ejpam-1773	519	39	1	1	NUM
ejpam-1773	519	40	+	+	NOUN
ejpam-1773	519	41	u2	u2	PROPN
ejpam-1773	519	42	1	1	NUM
ejpam-1773	519	43	+	+	NUM
ejpam-1773	519	44	u1	u1	NOUN
ejpam-1773	519	45	+	+	CCONJ
ejpam-1773	519	46	u2	u2	PROPN
ejpam-1773	519	47	+	+	CCONJ
ejpam-1773	519	48	u1u2	u1u2	PROPN
ejpam-1773	519	49	1	1	NUM
ejpam-1773	519	50	+	+	NUM
ejpam-1773	519	51	u2	u2	PROPN
ejpam-1773	519	52	1	1	NUM
ejpam-1773	519	53	+	+	NUM
ejpam-1773	519	54	u1	u1	NOUN
ejpam-1773	519	55	+	+	CCONJ
ejpam-1773	519	56	u2	u2	PROPN
ejpam-1773	519	57	+	+	CCONJ
ejpam-1773	519	58	u1u2	u1u2	PROPN
ejpam-1773	519	59	1	1	NUM
ejpam-1773	519	60	+	+	NOUN
ejpam-1773	519	61	u2	u2	NOUN
ejpam-1773	519	62	u1	u1	NOUN
ejpam-1773	519	63	1	1	NUM
ejpam-1773	519	64	+	+	NUM
ejpam-1773	519	65	u1	u1	NOUN
ejpam-1773	519	66	+	+	CCONJ
ejpam-1773	519	67	u2	u2	PROPN
ejpam-1773	519	68	+	+	CCONJ
ejpam-1773	519	69	u1u2	u1u2	PROPN
ejpam-1773	519	70	1	1	NUM
ejpam-1773	519	71	+	+	NUM
ejpam-1773	519	72	u1	u1	NOUN
ejpam-1773	519	73	u1	u1	NOUN
ejpam-1773	519	74	u1	u1	NOUN
ejpam-1773	519	75	+	+	CCONJ
ejpam-1773	519	76	u1u2	u1u2	PROPN
ejpam-1773	519	77	u2	u2	NOUN
ejpam-1773	519	78	1	1	NUM
ejpam-1773	519	79	+	+	NOUN
ejpam-1773	519	80	u2	u2	NOUN
ejpam-1773	519	81	u1	u1	NOUN
ejpam-1773	519	82	1	1	NUM
ejpam-1773	519	83	+	+	NOUN
ejpam-1773	519	84	u2	u2	NOUN
ejpam-1773	519	85	+	+	CCONJ
ejpam-1773	519	86	u1u2	u1u2	PROPN
ejpam-1773	519	87	1	1	NUM
ejpam-1773	519	88	+	+	NUM
ejpam-1773	519	89	u1u2	u1u2	SYM
ejpam-1773	519	90	0	0	NUM
ejpam-1773	519	91	u1	u1	NOUN
ejpam-1773	519	92	+	+	CCONJ
ejpam-1773	519	93	u2	u2	PROPN
ejpam-1773	519	94	1	1	NUM
ejpam-1773	519	95	+	+	NUM
ejpam-1773	519	96	u1	u1	NOUN
ejpam-1773	519	97	+	+	CCONJ
ejpam-1773	519	98	u1u2	u1u2	NOUN
ejpam-1773	519	99	0	0	NUM
ejpam-1773	519	100	1	1	NUM
ejpam-1773	519	101	u1	u1	NOUN
ejpam-1773	519	102	1	1	NUM
ejpam-1773	519	103	+	+	NUM
ejpam-1773	519	104	u1	u1	NOUN
ejpam-1773	519	105	u1	u1	NOUN
ejpam-1773	519	106	1	1	NUM
ejpam-1773	519	107	+	+	NOUN
ejpam-1773	519	108	u2	u2	PROPN
ejpam-1773	519	109	0	0	NUM
ejpam-1773	519	110	u2	u2	PROPN
ejpam-1773	519	111	+	+	CCONJ
ejpam-1773	519	112	u1u2	u1u2	X
ejpam-1773	519	113	u1u2	u1u2	X
ejpam-1773	519	114	0	0	NUM
ejpam-1773	519	115	u2	u2	PROPN
ejpam-1773	519	116	u2	u2	PROPN
ejpam-1773	519	117			PROPN
ejpam-1773	519	118			PROPN
ejpam-1773	519	119			PROPN
ejpam-1773	519	120	.	.	PUNCT
ejpam-1773	520	1	then	then	ADV
ejpam-1773	520	2	c6	c6	PROPN
ejpam-1773	520	3	is	be	AUX
ejpam-1773	520	4	a	a	DET
ejpam-1773	520	5	self	self	NOUN
ejpam-1773	520	6	-	-	PUNCT
ejpam-1773	520	7	dual	dual	ADJ
ejpam-1773	520	8	code	code	NOUN
ejpam-1773	520	9	over	over	ADP
ejpam-1773	520	10	r2	r2	PROPN
ejpam-1773	520	11	of	of	ADP
ejpam-1773	520	12	length	length	NOUN
ejpam-1773	520	13	11	11	NUM
ejpam-1773	520	14	with	with	ADP
ejpam-1773	520	15	weight	weight	NOUN
ejpam-1773	520	16	enumerator	enumerator	NOUN
ejpam-1773	520	17	1	1	NUM
ejpam-1773	520	18	+	+	SYM
ejpam-1773	520	19	104z8	104z8	NUM
ejpam-1773	520	20	+	+	NUM
ejpam-1773	520	21	512z10	512z10	NUM
ejpam-1773	520	22	+	+	NOUN
ejpam-1773	520	23	.	.	PUNCT
ejpam-1773	520	24	.	.	PUNCT
ejpam-1773	521	1	..	..	PUNCT
ejpam-1773	522	1	the	the	DET
ejpam-1773	522	2	binary	binary	PROPN
ejpam-1773	522	3	image	image	NOUN
ejpam-1773	522	4	ψ2(c6	ψ2(c6	PROPN
ejpam-1773	522	5	)	)	PUNCT
ejpam-1773	522	6	is	be	AUX
ejpam-1773	522	7	an	an	DET
ejpam-1773	522	8	extremal	extremal	ADJ
ejpam-1773	522	9	singly	singly	ADV
ejpam-1773	522	10	-	-	PUNCT
ejpam-1773	522	11	even	even	ADV
ejpam-1773	522	12	self	self	NOUN
ejpam-1773	522	13	-	-	PUNCT
ejpam-1773	522	14	dual	dual	ADJ
ejpam-1773	522	15	code	code	NOUN
ejpam-1773	522	16	with	with	ADP
ejpam-1773	522	17	parameters	parameter	NOUN
ejpam-1773	522	18	[	[	X
ejpam-1773	522	19	44,22,8	44,22,8	X
ejpam-1773	522	20	]	]	PUNCT
ejpam-1773	522	21	with	with	ADP
ejpam-1773	522	22	|aut(c)|	|aut(c)|	PROPN
ejpam-1773	522	23	=	=	SYM
ejpam-1773	522	24	216	216	NUM
ejpam-1773	522	25	·	·	SYM
ejpam-1773	522	26	32	32	NUM
ejpam-1773	522	27	·	·	SYM
ejpam-1773	522	28	52	52	NUM
ejpam-1773	522	29	.	.	PUNCT
ejpam-1773	523	1	s.	s.	PROPN
ejpam-1773	523	2	dougherty	dougherty	PROPN
ejpam-1773	523	3	,	,	PUNCT
ejpam-1773	523	4	b.yıldız	b.yıldız	NOUN
ejpam-1773	523	5	,	,	PUNCT
ejpam-1773	523	6	s.karadeniz	s.karadeniz	NOUN
ejpam-1773	523	7	/	/	SYM
ejpam-1773	523	8	eur	eur	NOUN
ejpam-1773	523	9	.	.	PUNCT
ejpam-1773	524	1	j.	j.	PROPN
ejpam-1773	524	2	pure	pure	PROPN
ejpam-1773	524	3	appl	appl	PROPN
ejpam-1773	524	4	.	.	PROPN
ejpam-1773	524	5	math	math	PROPN
ejpam-1773	524	6	,	,	PUNCT
ejpam-1773	524	7	6	6	NUM
ejpam-1773	524	8	(	(	PUNCT
ejpam-1773	524	9	2013	2013	NUM
ejpam-1773	524	10	)	)	PUNCT
ejpam-1773	524	11	,	,	PUNCT
ejpam-1773	524	12	89	89	NUM
ejpam-1773	524	13	-	-	SYM
ejpam-1773	524	14	106	106	NUM
ejpam-1773	524	15	104	104	NUM
ejpam-1773	524	16	7.8	7.8	NUM
ejpam-1773	524	17	.	.	PUNCT
ejpam-1773	525	1	binary	binary	ADJ
ejpam-1773	525	2	self	self	NOUN
ejpam-1773	525	3	-	-	PUNCT
ejpam-1773	525	4	dual	dual	ADJ
ejpam-1773	525	5	code	code	NOUN
ejpam-1773	525	6	with	with	ADP
ejpam-1773	525	7	parameters	parameter	NOUN
ejpam-1773	525	8	[	[	X
ejpam-1773	525	9	56	56	NUM
ejpam-1773	525	10	,	,	PUNCT
ejpam-1773	525	11	28	28	NUM
ejpam-1773	525	12	,	,	PUNCT
ejpam-1773	525	13	12	12	NUM
ejpam-1773	525	14	]	]	PUNCT
ejpam-1773	525	15	the	the	DET
ejpam-1773	525	16	existence	existence	NOUN
ejpam-1773	525	17	of	of	ADP
ejpam-1773	525	18	type	type	NOUN
ejpam-1773	525	19	i	i	PRON
ejpam-1773	525	20	extremal	extremal	VERB
ejpam-1773	525	21	self	self	NOUN
ejpam-1773	525	22	-	-	PUNCT
ejpam-1773	525	23	dual	dual	ADJ
ejpam-1773	525	24	code	code	NOUN
ejpam-1773	525	25	of	of	ADP
ejpam-1773	525	26	length	length	NOUN
ejpam-1773	525	27	56	56	NUM
ejpam-1773	525	28	is	be	AUX
ejpam-1773	525	29	not	not	PART
ejpam-1773	525	30	known	know	VERB
ejpam-1773	525	31	in	in	ADP
ejpam-1773	525	32	the	the	DET
ejpam-1773	525	33	literature	literature	NOUN
ejpam-1773	525	34	,	,	PUNCT
ejpam-1773	525	35	however	however	ADV
ejpam-1773	525	36	extremal	extremal	ADJ
ejpam-1773	525	37	type	type	PROPN
