id	sid	tid	token	lemma	pos
ejpam-3838	1	1	european	european	PROPN
ejpam-3838	1	2	journal	journal	PROPN
ejpam-3838	1	3	of	of	ADP
ejpam-3838	1	4	pure	pure	ADJ
ejpam-3838	1	5	and	and	CCONJ
ejpam-3838	1	6	applied	apply	VERB
ejpam-3838	1	7	mathematics	mathematic	NOUN
ejpam-3838	1	8	vol	vol	NOUN
ejpam-3838	1	9	.	.	PROPN
ejpam-3838	2	1	13	13	NUM
ejpam-3838	2	2	,	,	PUNCT
ejpam-3838	2	3	no	no	INTJ
ejpam-3838	2	4	.	.	NOUN
ejpam-3838	2	5	4	4	NUM
ejpam-3838	2	6	,	,	PUNCT
ejpam-3838	2	7	2020	2020	NUM
ejpam-3838	2	8	,	,	PUNCT
ejpam-3838	2	9	873	873	NUM
ejpam-3838	2	10	-	-	NUM
ejpam-3838	2	11	892	892	NUM
ejpam-3838	2	12	issn	issn	PROPN
ejpam-3838	2	13	1307	1307	NUM
ejpam-3838	2	14	-	-	SYM
ejpam-3838	2	15	5543	5543	NUM
ejpam-3838	2	16	–	–	PUNCT
ejpam-3838	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3838	2	18	published	publish	VERB
ejpam-3838	2	19	by	by	ADP
ejpam-3838	2	20	new	new	PROPN
ejpam-3838	2	21	york	york	PROPN
ejpam-3838	2	22	business	business	PROPN
ejpam-3838	2	23	global	global	ADJ
ejpam-3838	2	24	self	self	NOUN
ejpam-3838	2	25	-	-	PUNCT
ejpam-3838	2	26	orthogonal	orthogonal	ADJ
ejpam-3838	2	27	codes	code	NOUN
ejpam-3838	2	28	over	over	ADP
ejpam-3838	2	29	fq	fq	PROPN
ejpam-3838	2	30	+	+	CCONJ
ejpam-3838	2	31	ufq	ufq	PROPN
ejpam-3838	2	32	and	and	CCONJ
ejpam-3838	2	33	fq	fq	PROPN
ejpam-3838	2	34	+	+	CCONJ
ejpam-3838	2	35	ufq	ufq	PROPN
ejpam-3838	2	36	+	+	NUM
ejpam-3838	2	37	u2fq	u2fq	PUNCT
ejpam-3838	2	38	lucky	lucky	PROPN
ejpam-3838	2	39	erap	erap	PROPN
ejpam-3838	2	40	galvez1,∗	galvez1,∗	PROPN
ejpam-3838	2	41	,	,	PUNCT
ejpam-3838	2	42	rowena	rowena	PROPN
ejpam-3838	2	43	alma	alma	PROPN
ejpam-3838	2	44	betty1	betty1	PROPN
ejpam-3838	2	45	,	,	PUNCT
ejpam-3838	2	46	fidel	fidel	PROPN
ejpam-3838	2	47	nemenzo1	nemenzo1	PROPN
ejpam-3838	2	48	1	1	NUM
ejpam-3838	2	49	institute	institute	PROPN
ejpam-3838	2	50	of	of	ADP
ejpam-3838	2	51	mathematics	mathematics	PROPN
ejpam-3838	2	52	,	,	PUNCT
ejpam-3838	2	53	university	university	NOUN
ejpam-3838	2	54	of	of	ADP
ejpam-3838	2	55	the	the	DET
ejpam-3838	2	56	philippines	philippines	PROPN
ejpam-3838	2	57	diliman	diliman	PROPN
ejpam-3838	2	58	,	,	PUNCT
ejpam-3838	2	59	quezon	quezon	PROPN
ejpam-3838	2	60	city	city	PROPN
ejpam-3838	2	61	,	,	PUNCT
ejpam-3838	2	62	philippines	philippine	NOUN
ejpam-3838	2	63	abstract	abstract	ADJ
ejpam-3838	2	64	.	.	PUNCT
ejpam-3838	3	1	in	in	ADP
ejpam-3838	3	2	this	this	DET
ejpam-3838	3	3	paper	paper	NOUN
ejpam-3838	3	4	,	,	PUNCT
ejpam-3838	3	5	we	we	PRON
ejpam-3838	3	6	establish	establish	VERB
ejpam-3838	3	7	a	a	DET
ejpam-3838	3	8	mass	mass	ADJ
ejpam-3838	3	9	formula	formula	NOUN
ejpam-3838	3	10	for	for	ADP
ejpam-3838	3	11	euclidean	euclidean	ADJ
ejpam-3838	3	12	and	and	CCONJ
ejpam-3838	3	13	hermitian	hermitian	ADJ
ejpam-3838	3	14	self	self	NOUN
ejpam-3838	3	15	-	-	PUNCT
ejpam-3838	3	16	orthogonal	orthogonal	ADJ
ejpam-3838	3	17	codes	code	NOUN
ejpam-3838	3	18	over	over	ADP
ejpam-3838	3	19	the	the	DET
ejpam-3838	3	20	finite	finite	PROPN
ejpam-3838	3	21	ring	ring	NOUN
ejpam-3838	3	22	fq	fq	PROPN
ejpam-3838	3	23	+	+	CCONJ
ejpam-3838	3	24	ufq	ufq	PROPN
ejpam-3838	3	25	,	,	PUNCT
ejpam-3838	3	26	where	where	SCONJ
ejpam-3838	3	27	fq	fq	PROPN
ejpam-3838	3	28	is	be	AUX
ejpam-3838	3	29	the	the	DET
ejpam-3838	3	30	finite	finite	ADJ
ejpam-3838	3	31	field	field	NOUN
ejpam-3838	3	32	of	of	ADP
ejpam-3838	3	33	order	order	NOUN
ejpam-3838	3	34	q	q	NOUN
ejpam-3838	3	35	and	and	CCONJ
ejpam-3838	3	36	u2	u2	PROPN
ejpam-3838	3	37	=	=	NOUN
ejpam-3838	3	38	0	0	NUM
ejpam-3838	3	39	.	.	PUNCT
ejpam-3838	4	1	we	we	PRON
ejpam-3838	4	2	also	also	ADV
ejpam-3838	4	3	establish	establish	VERB
ejpam-3838	4	4	a	a	DET
ejpam-3838	4	5	mass	mass	ADJ
ejpam-3838	4	6	formula	formula	NOUN
ejpam-3838	4	7	for	for	ADP
ejpam-3838	4	8	euclidean	euclidean	ADJ
ejpam-3838	4	9	self	self	NOUN
ejpam-3838	4	10	-	-	PUNCT
ejpam-3838	4	11	orthogonal	orthogonal	ADJ
ejpam-3838	4	12	codes	code	NOUN
ejpam-3838	4	13	over	over	ADP
ejpam-3838	4	14	the	the	DET
ejpam-3838	4	15	finite	finite	PROPN
ejpam-3838	4	16	ring	ring	NOUN
ejpam-3838	4	17	fq	fq	PROPN
ejpam-3838	4	18	+	+	CCONJ
ejpam-3838	4	19	ufq	ufq	PROPN
ejpam-3838	4	20	+	+	NUM
ejpam-3838	4	21	u2fq	u2fq	PUNCT
ejpam-3838	4	22	,	,	PUNCT
ejpam-3838	4	23	with	with	ADP
ejpam-3838	4	24	u3	u3	NOUN
ejpam-3838	4	25	=	=	SYM
ejpam-3838	4	26	0	0	NUM
ejpam-3838	4	27	and	and	CCONJ
ejpam-3838	4	28	characteristic	characteristic	NOUN
ejpam-3838	4	29	of	of	ADP
ejpam-3838	4	30	fq	fq	PROPN
ejpam-3838	4	31	is	be	AUX
ejpam-3838	4	32	odd	odd	ADJ
ejpam-3838	4	33	.	.	PUNCT
ejpam-3838	5	1	these	these	DET
ejpam-3838	5	2	mass	mass	ADJ
ejpam-3838	5	3	formulas	formula	NOUN
ejpam-3838	5	4	are	be	AUX
ejpam-3838	5	5	used	use	VERB
ejpam-3838	5	6	to	to	PART
ejpam-3838	5	7	give	give	VERB
ejpam-3838	5	8	a	a	DET
ejpam-3838	5	9	classification	classification	NOUN
ejpam-3838	5	10	of	of	ADP
ejpam-3838	5	11	euclidean	euclidean	ADJ
ejpam-3838	5	12	and	and	CCONJ
ejpam-3838	5	13	hermitian	hermitian	ADJ
ejpam-3838	5	14	self	self	NOUN
ejpam-3838	5	15	-	-	PUNCT
ejpam-3838	5	16	orthogonal	orthogonal	ADJ
ejpam-3838	5	17	codes	code	NOUN
ejpam-3838	5	18	over	over	ADP
ejpam-3838	5	19	f2	f2	PROPN
ejpam-3838	5	20	+	+	CCONJ
ejpam-3838	5	21	uf2	uf2	NOUN
ejpam-3838	5	22	and	and	CCONJ
ejpam-3838	5	23	f3	f3	PROPN
ejpam-3838	5	24	+	+	CCONJ
ejpam-3838	5	25	uf3	uf3	NOUN
ejpam-3838	5	26	of	of	ADP
ejpam-3838	5	27	small	small	ADJ
ejpam-3838	5	28	lengths	length	NOUN
ejpam-3838	5	29	.	.	PUNCT
ejpam-3838	6	1	2020	2020	NUM
ejpam-3838	6	2	mathematics	mathematic	NOUN
ejpam-3838	6	3	subject	subject	NOUN
ejpam-3838	6	4	classifications	classification	NOUN
ejpam-3838	6	5	:	:	PUNCT
ejpam-3838	6	6	94b05	94b05	NUM
ejpam-3838	6	7	key	key	ADJ
ejpam-3838	6	8	words	word	NOUN
ejpam-3838	6	9	and	and	CCONJ
ejpam-3838	6	10	phrases	phrase	NOUN
ejpam-3838	6	11	:	:	PUNCT
ejpam-3838	6	12	codes	code	NOUN
ejpam-3838	6	13	over	over	ADP
ejpam-3838	6	14	rings	ring	NOUN
ejpam-3838	6	15	,	,	PUNCT
ejpam-3838	6	16	self	self	NOUN
ejpam-3838	6	17	-	-	PUNCT
ejpam-3838	6	18	orthogonal	orthogonal	ADJ
ejpam-3838	6	19	codes	code	NOUN
ejpam-3838	6	20	,	,	PUNCT
ejpam-3838	6	21	mass	mass	ADJ
ejpam-3838	6	22	formula	formula	NOUN
ejpam-3838	6	23	1	1	NUM
ejpam-3838	6	24	.	.	PUNCT
ejpam-3838	7	1	introduction	introduction	NOUN
ejpam-3838	7	2	self	self	NOUN
ejpam-3838	7	3	-	-	PUNCT
ejpam-3838	7	4	dual	dual	ADJ
ejpam-3838	7	5	codes	code	NOUN
ejpam-3838	7	6	have	have	VERB
ejpam-3838	7	7	rich	rich	ADJ
ejpam-3838	7	8	mathematical	mathematical	ADJ
ejpam-3838	7	9	theory	theory	NOUN
ejpam-3838	7	10	and	and	CCONJ
ejpam-3838	7	11	are	be	AUX
ejpam-3838	7	12	of	of	ADP
ejpam-3838	7	13	great	great	ADJ
ejpam-3838	7	14	interest	interest	NOUN
ejpam-3838	7	15	to	to	ADP
ejpam-3838	7	16	researchers	researcher	NOUN
ejpam-3838	7	17	because	because	SCONJ
ejpam-3838	7	18	many	many	ADJ
ejpam-3838	7	19	of	of	ADP
ejpam-3838	7	20	the	the	DET
ejpam-3838	7	21	best	well	ADV
ejpam-3838	7	22	known	know	VERB
ejpam-3838	7	23	codes	code	NOUN
ejpam-3838	7	24	are	be	AUX
ejpam-3838	7	25	self	self	NOUN
ejpam-3838	7	26	-	-	PUNCT
ejpam-3838	7	27	dual	dual	ADJ
ejpam-3838	7	28	.	.	PUNCT
ejpam-3838	8	1	a	a	DET
ejpam-3838	8	2	fundamental	fundamental	ADJ
ejpam-3838	8	3	problem	problem	NOUN
ejpam-3838	8	4	in	in	ADP
ejpam-3838	8	5	coding	code	VERB
ejpam-3838	8	6	theory	theory	NOUN
ejpam-3838	8	7	is	be	AUX
ejpam-3838	8	8	the	the	DET
ejpam-3838	8	9	classification	classification	NOUN
ejpam-3838	8	10	of	of	ADP
ejpam-3838	8	11	self	self	NOUN
ejpam-3838	8	12	-	-	PUNCT
ejpam-3838	8	13	dual	dual	ADJ
ejpam-3838	8	14	codes	code	NOUN
ejpam-3838	8	15	,	,	PUNCT
ejpam-3838	8	16	that	that	ADV
ejpam-3838	8	17	is	is	ADV
ejpam-3838	8	18	,	,	PUNCT
ejpam-3838	8	19	an	an	DET
ejpam-3838	8	20	enumeration	enumeration	NOUN
ejpam-3838	8	21	of	of	ADP
ejpam-3838	8	22	a	a	DET
ejpam-3838	8	23	complete	complete	ADJ
ejpam-3838	8	24	set	set	NOUN
ejpam-3838	8	25	of	of	ADP
ejpam-3838	8	26	representatives	representative	NOUN
ejpam-3838	8	27	for	for	ADP
ejpam-3838	8	28	the	the	DET
ejpam-3838	8	29	equivalence	equivalence	NOUN
ejpam-3838	8	30	classes	class	NOUN
ejpam-3838	8	31	of	of	ADP
ejpam-3838	8	32	self	self	NOUN
ejpam-3838	8	33	-	-	PUNCT
ejpam-3838	8	34	dual	dual	ADJ
ejpam-3838	8	35	codes	code	NOUN
ejpam-3838	8	36	.	.	PUNCT
ejpam-3838	9	1	in	in	ADP
ejpam-3838	9	2	the	the	DET
ejpam-3838	9	3	past	past	ADJ
ejpam-3838	9	4	years	year	NOUN
ejpam-3838	9	5	,	,	PUNCT
ejpam-3838	9	6	self	self	NOUN
ejpam-3838	9	7	-	-	PUNCT
ejpam-3838	9	8	dual	dual	ADJ
ejpam-3838	9	9	codes	code	NOUN
ejpam-3838	9	10	over	over	ADP
ejpam-3838	9	11	finite	finite	ADJ
ejpam-3838	9	12	fields	field	NOUN
ejpam-3838	9	13	have	have	AUX
ejpam-3838	9	14	been	be	AUX
ejpam-3838	9	15	extensively	extensively	ADV
ejpam-3838	9	16	studied	study	VERB
ejpam-3838	9	17	and	and	CCONJ
ejpam-3838	9	18	classified	classify	VERB
ejpam-3838	9	19	up	up	ADP
ejpam-3838	9	20	to	to	ADP
ejpam-3838	9	21	various	various	ADJ
ejpam-3838	9	22	lengths	length	NOUN
ejpam-3838	9	23	(	(	PUNCT
ejpam-3838	9	24	see	see	VERB
ejpam-3838	9	25	[	[	X
ejpam-3838	9	26	8	8	NUM
ejpam-3838	9	27	,	,	PUNCT
ejpam-3838	9	28	11	11	NUM
ejpam-3838	9	29	]	]	NUM
ejpam-3838	9	30	)	)	PUNCT
ejpam-3838	9	31	.	.	PUNCT
ejpam-3838	10	1	since	since	SCONJ
ejpam-3838	10	2	the	the	DET
ejpam-3838	10	3	discovery	discovery	NOUN
ejpam-3838	10	4	in	in	ADP
ejpam-3838	10	5	1994	1994	NUM
ejpam-3838	10	6	[	[	X
ejpam-3838	10	7	7	7	X
ejpam-3838	10	8	]	]	PUNCT
ejpam-3838	10	9	that	that	SCONJ
ejpam-3838	10	10	certain	certain	ADJ
ejpam-3838	10	11	non	non	ADJ
ejpam-3838	10	12	-	-	ADJ
ejpam-3838	10	13	linear	linear	ADJ
ejpam-3838	10	14	binary	binary	ADJ
ejpam-3838	10	15	codes	code	NOUN
ejpam-3838	10	16	can	can	AUX
ejpam-3838	10	17	be	be	AUX
ejpam-3838	10	18	viewed	view	VERB
ejpam-3838	10	19	as	as	ADP
ejpam-3838	10	20	linear	linear	ADJ
ejpam-3838	10	21	codes	code	NOUN
ejpam-3838	10	22	over	over	ADP
ejpam-3838	10	23	the	the	DET
ejpam-3838	10	24	ring	ring	NOUN
ejpam-3838	10	25	z4	z4	PROPN
ejpam-3838	10	26	,	,	PUNCT
ejpam-3838	10	27	there	there	PRON
ejpam-3838	10	28	has	have	AUX
ejpam-3838	10	29	been	be	AUX
ejpam-3838	10	30	much	much	ADJ
ejpam-3838	10	31	interest	interest	NOUN
ejpam-3838	10	32	in	in	ADP
ejpam-3838	10	33	the	the	DET
ejpam-3838	10	34	study	study	NOUN
ejpam-3838	10	35	of	of	ADP
ejpam-3838	10	36	self	self	NOUN
ejpam-3838	10	37	-	-	PUNCT
ejpam-3838	10	38	dual	dual	ADJ
ejpam-3838	10	39	codes	code	NOUN
ejpam-3838	10	40	over	over	ADP
ejpam-3838	10	41	various	various	ADJ
ejpam-3838	10	42	finite	finite	ADJ
ejpam-3838	10	43	rings	ring	NOUN
ejpam-3838	10	44	.	.	PUNCT
ejpam-3838	11	1	a	a	DET
ejpam-3838	11	2	key	key	ADJ
ejpam-3838	11	3	problem	problem	NOUN
ejpam-3838	11	4	is	be	AUX
ejpam-3838	11	5	to	to	PART
ejpam-3838	11	6	establish	establish	VERB
ejpam-3838	11	7	an	an	DET
ejpam-3838	11	8	explicit	explicit	ADJ
ejpam-3838	11	9	formula	formula	NOUN
ejpam-3838	11	10	for	for	ADP
ejpam-3838	11	11	the	the	DET
ejpam-3838	11	12	number	number	NOUN
ejpam-3838	11	13	of	of	ADP
ejpam-3838	11	14	distinct	distinct	ADJ
ejpam-3838	11	15	self	self	NOUN
ejpam-3838	11	16	-	-	PUNCT
ejpam-3838	11	17	dual	dual	ADJ
ejpam-3838	11	18	codes	code	NOUN
ejpam-3838	11	19	of	of	ADP
ejpam-3838	11	20	length	length	NOUN
ejpam-3838	11	21	n	n	NOUN
ejpam-3838	11	22	over	over	ADP
ejpam-3838	11	23	a	a	DET
ejpam-3838	11	24	ring	ring	NOUN
ejpam-3838	11	25	r	r	NOUN
ejpam-3838	11	26	,	,	PUNCT
ejpam-3838	11	27	given	give	VERB
ejpam-3838	11	28	by∑	by∑	PROPN
ejpam-3838	11	29	c	c	NOUN
ejpam-3838	11	30	|en|	|en|	PROPN
ejpam-3838	11	31	|aut(c)|	|aut(c)|	NOUN
ejpam-3838	11	32	where	where	SCONJ
ejpam-3838	11	33	c	c	NOUN
ejpam-3838	11	34	runs	run	VERB
ejpam-3838	11	35	through	through	ADP
ejpam-3838	11	36	the	the	DET
ejpam-3838	11	37	set	set	NOUN
ejpam-3838	11	38	of	of	ADP
ejpam-3838	11	39	all	all	DET
ejpam-3838	11	40	inequivalent	inequivalent	NOUN
ejpam-3838	11	41	self	self	NOUN
ejpam-3838	11	42	-	-	PUNCT
ejpam-3838	11	43	dual	dual	ADJ
ejpam-3838	11	44	codes	code	NOUN
ejpam-3838	11	45	of	of	ADP
ejpam-3838	11	46	length	length	NOUN
ejpam-3838	11	47	n	n	CCONJ
ejpam-3838	11	48	over	over	ADP
ejpam-3838	11	49	r	r	NOUN
ejpam-3838	11	50	,	,	PUNCT
ejpam-3838	11	51	en	en	X
ejpam-3838	11	52	is	be	AUX
ejpam-3838	11	53	the	the	DET
ejpam-3838	11	54	full	full	ADJ
ejpam-3838	11	55	group	group	NOUN
ejpam-3838	11	56	of	of	ADP
ejpam-3838	11	57	transformations	transformation	NOUN
ejpam-3838	11	58	allowed	allow	VERB
ejpam-3838	11	59	in	in	ADP
ejpam-3838	11	60	defining	define	VERB
ejpam-3838	11	61	the	the	DET
ejpam-3838	11	62	equivalence	equivalence	NOUN
ejpam-3838	11	63	for	for	ADP
ejpam-3838	11	64	code	code	NOUN
ejpam-3838	11	65	c	c	PROPN
ejpam-3838	11	66	and	and	CCONJ
ejpam-3838	11	67	aut(c	aut(c	PROPN
ejpam-3838	11	68	)	)	PUNCT
ejpam-3838	11	69	is	be	AUX
ejpam-3838	11	70	the	the	DET
ejpam-3838	11	71	automorphism	automorphism	NOUN
ejpam-3838	11	72	group	group	NOUN
ejpam-3838	11	73	.	.	PUNCT
ejpam-3838	12	1	this	this	PRON
ejpam-3838	12	2	is	be	AUX
ejpam-3838	12	3	called	call	VERB
ejpam-3838	12	4	the	the	DET
ejpam-3838	12	5	mass	mass	ADJ
ejpam-3838	12	6	formula	formula	NOUN
ejpam-3838	12	7	,	,	PUNCT
ejpam-3838	12	8	and	and	CCONJ
ejpam-3838	12	9	is	be	AUX
ejpam-3838	12	10	an	an	DET
ejpam-3838	12	11	important	important	ADJ
ejpam-3838	12	12	computational	computational	ADJ
ejpam-3838	12	13	tool	tool	NOUN
ejpam-3838	12	14	for	for	ADP
ejpam-3838	12	15	the	the	DET
ejpam-3838	12	16	classification	classification	NOUN
ejpam-3838	12	17	of	of	ADP
ejpam-3838	12	18	such	such	ADJ
ejpam-3838	12	19	codes	code	NOUN
ejpam-3838	12	20	.	.	PUNCT
ejpam-3838	13	1	mass	mass	ADJ
ejpam-3838	13	2	formulas	formula	NOUN
ejpam-3838	13	3	for	for	ADP
ejpam-3838	13	4	self	self	NOUN
ejpam-3838	13	5	-	-	PUNCT
ejpam-3838	13	6	dual	dual	ADJ
ejpam-3838	13	7	codes	code	NOUN
ejpam-3838	13	8	∗corresponding	∗corresponde	VERB
ejpam-3838	13	9	author	author	NOUN
ejpam-3838	13	10	.	.	PUNCT
ejpam-3838	14	1	doi	doi	NOUN
ejpam-3838	14	2	:	:	PUNCT
ejpam-3838	14	3	https://doi.org/10.29020/nybg.ejpam.v13i4.3838	https://doi.org/10.29020/nybg.ejpam.v13i4.3838	VERB
ejpam-3838	14	4	email	email	NOUN
ejpam-3838	14	5	addresses	address	NOUN
ejpam-3838	14	6	:	:	PUNCT
ejpam-3838	14	7	legalvez@math.upd.edu.ph	legalvez@math.upd.edu.ph	PROPN
ejpam-3838	14	8	(	(	PUNCT
ejpam-3838	14	9	l.	l.	PROPN
ejpam-3838	14	10	e.	e.	PROPN
ejpam-3838	14	11	galvez	galvez	PROPN
ejpam-3838	14	12	)	)	PUNCT
ejpam-3838	14	13	,	,	PUNCT
ejpam-3838	14	14	rabetty@math.upd.edu.ph	rabetty@math.upd.edu.ph	PROPN
ejpam-3838	14	15	(	(	PUNCT
ejpam-3838	14	16	r.	r.	PROPN
ejpam-3838	14	17	a.	a.	PROPN
ejpam-3838	14	18	betty	betty	PROPN
ejpam-3838	14	19	)	)	PUNCT
ejpam-3838	14	20	,	,	PUNCT
ejpam-3838	14	21	fidel@math.upd.edu.ph	fidel@math.upd.edu.ph	PROPN
ejpam-3838	14	22	(	(	PUNCT
ejpam-3838	14	23	f.	f.	PROPN
ejpam-3838	14	24	nemenzo	nemenzo	NOUN
ejpam-3838	14	25	)	)	PUNCT
ejpam-3838	14	26	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-3838	15	1	873	873	NUM
ejpam-3838	15	2	c	c	NOUN
ejpam-3838	15	3	©	©	PROPN
ejpam-3838	15	4	2020	2020	NUM
ejpam-3838	15	5	ejpam	ejpam	VERB
ejpam-3838	15	6	all	all	DET
ejpam-3838	15	7	rights	right	NOUN
ejpam-3838	15	8	reserved	reserve	VERB
ejpam-3838	15	9	.	.	PUNCT
ejpam-3838	16	1	l.e	l.e	PROPN
ejpam-3838	16	2	.	.	PROPN
ejpam-3838	16	3	galvez	galvez	PROPN
ejpam-3838	16	4	,	,	PUNCT
ejpam-3838	16	5	r.a	r.a	PROPN
ejpam-3838	16	6	.	.	PROPN
ejpam-3838	16	7	betty	betty	PROPN
ejpam-3838	16	8	,	,	PUNCT
ejpam-3838	16	9	f.	f.	PROPN
ejpam-3838	16	10	nemenzo	nemenzo	PROPN
ejpam-3838	16	11	/	/	SYM
ejpam-3838	16	12	eur	eur	NOUN
ejpam-3838	16	13	.	.	PUNCT
ejpam-3838	17	1	j.	j.	PROPN
ejpam-3838	17	2	pure	pure	PROPN
ejpam-3838	17	3	appl	appl	PROPN
ejpam-3838	17	4	.	.	PROPN
ejpam-3838	17	5	math	math	PROPN
ejpam-3838	17	6	,	,	PUNCT
ejpam-3838	17	7	13	13	NUM
ejpam-3838	17	8	(	(	PUNCT
ejpam-3838	17	9	4	4	NUM
ejpam-3838	17	10	)	)	PUNCT
ejpam-3838	17	11	(	(	PUNCT
ejpam-3838	17	12	2020	2020	NUM
ejpam-3838	17	13	)	)	PUNCT
ejpam-3838	17	14	,	,	PUNCT
ejpam-3838	17	15	873	873	NUM
ejpam-3838	17	16	-	-	NUM
ejpam-3838	17	17	892	892	NUM
ejpam-3838	17	18	874	874	NUM
ejpam-3838	17	19	over	over	ADP
ejpam-3838	17	20	finite	finite	PROPN
ejpam-3838	17	21	rings	ring	NOUN
ejpam-3838	17	22	rings	ring	NOUN
ejpam-3838	17	23	such	such	ADJ
ejpam-3838	17	24	as	as	ADP
ejpam-3838	17	25	z4	z4	PROPN
ejpam-3838	17	26	and	and	CCONJ