ejpam-1773	525	38	ii	ii	PROPN
ejpam-1773	525	39	code	code	NOUN
ejpam-1773	525	40	of	of	ADP
ejpam-1773	525	41	length	length	NOUN
ejpam-1773	525	42	56	56	NUM
ejpam-1773	525	43	is	be	AUX
ejpam-1773	525	44	known	know	VERB
ejpam-1773	525	45	and	and	CCONJ
ejpam-1773	525	46	there	there	PRON
ejpam-1773	525	47	is	be	VERB
ejpam-1773	525	48	only	only	ADV
ejpam-1773	525	49	one	one	NUM
ejpam-1773	525	50	possible	possible	ADJ
ejpam-1773	525	51	weight	weight	NOUN
ejpam-1773	525	52	enumerator	enumerator	NOUN
ejpam-1773	525	53	for	for	ADP
ejpam-1773	525	54	such	such	ADJ
ejpam-1773	525	55	codes	code	NOUN
ejpam-1773	525	56	,	,	PUNCT
ejpam-1773	525	57	that	that	PRON
ejpam-1773	525	58	starts	start	VERB
ejpam-1773	525	59	with	with	ADP
ejpam-1773	525	60	1	1	NUM
ejpam-1773	525	61	+	+	NOUN
ejpam-1773	525	62	8190z12	8190z12	NOUN
ejpam-1773	525	63	+	+	NOUN
ejpam-1773	525	64	.	.	PUNCT
ejpam-1773	525	65	.	.	PUNCT
ejpam-1773	526	1	..	..	PUNCT
ejpam-1773	527	1	we	we	PRON
ejpam-1773	527	2	are	be	AUX
ejpam-1773	527	3	going	go	VERB
ejpam-1773	527	4	to	to	PART
ejpam-1773	527	5	give	give	VERB
ejpam-1773	527	6	two	two	NUM
ejpam-1773	527	7	separate	separate	ADJ
ejpam-1773	527	8	constructions	construction	NOUN
ejpam-1773	527	9	for	for	ADP
ejpam-1773	527	10	this	this	DET
ejpam-1773	527	11	code	code	NOUN
ejpam-1773	527	12	,	,	PUNCT
ejpam-1773	527	13	one	one	NUM
ejpam-1773	527	14	from	from	ADP
ejpam-1773	527	15	r2	r2	PROPN
ejpam-1773	527	16	and	and	CCONJ
ejpam-1773	527	17	one	one	NUM
ejpam-1773	527	18	from	from	ADP
ejpam-1773	527	19	r3	r3	PROPN
ejpam-1773	527	20	with	with	ADP
ejpam-1773	527	21	different	different	ADJ
ejpam-1773	527	22	automorphism	automorphism	NOUN
ejpam-1773	527	23	groups	group	NOUN
ejpam-1773	527	24	:	:	PUNCT
ejpam-1773	527	25	from	from	ADP
ejpam-1773	527	26	r2	r2	PROPN
ejpam-1773	527	27	:	:	PUNCT
ejpam-1773	527	28	let	let	VERB
ejpam-1773	527	29	c7	c7	PROPN
ejpam-1773	527	30	be	be	AUX
ejpam-1773	527	31	the	the	DET
ejpam-1773	527	32	linear	linear	PROPN
ejpam-1773	527	33	code	code	NOUN
ejpam-1773	527	34	over	over	ADP
ejpam-1773	527	35	r2	r2	PROPN
ejpam-1773	527	36	of	of	ADP
ejpam-1773	527	37	length	length	NOUN
ejpam-1773	527	38	14	14	NUM
ejpam-1773	527	39	,	,	PUNCT
ejpam-1773	527	40	generated	generate	VERB
ejpam-1773	527	41	by	by	ADP
ejpam-1773	527	42	the	the	DET
ejpam-1773	527	43	matrix	matrix	NOUN
ejpam-1773	528	1	[	[	X
ejpam-1773	528	2	i7|a	i7|a	X
ejpam-1773	528	3	]	]	X
ejpam-1773	528	4	where	where	SCONJ
ejpam-1773	528	5	the	the	DET
ejpam-1773	528	6	rows	row	NOUN
ejpam-1773	528	7	of	of	ADP
ejpam-1773	528	8	a	a	PRON
ejpam-1773	528	9	are	be	AUX
ejpam-1773	528	10	given	give	VERB
ejpam-1773	528	11	by	by	ADP
ejpam-1773	528	12	{	{	PUNCT
ejpam-1773	528	13	(	(	PUNCT
ejpam-1773	528	14	1	1	NUM
ejpam-1773	528	15	+	+	NUM
ejpam-1773	528	16	u1	u1	NOUN
ejpam-1773	528	17	,	,	PUNCT
ejpam-1773	528	18	1	1	NUM
ejpam-1773	528	19	+	+	NUM
ejpam-1773	528	20	u2	u2	NOUN
ejpam-1773	528	21	,	,	PUNCT
ejpam-1773	528	22	1,u1,u1	1,u1,u1	NUM
ejpam-1773	528	23	,	,	PUNCT
ejpam-1773	528	24	1	1	NUM
ejpam-1773	528	25	+	+	NUM
ejpam-1773	528	26	u1	u1	NOUN
ejpam-1773	528	27	,	,	PUNCT
ejpam-1773	528	28	1	1	NUM
ejpam-1773	528	29	+	+	NUM
ejpam-1773	528	30	u1	u1	NOUN
ejpam-1773	528	31	+	+	CCONJ
ejpam-1773	528	32	u2	u2	NOUN
ejpam-1773	528	33	)	)	PUNCT
ejpam-1773	528	34	,	,	PUNCT
ejpam-1773	528	35	(	(	PUNCT
ejpam-1773	528	36	1	1	NUM
ejpam-1773	528	37	+	+	NUM
ejpam-1773	528	38	u1u2,u1	u1u2,u1	ADJ
ejpam-1773	528	39	+	+	X
ejpam-1773	528	40	u2	u2	NOUN
ejpam-1773	528	41	,	,	PUNCT
ejpam-1773	528	42	1	1	NUM
ejpam-1773	528	43	+	+	NUM
ejpam-1773	528	44	u1	u1	NOUN
ejpam-1773	528	45	+	+	CCONJ
ejpam-1773	528	46	u2	u2	PROPN
ejpam-1773	528	47	+	+	CCONJ
ejpam-1773	528	48	u1u2	u1u2	PROPN
ejpam-1773	528	49	,	,	PUNCT
ejpam-1773	528	50	1	1	NUM
ejpam-1773	528	51	+	+	NUM
ejpam-1773	528	52	u2	u2	PROPN
ejpam-1773	528	53	+	+	CCONJ
ejpam-1773	528	54	u1u2,u1	u1u2,u1	ADJ
ejpam-1773	528	55	+	+	CCONJ
ejpam-1773	528	56	u2	u2	PROPN
ejpam-1773	528	57	+	+	CCONJ
ejpam-1773	528	58	u1u2,u1	u1u2,u1	ADJ
ejpam-1773	528	59	+	+	CCONJ
ejpam-1773	528	60	u1u2,u1	u1u2,u1	ADJ
ejpam-1773	528	61	+	+	X
ejpam-1773	528	62	u1u2	u1u2	NOUN
ejpam-1773	528	63	)	)	PUNCT
ejpam-1773	528	64	,	,	PUNCT
ejpam-1773	528	65	(	(	PUNCT
ejpam-1773	528	66	0,1	0,1	NUM
ejpam-1773	528	67	+	+	NUM
ejpam-1773	528	68	u1	u1	NOUN
ejpam-1773	528	69	+	+	CCONJ
ejpam-1773	528	70	u2	u2	PROPN
ejpam-1773	528	71	+	+	CCONJ
ejpam-1773	528	72	u1u2	u1u2	PROPN
ejpam-1773	528	73	,	,	PUNCT
ejpam-1773	528	74	1	1	NUM
ejpam-1773	528	75	+	+	NUM
ejpam-1773	528	76	u1	u1	NOUN
ejpam-1773	528	77	,	,	PUNCT
ejpam-1773	528	78	1	1	NUM
ejpam-1773	528	79	+	+	NUM
ejpam-1773	528	80	u1	u1	NOUN
ejpam-1773	528	81	+	+	CCONJ
ejpam-1773	528	82	u2,u1	u2,u1	PROPN
ejpam-1773	528	83	+	+	NUM
ejpam-1773	528	84	u1u2	u1u2	PROPN
ejpam-1773	528	85	,	,	PUNCT
ejpam-1773	528	86	1	1	NUM
ejpam-1773	528	87	+	+	NUM
ejpam-1773	528	88	u2	u2	PROPN
ejpam-1773	528	89	+	+	CCONJ
ejpam-1773	528	90	u1u2	u1u2	PROPN
ejpam-1773	528	91	,	,	PUNCT
ejpam-1773	528	92	1	1	NUM
ejpam-1773	528	93	+	+	NUM
ejpam-1773	528	94	u1	u1	NOUN
ejpam-1773	528	95	)	)	PUNCT
ejpam-1773	528	96	,	,	PUNCT
ejpam-1773	528	97	(	(	PUNCT
ejpam-1773	528	98	1	1	NUM
ejpam-1773	528	99	+	+	SYM
ejpam-1773	528	100	u1u2	u1u2	NOUN
ejpam-1773	528	101	,	,	PUNCT
ejpam-1773	528	102	1	1	NUM
ejpam-1773	528	103	+	+	NUM
ejpam-1773	528	104	u1	u1	NOUN
ejpam-1773	528	105	+	+	CCONJ
ejpam-1773	528	106	u1u2,u2	u1u2,u2	X
ejpam-1773	528	107	+	+	SYM
ejpam-1773	528	108	u1u2	u1u2	NOUN
ejpam-1773	528	109	,	,	PUNCT
ejpam-1773	528	110	1	1	NUM
ejpam-1773	528	111	+	+	NUM
ejpam-1773	528	112	u1	u1	NOUN
ejpam-1773	528	113	+	+	CCONJ
ejpam-1773	528	114	u2	u2	PROPN
ejpam-1773	528	115	+	+	CCONJ
ejpam-1773	528	116	u1u2,u1	u1u2,u1	ADJ
ejpam-1773	528	117	,	,	PUNCT
ejpam-1773	528	118	1	1	NUM
ejpam-1773	528	119	+	+	NUM
ejpam-1773	528	120	u1	u1	NOUN
ejpam-1773	528	121	+	+	CCONJ
ejpam-1773	528	122	u2	u2	PROPN
ejpam-1773	528	123	+	+	CCONJ
ejpam-1773	528	124	u1u2	u1u2	PROPN
ejpam-1773	528	125	,	,	PUNCT
ejpam-1773	528	126	1	1	NUM