ejpam-3838	17	27	fq	fq	PROPN
ejpam-3838	17	28	+	+	CCONJ
ejpam-3838	17	29	ufq	ufq	PROPN
ejpam-3838	17	30	were	be	AUX
ejpam-3838	17	31	given	give	VERB
ejpam-3838	17	32	in	in	ADP
ejpam-3838	17	33	[	[	X
ejpam-3838	17	34	6	6	NUM
ejpam-3838	17	35	]	]	PUNCT
ejpam-3838	17	36	,	,	PUNCT
ejpam-3838	17	37	while	while	SCONJ
ejpam-3838	17	38	[	[	X
ejpam-3838	17	39	3	3	X
ejpam-3838	17	40	]	]	PUNCT
ejpam-3838	17	41	gave	give	VERB
ejpam-3838	17	42	the	the	DET
ejpam-3838	17	43	mass	mass	ADJ
ejpam-3838	17	44	formula	formula	NOUN
ejpam-3838	17	45	for	for	ADP
ejpam-3838	17	46	self	self	NOUN
ejpam-3838	17	47	-	-	PUNCT
ejpam-3838	17	48	dual	dual	ADJ
ejpam-3838	17	49	codes	code	NOUN
ejpam-3838	17	50	over	over	ADP
ejpam-3838	17	51	fq	fq	PROPN
ejpam-3838	17	52	+	+	CCONJ
ejpam-3838	17	53	ufq	ufq	PROPN
ejpam-3838	17	54	+	+	NUM
ejpam-3838	17	55	u2fq	u2fq	X
ejpam-3838	17	56	.	.	PUNCT
ejpam-3838	18	1	in	in	ADP
ejpam-3838	18	2	this	this	DET
ejpam-3838	18	3	paper	paper	NOUN
ejpam-3838	18	4	,	,	PUNCT
ejpam-3838	18	5	we	we	PRON
ejpam-3838	18	6	focus	focus	VERB
ejpam-3838	18	7	on	on	ADP
ejpam-3838	18	8	the	the	DET
ejpam-3838	18	9	more	more	ADV
ejpam-3838	18	10	general	general	ADJ
ejpam-3838	18	11	mass	mass	NOUN
ejpam-3838	18	12	formula	formula	NOUN
ejpam-3838	18	13	for	for	ADP
ejpam-3838	18	14	self	self	NOUN
ejpam-3838	18	15	-	-	PUNCT
ejpam-3838	18	16	orthogonal	orthogonal	ADJ
ejpam-3838	18	17	codes	code	NOUN
ejpam-3838	18	18	,	,	PUNCT
ejpam-3838	18	19	which	which	PRON
ejpam-3838	18	20	will	will	AUX
ejpam-3838	18	21	include	include	VERB
ejpam-3838	18	22	the	the	DET
ejpam-3838	18	23	mass	mass	ADJ
ejpam-3838	18	24	formula	formula	NOUN
ejpam-3838	18	25	for	for	ADP
ejpam-3838	18	26	self	self	NOUN
ejpam-3838	18	27	-	-	PUNCT
ejpam-3838	18	28	dual	dual	ADJ
ejpam-3838	18	29	codes	code	NOUN
ejpam-3838	18	30	as	as	ADP
ejpam-3838	18	31	a	a	DET
ejpam-3838	18	32	special	special	ADJ
ejpam-3838	18	33	case	case	NOUN
ejpam-3838	18	34	.	.	PUNCT
ejpam-3838	19	1	mass	mass	ADJ
ejpam-3838	19	2	formulas	formula	NOUN
ejpam-3838	19	3	for	for	ADP
ejpam-3838	19	4	self	self	NOUN
ejpam-3838	19	5	-	-	PUNCT
ejpam-3838	19	6	orthogonal	orthogonal	ADJ
ejpam-3838	19	7	codes	code	NOUN
ejpam-3838	19	8	over	over	ADP
ejpam-3838	19	9	zp2	zp2	PROPN
ejpam-3838	19	10	,	,	PUNCT
ejpam-3838	19	11	where	where	SCONJ
ejpam-3838	19	12	p	p	NOUN
ejpam-3838	19	13	is	be	AUX
ejpam-3838	19	14	a	a	DET
ejpam-3838	19	15	prime	prime	NOUN
ejpam-3838	19	16	,	,	PUNCT
ejpam-3838	19	17	were	be	AUX
ejpam-3838	19	18	given	give	VERB
ejpam-3838	19	19	in	in	ADP
ejpam-3838	19	20	[	[	X
ejpam-3838	19	21	2	2	NUM
ejpam-3838	19	22	]	]	PUNCT
ejpam-3838	19	23	,	,	PUNCT
ejpam-3838	19	24	while	while	SCONJ
ejpam-3838	19	25	the	the	DET
ejpam-3838	19	26	mass	mass	ADJ
ejpam-3838	19	27	formula	formula	NOUN
ejpam-3838	19	28	for	for	ADP
ejpam-3838	19	29	even	even	ADV
ejpam-3838	19	30	codes	code	NOUN
ejpam-3838	19	31	over	over	ADP
ejpam-3838	19	32	z8	z8	NOUN
ejpam-3838	19	33	,	,	PUNCT
ejpam-3838	19	34	i.e.	i.e.	X
ejpam-3838	19	35	,	,	PUNCT
ejpam-3838	19	36	self	self	NOUN
ejpam-3838	19	37	-	-	PUNCT
ejpam-3838	19	38	orthogonal	orthogonal	ADJ
ejpam-3838	19	39	codes	code	NOUN
ejpam-3838	19	40	whose	whose	DET
ejpam-3838	19	41	codewords	codeword	NOUN
ejpam-3838	19	42	have	have	VERB
ejpam-3838	19	43	euclidean	euclidean	ADJ
ejpam-3838	19	44	weights	weight	NOUN
ejpam-3838	19	45	divisible	divisible	ADJ
ejpam-3838	19	46	by	by	ADP
ejpam-3838	19	47	16	16	NUM
ejpam-3838	19	48	,	,	PUNCT
ejpam-3838	19	49	was	be	AUX
ejpam-3838	19	50	computed	compute	VERB
ejpam-3838	19	51	in	in	ADP
ejpam-3838	19	52	[	[	X
ejpam-3838	19	53	1	1	NUM
ejpam-3838	19	54	]	]	PUNCT
ejpam-3838	19	55	.	.	PUNCT
ejpam-3838	20	1	codes	code	NOUN
ejpam-3838	20	2	over	over	ADP
ejpam-3838	20	3	fq	fq	PROPN
ejpam-3838	20	4	+	+	CCONJ
ejpam-3838	20	5	ufq	ufq	PROPN
ejpam-3838	20	6	and	and	CCONJ
ejpam-3838	20	7	fq	fq	PROPN
ejpam-3838	20	8	+	+	CCONJ
ejpam-3838	20	9	ufq	ufq	PROPN
ejpam-3838	20	10	+	+	CCONJ
ejpam-3838	20	11	u2fq	u2fq	PUNCT
ejpam-3838	20	12	have	have	VERB
ejpam-3838	20	13	an	an	DET
ejpam-3838	20	14	invariant	invariant	ADJ
ejpam-3838	20	15	called	call	VERB
ejpam-3838	20	16	type	type	NOUN
ejpam-3838	20	17	,	,	PUNCT
ejpam-3838	20	18	denoted	denote	VERB
ejpam-3838	20	19	by	by	ADP
ejpam-3838	20	20	{	{	PUNCT
ejpam-3838	20	21	k0	k0	PROPN
ejpam-3838	20	22	,	,	PUNCT
ejpam-3838	20	23	k1	k1	NOUN
ejpam-3838	20	24	}	}	PUNCT
ejpam-3838	20	25	and	and	CCONJ
ejpam-3838	20	26	{	{	PUNCT
ejpam-3838	20	27	k0	k0	PROPN
ejpam-3838	20	28	,	,	PUNCT
ejpam-3838	20	29	k1	k1	NOUN
ejpam-3838	20	30	,	,	PUNCT
ejpam-3838	20	31	k2	k2	NOUN
ejpam-3838	20	32	}	}	PUNCT
ejpam-3838	20	33	,	,	PUNCT
ejpam-3838	20	34	respectively	respectively	ADV
ejpam-3838	20	35	,	,	PUNCT
ejpam-3838	20	36	where	where	SCONJ
ejpam-3838	20	37	k0	k0	PROPN
ejpam-3838	20	38	,	,	PUNCT
ejpam-3838	20	39	k1	k1	PROPN
ejpam-3838	20	40	and	and	CCONJ
ejpam-3838	20	41	k2	k2	PROPN
ejpam-3838	20	42	are	be	AUX
ejpam-3838	20	43	nonnegative	nonnegative	ADJ
ejpam-3838	20	44	integers	integer	NOUN
ejpam-3838	20	45	.	.	PUNCT
ejpam-3838	21	1	the	the	DET
ejpam-3838	21	2	type	type	NOUN
ejpam-3838	21	3	of	of	ADP
ejpam-3838	21	4	a	a	DET
ejpam-3838	21	5	code	code	NOUN
ejpam-3838	21	6	is	be	AUX
ejpam-3838	21	7	determined	determine	VERB
ejpam-3838	21	8	by	by	ADP
ejpam-3838	21	9	its	its	PRON
ejpam-3838	21	10	residue	residue	NOUN
ejpam-3838	21	11	and	and	CCONJ
ejpam-3838	21	12	torsion	torsion	NOUN
ejpam-3838	21	13	codes	code	NOUN
ejpam-3838	21	14	.	.	PUNCT
ejpam-3838	22	1	to	to	PART
ejpam-3838	22	2	obtain	obtain	VERB
ejpam-3838	22	3	the	the	DET
ejpam-3838	22	4	mass	mass	ADJ
ejpam-3838	22	5	formula	formula	NOUN
ejpam-3838	22	6	,	,	PUNCT
ejpam-3838	22	7	we	we	PRON
ejpam-3838	22	8	will	will	AUX
ejpam-3838	22	9	determine	determine	VERB
ejpam-3838	22	10	the	the	DET
ejpam-3838	22	11	number	number	NOUN
ejpam-3838	22	12	of	of	ADP
ejpam-3838	22	13	self	self	NOUN
ejpam-3838	22	14	-	-	PUNCT
ejpam-3838	22	15	orthogonal	orthogonal	ADJ
ejpam-3838	22	16	codes	code	NOUN
ejpam-3838	22	17	of	of	ADP
ejpam-3838	22	18	length	length	NOUN
ejpam-3838	22	19	n	n	PROPN
ejpam-3838	22	20	over	over	ADP
ejpam-3838	22	21	fq	fq	PROPN
ejpam-3838	22	22	+	+	CCONJ
ejpam-3838	22	23	ufq	ufq	VERB
ejpam-3838	22	24	with	with	ADP
ejpam-3838	22	25	given	give	VERB
ejpam-3838	22	26	residue	residue	NOUN
ejpam-3838	22	27	and	and	CCONJ
ejpam-3838	22	28	torsion	torsion	NOUN
ejpam-3838	22	29	,	,	PUNCT
ejpam-3838	22	30	and	and	CCONJ
ejpam-3838	22	31	compute	compute	VERB
ejpam-3838	22	32	the	the	DET
ejpam-3838	22	33	number	number	NOUN
ejpam-3838	22	34	of	of	ADP
ejpam-3838	22	35	self	self	NOUN
ejpam-3838	22	36	-	-	PUNCT
ejpam-3838	22	37	orthogonal	orthogonal	ADJ
ejpam-3838	22	38	codes	code	NOUN
ejpam-3838	22	39	of	of	ADP
ejpam-3838	22	40	length	length	NOUN
ejpam-3838	22	41	n	n	PROPN
ejpam-3838	22	42	over	over	ADP
ejpam-3838	22	43	fq	fq	PROPN
ejpam-3838	22	44	+	+	CCONJ
ejpam-3838	22	45	ufq	ufq	PROPN
ejpam-3838	22	46	+	+	NUM
ejpam-3838	22	47	u2fq	u2fq	PUNCT
ejpam-3838	22	48	,	,	PUNCT
ejpam-3838	22	49	for	for	ADP
ejpam-3838	22	50	odd	odd	ADJ
ejpam-3838	22	51	q	q	NOUN
ejpam-3838	22	52	,	,	PUNCT
ejpam-3838	22	53	with	with	ADP
ejpam-3838	22	54	given	give	VERB
ejpam-3838	22	55	u2	u2	NOUN
ejpam-3838	22	56	-	-	PUNCT
ejpam-3838	22	57	residue	residue	NOUN
ejpam-3838	22	58	and	and	CCONJ
ejpam-3838	22	59	torsion	torsion	NOUN
ejpam-3838	22	60	.	.	PUNCT
ejpam-3838	23	1	we	we	PRON
ejpam-3838	23	2	also	also	ADV
ejpam-3838	23	3	give	give	VERB
ejpam-3838	23	4	a	a	DET
ejpam-3838	23	5	classification	classification	NOUN
ejpam-3838	23	6	of	of	ADP
ejpam-3838	23	7	euclidean	euclidean	ADJ
ejpam-3838	23	8	and	and	CCONJ
ejpam-3838	23	9	hermitian	hermitian	ADJ
ejpam-3838	23	10	self	self	NOUN
ejpam-3838	23	11	-	-	PUNCT
ejpam-3838	23	12	orthogonal	orthogonal	ADJ
ejpam-3838	23	13	codes	code	NOUN
ejpam-3838	23	14	over	over	ADP
ejpam-3838	23	15	f2+uf2	f2+uf2	NOUN
ejpam-3838	23	16	and	and	CCONJ
ejpam-3838	23	17	f3+uf3	f3+uf3	NOUN
ejpam-3838	23	18	,	,	PUNCT
ejpam-3838	23	19	up	up	ADP
ejpam-3838	23	20	to	to	ADP
ejpam-3838	23	21	some	some	DET
ejpam-3838	23	22	short	short	ADJ
ejpam-3838	23	23	lengths	length	NOUN
ejpam-3838	23	24	.	.	PUNCT