ejpam-1773	528	127	+	+	NUM
ejpam-1773	528	128	u1u2	u1u2	NOUN
ejpam-1773	528	129	)	)	PUNCT
ejpam-1773	528	130	,	,	PUNCT
ejpam-1773	528	131	(	(	PUNCT
ejpam-1773	528	132	u2	u2	NOUN
ejpam-1773	528	133	+	+	CCONJ
ejpam-1773	528	134	u1u2,u2	u1u2,u2	X
ejpam-1773	529	1	+	+	CCONJ
ejpam-1773	529	2	u1u2,u2,u1	u1u2,u2,u1	PROPN
ejpam-1773	530	1	+	+	NUM
ejpam-1773	530	2	u2	u2	PROPN
ejpam-1773	530	3	+	+	CCONJ
ejpam-1773	530	4	u1u2	u1u2	PROPN
ejpam-1773	530	5	,	,	PUNCT
ejpam-1773	530	6	1	1	NUM
ejpam-1773	530	7	+	+	NUM
ejpam-1773	530	8	u1	u1	NOUN
ejpam-1773	530	9	+	+	CCONJ
ejpam-1773	530	10	u2	u2	PROPN
ejpam-1773	530	11	+	+	CCONJ
ejpam-1773	530	12	u1u2	u1u2	PROPN
ejpam-1773	530	13	,	,	PUNCT
ejpam-1773	530	14	1	1	NUM
ejpam-1773	530	15	+	+	NUM
ejpam-1773	530	16	u1	u1	NOUN
ejpam-1773	530	17	+	+	CCONJ
ejpam-1773	530	18	u2	u2	PROPN
ejpam-1773	530	19	+	+	CCONJ
ejpam-1773	530	20	u1u2	u1u2	PROPN
ejpam-1773	530	21	,	,	PUNCT
ejpam-1773	530	22	1	1	NUM
ejpam-1773	530	23	+	+	NUM
ejpam-1773	530	24	u2	u2	NOUN
ejpam-1773	530	25	)	)	PUNCT
ejpam-1773	530	26	,	,	PUNCT
ejpam-1773	530	27	(	(	PUNCT
ejpam-1773	531	1	0,1,u1	0,1,u1	ADJ
ejpam-1773	531	2	+	+	NUM
ejpam-1773	531	3	u2	u2	NOUN
ejpam-1773	531	4	+	+	CCONJ
ejpam-1773	531	5	u1u2,u2	u1u2,u2	NOUN
ejpam-1773	531	6	+	+	SYM
ejpam-1773	531	7	u1u2	u1u2	NOUN
ejpam-1773	531	8	,	,	PUNCT
ejpam-1773	531	9	1	1	NUM
ejpam-1773	531	10	+	+	NUM
ejpam-1773	531	11	u1	u1	NOUN
ejpam-1773	531	12	+	+	CCONJ
ejpam-1773	531	13	u2	u2	NOUN
ejpam-1773	531	14	,	,	PUNCT
ejpam-1773	531	15	1,u1	1,u1	NUM
ejpam-1773	531	16	)	)	PUNCT
ejpam-1773	531	17	,	,	PUNCT
ejpam-1773	531	18	(	(	PUNCT
ejpam-1773	531	19	u1	u1	NOUN
ejpam-1773	531	20	,	,	PUNCT
ejpam-1773	531	21	1	1	NUM
ejpam-1773	531	22	+	+	NUM
ejpam-1773	531	23	u2,u2	u2,u2	PROPN
ejpam-1773	531	24	+	+	CCONJ
ejpam-1773	531	25	u1u2,u2	u1u2,u2	PROPN
ejpam-1773	531	26	,	,	PUNCT
ejpam-1773	531	27	1	1	NUM
ejpam-1773	532	1	+	+	NUM
ejpam-1773	532	2	u2	u2	PROPN
ejpam-1773	532	3	+	+	CCONJ
ejpam-1773	532	4	u1u2,u2	u1u2,u2	PROPN
ejpam-1773	532	5	,	,	PUNCT
ejpam-1773	532	6	1	1	NUM
ejpam-1773	532	7	+	+	NUM
ejpam-1773	532	8	u1	u1	NOUN
ejpam-1773	532	9	+	+	CCONJ
ejpam-1773	532	10	u2	u2	NOUN
ejpam-1773	532	11	)	)	PUNCT
ejpam-1773	532	12	}	}	PUNCT
ejpam-1773	532	13	then	then	ADV
ejpam-1773	532	14	ψ2(c7	ψ2(c7	VERB
ejpam-1773	532	15	)	)	PUNCT
ejpam-1773	532	16	is	be	AUX
ejpam-1773	532	17	an	an	DET
ejpam-1773	532	18	extremal	extremal	ADJ
ejpam-1773	532	19	binary	binary	ADJ
ejpam-1773	532	20	type	type	NOUN
ejpam-1773	532	21	ii	ii	PROPN
ejpam-1773	532	22	self	self	NOUN
ejpam-1773	532	23	-	-	PUNCT
ejpam-1773	532	24	dual	dual	ADJ
ejpam-1773	532	25	code	code	NOUN
ejpam-1773	532	26	of	of	ADP
ejpam-1773	532	27	parameters	parameter	NOUN
ejpam-1773	532	28	[	[	X
ejpam-1773	532	29	56,28,12	56,28,12	X
ejpam-1773	532	30	]	]	PUNCT
ejpam-1773	532	31	with	with	ADP
ejpam-1773	532	32	an	an	DET
ejpam-1773	532	33	automorphism	automorphism	NOUN
ejpam-1773	532	34	group	group	NOUN
ejpam-1773	532	35	of	of	ADP
ejpam-1773	532	36	order	order	NOUN
ejpam-1773	532	37	4	4	NUM
ejpam-1773	532	38	.	.	NOUN
ejpam-1773	532	39	from	from	ADP
ejpam-1773	532	40	r3	r3	PROPN
ejpam-1773	532	41	:	:	PUNCT
ejpam-1773	532	42	let	let	VERB
ejpam-1773	532	43	c	c	NOUN
ejpam-1773	532	44	′7	′7	ADV
ejpam-1773	532	45	be	be	AUX
ejpam-1773	532	46	the	the	DET
ejpam-1773	532	47	linear	linear	PROPN
ejpam-1773	532	48	code	code	NOUN
ejpam-1773	532	49	over	over	ADP
ejpam-1773	532	50	r3	r3	PROPN
ejpam-1773	532	51	of	of	ADP
ejpam-1773	532	52	length	length	NOUN
ejpam-1773	532	53	7	7	NUM
ejpam-1773	532	54	generated	generate	VERB
ejpam-1773	532	55	by	by	ADP
ejpam-1773	532	56	the	the	DET
ejpam-1773	532	57	matrix	matrix	NOUN
ejpam-1773	532	58	�	�	PROPN
ejpam-1773	532	59	i3	i3	NOUN
ejpam-1773	532	60	|	|	ADV
ejpam-1773	532	61	0	0	NUM
ejpam-1773	533	1	|	|	ADV
ejpam-1773	533	2	a	a	DET
ejpam-1773	533	3	�	�	PROPN
ejpam-1773	533	4	,	,	PUNCT
ejpam-1773	533	5	where	where	SCONJ
ejpam-1773	533	6	a	a	PRON
ejpam-1773	533	7	is	be	AUX
ejpam-1773	533	8	a	a	DET
ejpam-1773	533	9	4×	4×	NOUN
ejpam-1773	533	10	4	4	NUM
ejpam-1773	533	11	matrix	matrix	NOUN
ejpam-1773	533	12	over	over	ADP
ejpam-1773	533	13	r3	r3	PROPN
ejpam-1773	533	14	whose	whose	DET
ejpam-1773	533	15	rows	row	NOUN
ejpam-1773	533	16	are	be	AUX
ejpam-1773	533	17	{	{	PUNCT
ejpam-1773	533	18	(	(	PUNCT
ejpam-1773	533	19	1	1	NUM
ejpam-1773	533	20	+	+	NUM
ejpam-1773	533	21	u3	u3	NOUN
ejpam-1773	533	22	+	+	CCONJ
ejpam-1773	533	23	u1u3	u1u3	NOUN
ejpam-1773	533	24	+	+	CCONJ
ejpam-1773	533	25	u1u2u3	u1u2u3	ADJ
ejpam-1773	533	26	,	,	PUNCT
ejpam-1773	533	27	1	1	NUM
ejpam-1773	533	28	+	+	NUM
ejpam-1773	533	29	u1	u1	NOUN
ejpam-1773	533	30	+	+	CCONJ
ejpam-1773	533	31	u2	u2	PROPN
ejpam-1773	533	32	+	+	CCONJ
ejpam-1773	533	33	u2u3	u2u3	X
ejpam-1773	533	34	+	+	CCONJ
ejpam-1773	533	35	u1u2u3	u1u2u3	ADJ
ejpam-1773	533	36	,	,	PUNCT
ejpam-1773	533	37	1	1	NUM
ejpam-1773	533	38	+	+	NUM
ejpam-1773	533	39	u2	u2	NOUN
ejpam-1773	533	40	+	+	CCONJ
ejpam-1773	533	41	u3	u3	NOUN
ejpam-1773	533	42	+	+	CCONJ
ejpam-1773	533	43	u1u2	u1u2	X
ejpam-1773	533	44	+	+	NUM
ejpam-1773	533	45	u1u3	u1u3	NOUN
ejpam-1773	533	46	+	+	X
ejpam-1773	533	47	u2u3	u2u3	X
ejpam-1773	533	48	+	+	CCONJ
ejpam-1773	533	49	u1u2u3,u1	u1u2u3,u1	ADJ
ejpam-1773	533	50	+	+	CCONJ
ejpam-1773	533	51	u3	u3	NOUN
ejpam-1773	533	52	+	+	CCONJ
ejpam-1773	533	53	u1u2u3	u1u2u3	PROPN
ejpam-1773	533	54	)	)	PUNCT
ejpam-1773	533	55	,	,	PUNCT
ejpam-1773	533	56	(	(	PUNCT
ejpam-1773	533	57	u2	u2	NOUN
ejpam-1773	533	58	+	+	CCONJ
ejpam-1773	533	59	u1u2	u1u2	X
ejpam-1773	533	60	+	+	NUM
ejpam-1773	533	61	u1u3	u1u3	NOUN
ejpam-1773	533	62	,	,	PUNCT
ejpam-1773	533	63	1	1	NUM
ejpam-1773	533	64	+	+	NUM
ejpam-1773	533	65	u2	u2	NOUN
ejpam-1773	533	66	+	+	CCONJ
ejpam-1773	533	67	u3	u3	NOUN
ejpam-1773	533	68	+	+	CCONJ
ejpam-1773	533	69	u1u2	u1u2	X
ejpam-1773	533	70	+	+	X
ejpam-1773	533	71	u2u3	u2u3	ADJ
ejpam-1773	533	72	,	,	PUNCT
ejpam-1773	533	73	1	1	NUM
ejpam-1773	533	74	+	+	NUM
ejpam-1773	533	75	u3	u3	NOUN
ejpam-1773	533	76	+	+	CCONJ
ejpam-1773	533	77	u1u2	u1u2	X
ejpam-1773	534	1	+	+	X
ejpam-1773	534	2	u2u3	u2u3	X
ejpam-1773	534	3	,	,	PUNCT
ejpam-1773	534	4	1	1	NUM
ejpam-1773	534	5	+	+	NUM
ejpam-1773	534	6	u1	u1	NOUN
ejpam-1773	534	7	+	+	CCONJ
ejpam-1773	534	8	u2	u2	PROPN
ejpam-1773	534	9	+	+	CCONJ
ejpam-1773	534	10	u1u2	u1u2	NOUN
ejpam-1773	534	11	+	+	NUM
ejpam-1773	534	12	u1u3	u1u3	NOUN
ejpam-1773	534	13	+	+	X
ejpam-1773	534	14	u2u3	u2u3	X
ejpam-1773	534	15	+	+	CCONJ
ejpam-1773	534	16	u1u2u3	u1u2u3	ADJ
ejpam-1773	534	17	)	)	PUNCT
ejpam-1773	534	18	,	,	PUNCT
ejpam-1773	534	19	(	(	PUNCT
ejpam-1773	534	20	1	1	NUM
ejpam-1773	534	21	+	+	NUM
ejpam-1773	534	22	u1	u1	NOUN
ejpam-1773	534	23	+	+	CCONJ
ejpam-1773	534	24	u3	u3	NOUN
ejpam-1773	534	25	+	+	CCONJ
ejpam-1773	534	26	u1u2u3,u2	u1u2u3,u2	X
ejpam-1773	534	27	+	+	SYM
ejpam-1773	534	28	u2u3	u2u3	PROPN
ejpam-1773	534	29	,	,	PUNCT
ejpam-1773	534	30	1	1	NUM
ejpam-1773	534	31	+	+	NUM
ejpam-1773	534	32	u1u2	u1u2	X
ejpam-1773	534	33	+	+	NUM
ejpam-1773	534	34	u1u3	u1u3	NOUN
ejpam-1773	534	35	+	+	X
ejpam-1773	534	36	u2u3	u2u3	X
ejpam-1773	534	37	+	+	CCONJ
ejpam-1773	534	38	u1u2u3	u1u2u3	ADJ
ejpam-1773	534	39	,	,	PUNCT
ejpam-1773	534	40	1	1	NUM
ejpam-1773	534	41	+	+	NUM
ejpam-1773	534	42	u1	u1	NOUN
ejpam-1773	534	43	+	+	CCONJ
ejpam-1773	534	44	u2	u2	NOUN
ejpam-1773	534	45	+	+	CCONJ
ejpam-1773	534	46	u3	u3	NOUN
ejpam-1773	534	47	+	+	CCONJ
ejpam-1773	534	48	u1u2	u1u2	X
ejpam-1773	534	49	+	+	X
ejpam-1773	534	50	u2u3	u2u3	X
ejpam-1773	534	51	)	)	PUNCT
ejpam-1773	534	52	,	,	PUNCT
ejpam-1773	534	53	(	(	PUNCT
ejpam-1773	534	54	u1	u1	NOUN
ejpam-1773	534	55	+	+	CCONJ
ejpam-1773	534	56	u3	u3	NOUN
ejpam-1773	534	57	+	+	CCONJ
ejpam-1773	534	58	u1u2	u1u2	X
ejpam-1773	534	59	+	+	NUM
ejpam-1773	534	60	u1u3	u1u3	X
ejpam-1773	534	61	+	+	CCONJ
ejpam-1773	534	62	u1u2u3,u1	u1u2u3,u1	ADJ
ejpam-1773	534	63	+	+	CCONJ
ejpam-1773	534	64	u3	u3	NOUN
ejpam-1773	534	65	+	+	CCONJ
ejpam-1773	534	66	u1u3	u1u3	NOUN
ejpam-1773	534	67	+	+	CCONJ
ejpam-1773	534	68	u1u2u3	u1u2u3	ADJ
ejpam-1773	534	69	,	,	PUNCT
ejpam-1773	534	70	0,u1	0,u1	ADJ
ejpam-1773	534	71	+	+	CCONJ
ejpam-1773	534	72	u3	u3	NOUN
ejpam-1773	534	73	+	+	CCONJ
ejpam-1773	534	74	u1u2	u1u2	X
ejpam-1773	534	75	+	+	NUM
ejpam-1773	534	76	u1u3	u1u3	NOUN
ejpam-1773	534	77	)	)	PUNCT
ejpam-1773	534	78	}	}	PUNCT
ejpam-1773	534	79	.	.	PUNCT
ejpam-1773	535	1	ψ3(c	ψ3(c	PUNCT
ejpam-1773	535	2	′	′	NUM
ejpam-1773	535	3	7	7	NUM
ejpam-1773	535	4	)	)	PUNCT
ejpam-1773	535	5	is	be	AUX
ejpam-1773	535	6	an	an	DET
ejpam-1773	535	7	extremal	extremal	ADJ
ejpam-1773	535	8	binary	binary	NOUN
ejpam-1773	535	9	self	self	NOUN
ejpam-1773	535	10	-	-	PUNCT
ejpam-1773	535	11	dual	dual	ADJ
ejpam-1773	535	12	code	code	NOUN
ejpam-1773	535	13	of	of	ADP
ejpam-1773	535	14	parameters	parameter	NOUN
ejpam-1773	536	1	[	[	X
ejpam-1773	536	2	56,28,12	56,28,12	X
ejpam-1773	536	3	]	]	PUNCT
ejpam-1773	536	4	with	with	ADP
ejpam-1773	536	5	an	an	DET
ejpam-1773	536	6	automorphism	automorphism	NOUN
ejpam-1773	536	7	group	group	NOUN
ejpam-1773	536	8	of	of	ADP
ejpam-1773	536	9	order	order	NOUN
ejpam-1773	536	10	8	8	NUM
ejpam-1773	536	11	.	.	PUNCT
ejpam-1773	537	1	7.9	7.9	NUM
ejpam-1773	537	2	.	.	PUNCT
ejpam-1773	538	1	binary	binary	ADJ
ejpam-1773	538	2	self	self	NOUN
ejpam-1773	538	3	-	-	PUNCT
ejpam-1773	538	4	dual	dual	ADJ
ejpam-1773	538	5	code	code	NOUN
ejpam-1773	538	6	with	with	ADP
ejpam-1773	538	7	parameters	parameter	NOUN
ejpam-1773	538	8	[	[	X
ejpam-1773	538	9	64	64	NUM
ejpam-1773	538	10	,	,	PUNCT
ejpam-1773	538	11	32	32	NUM
ejpam-1773	538	12	,	,	PUNCT
ejpam-1773	538	13	12	12	NUM
ejpam-1773	538	14	]	]	PUNCT
ejpam-1773	538	15	let	let	VERB
ejpam-1773	538	16	c8	c8	PROPN
ejpam-1773	538	17	be	be	AUX
ejpam-1773	538	18	the	the	DET
ejpam-1773	538	19	linear	linear	PROPN
ejpam-1773	538	20	code	code	NOUN
ejpam-1773	538	21	over	over	ADP
ejpam-1773	538	22	r3	r3	PROPN
ejpam-1773	538	23	of	of	ADP
ejpam-1773	538	24	length	length	NOUN
ejpam-1773	538	25	8	8	NUM
ejpam-1773	538	26	generated	generate	VERB
ejpam-1773	538	27	by	by	ADP
ejpam-1773	538	28	the	the	DET
ejpam-1773	538	29	matrix	matrix	NOUN
ejpam-1773	538	30	[	[	X
ejpam-1773	538	31	i4|a	i4|a	NOUN
ejpam-1773	538	32	]	]	PUNCT
ejpam-1773	538	33	where	where	SCONJ
ejpam-1773	538	34	a	a	PRON
ejpam-1773	538	35	is	be	AUX
ejpam-1773	538	36	a	a	DET
ejpam-1773	538	37	4×	4×	NOUN
ejpam-1773	538	38	4	4	NUM
ejpam-1773	538	39	matrix	matrix	NOUN
ejpam-1773	538	40	over	over	ADP
ejpam-1773	538	41	r3	r3	PROPN
ejpam-1773	538	42	whose	whose	DET
ejpam-1773	538	43	rows	row	NOUN
ejpam-1773	538	44	are	be	AUX
ejpam-1773	538	45	{	{	PUNCT
ejpam-1773	538	46	(	(	PUNCT
ejpam-1773	538	47	1	1	NUM
ejpam-1773	538	48	+	+	NUM
ejpam-1773	538	49	u1	u1	NOUN
ejpam-1773	538	50	+	+	CCONJ
ejpam-1773	538	51	u1u2	u1u2	X
ejpam-1773	538	52	+	+	NUM
ejpam-1773	538	53	u1u3	u1u3	NOUN
ejpam-1773	538	54	+	+	CCONJ
ejpam-1773	538	55	u1u2u3	u1u2u3	ADJ
ejpam-1773	538	56	,	,	PUNCT
ejpam-1773	538	57	1	1	NUM
ejpam-1773	538	58	+	+	NUM
ejpam-1773	538	59	u1	u1	NOUN
ejpam-1773	538	60	+	+	CCONJ
ejpam-1773	538	61	u2	u2	PROPN
ejpam-1773	538	62	+	+	CCONJ
ejpam-1773	538	63	u1u2	u1u2	NOUN
ejpam-1773	538	64	+	+	NUM
ejpam-1773	538	65	u1u3	u1u3	NOUN
ejpam-1773	538	66	+	+	X
ejpam-1773	538	67	u2u3	u2u3	X
ejpam-1773	538	68	+	+	CCONJ
ejpam-1773	538	69	u1u2u3	u1u2u3	ADJ
ejpam-1773	538	70	,	,	PUNCT
ejpam-1773	538	71	1	1	NUM
ejpam-1773	538	72	+	+	NUM
ejpam-1773	538	73	u3	u3	NOUN
ejpam-1773	538	74	+	+	CCONJ
ejpam-1773	538	75	u1u2u3,u3	u1u2u3,u3	NOUN
ejpam-1773	538	76	+	+	CCONJ
ejpam-1773	538	77	u2u3	u2u3	X
ejpam-1773	538	78	)	)	PUNCT
ejpam-1773	538	79	,	,	PUNCT
ejpam-1773	538	80	(	(	PUNCT
ejpam-1773	538	81	u3	u3	NOUN
ejpam-1773	538	82	+	+	CCONJ
ejpam-1773	538	83	u1u2	u1u2	X
ejpam-1773	538	84	+	+	X
ejpam-1773	538	85	u2u3	u2u3	ADJ
ejpam-1773	538	86	,	,	PUNCT
ejpam-1773	538	87	1	1	NUM
ejpam-1773	538	88	+	+	NUM
ejpam-1773	538	89	u2	u2	PROPN
ejpam-1773	538	90	+	+	CCONJ
ejpam-1773	538	91	u1u2	u1u2	PROPN
ejpam-1773	538	92	,	,	PUNCT
ejpam-1773	538	93	1	1	NUM
ejpam-1773	538	94	+	+	NUM
ejpam-1773	538	95	u1	u1	NOUN
ejpam-1773	538	96	+	+	CCONJ
ejpam-1773	538	97	u3	u3	NOUN