ejpam-3838	24	1	2	2	X
ejpam-3838	24	2	.	.	X
ejpam-3838	24	3	codes	code	NOUN
ejpam-3838	24	4	over	over	ADP
ejpam-3838	24	5	fq	fq	PROPN
ejpam-3838	24	6	+	+	CCONJ
ejpam-3838	24	7	ufq	ufq	NOUN
ejpam-3838	24	8	we	we	PRON
ejpam-3838	24	9	begin	begin	VERB
ejpam-3838	24	10	with	with	ADP
ejpam-3838	24	11	some	some	DET
ejpam-3838	24	12	basic	basic	ADJ
ejpam-3838	24	13	concepts	concept	NOUN
ejpam-3838	24	14	about	about	ADP
ejpam-3838	24	15	codes	code	NOUN
ejpam-3838	24	16	over	over	ADP
ejpam-3838	24	17	rings	ring	NOUN
ejpam-3838	24	18	.	.	PUNCT
ejpam-3838	25	1	a	a	DET
ejpam-3838	25	2	linear	linear	ADJ
ejpam-3838	25	3	code	code	NOUN
ejpam-3838	25	4	c	c	NOUN
ejpam-3838	25	5	of	of	ADP
ejpam-3838	25	6	length	length	NOUN
ejpam-3838	25	7	n	n	CCONJ
ejpam-3838	25	8	over	over	ADP
ejpam-3838	25	9	a	a	DET
ejpam-3838	25	10	ring	ring	NOUN
ejpam-3838	25	11	r	r	NOUN
ejpam-3838	25	12	is	be	AUX
ejpam-3838	25	13	a	a	DET
ejpam-3838	25	14	submodule	submodule	NOUN
ejpam-3838	25	15	of	of	ADP
ejpam-3838	25	16	the	the	DET
ejpam-3838	25	17	rn	rn	PROPN
ejpam-3838	25	18	module	module	NOUN
ejpam-3838	25	19	.	.	PUNCT
ejpam-3838	26	1	a	a	DET
ejpam-3838	26	2	generator	generator	NOUN
ejpam-3838	26	3	matrix	matrix	NOUN
ejpam-3838	26	4	for	for	ADP
ejpam-3838	26	5	c	c	PROPN
ejpam-3838	26	6	is	be	AUX
ejpam-3838	26	7	a	a	DET
ejpam-3838	26	8	matrix	matrix	NOUN
ejpam-3838	26	9	g	g	ADP
ejpam-3838	26	10	∈	∈	PROPN
ejpam-3838	26	11	mk×n(r	mk×n(r	NOUN
ejpam-3838	26	12	)	)	PUNCT
ejpam-3838	26	13	whose	whose	DET
ejpam-3838	26	14	rows	row	NOUN
ejpam-3838	26	15	generate	generate	VERB
ejpam-3838	26	16	the	the	DET
ejpam-3838	26	17	code	code	NOUN
ejpam-3838	26	18	.	.	PUNCT
ejpam-3838	27	1	for	for	ADP
ejpam-3838	27	2	a	a	DET
ejpam-3838	27	3	matrix	matrix	NOUN
ejpam-3838	27	4	g	g	ADP
ejpam-3838	27	5	∈	∈	PROPN
ejpam-3838	27	6	mk×n(r	mk×n(r	NOUN
ejpam-3838	27	7	)	)	PUNCT
ejpam-3838	27	8	,	,	PUNCT
ejpam-3838	27	9	we	we	PRON
ejpam-3838	27	10	denote	denote	VERB
ejpam-3838	27	11	by	by	ADP
ejpam-3838	27	12	rkg	rkg	NOUN
ejpam-3838	27	13	the	the	DET
ejpam-3838	27	14	code	code	NOUN
ejpam-3838	27	15	{	{	PUNCT
ejpam-3838	27	16	ag	ag	PROPN
ejpam-3838	27	17	|	|	ADV
ejpam-3838	27	18	a	a	DET
ejpam-3838	27	19	∈	∈	PROPN
ejpam-3838	27	20	rk	rk	NOUN
ejpam-3838	27	21	}	}	PUNCT
ejpam-3838	27	22	of	of	ADP
ejpam-3838	27	23	length	length	NOUN
ejpam-3838	27	24	n	n	CCONJ
ejpam-3838	27	25	over	over	ADP
ejpam-3838	27	26	r.	r.	PROPN
ejpam-3838	27	27	let	let	VERB
ejpam-3838	27	28	q	q	NOUN
ejpam-3838	27	29	be	be	AUX
ejpam-3838	27	30	a	a	DET
ejpam-3838	27	31	power	power	NOUN
ejpam-3838	27	32	of	of	ADP
ejpam-3838	27	33	a	a	DET
ejpam-3838	27	34	prime	prime	NOUN
ejpam-3838	27	35	and	and	CCONJ
ejpam-3838	27	36	fq	fq	PROPN
ejpam-3838	27	37	denote	denote	VERB
ejpam-3838	27	38	the	the	DET
ejpam-3838	27	39	finite	finite	ADJ
ejpam-3838	27	40	field	field	NOUN
ejpam-3838	27	41	of	of	ADP
ejpam-3838	27	42	q	q	NOUN
ejpam-3838	27	43	elements	element	NOUN
ejpam-3838	27	44	.	.	PUNCT
ejpam-3838	28	1	let	let	VERB
ejpam-3838	28	2	r1	r1	PROPN
ejpam-3838	28	3	be	be	AUX
ejpam-3838	28	4	the	the	DET
ejpam-3838	28	5	commutative	commutative	ADJ
ejpam-3838	28	6	ring	ring	NOUN
ejpam-3838	28	7	fq[u]/(u2	fq[u]/(u2	PROPN
ejpam-3838	28	8	)	)	PUNCT
ejpam-3838	29	1	=	=	SYM
ejpam-3838	29	2	fq	fq	PROPN
ejpam-3838	29	3	+	+	CCONJ
ejpam-3838	29	4	ufq	ufq	PROPN
ejpam-3838	29	5	,	,	PUNCT
ejpam-3838	29	6	where	where	SCONJ
ejpam-3838	29	7	u2	u2	NOUN
ejpam-3838	29	8	=	=	NOUN
ejpam-3838	29	9	0	0	PROPN
ejpam-3838	29	10	.	.	PUNCT
ejpam-3838	30	1	this	this	DET
ejpam-3838	30	2	finite	finite	PROPN
ejpam-3838	30	3	chain	chain	NOUN
ejpam-3838	30	4	ring	ring	NOUN
ejpam-3838	30	5	is	be	AUX
ejpam-3838	30	6	a	a	DET
ejpam-3838	30	7	local	local	ADJ
ejpam-3838	30	8	ring	ring	NOUN
ejpam-3838	30	9	with	with	ADP
ejpam-3838	30	10	unique	unique	ADJ
ejpam-3838	30	11	maximal	maximal	ADJ
ejpam-3838	30	12	ideal	ideal	NOUN
ejpam-3838	30	13	(	(	PUNCT
ejpam-3838	30	14	u	u	NOUN
ejpam-3838	30	15	)	)	PUNCT
ejpam-3838	30	16	and	and	CCONJ
ejpam-3838	30	17	residue	residue	NOUN
ejpam-3838	30	18	field	field	NOUN
ejpam-3838	30	19	fq	fq	PROPN
ejpam-3838	30	20	+	+	CCONJ
ejpam-3838	30	21	ufq/(u	ufq/(u	PROPN
ejpam-3838	30	22	)	)	PUNCT
ejpam-3838	31	1	=	=	SYM
ejpam-3838	31	2	fq	fq	PROPN
ejpam-3838	31	3	.	.	PUNCT
ejpam-3838	32	1	every	every	DET
ejpam-3838	32	2	code	code	NOUN
ejpam-3838	32	3	c	c	NOUN
ejpam-3838	32	4	of	of	ADP
ejpam-3838	32	5	length	length	NOUN
ejpam-3838	32	6	n	n	CCONJ
ejpam-3838	32	7	over	over	ADP
ejpam-3838	32	8	r1	r1	PROPN
ejpam-3838	32	9	is	be	AUX
ejpam-3838	32	10	permutation	permutation	NOUN
ejpam-3838	32	11	-	-	PUNCT
ejpam-3838	32	12	equivalent	equivalent	ADJ
ejpam-3838	32	13	to	to	ADP
ejpam-3838	32	14	a	a	DET
ejpam-3838	32	15	code	code	NOUN
ejpam-3838	32	16	with	with	ADP
ejpam-3838	32	17	the	the	DET
ejpam-3838	32	18	following	follow	VERB
ejpam-3838	32	19	generator	generator	NOUN
ejpam-3838	32	20	matrix	matrix	NOUN
ejpam-3838	32	21	,	,	PUNCT
ejpam-3838	32	22	[	[	PUNCT
ejpam-3838	32	23	ik0	ik0	NOUN
ejpam-3838	32	24	a+	a+	PUNCT
ejpam-3838	32	25	ub	ub	INTJ
ejpam-3838	32	26	0	0	NUM
ejpam-3838	32	27	ud	ud	NOUN
ejpam-3838	32	28	]	]	PUNCT
ejpam-3838	32	29	,	,	PUNCT
ejpam-3838	32	30	(	(	PUNCT
ejpam-3838	32	31	1	1	X
ejpam-3838	32	32	)	)	PUNCT
ejpam-3838	32	33	where	where	SCONJ
ejpam-3838	32	34	ik0	ik0	NOUN
ejpam-3838	32	35	is	be	AUX
ejpam-3838	32	36	the	the	DET
ejpam-3838	32	37	k0	k0	PROPN
ejpam-3838	32	38	×	×	PROPN
ejpam-3838	32	39	k0	k0	PROPN
ejpam-3838	32	40	identity	identity	NOUN
ejpam-3838	32	41	matrix	matrix	NOUN
ejpam-3838	32	42	,	,	PUNCT
ejpam-3838	32	43	a	a	DET
ejpam-3838	32	44	,	,	PUNCT
ejpam-3838	32	45	b	b	NOUN
ejpam-3838	32	46	∈mk0×(n−k0)(fq	∈mk0×(n−k0)(fq	NOUN
ejpam-3838	32	47	)	)	PUNCT
ejpam-3838	32	48	and	and	CCONJ
ejpam-3838	32	49	d	d	ADP
ejpam-3838	32	50	∈mk1×(n−k0)(fq	∈mk1×(n−k0)(fq	PROPN
ejpam-3838	32	51	)	)	PUNCT
ejpam-3838	32	52	.	.	PUNCT
ejpam-3838	33	1	such	such	DET
ejpam-3838	33	2	a	a	DET
ejpam-3838	33	3	code	code	NOUN
ejpam-3838	33	4	c	c	NOUN
ejpam-3838	33	5	is	be	AUX
ejpam-3838	33	6	said	say	VERB
ejpam-3838	33	7	to	to	PART
ejpam-3838	33	8	be	be	AUX
ejpam-3838	33	9	of	of	ADP
ejpam-3838	33	10	type	type	NOUN
ejpam-3838	33	11	{	{	PUNCT
ejpam-3838	33	12	k0	k0	PROPN
ejpam-3838	33	13	,	,	PUNCT
ejpam-3838	33	14	k1	k1	PROPN
ejpam-3838	33	15	}	}	PUNCT
ejpam-3838	33	16	.	.	PUNCT
ejpam-3838	34	1	the	the	DET
ejpam-3838	34	2	code	code	NOUN
ejpam-3838	34	3	c	c	PROPN
ejpam-3838	34	4	is	be	AUX
ejpam-3838	34	5	said	say	VERB
ejpam-3838	34	6	to	to	PART
ejpam-3838	34	7	be	be	AUX
ejpam-3838	34	8	free	free	ADJ
ejpam-3838	34	9	if	if	SCONJ
ejpam-3838	34	10	k1	k1	NOUN
ejpam-3838	34	11	=	=	SYM
ejpam-3838	34	12	0	0	NUM
ejpam-3838	34	13	.	.	PUNCT
ejpam-3838	35	1	the	the	DET
ejpam-3838	35	2	type	type	NOUN
ejpam-3838	35	3	is	be	AUX
ejpam-3838	35	4	the	the	DET
ejpam-3838	35	5	analog	analog	NOUN
ejpam-3838	35	6	of	of	ADP
ejpam-3838	35	7	the	the	DET
ejpam-3838	35	8	dimension	dimension	NOUN
ejpam-3838	35	9	of	of	ADP
ejpam-3838	35	10	a	a	DET
ejpam-3838	35	11	code	code	NOUN
ejpam-3838	35	12	over	over	ADP
ejpam-3838	35	13	a	a	DET
ejpam-3838	35	14	finite	finite	ADJ
ejpam-3838	35	15	field	field	NOUN
ejpam-3838	35	16	.	.	PUNCT
ejpam-3838	36	1	such	such	DET
ejpam-3838	36	2	a	a	DET
ejpam-3838	36	3	code	code	NOUN
ejpam-3838	36	4	c	c	NOUN
ejpam-3838	36	5	contains	contain	VERB
ejpam-3838	36	6	q2k0+k1	q2k0+k1	NOUN
ejpam-3838	36	7	codewords	codeword	NOUN
ejpam-3838	36	8	.	.	PUNCT
ejpam-3838	37	1	let	let	VERB
ejpam-3838	37	2	x	x	PUNCT
ejpam-3838	37	3	=	=	SYM
ejpam-3838	37	4	(	(	PUNCT
ejpam-3838	37	5	x1	x1	PROPN
ejpam-3838	37	6	,	,	PUNCT
ejpam-3838	37	7	x2	x2	PROPN
ejpam-3838	37	8	,	,	PUNCT
ejpam-3838	37	9	.	.	PUNCT
ejpam-3838	37	10	.	.	PUNCT
ejpam-3838	37	11	.	.	PUNCT
ejpam-3838	38	1	,	,	PUNCT
ejpam-3838	38	2	xn	xn	X
ejpam-3838	38	3	)	)	PUNCT
ejpam-3838	38	4	and	and	CCONJ
ejpam-3838	38	5	y	y	PROPN
ejpam-3838	38	6	=	=	SYM
ejpam-3838	38	7	(	(	PUNCT
ejpam-3838	38	8	y1	y1	PROPN
ejpam-3838	38	9	,	,	PUNCT
ejpam-3838	38	10	y2	y2	PROPN
ejpam-3838	38	11	,	,	PUNCT
ejpam-3838	38	12	.	.	PUNCT
ejpam-3838	38	13	.	.	PUNCT
ejpam-3838	38	14	.	.	PUNCT
ejpam-3838	39	1	,	,	PUNCT
ejpam-3838	39	2	yn	yn	X
ejpam-3838	39	3	)	)	PUNCT
ejpam-3838	39	4	be	be	VERB
ejpam-3838	39	5	elements	element	NOUN
ejpam-3838	39	6	of	of	ADP
ejpam-3838	39	7	rn1	rn1	PROPN
ejpam-3838	39	8	.	.	PUNCT
ejpam-3838	40	1	define	define	VERB
ejpam-3838	40	2	the	the	DET
ejpam-3838	40	3	euclidean	euclidean	ADJ
ejpam-3838	40	4	inner	inner	ADJ
ejpam-3838	40	5	product	product	NOUN
ejpam-3838	40	6	on	on	ADP
ejpam-3838	40	7	rn1	rn1	PROPN
ejpam-3838	40	8	as	as	ADP
ejpam-3838	40	9	〈	〈	PROPN
ejpam-3838	40	10	x	x	X
ejpam-3838	40	11	,	,	PUNCT
ejpam-3838	41	1	y〉e	y〉e	NOUN
ejpam-3838	41	2	=	=	PUNCT
ejpam-3838	41	3	∑n	∑n	PROPN
ejpam-3838	41	4	i=1	i=1	PROPN
ejpam-3838	41	5	xiyi	xiyi	PROPN
ejpam-3838	41	6	.	.	PUNCT
ejpam-3838	42	1	now	now	ADV
ejpam-3838	42	2	,	,	PUNCT
ejpam-3838	42	3	let	let	VERB
ejpam-3838	42	4	z	z	NOUN
ejpam-3838	42	5	=	=	PRON
ejpam-3838	42	6	a+	a+	PUNCT
ejpam-3838	42	7	ub	ub	PROPN
ejpam-3838	42	8	∈	∈	PROPN
ejpam-3838	42	9	r1	r1	NOUN
ejpam-3838	42	10	and	and	CCONJ
ejpam-3838	42	11	define	define	VERB
ejpam-3838	42	12	z	z	NOUN
ejpam-3838	42	13	=	=	SYM
ejpam-3838	42	14	a−ub	a−ub	PROPN
ejpam-3838	42	15	.	.	PUNCT
ejpam-3838	43	1	the	the	DET
ejpam-3838	43	2	hermitian	hermitian	ADJ
ejpam-3838	43	3	inner	inner	ADJ
ejpam-3838	43	4	product	product	NOUN
ejpam-3838	43	5	on	on	ADP
ejpam-3838	43	6	rn1	rn1	PROPN
ejpam-3838	43	7	is	be	AUX
ejpam-3838	43	8	defined	define	VERB
ejpam-3838	43	9	as	as	ADP
ejpam-3838	43	10	〈	〈	PROPN
ejpam-3838	43	11	x	x	X
ejpam-3838	43	12	,	,	PUNCT
ejpam-3838	43	13	y〉h	y〉h	PROPN
ejpam-3838	43	14	=	=	SYM
ejpam-3838	44	1	∑n	∑n	PROPN
ejpam-3838	44	2	i=1	i=1	PROPN
ejpam-3838	44	3	xiyi	xiyi	PROPN
ejpam-3838	44	4	.	.	PUNCT
ejpam-3838	45	1	the	the	DET
ejpam-3838	45	2	set	set	NOUN
ejpam-3838	45	3	c⊥	c⊥	X
ejpam-3838	45	4	=	=	SYM
ejpam-3838	45	5	{	{	PUNCT
ejpam-3838	45	6	x	x	PUNCT
ejpam-3838	45	7	∈	∈	PROPN
ejpam-3838	45	8	rn1	rn1	VERB
ejpam-3838	45	9	|	|	ADV
ejpam-3838	45	10	〈	〈	PROPN
ejpam-3838	45	11	x	x	X
ejpam-3838	45	12	,	,	PUNCT
ejpam-3838	45	13	y	y	PROPN
ejpam-3838	46	1	〉	〉	NUM
ejpam-3838	46	2	=	=	SYM
ejpam-3838	46	3	0	0	NUM
ejpam-3838	46	4	∀y	∀y	PROPN
ejpam-3838	46	5	∈	∈	NOUN
ejpam-3838	46	6	c	c	AUX
ejpam-3838	46	7	}	}	PUNCT
ejpam-3838	46	8	is	be	AUX
ejpam-3838	46	9	called	call	VERB
ejpam-3838	46	10	the	the	DET
ejpam-3838	46	11	euclidean	euclidean	ADJ
ejpam-3838	46	12	or	or	CCONJ
ejpam-3838	46	13	hermitian	hermitian	NOUN
ejpam-3838	46	14	dual	dual	ADJ
ejpam-3838	46	15	of	of	ADP
ejpam-3838	46	16	c	c	NOUN
ejpam-3838	46	17	,	,	PUNCT
ejpam-3838	46	18	depending	depend	VERB
ejpam-3838	46	19	on	on	ADP
ejpam-3838	46	20	which	which	PRON
ejpam-3838	46	21	inner	inner	ADJ
ejpam-3838	46	22	product	product	NOUN
ejpam-3838	46	23	〈	〈	PROPN
ejpam-3838	46	24	x	x	X
ejpam-3838	46	25	,	,	PUNCT
ejpam-3838	46	26	y	y	PROPN
ejpam-3838	46	27	〉	〉	PROPN
ejpam-3838	46	28	is	be	AUX
ejpam-3838	46	29	used	use	VERB
ejpam-3838	46	30	.	.	PUNCT
ejpam-3838	47	1	the	the	DET
ejpam-3838	47	2	code	code	NOUN
ejpam-3838	47	3	c	c	PROPN
ejpam-3838	47	4	is	be	AUX
ejpam-3838	47	5	said	say	VERB
ejpam-3838	47	6	to	to	PART
ejpam-3838	47	7	be	be	AUX
ejpam-3838	47	8	euclidean	euclidean	ADJ
ejpam-3838	47	9	or	or	CCONJ
ejpam-3838	47	10	hermitian	hermitian	ADJ
ejpam-3838	47	11	self	self	NOUN
ejpam-3838	47	12	-	-	PUNCT
ejpam-3838	47	13	orthogonal	orthogonal	ADJ
ejpam-3838	47	14	if	if	SCONJ
ejpam-3838	47	15	c	c	PROPN
ejpam-3838	47	16	⊆	⊆	NUM
ejpam-3838	47	17	c⊥.	c⊥.	NOUN
ejpam-3838	47	18	if	if	SCONJ
ejpam-3838	47	19	c	c	NOUN
ejpam-3838	47	20	=	=	SYM
ejpam-3838	47	21	c⊥	c⊥	PROPN
ejpam-3838	47	22	,	,	PUNCT
ejpam-3838	47	23	then	then	ADV
ejpam-3838	47	24	we	we	PRON
ejpam-3838	47	25	say	say	VERB
ejpam-3838	47	26	c	c	NOUN
ejpam-3838	47	27	is	be	AUX
ejpam-3838	47	28	euclidean	euclidean	ADJ
ejpam-3838	47	29	or	or	CCONJ
ejpam-3838	47	30	hermitian	hermitian	ADJ
ejpam-3838	47	31	self	self	NOUN
ejpam-3838	47	32	-	-	PUNCT
ejpam-3838	47	33	dual	dual	ADJ
ejpam-3838	47	34	.	.	PUNCT
ejpam-3838	48	1	let	let	VERB
ejpam-3838	48	2	c	c	PRON
ejpam-3838	48	3	be	be	AUX
ejpam-3838	48	4	a	a	DET
ejpam-3838	48	5	code	code	NOUN
ejpam-3838	48	6	over	over	ADP
ejpam-3838	48	7	r1	r1	PROPN
ejpam-3838	48	8	.	.	PUNCT
ejpam-3838	49	1	the	the	DET
ejpam-3838	49	2	code	code	NOUN
ejpam-3838	49	3	{	{	PUNCT
ejpam-3838	49	4	v	v	NOUN
ejpam-3838	49	5	∈	∈	PRON
ejpam-3838	49	6	fnq	fnq	NOUN
ejpam-3838	50	1	|	|	ADV
ejpam-3838	50	2	∃w	∃w	PROPN
ejpam-3838	50	3	∈	∈	PROPN
ejpam-3838	50	4	fnq	fnq	PROPN
ejpam-3838	50	5	,	,	PUNCT
ejpam-3838	50	6	v	v	PROPN
ejpam-3838	50	7	+	+	CCONJ
ejpam-3838	50	8	uw	uw	PROPN
ejpam-3838	50	9	∈	∈	PROPN
ejpam-3838	50	10	c	c	PROPN
ejpam-3838	50	11	}	}	PUNCT
ejpam-3838	50	12	is	be	AUX
ejpam-3838	50	13	called	call	VERB
ejpam-3838	50	14	the	the	DET
ejpam-3838	50	15	residue	residue	NOUN
ejpam-3838	50	16	code	code	NOUN
ejpam-3838	50	17	of	of	ADP
ejpam-3838	50	18	c	c	PROPN
ejpam-3838	50	19	and	and	CCONJ
ejpam-3838	50	20	is	be	AUX
ejpam-3838	50	21	denoted	denote	VERB
ejpam-3838	50	22	by	by	ADP
ejpam-3838	50	23	res(c	res(c	PROPN
ejpam-3838	50	24	)	)	PUNCT
ejpam-3838	50	25	.	.	PUNCT
ejpam-3838	51	1	the	the	DET
ejpam-3838	51	2	code	code	NOUN
ejpam-3838	51	3	{	{	PUNCT
ejpam-3838	51	4	v	v	PROPN
ejpam-3838	51	5	∈	∈	PRON
ejpam-3838	51	6	fnq	fnq	NOUN
ejpam-3838	51	7	|uv	|uv	X
ejpam-3838	51	8	∈	∈	PROPN
ejpam-3838	51	9	c	c	X
ejpam-3838	51	10	}	}	PUNCT
ejpam-3838	51	11	is	be	AUX
ejpam-3838	51	12	called	call	VERB
ejpam-3838	51	13	the	the	DET
ejpam-3838	51	14	l.e	l.e	PROPN
ejpam-3838	51	15	.	.	PROPN
ejpam-3838	51	16	galvez	galvez	PROPN
ejpam-3838	51	17	,	,	PUNCT
ejpam-3838	51	18	r.a	r.a	PROPN
ejpam-3838	51	19	.	.	PROPN
ejpam-3838	51	20	betty	betty	PROPN
ejpam-3838	51	21	,	,	PUNCT
ejpam-3838	51	22	f.	f.	PROPN
ejpam-3838	51	23	nemenzo	nemenzo	PROPN
ejpam-3838	51	24	/	/	SYM
ejpam-3838	51	25	eur	eur	NOUN
ejpam-3838	51	26	.	.	PUNCT
ejpam-3838	52	1	j.	j.	PROPN
ejpam-3838	52	2	pure	pure	PROPN
ejpam-3838	52	3	appl	appl	PROPN
ejpam-3838	52	4	.	.	PROPN
ejpam-3838	52	5	math	math	PROPN
ejpam-3838	52	6	,	,	PUNCT
ejpam-3838	52	7	13	13	NUM
ejpam-3838	52	8	(	(	PUNCT
ejpam-3838	52	9	4	4	NUM
ejpam-3838	52	10	)	)	PUNCT
ejpam-3838	52	11	(	(	PUNCT
ejpam-3838	52	12	2020	2020	NUM
ejpam-3838	52	13	)	)	PUNCT
ejpam-3838	52	14	,	,	PUNCT
ejpam-3838	52	15	873	873	NUM
ejpam-3838	52	16	-	-	NUM
ejpam-3838	52	17	892	892	NUM
ejpam-3838	52	18	875	875	NUM
ejpam-3838	52	19	torsion	torsion	NOUN
ejpam-3838	52	20	code	code	NOUN
ejpam-3838	52	21	of	of	ADP
ejpam-3838	52	22	c	c	PROPN
ejpam-3838	52	23	and	and	CCONJ
ejpam-3838	52	24	is	be	AUX
ejpam-3838	52	25	denoted	denote	VERB
ejpam-3838	52	26	by	by	ADP
ejpam-3838	52	27	tor(c	tor(c	PROPN
ejpam-3838	52	28	)	)	PUNCT
ejpam-3838	52	29	.	.	PUNCT
ejpam-3838	53	1	if	if	SCONJ
ejpam-3838	53	2	c	c	PROPN
ejpam-3838	53	3	has	have	AUX
ejpam-3838	53	4	generator	generator	NOUN
ejpam-3838	53	5	matrix	matrix	NOUN
ejpam-3838	53	6	(	(	PUNCT
ejpam-3838	53	7	1	1	NUM
ejpam-3838	53	8	)	)	PUNCT
ejpam-3838	53	9	,	,	PUNCT
ejpam-3838	53	10	then	then	ADV
ejpam-3838	53	11	res(c	res(c	ADJ
ejpam-3838	53	12	)	)	PUNCT
ejpam-3838	53	13	and	and	CCONJ
ejpam-3838	53	14	tor(c	tor(c	PROPN
ejpam-3838	53	15	)	)	PUNCT
ejpam-3838	53	16	are	be	AUX
ejpam-3838	53	17	[	[	X
ejpam-3838	53	18	n	n	X
ejpam-3838	53	19	,	,	PUNCT
ejpam-3838	53	20	k0	k0	PROPN
ejpam-3838	53	21	]	]	PUNCT
ejpam-3838	53	22	and	and	CCONJ
ejpam-3838	53	23	[	[	X
ejpam-3838	53	24	n	n	CCONJ
ejpam-3838	53	25	,	,	PUNCT
ejpam-3838	53	26	k0	k0	PROPN
ejpam-3838	53	27	+	+	CCONJ
ejpam-3838	53	28	k1	k1	NOUN
ejpam-3838	53	29	]	]	PUNCT
ejpam-3838	53	30	codes	code	NOUN
ejpam-3838	53	31	over	over	ADP
ejpam-3838	53	32	fq	fq	PROPN
ejpam-3838	53	33	,	,	PUNCT
ejpam-3838	53	34	with	with	ADP
ejpam-3838	53	35	generator	generator	NOUN
ejpam-3838	53	36	matrices	matrix	NOUN
ejpam-3838	53	37	[	[	PUNCT
ejpam-3838	53	38	ik0	ik0	VERB
ejpam-3838	53	39	a	a	PRON
ejpam-3838	53	40	]	]	X
ejpam-3838	53	41	and	and	CCONJ
ejpam-3838	53	42	[	[	PUNCT
ejpam-3838	53	43	ik0	ik0	VERB
ejpam-3838	53	44	a	a	DET
ejpam-3838	53	45	0	0	NUM
ejpam-3838	53	46	d	d	NOUN
ejpam-3838	53	47	]	]	X
ejpam-3838	53	48	,	,	PUNCT
ejpam-3838	53	49	respectively	respectively	ADV
ejpam-3838	53	50	.	.	PUNCT
ejpam-3838	54	1	clearly	clearly	ADV
ejpam-3838	54	2	,	,	PUNCT