ejpam-1773	538	98	+	+	CCONJ
ejpam-1773	538	99	u2u3	u2u3	PROPN
ejpam-1773	538	100	,	,	PUNCT
ejpam-1773	538	101	1	1	NUM
ejpam-1773	538	102	+	+	NUM
ejpam-1773	538	103	u1	u1	NOUN
ejpam-1773	538	104	+	+	CCONJ
ejpam-1773	538	105	u3	u3	NOUN
ejpam-1773	538	106	+	+	CCONJ
ejpam-1773	538	107	u1u3	u1u3	NOUN
ejpam-1773	538	108	+	+	CCONJ
ejpam-1773	538	109	u1u2u3	u1u2u3	ADJ
ejpam-1773	538	110	)	)	PUNCT
ejpam-1773	538	111	,	,	PUNCT
ejpam-1773	538	112	(	(	PUNCT
ejpam-1773	538	113	1	1	NUM
ejpam-1773	538	114	+	+	NUM
ejpam-1773	538	115	u1	u1	NOUN
ejpam-1773	538	116	+	+	CCONJ
ejpam-1773	538	117	u3	u3	NOUN
ejpam-1773	538	118	+	+	CCONJ
ejpam-1773	538	119	u1u3	u1u3	NOUN
ejpam-1773	538	120	+	+	X
ejpam-1773	538	121	u2u3	u2u3	X
ejpam-1773	538	122	+	+	CCONJ
ejpam-1773	538	123	u1u2u3,u1	u1u2u3,u1	X
ejpam-1773	538	124	+	+	X
ejpam-1773	538	125	u1u2	u1u2	X
ejpam-1773	538	126	+	+	X
ejpam-1773	538	127	u2u3	u2u3	ADJ
ejpam-1773	538	128	,	,	PUNCT
ejpam-1773	538	129	1	1	NUM
ejpam-1773	538	130	+	+	NUM
ejpam-1773	538	131	u1	u1	NOUN
ejpam-1773	538	132	+	+	CCONJ
ejpam-1773	538	133	u3	u3	NOUN
ejpam-1773	538	134	+	+	CCONJ
ejpam-1773	538	135	u1u2	u1u2	X
ejpam-1773	538	136	+	+	X
ejpam-1773	538	137	u2u3	u2u3	ADJ
ejpam-1773	538	138	,	,	PUNCT
ejpam-1773	538	139	1	1	NUM
ejpam-1773	538	140	)	)	PUNCT
ejpam-1773	538	141	,	,	PUNCT
ejpam-1773	538	142	(	(	PUNCT
ejpam-1773	538	143	1	1	NUM
ejpam-1773	538	144	+	+	NUM
ejpam-1773	538	145	u2	u2	NOUN
ejpam-1773	538	146	+	+	CCONJ
ejpam-1773	538	147	u3	u3	NOUN
ejpam-1773	538	148	+	+	CCONJ
ejpam-1773	538	149	u2u3	u2u3	X
ejpam-1773	538	150	+	+	CCONJ
ejpam-1773	538	151	u1u2u3	u1u2u3	ADJ
ejpam-1773	538	152	,	,	PUNCT
ejpam-1773	538	153	1	1	NUM
ejpam-1773	538	154	+	+	NUM
ejpam-1773	538	155	u1	u1	NOUN
ejpam-1773	538	156	+	+	CCONJ
ejpam-1773	538	157	u2u3,u1	u2u3,u1	ADJ
ejpam-1773	538	158	+	+	NUM
ejpam-1773	538	159	u1u3	u1u3	X
ejpam-1773	538	160	+	+	X
ejpam-1773	538	161	u2u3	u2u3	X
ejpam-1773	539	1	+	+	CCONJ
ejpam-1773	539	2	u1u2u3	u1u2u3	ADJ
ejpam-1773	539	3	,	,	PUNCT
ejpam-1773	539	4	1	1	NUM
ejpam-1773	539	5	+	+	NUM
ejpam-1773	539	6	u1	u1	NOUN
ejpam-1773	539	7	+	+	CCONJ
ejpam-1773	539	8	u2	u2	PROPN
ejpam-1773	539	9	+	+	CCONJ
ejpam-1773	539	10	u2u3	u2u3	X
ejpam-1773	539	11	+	+	CCONJ
ejpam-1773	539	12	u1u2u3	u1u2u3	ADJ
ejpam-1773	539	13	)	)	PUNCT
ejpam-1773	539	14	}	}	PUNCT
ejpam-1773	539	15	.	.	PUNCT
ejpam-1773	540	1	references	reference	NOUN
ejpam-1773	540	2	105	105	NUM
ejpam-1773	540	3	then	then	ADV
ejpam-1773	540	4	c8	c8	PROPN
ejpam-1773	540	5	turns	turn	VERB
ejpam-1773	540	6	out	out	ADP
ejpam-1773	540	7	to	to	PART
ejpam-1773	540	8	be	be	AUX
ejpam-1773	540	9	a	a	DET
ejpam-1773	540	10	type	type	NOUN
ejpam-1773	540	11	i	i	PRON
ejpam-1773	540	12	code	code	VERB
ejpam-1773	540	13	with	with	ADP
ejpam-1773	540	14	lee	lee	PROPN
ejpam-1773	540	15	weight	weight	NOUN
ejpam-1773	540	16	distribution	distribution	NOUN
ejpam-1773	540	17	1	1	NUM
ejpam-1773	540	18	+	+	NOUN
ejpam-1773	540	19	1888z12	1888z12	NUM
ejpam-1773	540	20	+	+	ADJ
ejpam-1773	540	21	20736z14	20736z14	PROPN
ejpam-1773	540	22	+	+	NOUN
ejpam-1773	540	23	.	.	PUNCT
ejpam-1773	540	24	.	.	PUNCT
ejpam-1773	540	25	.	.	PUNCT
ejpam-1773	541	1	we	we	PRON
ejpam-1773	541	2	see	see	VERB
ejpam-1773	541	3	that	that	DET
ejpam-1773	541	4	ψ3(c8	ψ3(c8	NOUN
ejpam-1773	541	5	)	)	PUNCT
ejpam-1773	541	6	is	be	AUX
ejpam-1773	541	7	an	an	DET
ejpam-1773	541	8	extremal	extremal	ADJ
ejpam-1773	541	9	binary	binary	ADJ
ejpam-1773	541	10	type	type	PROPN
ejpam-1773	541	11	ii	ii	PROPN
ejpam-1773	541	12	code	code	NOUN
ejpam-1773	541	13	with	with	ADP
ejpam-1773	541	14	parameters	parameter	NOUN
ejpam-1773	541	15	[	[	X
ejpam-1773	541	16	64,32,12	64,32,12	NUM
ejpam-1773	541	17	]	]	PUNCT
ejpam-1773	541	18	and	and	CCONJ
ejpam-1773	541	19	an	an	DET
ejpam-1773	541	20	automorphism	automorphism	NOUN
ejpam-1773	541	21	group	group	NOUN
ejpam-1773	541	22	of	of	ADP
ejpam-1773	541	23	order	order	NOUN
ejpam-1773	541	24	8	8	NUM
ejpam-1773	541	25	.	.	NOUN
ejpam-1773	541	26	8	8	NUM
ejpam-1773	541	27	.	.	X
ejpam-1773	541	28	conclusion	conclusion	NOUN
ejpam-1773	541	29	binary	binary	ADJ
ejpam-1773	541	30	self	self	NOUN
ejpam-1773	541	31	-	-	PUNCT
ejpam-1773	541	32	dual	dual	ADJ
ejpam-1773	541	33	codes	code	NOUN
ejpam-1773	541	34	are	be	AUX
ejpam-1773	541	35	a	a	DET
ejpam-1773	541	36	rich	rich	ADJ
ejpam-1773	541	37	source	source	NOUN
ejpam-1773	541	38	of	of	ADP
ejpam-1773	541	39	research	research	NOUN
ejpam-1773	541	40	in	in	ADP
ejpam-1773	541	41	coding	code	VERB
ejpam-1773	541	42	theory	theory	NOUN
ejpam-1773	541	43	.	.	PUNCT
ejpam-1773	542	1	there	there	PRON
ejpam-1773	542	2	are	be	VERB
ejpam-1773	542	3	numerous	numerous	ADJ
ejpam-1773	542	4	methods	method	NOUN
ejpam-1773	542	5	of	of	ADP
ejpam-1773	542	6	constructing	construct	VERB
ejpam-1773	542	7	good	good	ADJ
ejpam-1773	542	8	self	self	NOUN
ejpam-1773	542	9	-	-	PUNCT
ejpam-1773	542	10	dual	dual	ADJ
ejpam-1773	542	11	codes	code	NOUN
ejpam-1773	542	12	,	,	PUNCT
ejpam-1773	542	13	in	in	ADP
ejpam-1773	542	14	particular	particular	ADJ
ejpam-1773	542	15	extremal	extremal	ADJ
ejpam-1773	542	16	binary	binary	NOUN
ejpam-1773	542	17	self	self	NOUN
ejpam-1773	542	18	-	-	PUNCT
ejpam-1773	542	19	dual	dual	ADJ
ejpam-1773	542	20	codes	code	NOUN
ejpam-1773	542	21	,	,	PUNCT
ejpam-1773	542	22	which	which	PRON
ejpam-1773	542	23	are	be	AUX
ejpam-1773	542	24	self	self	NOUN
ejpam-1773	542	25	-	-	PUNCT
ejpam-1773	542	26	dual	dual	ADJ
ejpam-1773	542	27	codes	code	NOUN
ejpam-1773	542	28	that	that	PRON
ejpam-1773	542	29	attain	attain	VERB
ejpam-1773	542	30	the	the	DET
ejpam-1773	542	31	upper	upper	ADJ
ejpam-1773	542	32	bounds	bound	NOUN
ejpam-1773	542	33	.	.	PUNCT
ejpam-1773	543	1	recently	recently	ADV
ejpam-1773	543	2	,	,	PUNCT
ejpam-1773	543	3	the	the	DET
ejpam-1773	543	4	family	family	NOUN
ejpam-1773	543	5	of	of	ADP
ejpam-1773	543	6	rings	ring	NOUN
ejpam-1773	543	7	that	that	PRON
ejpam-1773	543	8	are	be	AUX
ejpam-1773	543	9	called	call	VERB
ejpam-1773	543	10	rk	rk	NOUN
ejpam-1773	543	11	have	have	AUX
ejpam-1773	543	12	been	be	AUX