ejpam-3838	54	3	res(c	res(c	ADJ
ejpam-3838	54	4	)	)	PUNCT
ejpam-3838	54	5	⊆	⊆	NUM
ejpam-3838	54	6	tor(c	tor(c	PROPN
ejpam-3838	54	7	)	)	PUNCT
ejpam-3838	54	8	and	and	CCONJ
ejpam-3838	54	9	|c|	|c|	PROPN
ejpam-3838	54	10	=	=	SYM
ejpam-3838	54	11	q2k0+k1	q2k0+k1	NOUN
ejpam-3838	54	12	=	=	PUNCT
ejpam-3838	54	13	|res(c)||tor(c)|	|res(c)||tor(c)|	NOUN
ejpam-3838	54	14	.	.	PUNCT
ejpam-3838	55	1	the	the	DET
ejpam-3838	55	2	following	follow	VERB
ejpam-3838	55	3	lemma	lemma	PROPN
ejpam-3838	55	4	shows	show	VERB
ejpam-3838	55	5	the	the	DET
ejpam-3838	55	6	relationship	relationship	NOUN
ejpam-3838	55	7	between	between	ADP
ejpam-3838	55	8	the	the	DET
ejpam-3838	55	9	residue	residue	NOUN
ejpam-3838	55	10	and	and	CCONJ
ejpam-3838	55	11	torsion	torsion	NOUN
ejpam-3838	55	12	codes	code	NOUN
ejpam-3838	55	13	of	of	ADP
ejpam-3838	55	14	a	a	DET
ejpam-3838	55	15	self	self	NOUN
ejpam-3838	55	16	-	-	PUNCT
ejpam-3838	55	17	orthogonal	orthogonal	ADJ
ejpam-3838	55	18	code	code	NOUN
ejpam-3838	55	19	over	over	ADP
ejpam-3838	55	20	r1	r1	PROPN
ejpam-3838	55	21	.	.	PUNCT
ejpam-3838	56	1	the	the	DET
ejpam-3838	56	2	proof	proof	NOUN
ejpam-3838	56	3	is	be	AUX
ejpam-3838	56	4	given	give	VERB
ejpam-3838	56	5	in	in	ADP
ejpam-3838	56	6	[	[	X
ejpam-3838	56	7	6	6	NUM
ejpam-3838	56	8	]	]	PUNCT
ejpam-3838	56	9	.	.	PUNCT
ejpam-3838	57	1	lemma	lemma	PROPN
ejpam-3838	57	2	1	1	X
ejpam-3838	57	3	.	.	PUNCT
ejpam-3838	58	1	let	let	VERB
ejpam-3838	58	2	c	c	PRON
ejpam-3838	58	3	be	be	AUX
ejpam-3838	58	4	a	a	DET
ejpam-3838	58	5	(	(	PUNCT
ejpam-3838	58	6	euclidean	euclidean	ADJ
ejpam-3838	58	7	or	or	CCONJ
ejpam-3838	58	8	hermitian	hermitian	ADJ
ejpam-3838	58	9	)	)	PUNCT
ejpam-3838	58	10	self	self	NOUN
ejpam-3838	58	11	-	-	PUNCT
ejpam-3838	58	12	orthogonal	orthogonal	ADJ
ejpam-3838	58	13	code	code	NOUN
ejpam-3838	58	14	over	over	ADP
ejpam-3838	58	15	r1	r1	PROPN
ejpam-3838	58	16	.	.	PUNCT
ejpam-3838	59	1	then	then	ADV
ejpam-3838	59	2	(	(	PUNCT
ejpam-3838	59	3	i	i	NOUN
ejpam-3838	59	4	)	)	PUNCT
ejpam-3838	59	5	res(c	res(c	PROPN
ejpam-3838	59	6	)	)	PUNCT
ejpam-3838	59	7	is	be	AUX
ejpam-3838	59	8	self	self	NOUN
ejpam-3838	59	9	-	-	PUNCT
ejpam-3838	59	10	orthogonal	orthogonal	ADJ
ejpam-3838	59	11	,	,	PUNCT
ejpam-3838	59	12	i.e.	i.e.	X
ejpam-3838	59	13	res(c	res(c	ADJ
ejpam-3838	59	14	)	)	PUNCT
ejpam-3838	59	15	⊆	⊆	NUM
ejpam-3838	59	16	res(c)⊥	res(c)⊥	NOUN
ejpam-3838	59	17	;	;	PUNCT
ejpam-3838	59	18	(	(	PUNCT
ejpam-3838	59	19	ii	ii	NOUN
ejpam-3838	59	20	)	)	PUNCT
ejpam-3838	59	21	tor(c	tor(c	PROPN
ejpam-3838	59	22	)	)	PUNCT
ejpam-3838	59	23	⊆	⊆	NUM
ejpam-3838	59	24	res(c)⊥.	res(c)⊥.	NOUN
ejpam-3838	59	25	in	in	ADP
ejpam-3838	59	26	particular	particular	ADJ
ejpam-3838	59	27	,	,	PUNCT
ejpam-3838	59	28	if	if	SCONJ
ejpam-3838	59	29	c	c	PROPN
ejpam-3838	59	30	is	be	AUX
ejpam-3838	59	31	(	(	PUNCT
ejpam-3838	59	32	euclidean	euclidean	ADJ
ejpam-3838	59	33	or	or	CCONJ
ejpam-3838	59	34	hermitian	hermitian	ADJ
ejpam-3838	59	35	)	)	PUNCT
ejpam-3838	59	36	self	self	NOUN
ejpam-3838	59	37	-	-	PUNCT
ejpam-3838	59	38	dual	dual	ADJ
ejpam-3838	59	39	,	,	PUNCT
ejpam-3838	59	40	tor(c	tor(c	PROPN
ejpam-3838	59	41	)	)	PUNCT
ejpam-3838	59	42	=	=	SYM
ejpam-3838	59	43	res(c)⊥.	res(c)⊥.	PROPN
ejpam-3838	59	44	3	3	NUM
ejpam-3838	59	45	.	.	PUNCT
ejpam-3838	59	46	codes	code	NOUN
ejpam-3838	59	47	over	over	ADP
ejpam-3838	59	48	fq	fq	PROPN
ejpam-3838	59	49	+	+	CCONJ
ejpam-3838	59	50	ufq	ufq	NOUN
ejpam-3838	59	51	with	with	ADP
ejpam-3838	59	52	prescribed	prescribed	ADJ
ejpam-3838	59	53	residue	residue	NOUN
ejpam-3838	59	54	and	and	CCONJ
ejpam-3838	59	55	torsion	torsion	NOUN
ejpam-3838	59	56	let	let	AUX
ejpam-3838	59	57	c1	c1	PROPN
ejpam-3838	59	58	be	be	AUX
ejpam-3838	59	59	a	a	DET
ejpam-3838	59	60	code	code	NOUN
ejpam-3838	59	61	of	of	ADP
ejpam-3838	59	62	length	length	NOUN
ejpam-3838	59	63	n	n	PROPN
ejpam-3838	59	64	over	over	ADP
ejpam-3838	59	65	fq	fq	PROPN
ejpam-3838	59	66	with	with	ADP
ejpam-3838	59	67	dimension	dimension	NOUN
ejpam-3838	59	68	k0	k0	PROPN
ejpam-3838	59	69	and	and	CCONJ
ejpam-3838	59	70	generator	generator	PROPN
ejpam-3838	59	71	matrix	matrix	NOUN
ejpam-3838	59	72	[	[	PUNCT
ejpam-3838	59	73	ik0	ik0	VERB
ejpam-3838	59	74	a	a	PRON
ejpam-3838	59	75	]	]	PUNCT
ejpam-3838	59	76	,	,	PUNCT
ejpam-3838	59	77	(	(	PUNCT
ejpam-3838	59	78	2	2	X
ejpam-3838	59	79	)	)	PUNCT
ejpam-3838	59	80	and	and	CCONJ
ejpam-3838	59	81	c2	c2	VERB
ejpam-3838	59	82	a	a	DET
ejpam-3838	59	83	code	code	NOUN
ejpam-3838	59	84	of	of	ADP
ejpam-3838	59	85	length	length	NOUN
ejpam-3838	59	86	n	n	PROPN
ejpam-3838	59	87	over	over	ADP
ejpam-3838	59	88	fq	fq	PROPN
ejpam-3838	59	89	of	of	ADP
ejpam-3838	59	90	dimension	dimension	PROPN
ejpam-3838	59	91	k0	k0	PROPN
ejpam-3838	59	92	+	+	CCONJ
ejpam-3838	59	93	k1	k1	PROPN
ejpam-3838	59	94	and	and	CCONJ
ejpam-3838	59	95	has	have	VERB
ejpam-3838	59	96	generator	generator	NOUN
ejpam-3838	59	97	matrix	matrix	NOUN
ejpam-3838	59	98	[	[	PUNCT
ejpam-3838	59	99	ik0	ik0	VERB
ejpam-3838	59	100	a	a	DET
ejpam-3838	59	101	0	0	NUM
ejpam-3838	59	102	d	d	NOUN
ejpam-3838	59	103	]	]	PUNCT
ejpam-3838	59	104	,	,	PUNCT
ejpam-3838	59	105	(	(	PUNCT
ejpam-3838	59	106	3	3	X
ejpam-3838	59	107	)	)	PUNCT
ejpam-3838	59	108	where	where	SCONJ
ejpam-3838	59	109	a	a	DET
ejpam-3838	59	110	∈mk0×(n−k0)(fq	∈mk0×(n−k0)(fq	NOUN
ejpam-3838	59	111	)	)	PUNCT
ejpam-3838	59	112	,	,	PUNCT
ejpam-3838	59	113	and	and	CCONJ
ejpam-3838	59	114	d	d	ADP
ejpam-3838	59	115	∈mk1×(n−k0)(fq	∈mk1×(n−k0)(fq	PROPN
ejpam-3838	59	116	)	)	PUNCT
ejpam-3838	59	117	is	be	AUX
ejpam-3838	59	118	of	of	ADP
ejpam-3838	59	119	full	full	ADJ
ejpam-3838	59	120	row	row	NOUN
ejpam-3838	59	121	rank	rank	NOUN
ejpam-3838	59	122	.	.	PUNCT
ejpam-3838	60	1	lemma	lemma	PROPN
ejpam-3838	60	2	2	2	X
ejpam-3838	60	3	.	.	PUNCT
ejpam-3838	61	1	if	if	SCONJ
ejpam-3838	61	2	c	c	PROPN
ejpam-3838	61	3	is	be	AUX
ejpam-3838	61	4	a	a	DET
ejpam-3838	61	5	code	code	NOUN
ejpam-3838	61	6	of	of	ADP
ejpam-3838	61	7	length	length	NOUN
ejpam-3838	61	8	n	n	CCONJ
ejpam-3838	61	9	over	over	ADP
ejpam-3838	61	10	r1	r1	NOUN
ejpam-3838	61	11	with	with	ADP
ejpam-3838	61	12	res(c	res(c	PROPN
ejpam-3838	61	13	)	)	PUNCT
ejpam-3838	61	14	=	=	SYM
ejpam-3838	61	15	c1	c1	PROPN
ejpam-3838	61	16	and	and	CCONJ
ejpam-3838	61	17	tor(c	tor(c	PROPN
ejpam-3838	61	18	)	)	PUNCT
ejpam-3838	61	19	=	=	SYM
ejpam-3838	61	20	c2	c2	PROPN
ejpam-3838	61	21	,	,	PUNCT
ejpam-3838	61	22	then	then	ADV
ejpam-3838	61	23	there	there	PRON
ejpam-3838	61	24	exists	exist	VERB
ejpam-3838	61	25	a	a	DET
ejpam-3838	61	26	matrix	matrix	NOUN
ejpam-3838	61	27	n	n	CCONJ
ejpam-3838	61	28	∈mk0×(n−k0)(fq	∈mk0×(n−k0)(fq	NOUN
ejpam-3838	61	29	)	)	PUNCT
ejpam-3838	61	30	such	such	ADJ
ejpam-3838	61	31	that	that	SCONJ
ejpam-3838	61	32	the	the	DET
ejpam-3838	61	33	matrix	matrix	NOUN
ejpam-3838	61	34	[	[	PUNCT
ejpam-3838	61	35	ik0	ik0	NOUN
ejpam-3838	61	36	a+	a+	PUNCT
ejpam-3838	61	37	un	un	PROPN
ejpam-3838	61	38	0	0	PROPN
ejpam-3838	61	39	ud	ud	PROPN
ejpam-3838	61	40	]	]	X
ejpam-3838	61	41	(	(	PUNCT
ejpam-3838	61	42	4	4	X
ejpam-3838	61	43	)	)	PUNCT
ejpam-3838	61	44	is	be	AUX
ejpam-3838	61	45	a	a	DET
ejpam-3838	61	46	generator	generator	NOUN
ejpam-3838	61	47	matrix	matrix	NOUN
ejpam-3838	61	48	of	of	ADP
ejpam-3838	61	49	c.	c.	PROPN
ejpam-3838	61	50	such	such	ADJ
ejpam-3838	61	51	matrix	matrix	NOUN
ejpam-3838	61	52	n	n	VERB
ejpam-3838	61	53	is	be	AUX
ejpam-3838	61	54	unique	unique	ADJ
ejpam-3838	61	55	if	if	SCONJ
ejpam-3838	61	56	c	c	PROPN
ejpam-3838	61	57	is	be	AUX
ejpam-3838	61	58	a	a	DET
ejpam-3838	61	59	free	free	ADJ
ejpam-3838	61	60	code	code	NOUN
ejpam-3838	61	61	.	.	PUNCT
ejpam-3838	62	1	proof	proof	NOUN
ejpam-3838	62	2	.	.	PUNCT
ejpam-3838	63	1	since	since	SCONJ
ejpam-3838	63	2	the	the	DET
ejpam-3838	63	3	residue	residue	NOUN
ejpam-3838	63	4	and	and	CCONJ
ejpam-3838	63	5	torsion	torsion	NOUN
ejpam-3838	63	6	codes	code	NOUN
ejpam-3838	63	7	of	of	ADP
ejpam-3838	63	8	c	c	PROPN
ejpam-3838	63	9	are	be	AUX
ejpam-3838	63	10	c1	c1	PROPN
ejpam-3838	63	11	and	and	CCONJ
ejpam-3838	63	12	c2	c2	PROPN
ejpam-3838	63	13	,	,	PUNCT
ejpam-3838	63	14	respectively	respectively	ADV
ejpam-3838	63	15	,	,	PUNCT
ejpam-3838	63	16	then	then	ADV
ejpam-3838	63	17	for	for	ADP
ejpam-3838	63	18	some	some	DET
ejpam-3838	63	19	m1	m1	NOUN
ejpam-3838	63	20	∈mk0(fq	∈mk0(fq	NOUN
ejpam-3838	63	21	)	)	PUNCT
ejpam-3838	63	22	and	and	CCONJ
ejpam-3838	63	23	m2	m2	PROPN
ejpam-3838	63	24	∈mk0×(n−k0)(fq	∈mk0×(n−k0)(fq	PROPN
ejpam-3838	63	25	)	)	PUNCT
ejpam-3838	63	26	,	,	PUNCT
ejpam-3838	63	27	rk0+k11	rk0+k11	VERB
ejpam-3838	63	28	[	[	PUNCT
ejpam-3838	63	29	ik0	ik0	VERB
ejpam-3838	63	30	+	+	CCONJ
ejpam-3838	63	31	um1	um1	X
ejpam-3838	63	32	a+	a+	X
ejpam-3838	63	33	um2	um2	PROPN
ejpam-3838	63	34	0	0	PROPN
ejpam-3838	64	1	ud	ud	NOUN
ejpam-3838	64	2	]	]	PUNCT
ejpam-3838	64	3	⊆	⊆	NUM
ejpam-3838	64	4	c.	c.	NOUN
ejpam-3838	64	5	by	by	ADP
ejpam-3838	64	6	an	an	DET
ejpam-3838	64	7	elementary	elementary	PROPN
ejpam-3838	64	8	row	row	NOUN
ejpam-3838	64	9	operation	operation	NOUN
ejpam-3838	64	10	,	,	PUNCT
ejpam-3838	64	11	l.e	l.e	PROPN
ejpam-3838	64	12	.	.	PROPN
ejpam-3838	64	13	galvez	galvez	PROPN
ejpam-3838	64	14	,	,	PUNCT
ejpam-3838	64	15	r.a	r.a	PROPN
ejpam-3838	64	16	.	.	PROPN
ejpam-3838	64	17	betty	betty	PROPN
ejpam-3838	64	18	,	,	PUNCT
ejpam-3838	64	19	f.	f.	PROPN
ejpam-3838	64	20	nemenzo	nemenzo	PROPN
ejpam-3838	64	21	/	/	SYM
ejpam-3838	64	22	eur	eur	NOUN
ejpam-3838	64	23	.	.	PUNCT
ejpam-3838	65	1	j.	j.	PROPN
ejpam-3838	65	2	pure	pure	PROPN
ejpam-3838	65	3	appl	appl	PROPN
ejpam-3838	65	4	.	.	PROPN
ejpam-3838	65	5	math	math	PROPN
ejpam-3838	65	6	,	,	PUNCT
ejpam-3838	65	7	13	13	NUM
ejpam-3838	65	8	(	(	PUNCT
ejpam-3838	65	9	4	4	NUM
ejpam-3838	65	10	)	)	PUNCT
ejpam-3838	65	11	(	(	PUNCT
ejpam-3838	65	12	2020	2020	NUM
ejpam-3838	65	13	)	)	PUNCT
ejpam-3838	65	14	,	,	PUNCT
ejpam-3838	65	15	873	873	NUM
ejpam-3838	65	16	-	-	SYM
ejpam-3838	65	17	892	892	NUM
ejpam-3838	65	18	876	876	NUM
ejpam-3838	65	19	c	c	PROPN
ejpam-3838	65	20	⊇	⊇	PROPN
ejpam-3838	65	21	rk0+k11	rk0+k11	VERB
ejpam-3838	65	22	[	[	PUNCT
ejpam-3838	65	23	ik0	ik0	NOUN
ejpam-3838	65	24	−	−	PROPN
ejpam-3838	65	25	um1	um1	X
ejpam-3838	65	26	0	0	NUM
ejpam-3838	65	27	0	0	NUM
ejpam-3838	66	1	ik1	ik1	ADJ
ejpam-3838	66	2	]	]	X
ejpam-3838	66	3	[	[	PUNCT
ejpam-3838	66	4	ik0	ik0	VERB
ejpam-3838	66	5	+	+	CCONJ
ejpam-3838	66	6	um1	um1	X
ejpam-3838	66	7	a+	a+	X
ejpam-3838	66	8	um2	um2	PROPN
ejpam-3838	66	9	0	0	PROPN
ejpam-3838	67	1	ud	ud	NOUN
ejpam-3838	67	2	]	]	X
ejpam-3838	67	3	=	=	SYM
ejpam-3838	67	4	rk0+k11	rk0+k11	VERB
ejpam-3838	67	5	[	[	PUNCT
ejpam-3838	67	6	ik0	ik0	NOUN
ejpam-3838	67	7	a+	a+	PUNCT
ejpam-3838	67	8	u(m2	u(m2	ADJ
ejpam-3838	67	9	−m1a	−m1a	NOUN
ejpam-3838	67	10	)	)	PUNCT
ejpam-3838	67	11	0	0	PUNCT
ejpam-3838	68	1	ud	ud	INTJ
ejpam-3838	68	2	]	]	PUNCT
ejpam-3838	68	3	.	.	PUNCT
ejpam-3838	69	1	taking	take	VERB
ejpam-3838	69	2	n	n	NOUN
ejpam-3838	69	3	=	=	SYM
ejpam-3838	69	4	m2	m2	PROPN
ejpam-3838	69	5	−m1a	−m1a	PROPN
ejpam-3838	69	6	,	,	PUNCT
ejpam-3838	69	7	we	we	PRON
ejpam-3838	69	8	have	have	VERB
ejpam-3838	69	9	|c|	|c|	PROPN
ejpam-3838	69	10	≥	≥	NOUN
ejpam-3838	69	11	∣∣∣∣rk0+k11	∣∣∣∣rk0+k11	ADP
ejpam-3838	69	12	[	[	PUNCT
ejpam-3838	69	13	ik0	ik0	NOUN
ejpam-3838	69	14	a+	a+	PUNCT
ejpam-3838	69	15	un	un	PROPN
ejpam-3838	69	16	0	0	PROPN
ejpam-3838	69	17	ud	ud	PROPN
ejpam-3838	69	18	]	]	X
ejpam-3838	69	19	∣∣∣∣	∣∣∣∣	NOUN
ejpam-3838	69	20	=	=	PUNCT
ejpam-3838	69	21	q2k0+k1	q2k0+k1	NOUN
ejpam-3838	69	22	=	=	PUNCT
ejpam-3838	69	23	|c1|	|c1|	PROPN
ejpam-3838	69	24	|c2|	|c2|	NOUN
ejpam-3838	69	25	=	=	PROPN
ejpam-3838	69	26	|c|	|c|	PROPN
ejpam-3838	69	27	.	.	PUNCT
ejpam-3838	70	1	thus	thus	ADV
ejpam-3838	70	2	,	,	PUNCT
ejpam-3838	70	3	c	c	PROPN
ejpam-3838	70	4	has	have	VERB
ejpam-3838	70	5	a	a	DET
ejpam-3838	70	6	generator	generator	NOUN
ejpam-3838	70	7	matrix	matrix	NOUN
ejpam-3838	70	8	(	(	PUNCT
ejpam-3838	70	9	4	4	NUM
ejpam-3838	70	10	)	)	PUNCT
ejpam-3838	70	11	.	.	PUNCT
ejpam-3838	71	1	suppose	suppose	VERB
ejpam-3838	71	2	c	c	NOUN
ejpam-3838	71	3	is	be	AUX
ejpam-3838	71	4	a	a	DET
ejpam-3838	71	5	free	free	ADJ
ejpam-3838	71	6	code	code	NOUN
ejpam-3838	71	7	and	and	CCONJ
ejpam-3838	71	8	there	there	PRON
ejpam-3838	71	9	exist	exist	VERB
ejpam-3838	71	10	n1	n1	NOUN
ejpam-3838	71	11	,	,	PUNCT
ejpam-3838	71	12	n2	n2	ADJ
ejpam-3838	71	13	∈mk0×(n−k0)(fq	∈mk0×(n−k0)(fq	NOUN
ejpam-3838	71	14	)	)	PUNCT
ejpam-3838	72	1	such	such	ADJ
ejpam-3838	72	2	that	that	SCONJ
ejpam-3838	72	3	rk01	rk01	PROPN
ejpam-3838	73	1	[	[	X
ejpam-3838	73	2	i	i	PRON
ejpam-3838	73	3	a+	a+	PUNCT
ejpam-3838	73	4	un1	un1	NOUN
ejpam-3838	73	5	]	]	X
ejpam-3838	74	1	=	=	PUNCT
ejpam-3838	75	1	rk01	rk01	PROPN
ejpam-3838	75	2	[	[	X
ejpam-3838	75	3	i	i	PRON
ejpam-3838	75	4	a+	a+	PUNCT
ejpam-3838	75	5	un2	un2	NOUN
ejpam-3838	75	6	]	]	PUNCT
ejpam-3838	75	7	.	.	PUNCT
ejpam-3838	76	1	then	then	ADV
ejpam-3838	76	2	a+	a+	PUNCT
ejpam-3838	76	3	un1	un1	PROPN
ejpam-3838	76	4	≡	≡	PROPN
ejpam-3838	76	5	a+	a+	PUNCT
ejpam-3838	76	6	un2	un2	PROPN
ejpam-3838	76	7	(	(	PUNCT
ejpam-3838	76	8	u2	u2	PROPN
ejpam-3838	76	9	)	)	PUNCT
ejpam-3838	76	10	,	,	PUNCT
ejpam-3838	76	11	which	which	PRON
ejpam-3838	76	12	implies	imply	VERB
ejpam-3838	76	13	that	that	SCONJ
ejpam-3838	76	14	n1	n1	PROPN
ejpam-3838	76	15	≡	≡	PROPN
ejpam-3838	76	16	n2	n2	NOUN
ejpam-3838	76	17	(	(	PUNCT
ejpam-3838	76	18	u	u	NOUN
ejpam-3838	76	19	)	)	PUNCT
ejpam-3838	76	20	.	.	PUNCT
ejpam-3838	77	1	�	�	PROPN
ejpam-3838	77	2	for	for	ADP
ejpam-3838	77	3	the	the	DET
ejpam-3838	77	4	remainder	remainder	NOUN
ejpam-3838	77	5	of	of	ADP
ejpam-3838	77	6	this	this	DET
ejpam-3838	77	7	section	section	NOUN
ejpam-3838	77	8	,	,	PUNCT
ejpam-3838	77	9	assume	assume	VERB
ejpam-3838	77	10	that	that	SCONJ
ejpam-3838	77	11	c1	c1	PROPN
ejpam-3838	77	12	⊆	⊆	NUM
ejpam-3838	77	13	c2	c2	PROPN
ejpam-3838	77	14	⊆	⊆	NUM
ejpam-3838	77	15	c⊥1	c⊥1	NOUN
ejpam-3838	77	16	.	.	PUNCT
ejpam-3838	78	1	then	then	ADV
ejpam-3838	78	2	ik0	ik0	VERB
ejpam-3838	79	1	+	+	ADV
ejpam-3838	79	2	aat	aat	ADJ
ejpam-3838	79	3	≡	≡	PROPN
ejpam-3838	79	4	0	0	PUNCT
ejpam-3838	79	5	(	(	PUNCT
ejpam-3838	79	6	u	u	NOUN
ejpam-3838	79	7	)	)	PUNCT
ejpam-3838	79	8	,	,	PUNCT
ejpam-3838	79	9	(	(	PUNCT
ejpam-3838	79	10	5	5	X
ejpam-3838	79	11	)	)	PUNCT
ejpam-3838	79	12	dat	dat	NOUN
ejpam-3838	79	13	≡	≡	PROPN
ejpam-3838	79	14	0	0	PUNCT
ejpam-3838	79	15	(	(	PUNCT
ejpam-3838	79	16	u	u	NOUN
ejpam-3838	79	17	)	)	PUNCT
ejpam-3838	79	18	.	.	PUNCT
ejpam-3838	80	1	(	(	PUNCT
ejpam-3838	80	2	6	6	X
ejpam-3838	80	3	)	)	PUNCT
ejpam-3838	80	4	it	it	PRON
ejpam-3838	80	5	follows	follow	VERB
ejpam-3838	80	6	from	from	ADP
ejpam-3838	80	7	(	(	PUNCT
ejpam-3838	80	8	5	5	NUM
ejpam-3838	80	9	)	)	PUNCT
ejpam-3838	80	10	that	that	SCONJ
ejpam-3838	80	11	a	a	PRON
ejpam-3838	80	12	is	be	AUX
ejpam-3838	80	13	of	of	ADP
ejpam-3838	80	14	full	full	ADJ
ejpam-3838	80	15	row	row	NOUN
ejpam-3838	80	16	rank	rank	NOUN
ejpam-3838	80	17	.	.	PUNCT
ejpam-3838	81	1	denote	denote	VERB
ejpam-3838	81	2	by	by	ADP
ejpam-3838	81	3	symk0(fq	symk0(fq	PROPN
ejpam-3838	81	4	)	)	PUNCT
ejpam-3838	81	5	the	the	DET
ejpam-3838	81	6	set	set	NOUN
ejpam-3838	81	7	of	of	ADP
ejpam-3838	81	8	k0×k0	k0×k0	PROPN
ejpam-3838	81	9	symmetric	symmetric	ADJ
ejpam-3838	81	10	matrices	matrix	NOUN
ejpam-3838	81	11	,	,	PUNCT
ejpam-3838	81	12	altk0(fq	altk0(fq	PROPN
ejpam-3838	81	13	)	)	PUNCT
ejpam-3838	81	14	the	the	DET
ejpam-3838	81	15	set	set	NOUN
ejpam-3838	81	16	of	of	ADP
ejpam-3838	81	17	k0×k0	k0×k0	NOUN
ejpam-3838	81	18	alternating	alternate	VERB
ejpam-3838	81	19	matrices	matrix	NOUN
ejpam-3838	81	20	,	,	PUNCT
ejpam-3838	81	21	and	and	CCONJ
ejpam-3838	81	22	skewk0(fq	skewk0(fq	NOUN
ejpam-3838	81	23	)	)	PUNCT
ejpam-3838	81	24	the	the	DET
ejpam-3838	81	25	set	set	NOUN
ejpam-3838	81	26	of	of	ADP
ejpam-3838	81	27	k0	k0	PROPN
ejpam-3838	81	28	×	×	PROPN
ejpam-3838	81	29	k0	k0	PROPN
ejpam-3838	81	30	skew	skew	ADJ
ejpam-3838	81	31	-	-	PUNCT
ejpam-3838	81	32	symmetric	symmetric	ADJ
ejpam-3838	81	33	matrices	matrix	NOUN
ejpam-3838	81	34	over	over	ADP
ejpam-3838	81	35	fq	fq	PROPN
ejpam-3838	81	36	.	.	PUNCT
ejpam-3838	82	1	lemma	lemma	PROPN
ejpam-3838	82	2	3	3	X
ejpam-3838	82	3	.	.	PUNCT
ejpam-3838	83	1	let	let	VERB
ejpam-3838	83	2	a	a	DET
ejpam-3838	83	3	∈mm×n(fq	∈mm×n(fq	PROPN
ejpam-3838	83	4	)	)	PUNCT
ejpam-3838	83	5	where	where	SCONJ