ejpam-1773	543	13	introduced	introduce	VERB
ejpam-1773	543	14	in	in	ADP
ejpam-1773	543	15	coding	code	VERB
ejpam-1773	543	16	theory	theory	NOUN
ejpam-1773	543	17	and	and	CCONJ
ejpam-1773	543	18	have	have	AUX
ejpam-1773	543	19	proved	prove	VERB
ejpam-1773	543	20	to	to	PART
ejpam-1773	543	21	be	be	AUX
ejpam-1773	543	22	useful	useful	ADJ
ejpam-1773	543	23	in	in	ADP
ejpam-1773	543	24	constructing	construct	VERB
ejpam-1773	543	25	binary	binary	ADJ
ejpam-1773	543	26	codes	code	NOUN
ejpam-1773	543	27	with	with	ADP
ejpam-1773	543	28	good	good	ADJ
ejpam-1773	543	29	parameters	parameter	NOUN
ejpam-1773	543	30	.	.	PUNCT
ejpam-1773	544	1	in	in	ADP
ejpam-1773	544	2	this	this	DET
ejpam-1773	544	3	work	work	NOUN
ejpam-1773	544	4	,	,	PUNCT
ejpam-1773	544	5	we	we	PRON
ejpam-1773	544	6	worked	work	VERB
ejpam-1773	544	7	our	our	PRON
ejpam-1773	544	8	the	the	DET
ejpam-1773	544	9	general	general	ADJ
ejpam-1773	544	10	properties	property	NOUN
ejpam-1773	544	11	of	of	ADP
ejpam-1773	544	12	self	self	NOUN
ejpam-1773	544	13	-	-	PUNCT
ejpam-1773	544	14	dual	dual	ADJ
ejpam-1773	544	15	codes	code	NOUN
ejpam-1773	544	16	over	over	ADP
ejpam-1773	544	17	rk	rk	NOUN
ejpam-1773	544	18	and	and	CCONJ
ejpam-1773	544	19	used	use	VERB
ejpam-1773	544	20	these	these	DET
ejpam-1773	544	21	codes	code	NOUN
ejpam-1773	544	22	to	to	PART
ejpam-1773	544	23	obtain	obtain	VERB
ejpam-1773	544	24	binary	binary	ADJ
ejpam-1773	544	25	self	self	NOUN
ejpam-1773	544	26	-	-	PUNCT
ejpam-1773	544	27	dual	dual	ADJ
ejpam-1773	544	28	codes	code	NOUN
ejpam-1773	544	29	.	.	PUNCT
ejpam-1773	545	1	we	we	PRON
ejpam-1773	545	2	gave	give	VERB
ejpam-1773	545	3	alternate	alternate	ADJ
ejpam-1773	545	4	constructions	construction	NOUN
ejpam-1773	545	5	for	for	ADP
ejpam-1773	545	6	some	some	PRON
ejpam-1773	545	7	of	of	ADP
ejpam-1773	545	8	the	the	DET
ejpam-1773	545	9	well	well	ADV
ejpam-1773	545	10	known	know	VERB
ejpam-1773	545	11	good	good	ADJ
ejpam-1773	545	12	self	self	NOUN
ejpam-1773	545	13	-	-	PUNCT
ejpam-1773	545	14	dual	dual	ADJ
ejpam-1773	545	15	binary	binary	ADJ
ejpam-1773	545	16	codes	code	NOUN
ejpam-1773	545	17	such	such	ADJ
ejpam-1773	545	18	as	as	ADP
ejpam-1773	545	19	the	the	DET
ejpam-1773	545	20	extended	extended	ADJ
ejpam-1773	545	21	hamming	hamming	NOUN
ejpam-1773	545	22	code	code	NOUN
ejpam-1773	545	23	and	and	CCONJ
ejpam-1773	545	24	the	the	DET
ejpam-1773	545	25	extended	extended	ADJ
ejpam-1773	545	26	binary	binary	NOUN
ejpam-1773	545	27	golay	golay	PROPN
ejpam-1773	545	28	code	code	PROPN
ejpam-1773	545	29	.	.	PUNCT
ejpam-1773	546	1	binary	binary	ADJ
ejpam-1773	546	2	codes	code	NOUN
ejpam-1773	546	3	that	that	PRON
ejpam-1773	546	4	are	be	AUX
ejpam-1773	546	5	images	image	NOUN
ejpam-1773	546	6	of	of	ADP
ejpam-1773	546	7	codes	code	NOUN
ejpam-1773	546	8	over	over	ADP
ejpam-1773	546	9	rk	rk	NOUN
ejpam-1773	546	10	have	have	VERB
ejpam-1773	546	11	automorphism	automorphism	NOUN
ejpam-1773	546	12	groups	group	NOUN
ejpam-1773	546	13	of	of	ADP
ejpam-1773	546	14	size	size	NOUN
ejpam-1773	546	15	that	that	PRON
ejpam-1773	546	16	are	be	AUX
ejpam-1773	546	17	multiple	multiple	ADJ
ejpam-1773	546	18	of	of	ADP
ejpam-1773	546	19	2k	2k	NUM
ejpam-1773	546	20	.	.	PUNCT
ejpam-1773	547	1	that	that	PRON
ejpam-1773	547	2	is	be	AUX
ejpam-1773	547	3	why	why	SCONJ
ejpam-1773	547	4	working	work	VERB
ejpam-1773	547	5	over	over	ADP
ejpam-1773	547	6	rk	rk	NOUN
ejpam-1773	547	7	helps	help	VERB
ejpam-1773	547	8	construct	construct	VERB
ejpam-1773	547	9	binary	binary	ADJ
ejpam-1773	547	10	self	self	NOUN
ejpam-1773	547	11	-	-	PUNCT
ejpam-1773	547	12	dual	dual	ADJ
ejpam-1773	547	13	codes	code	NOUN
ejpam-1773	547	14	of	of	ADP
ejpam-1773	547	15	high	high	ADJ
ejpam-1773	547	16	automorphism	automorphism	NOUN
ejpam-1773	547	17	groups	group	NOUN
ejpam-1773	547	18	.	.	PUNCT
ejpam-1773	548	1	the	the	DET
ejpam-1773	548	2	rich	rich	ADJ
ejpam-1773	548	3	algebraic	algebraic	ADJ
ejpam-1773	548	4	structure	structure	NOUN
ejpam-1773	548	5	of	of	ADP
ejpam-1773	548	6	rk	rk	PRON
ejpam-1773	548	7	can	can	AUX
ejpam-1773	548	8	prove	prove	VERB
ejpam-1773	548	9	to	to	PART
ejpam-1773	548	10	be	be	AUX
ejpam-1773	548	11	useful	useful	ADJ
ejpam-1773	548	12	in	in	ADP
ejpam-1773	548	13	obtaining	obtain	VERB
ejpam-1773	548	14	better	well	ADJ
ejpam-1773	548	15	codes	code	NOUN
ejpam-1773	548	16	in	in	ADP
ejpam-1773	548	17	the	the	DET
ejpam-1773	548	18	future	future	NOUN
ejpam-1773	548	19	.	.	PUNCT
ejpam-1773	549	1	the	the	DET
ejpam-1773	549	2	connection	connection	NOUN
ejpam-1773	549	3	of	of	ADP
ejpam-1773	549	4	codes	code	NOUN
ejpam-1773	549	5	over	over	ADP
ejpam-1773	549	6	rk	rk	NOUN
ejpam-1773	549	7	with	with	ADP
ejpam-1773	549	8	some	some	DET
ejpam-1773	549	9	other	other	ADJ
ejpam-1773	549	10	structures	structure	NOUN
ejpam-1773	549	11	such	such	ADJ
ejpam-1773	549	12	as	as	ADP
ejpam-1773	549	13	lattices	lattice	NOUN
ejpam-1773	549	14	and	and	CCONJ
ejpam-1773	549	15	designs	design	NOUN
ejpam-1773	549	16	can	can	AUX
ejpam-1773	549	17	further	far	ADV
ejpam-1773	549	18	be	be	AUX
ejpam-1773	549	19	explored	explore	VERB
ejpam-1773	549	20	.	.	PUNCT
ejpam-1773	550	1	we	we	PRON
ejpam-1773	550	2	obtained	obtain	VERB
ejpam-1773	550	3	a	a	DET
ejpam-1773	550	4	number	number	NOUN
ejpam-1773	550	5	of	of	ADP
ejpam-1773	550	6	extremal	extremal	ADJ
ejpam-1773	550	7	binary	binary	ADJ
ejpam-1773	550	8	self	self	NOUN
ejpam-1773	550	9	-	-	PUNCT
ejpam-1773	550	10	dual	dual	ADJ
ejpam-1773	550	11	codes	code	NOUN
ejpam-1773	550	12	of	of	ADP
ejpam-1773	550	13	certain	certain	ADJ
ejpam-1773	550	14	lengths	length	NOUN
ejpam-1773	550	15	from	from	ADP
ejpam-1773	550	16	rk	rk	NOUN
ejpam-1773	550	17	.	.	PUNCT
ejpam-1773	551	1	this	this	PRON
ejpam-1773	551	2	can	can	AUX
ejpam-1773	551	3	be	be	AUX
ejpam-1773	551	4	done	do	VERB
ejpam-1773	551	5	for	for	ADP
ejpam-1773	551	6	more	more	ADJ
ejpam-1773	551	7	lengths	length	NOUN
ejpam-1773	551	8	and	and	CCONJ
ejpam-1773	551	9	to	to	ADP
ejpam-1773	551	10	a	a	DET
ejpam-1773	551	11	further	further	ADJ
ejpam-1773	551	12	extent	extent	NOUN
ejpam-1773	551	13	.	.	PUNCT
ejpam-1773	552	1	acknowledgements	acknowledgement	NOUN
ejpam-1773	552	2	the	the	DET
ejpam-1773	552	3	authors	author	NOUN
ejpam-1773	552	4	wish	wish	VERB
ejpam-1773	552	5	to	to	PART
ejpam-1773	552	6	thank	thank	VERB
ejpam-1773	552	7	the	the	DET
ejpam-1773	552	8	anonymous	anonymous	ADJ
ejpam-1773	552	9	referees	referee	NOUN
ejpam-1773	552	10	for	for	ADP
ejpam-1773	552	11	their	their	PRON
ejpam-1773	552	12	useful	useful	ADJ
ejpam-1773	552	13	comments	comment	NOUN
ejpam-1773	552	14	and	and	CCONJ
ejpam-1773	552	15	suggestions	suggestion	NOUN
ejpam-1773	552	16	.	.	PUNCT
ejpam-1773	553	1	references	reference	NOUN
ejpam-1773	553	2	[	[	X
ejpam-1773	553	3	1	1	NUM
ejpam-1773	553	4	]	]	X
ejpam-1773	553	5	e.	e.	PROPN
ejpam-1773	553	6	bannai	bannai	PROPN
ejpam-1773	553	7	,	,	PUNCT
ejpam-1773	553	8	s.t	s.t	PROPN
ejpam-1773	553	9	.	.	PROPN
ejpam-1773	553	10	dougherty	dougherty	PROPN
ejpam-1773	553	11	,	,	PUNCT
ejpam-1773	553	12	m.	m.	NOUN
ejpam-1773	553	13	harada	harada	PROPN
ejpam-1773	553	14	,	,	PUNCT
ejpam-1773	553	15	and	and	CCONJ
ejpam-1773	553	16	m.	m.	NOUN
ejpam-1773	553	17	oura	oura	PROPN
ejpam-1773	553	18	.	.	PUNCT
ejpam-1773	554	1	type	type	PROPN
ejpam-1773	554	2	ii	ii	PROPN
ejpam-1773	554	3	codes	code	NOUN
ejpam-1773	554	4	,	,	PUNCT
ejpam-1773	554	5	even	even	ADV
ejpam-1773	554	6	unimodular	unimodular	ADJ
ejpam-1773	554	7	lattices	lattice	NOUN
ejpam-1773	554	8	,	,	PUNCT
ejpam-1773	554	9	and	and	CCONJ
ejpam-1773	554	10	invariant	invariant	ADJ
ejpam-1773	554	11	rings	ring	NOUN
ejpam-1773	554	12	,	,	PUNCT
ejpam-1773	554	13	ieee	ieee	NOUN
ejpam-1773	554	14	transactions	transaction	NOUN
ejpam-1773	554	15	on	on	ADP
ejpam-1773	554	16	information	information	NOUN
ejpam-1773	554	17	theory	theory	NOUN
ejpam-1773	554	18	,	,	PUNCT
ejpam-1773	554	19	45:1194	45:1194	NOUN
ejpam-1773	554	20	-	-	SYM
ejpam-1773	554	21	1205	1205	NUM
ejpam-1773	554	22	,	,	PUNCT
ejpam-1773	554	23	1999	1999	NUM
ejpam-1773	554	24	.	.	PUNCT
ejpam-1773	555	1	[	[	X
ejpam-1773	555	2	2	2	NUM
ejpam-1773	555	3	]	]	X
ejpam-1773	555	4	y.j	y.j	PROPN
ejpam-1773	555	5	.	.	PROPN
ejpam-1773	555	6	choie	choie	NOUN
ejpam-1773	555	7	and	and	CCONJ
ejpam-1773	555	8	s.t	s.t	PROPN
ejpam-1773	555	9	.	.	PROPN
ejpam-1773	555	10	dougherty	dougherty	PROPN
ejpam-1773	555	11	.	.	PUNCT
ejpam-1773	556	1	codes	code	NOUN
ejpam-1773	556	2	over	over	ADP
ejpam-1773	556	3	σ2	σ2	PROPN
ejpam-1773	556	4	m	m	PROPN
ejpam-1773	556	5	and	and	CCONJ
ejpam-1773	556	6	jacobi	jacobi	PROPN
ejpam-1773	556	7	forms	form	NOUN
ejpam-1773	556	8	over	over	ADP
ejpam-1773	556	9	the	the	DET
ejpam-1773	556	10	quaternions	quaternion	NOUN
ejpam-1773	556	11	,	,	PUNCT
ejpam-1773	556	12	applicable	applicable	ADJ
ejpam-1773	556	13	algebra	algebra	NOUN
ejpam-1773	556	14	in	in	ADP
ejpam-1773	556	15	engineering	engineering	NOUN
ejpam-1773	556	16	,	,	PUNCT
ejpam-1773	556	17	communications	communication	NOUN
ejpam-1773	556	18	and	and	CCONJ
ejpam-1773	556	19	computing	compute	VERB
ejpam-1773	556	20	15:129	15:129	NUM
ejpam-1773	556	21	-	-	SYM
ejpam-1773	556	22	147	147	NUM
ejpam-1773	556	23	,	,	PUNCT
ejpam-1773	556	24	2004	2004	NUM
ejpam-1773	556	25	.	.	PUNCT
ejpam-1773	557	1	[	[	X
ejpam-1773	557	2	3	3	X
ejpam-1773	557	3	]	]	X
ejpam-1773	557	4	j.h	j.h	PROPN
ejpam-1773	557	5	.	.	PROPN
ejpam-1773	557	6	conway	conway	PROPN
ejpam-1773	557	7	and	and	CCONJ
ejpam-1773	557	8	n.j.a	n.j.a	PROPN
ejpam-1773	557	9	.	.	PROPN
ejpam-1773	557	10	sloane	sloane	NOUN
ejpam-1773	557	11	.	.	PUNCT
ejpam-1773	558	1	sphere	sphere	NOUN
ejpam-1773	558	2	packing	packing	NOUN
ejpam-1773	558	3	,	,	PUNCT
ejpam-1773	558	4	lattices	lattice	NOUN
ejpam-1773	558	5	and	and	CCONJ
ejpam-1773	558	6	groups	group	NOUN
ejpam-1773	558	7	(	(	PUNCT
ejpam-1773	558	8	2nd	2nd	ADJ
ejpam-1773	558	9	ed	ed	NOUN
ejpam-1773	558	10	.	.	PUNCT
ejpam-1773	558	11	)	)	PUNCT
ejpam-1773	558	12	,	,	PUNCT
ejpam-1773	558	13	new	new	PROPN
ejpam-1773	558	14	york	york	PROPN
ejpam-1773	558	15	:	:	PUNCT
ejpam-1773	558	16	springer	springer	NOUN
ejpam-1773	558	17	-	-	PUNCT
ejpam-1773	558	18	verlag	verlag	PROPN
ejpam-1773	558	19	,	,	PUNCT
ejpam-1773	558	20	1993	1993	NUM
ejpam-1773	558	21	.	.	PUNCT
ejpam-1773	559	1	[	[	X
ejpam-1773	559	2	4	4	X
ejpam-1773	559	3	]	]	PUNCT
ejpam-1773	559	4	j.	j.	PROPN
ejpam-1773	559	5	h.	h.	PROPN
ejpam-1773	559	6	conway	conway	PROPN
ejpam-1773	559	7	and	and	CCONJ
ejpam-1773	559	8	n.	n.	PROPN
ejpam-1773	559	9	j.	j.	PROPN
ejpam-1773	559	10	a.	a.	PROPN
ejpam-1773	559	11	sloane	sloane	PROPN
ejpam-1773	559	12	.	.	PUNCT
ejpam-1773	560	1	sphere	sphere	NOUN
ejpam-1773	560	2	packings	packing	NOUN
ejpam-1773	560	3	,	,	PUNCT
ejpam-1773	560	4	lattices	lattice	NOUN
ejpam-1773	560	5	and	and	CCONJ
ejpam-1773	560	6	groups	group	NOUN
ejpam-1773	560	7	.	.	PUNCT
ejpam-1773	561	1	springer	springer	NOUN
ejpam-1773	561	2	-	-	PUNCT
ejpam-1773	561	3	verlag	verlag	PROPN
ejpam-1773	561	4	,	,	PUNCT
ejpam-1773	561	5	ny	ny	PROPN
ejpam-1773	561	6	,	,	PUNCT
ejpam-1773	561	7	3rd	3rd	ADJ
ejpam-1773	561	8	ed	ed	NOUN
ejpam-1773	561	9	.	.	PROPN
ejpam-1773	561	10	,	,	PUNCT
ejpam-1773	561	11	1998	1998	NUM
ejpam-1773	561	12	.	.	PUNCT
ejpam-1773	562	1	[	[	X
ejpam-1773	562	2	5	5	NUM
ejpam-1773	562	3	]	]	X
ejpam-1773	562	4	s.t	s.t	PROPN
ejpam-1773	562	5	.	.	PROPN
ejpam-1773	562	6	dougherty	dougherty	PROPN
ejpam-1773	562	7	,	,	PUNCT
ejpam-1773	562	8	t.	t.	NOUN
ejpam-1773	562	9	a.	a.	NOUN
ejpam-1773	562	10	gulliver	gulliver	NOUN
ejpam-1773	562	11	and	and	CCONJ
ejpam-1773	562	12	m.	m.	NOUN
ejpam-1773	562	13	harada	harada	PROPN
ejpam-1773	562	14	.	.	PUNCT
ejpam-1773	563	1	type	type	PROPN
ejpam-1773	563	2	ii	ii	PROPN
ejpam-1773	563	3	self	self	NOUN
ejpam-1773	563	4	-	-	PUNCT
ejpam-1773	563	5	dual	dual	ADJ
ejpam-1773	563	6	codes	code	NOUN
ejpam-1773	563	7	over	over	ADP
ejpam-1773	563	8	finite	finite	ADJ
ejpam-1773	563	9	rings	ring	NOUN
ejpam-1773	563	10	and	and	CCONJ
ejpam-1773	563	11	even	even	ADV
ejpam-1773	563	12	unimodular	unimodular	ADJ
ejpam-1773	563	13	lattices	lattice	NOUN
ejpam-1773	563	14	,	,	PUNCT
ejpam-1773	563	15	journal	journal	NOUN
ejpam-1773	563	16	of	of	ADP
ejpam-1773	563	17	algebraic	algebraic	PROPN
ejpam-1773	563	18	combinatorics	combinatoric	NOUN
ejpam-1773	563	19	,	,	PUNCT
ejpam-1773	563	20	9:233–250	9:233–250	NUM
ejpam-1773	563	21	,	,	PUNCT
ejpam-1773	563	22	1999	1999	NUM
ejpam-1773	563	23	.	.	PUNCT
ejpam-1773	564	1	references	reference	NOUN
ejpam-1773	564	2	106	106	NUM
ejpam-1773	565	1	[	[	X
ejpam-1773	565	2	6	6	NUM
ejpam-1773	565	3	]	]	X
ejpam-1773	565	4	s.t	s.t	PROPN
ejpam-1773	565	5	.	.	PROPN
ejpam-1773	565	6	dougherty	dougherty	PROPN
ejpam-1773	565	7	,	,	PUNCT
ejpam-1773	565	8	m.	m.	NOUN
ejpam-1773	565	9	harada	harada	PROPN
ejpam-1773	565	10	,	,	PUNCT
ejpam-1773	565	11	p.	p.	NOUN
ejpam-1773	565	12	gaborit	gaborit	NOUN
ejpam-1773	565	13	,	,	PUNCT
ejpam-1773	565	14	and	and	CCONJ
ejpam-1773	565	15	p.	p.	PROPN
ejpam-1773	565	16	solé	solé	NOUN
ejpam-1773	565	17	.	.	PUNCT
ejpam-1773	566	1	type	type	PROPN
ejpam-1773	566	2	ii	ii	PROPN
ejpam-1773	566	3	codes	code	NOUN
ejpam-1773	566	4	over	over	ADP
ejpam-1773	566	5	f2	f2	PROPN
ejpam-1773	566	6	+	+	CCONJ
ejpam-1773	566	7	uf2	uf2	NOUN
ejpam-1773	566	8	,	,	PUNCT
ejpam-1773	566	9	ieee	ieee	NOUN
ejpam-1773	566	10	transactions	transaction	NOUN
ejpam-1773	566	11	on	on	ADP
ejpam-1773	566	12	information	information	NOUN
ejpam-1773	566	13	theory	theory	NOUN
ejpam-1773	566	14	,	,	PUNCT
ejpam-1773	566	15	45:32	45:32	NUM
ejpam-1773	566	16	-	-	SYM
ejpam-1773	566	17	45	45	NUM
ejpam-1773	566	18	,	,	PUNCT
ejpam-1773	566	19	1999	1999	NUM
ejpam-1773	566	20	.	.	PUNCT
ejpam-1773	567	1	[	[	X
ejpam-1773	567	2	7	7	X
ejpam-1773	567	3	]	]	X
ejpam-1773	567	4	s.t	s.t	PROPN
ejpam-1773	567	5	.	.	PROPN
ejpam-1773	567	6	dougherty	dougherty	PROPN
ejpam-1773	567	7	,	,	PUNCT
ejpam-1773	567	8	b.	b.	PROPN
ejpam-1773	567	9	yıldız	yıldız	PROPN
ejpam-1773	567	10	and	and	CCONJ
ejpam-1773	567	11	s.	s.	PROPN
ejpam-1773	567	12	karadeniz	karadeniz	PROPN
ejpam-1773	567	13	.	.	PUNCT
ejpam-1773	567	14	codes	code	NOUN
ejpam-1773	567	15	over	over	ADP
ejpam-1773	567	16	rk	rk	NOUN
ejpam-1773	567	17	,	,	PUNCT
ejpam-1773	567	18	gray	gray	ADJ
ejpam-1773	567	19	maps	map	NOUN
ejpam-1773	567	20	and	and	CCONJ
ejpam-1773	567	21	their	their	PRON
ejpam-1773	567	22	binary	binary	ADJ
ejpam-1773	567	23	images	image	NOUN
ejpam-1773	567	24	,	,	PUNCT
ejpam-1773	567	25	finite	finite	ADJ
ejpam-1773	567	26	fields	field	NOUN
ejpam-1773	567	27	and	and	CCONJ
ejpam-1773	567	28	their	their	PRON
ejpam-1773	567	29	applications	application	NOUN
ejpam-1773	567	30	17:205–219	17:205–219	NUM
ejpam-1773	567	31	,	,	PUNCT
ejpam-1773	567	32	2011	2011	NUM
ejpam-1773	567	33	.	.	PUNCT
ejpam-1773	568	1	[	[	X
ejpam-1773	568	2	8	8	NUM
ejpam-1773	568	3	]	]	X
ejpam-1773	568	4	f.j	f.j	PROPN
ejpam-1773	568	5	.	.	PROPN
ejpam-1773	568	6	macwilliams	macwilliam	NOUN
ejpam-1773	568	7	and	and	CCONJ
ejpam-1773	568	8	n.j.a	n.j.a	PROPN
ejpam-1773	568	9	.	.	PROPN
ejpam-1773	568	10	sloane	sloane	NOUN
ejpam-1773	568	11	.	.	PUNCT
ejpam-1773	569	1	the	the	DET
ejpam-1773	569	2	theory	theory	NOUN
ejpam-1773	569	3	of	of	ADP
ejpam-1773	569	4	error	error	NOUN
ejpam-1773	569	5	-	-	PUNCT
ejpam-1773	569	6	correcting	correct	VERB
ejpam-1773	569	7	codes	code	NOUN
ejpam-1773	569	8	.	.	PUNCT
ejpam-1773	570	1	amsterdam	amsterdam	ADJ
ejpam-1773	570	2	:	:	PUNCT
ejpam-1773	570	3	north	north	NOUN
ejpam-1773	570	4	-	-	PUNCT
ejpam-1773	570	5	holland	holland	PROPN
ejpam-1773	570	6	,	,	PUNCT
ejpam-1773	570	7	1977	1977	NUM
ejpam-1773	570	8	.	.	PUNCT
ejpam-1773	571	1	[	[	X
ejpam-1773	571	2	9	9	NUM
ejpam-1773	571	3	]	]	X
ejpam-1773	571	4	g.	g.	PROPN
ejpam-1773	571	5	nebe	nebe	PROPN
ejpam-1773	571	6	,	,	PUNCT
ejpam-1773	571	7	e.	e.	PROPN
ejpam-1773	571	8	m.	m.	PROPN
ejpam-1773	571	9	rains	rain	NOUN
ejpam-1773	571	10	and	and	CCONJ
ejpam-1773	571	11	n.	n.	PROPN
ejpam-1773	571	12	j.	j.	PROPN
ejpam-1773	571	13	a.	a.	PROPN
ejpam-1773	571	14	sloane	sloane	PROPN
ejpam-1773	571	15	.	.	PUNCT
ejpam-1773	572	1	self	self	NOUN
ejpam-1773	572	2	-	-	PUNCT
ejpam-1773	572	3	dual	dual	ADJ
ejpam-1773	572	4	codes	code	NOUN
ejpam-1773	572	5	and	and	CCONJ
ejpam-1773	572	6	invariant	invariant	ADJ
ejpam-1773	572	7	theory	theory	NOUN
ejpam-1773	572	8	.	.	PUNCT
ejpam-1773	573	1	springerverlag	springerverlag	PROPN
ejpam-1773	573	2	,	,	PUNCT
ejpam-1773	573	3	2006	2006	NUM
ejpam-1773	573	4	.	.	PUNCT
ejpam-1773	574	1	[	[	X
ejpam-1773	574	2	10	10	NUM
ejpam-1773	574	3	]	]	X
ejpam-1773	574	4	e.m	e.m	PROPN
ejpam-1773	574	5	.	.	PROPN
ejpam-1773	574	6	rains	rain	NOUN
ejpam-1773	574	7	.	.	PUNCT
ejpam-1773	575	1	shadow	shadow	NOUN
ejpam-1773	575	2	bounds	bound	VERB
ejpam-1773	575	3	for	for	ADP
ejpam-1773	575	4	self	self	NOUN
ejpam-1773	575	5	-	-	PUNCT
ejpam-1773	575	6	dual	dual	ADJ
ejpam-1773	575	7	codes	code	NOUN
ejpam-1773	575	8	,	,	PUNCT
ejpam-1773	575	9	ieee	ieee	NOUN
ejpam-1773	575	10	transactions	transaction	NOUN
ejpam-1773	575	11	on	on	ADP
ejpam-1773	575	12	information	information	NOUN
ejpam-1773	575	13	theory	theory	NOUN
ejpam-1773	575	14	,	,	PUNCT
ejpam-1773	575	15	44:134–139	44:134–139	PROPN
ejpam-1773	575	16	,	,	PUNCT
ejpam-1773	575	17	1998	1998	NUM
ejpam-1773	575	18	.	.	PUNCT
ejpam-1773	576	1	[	[	X
ejpam-1773	576	2	11	11	NUM
ejpam-1773	576	3	]	]	PUNCT
ejpam-1773	576	4	j.	j.	PROPN
ejpam-1773	576	5	wood	wood	PROPN
ejpam-1773	576	6	.	.	PUNCT
ejpam-1773	577	1	duality	duality	NOUN
ejpam-1773	577	2	for	for	ADP
ejpam-1773	577	3	modules	module	NOUN
ejpam-1773	577	4	over	over	ADP
ejpam-1773	577	5	finite	finite	ADJ
ejpam-1773	577	6	rings	ring	NOUN
ejpam-1773	577	7	and	and	CCONJ
ejpam-1773	577	8	applications	application	NOUN
ejpam-1773	577	9	to	to	ADP
ejpam-1773	577	10	coding	code	VERB
ejpam-1773	577	11	theory	theory	NOUN
ejpam-1773	577	12	.	.	PUNCT
ejpam-1773	578	1	american	american	PROPN
ejpam-1773	578	2	journal	journal	PROPN
ejpam-1773	578	3	of	of	ADP
ejpam-1773	578	4	mathematics	mathematic	NOUN
ejpam-1773	578	5	,	,	PUNCT
ejpam-1773	578	6	121:555	121:555	NOUN
ejpam-1773	578	7	-	-	PUNCT
ejpam-1773	578	8	575	575	NUM
ejpam-1773	578	9	,	,	PUNCT
ejpam-1773	578	10	1999	1999	NUM
ejpam-1773	578	11	.	.	PUNCT