ejpam-3838	83	6	rank	rank	VERB
ejpam-3838	83	7	a	a	DET
ejpam-3838	83	8	=	=	NOUN
ejpam-3838	83	9	m.	m.	NOUN
ejpam-3838	83	10	we	we	PRON
ejpam-3838	83	11	define	define	VERB
ejpam-3838	83	12	the	the	DET
ejpam-3838	83	13	mappings	mapping	NOUN
ejpam-3838	83	14	ψa	ψa	ADP
ejpam-3838	83	15	:	:	PUNCT
ejpam-3838	83	16	mm×n(fq	mm×n(fq	ADJ
ejpam-3838	83	17	)	)	PUNCT
ejpam-3838	83	18	−→	−→	NOUN
ejpam-3838	83	19	mm(fq	mm(fq	NOUN
ejpam-3838	83	20	)	)	PUNCT
ejpam-3838	84	1	n	n	NOUN
ejpam-3838	84	2	7−→	7−→	NOUN
ejpam-3838	84	3	ant	ant	NOUN
ejpam-3838	84	4	+	+	CCONJ
ejpam-3838	84	5	nat	nat	NOUN
ejpam-3838	84	6	,	,	PUNCT
ejpam-3838	84	7	and	and	CCONJ
ejpam-3838	84	8	φa	φa	INTJ
ejpam-3838	84	9	:	:	PUNCT
ejpam-3838	84	10	mm×n(fq	mm×n(fq	ADJ
ejpam-3838	84	11	)	)	PUNCT
ejpam-3838	84	12	−→	−→	NOUN
ejpam-3838	84	13	mm(fq	mm(fq	NOUN
ejpam-3838	84	14	)	)	PUNCT
ejpam-3838	84	15	n	n	CCONJ
ejpam-3838	84	16	7−→	7−→	PROPN
ejpam-3838	84	17	ant	ant	NOUN
ejpam-3838	84	18	−nat	−nat	NOUN
ejpam-3838	84	19	.	.	PUNCT
ejpam-3838	85	1	then	then	ADV
ejpam-3838	85	2	ψa	ψa	X
ejpam-3838	85	3	(	(	PUNCT
ejpam-3838	85	4	mm×n(fq	mm×n(fq	PROPN
ejpam-3838	85	5	)	)	PUNCT
ejpam-3838	85	6	)	)	PUNCT
ejpam-3838	86	1	=	=	PRON
ejpam-3838	86	2	{	{	PUNCT
ejpam-3838	86	3	symm(fq	symm(fq	NOUN
ejpam-3838	86	4	)	)	PUNCT
ejpam-3838	86	5	,	,	PUNCT
ejpam-3838	86	6	if	if	SCONJ
ejpam-3838	86	7	q	q	NOUN
ejpam-3838	86	8	is	be	AUX
ejpam-3838	86	9	odd	odd	ADJ
ejpam-3838	86	10	altm(fq	altm(fq	NOUN
ejpam-3838	86	11	)	)	PUNCT
ejpam-3838	86	12	,	,	PUNCT
ejpam-3838	86	13	if	if	SCONJ
ejpam-3838	86	14	q	q	NOUN
ejpam-3838	86	15	is	be	AUX
ejpam-3838	86	16	even	even	ADV
ejpam-3838	86	17	,	,	PUNCT
ejpam-3838	86	18	and	and	CCONJ
ejpam-3838	86	19	the	the	DET
ejpam-3838	86	20	image	image	NOUN
ejpam-3838	86	21	of	of	ADP
ejpam-3838	86	22	the	the	DET
ejpam-3838	86	23	map	map	NOUN
ejpam-3838	86	24	φa	φa	ADP
ejpam-3838	86	25	is	be	AUX
ejpam-3838	86	26	skewm(fq	skewm(fq	NOUN
ejpam-3838	86	27	)	)	PUNCT
ejpam-3838	86	28	.	.	PUNCT
ejpam-3838	87	1	l.e	l.e	PROPN
ejpam-3838	87	2	.	.	PROPN
ejpam-3838	87	3	galvez	galvez	PROPN
ejpam-3838	87	4	,	,	PUNCT
ejpam-3838	87	5	r.a	r.a	PROPN
ejpam-3838	87	6	.	.	PROPN
ejpam-3838	87	7	betty	betty	PROPN
ejpam-3838	87	8	,	,	PUNCT
ejpam-3838	87	9	f.	f.	PROPN
ejpam-3838	87	10	nemenzo	nemenzo	PROPN
ejpam-3838	87	11	/	/	SYM
ejpam-3838	87	12	eur	eur	NOUN
ejpam-3838	87	13	.	.	PUNCT
ejpam-3838	88	1	j.	j.	PROPN
ejpam-3838	88	2	pure	pure	PROPN
ejpam-3838	88	3	appl	appl	PROPN
ejpam-3838	88	4	.	.	PROPN
ejpam-3838	88	5	math	math	PROPN
ejpam-3838	88	6	,	,	PUNCT
ejpam-3838	88	7	13	13	NUM
ejpam-3838	88	8	(	(	PUNCT
ejpam-3838	88	9	4	4	NUM
ejpam-3838	88	10	)	)	PUNCT
ejpam-3838	88	11	(	(	PUNCT
ejpam-3838	88	12	2020	2020	NUM
ejpam-3838	88	13	)	)	PUNCT
ejpam-3838	88	14	,	,	PUNCT
ejpam-3838	88	15	873	873	NUM
ejpam-3838	88	16	-	-	SYM
ejpam-3838	88	17	892	892	NUM
ejpam-3838	88	18	877	877	NUM
ejpam-3838	88	19	proof	proof	NOUN
ejpam-3838	88	20	.	.	PUNCT
ejpam-3838	89	1	the	the	DET
ejpam-3838	89	2	image	image	NOUN
ejpam-3838	89	3	of	of	ADP
ejpam-3838	89	4	ψa	ψa	PROPN
ejpam-3838	89	5	was	be	AUX
ejpam-3838	89	6	shown	show	VERB
ejpam-3838	89	7	in	in	ADP
ejpam-3838	89	8	[	[	X
ejpam-3838	89	9	2	2	NUM
ejpam-3838	89	10	]	]	PUNCT
ejpam-3838	89	11	.	.	PUNCT
ejpam-3838	90	1	since	since	SCONJ
ejpam-3838	90	2	rank	rank	VERB
ejpam-3838	90	3	a	a	DET
ejpam-3838	90	4	=	=	PROPN
ejpam-3838	90	5	m	m	PROPN
ejpam-3838	90	6	,	,	PUNCT
ejpam-3838	90	7	a(mn×m(fq	a(mn×m(fq	PROPN
ejpam-3838	90	8	)	)	PUNCT
ejpam-3838	90	9	)	)	PUNCT
ejpam-3838	91	1	=	=	PUNCT
ejpam-3838	91	2	mm(fq	mm(fq	NOUN
ejpam-3838	91	3	)	)	PUNCT
ejpam-3838	91	4	.	.	PUNCT
ejpam-3838	92	1	indeed	indeed	ADV
ejpam-3838	92	2	,	,	PUNCT
ejpam-3838	92	3	φa(mk0×(n−k0)(fq	φa(mk0×(n−k0)(fq	NOUN
ejpam-3838	92	4	)	)	PUNCT
ejpam-3838	92	5	)	)	PUNCT
ejpam-3838	93	1	=	=	PRON
ejpam-3838	93	2	{	{	PUNCT
ejpam-3838	93	3	ant	ant	PROPN
ejpam-3838	93	4	−nat	−nat	NOUN
ejpam-3838	93	5	|	|	ADV
ejpam-3838	93	6	n	n	CCONJ
ejpam-3838	93	7	∈mk0×(n−k0)(fq	∈mk0×(n−k0)(fq	NOUN
ejpam-3838	93	8	)	)	PUNCT
ejpam-3838	93	9	}	}	PUNCT
ejpam-3838	94	1	=	=	PUNCT
ejpam-3838	94	2	{	{	PUNCT
ejpam-3838	94	3	s	s	PROPN
ejpam-3838	94	4	−	−	PROPN
ejpam-3838	94	5	st	st	PROPN
ejpam-3838	95	1	|	|	NOUN
ejpam-3838	95	2	s	s	NOUN
ejpam-3838	95	3	∈mk0(fq	∈mk0(fq	NOUN
ejpam-3838	95	4	)	)	PUNCT
ejpam-3838	95	5	}	}	PUNCT
ejpam-3838	95	6	=	=	SYM
ejpam-3838	95	7	skewk0(fq	skewk0(fq	NOUN
ejpam-3838	95	8	)	)	PUNCT
ejpam-3838	95	9	.	.	PUNCT
ejpam-3838	96	1	�	�	PROPN
ejpam-3838	96	2	lemma	lemma	PROPN
ejpam-3838	96	3	4	4	X
ejpam-3838	96	4	.	.	PUNCT
ejpam-3838	97	1	the	the	DET
ejpam-3838	97	2	number	number	NOUN
ejpam-3838	97	3	of	of	ADP
ejpam-3838	97	4	free	free	ADJ
ejpam-3838	97	5	euclidean	euclidean	ADJ
ejpam-3838	97	6	self	self	NOUN
ejpam-3838	97	7	-	-	PUNCT
ejpam-3838	97	8	orthogonal	orthogonal	ADJ
ejpam-3838	97	9	codes	code	NOUN
ejpam-3838	97	10	over	over	ADP
ejpam-3838	97	11	r1	r1	NOUN
ejpam-3838	97	12	with	with	ADP
ejpam-3838	97	13	residue	residue	NOUN
ejpam-3838	97	14	code	code	NOUN
ejpam-3838	97	15	c1	c1	PROPN
ejpam-3838	97	16	is	be	AUX
ejpam-3838	97	17	qk0(2n−3k0+ε)/2	qk0(2n−3k0+ε)/2	PROPN
ejpam-3838	97	18	,	,	PUNCT
ejpam-3838	97	19	where	where	SCONJ
ejpam-3838	97	20	ε	ε	PROPN
ejpam-3838	97	21	=	=	VERB
ejpam-3838	97	22	−1	−1	NOUN
ejpam-3838	97	23	if	if	SCONJ
ejpam-3838	97	24	q	q	NOUN
ejpam-3838	97	25	is	be	AUX
ejpam-3838	97	26	odd	odd	ADJ
ejpam-3838	97	27	and	and	CCONJ
ejpam-3838	97	28	ε	ε	PROPN
ejpam-3838	97	29	=	=	SYM
ejpam-3838	97	30	1	1	NUM
ejpam-3838	97	31	if	if	SCONJ
ejpam-3838	97	32	q	q	NOUN
ejpam-3838	97	33	is	be	AUX
ejpam-3838	97	34	even	even	ADV
ejpam-3838	97	35	.	.	PUNCT
ejpam-3838	98	1	the	the	DET
ejpam-3838	98	2	number	number	NOUN
ejpam-3838	98	3	of	of	ADP
ejpam-3838	98	4	free	free	ADJ
ejpam-3838	98	5	hermitian	hermitian	ADJ
ejpam-3838	98	6	self	self	NOUN
ejpam-3838	98	7	-	-	PUNCT
ejpam-3838	98	8	orthogonal	orthogonal	ADJ
ejpam-3838	98	9	codes	code	NOUN
ejpam-3838	98	10	over	over	ADP
ejpam-3838	98	11	r1	r1	NOUN
ejpam-3838	98	12	with	with	ADP
ejpam-3838	98	13	residue	residue	NOUN
ejpam-3838	98	14	code	code	NOUN
ejpam-3838	98	15	c1	c1	PROPN
ejpam-3838	98	16	is	be	AUX
ejpam-3838	98	17	qk0(2n−3k0	qk0(2n−3k0	PROPN
ejpam-3838	98	18	+	+	NOUN
ejpam-3838	98	19	1)/2	1)/2	NOUN
ejpam-3838	98	20	.	.	PUNCT
ejpam-3838	99	1	proof	proof	NOUN
ejpam-3838	99	2	.	.	PUNCT
ejpam-3838	100	1	if	if	SCONJ
ejpam-3838	100	2	c	c	PROPN
ejpam-3838	100	3	is	be	AUX
ejpam-3838	100	4	a	a	DET
ejpam-3838	100	5	free	free	ADJ
ejpam-3838	100	6	code	code	NOUN
ejpam-3838	100	7	with	with	ADP
ejpam-3838	100	8	residue	residue	NOUN
ejpam-3838	100	9	code	code	NOUN
ejpam-3838	100	10	c1	c1	NOUN
ejpam-3838	100	11	,	,	PUNCT
ejpam-3838	100	12	then	then	ADV
ejpam-3838	100	13	by	by	ADP
ejpam-3838	100	14	lemma	lemma	PROPN
ejpam-3838	100	15	2	2	NUM
ejpam-3838	100	16	,	,	PUNCT
ejpam-3838	100	17	c	c	PROPN
ejpam-3838	100	18	has	have	VERB
ejpam-3838	100	19	generator	generator	NOUN
ejpam-3838	100	20	matrix	matrix	NOUN
ejpam-3838	100	21	[	[	PUNCT
ejpam-3838	100	22	ik0	ik0	NOUN
ejpam-3838	100	23	a+	a+	PUNCT
ejpam-3838	100	24	un	un	PROPN
ejpam-3838	100	25	]	]	PUNCT
ejpam-3838	100	26	,	,	PUNCT
ejpam-3838	100	27	for	for	ADP
ejpam-3838	100	28	some	some	DET
ejpam-3838	100	29	unique	unique	ADJ
ejpam-3838	100	30	n	n	PRON
ejpam-3838	100	31	∈mk0×(n−k0)(fq	∈mk0×(n−k0)(fq	NOUN
ejpam-3838	100	32	)	)	PUNCT
ejpam-3838	100	33	.	.	PUNCT
ejpam-3838	101	1	observe	observe	VERB
ejpam-3838	101	2	that	that	SCONJ
ejpam-3838	101	3	c	c	PROPN
ejpam-3838	101	4	is	be	AUX
ejpam-3838	101	5	euclidean	euclidean	ADJ
ejpam-3838	101	6	self	self	NOUN
ejpam-3838	101	7	-	-	PUNCT
ejpam-3838	101	8	orthogonal	orthogonal	ADJ
ejpam-3838	101	9	if	if	SCONJ
ejpam-3838	101	10	and	and	CCONJ
ejpam-3838	101	11	only	only	ADV
ejpam-3838	101	12	if	if	SCONJ
ejpam-3838	101	13	ik0	ik0	VERB
ejpam-3838	101	14	+	+	ADV
ejpam-3838	101	15	aat	aat	VERB
ejpam-3838	101	16	+	+	CCONJ
ejpam-3838	102	1	u(ant	u(ant	PROPN
ejpam-3838	102	2	+	+	ADJ
ejpam-3838	102	3	nat	nat	NOUN
ejpam-3838	102	4	)	)	PUNCT
ejpam-3838	102	5	≡	≡	PROPN
ejpam-3838	102	6	0	0	PUNCT
ejpam-3838	103	1	(	(	PUNCT
ejpam-3838	103	2	u2	u2	NOUN
ejpam-3838	103	3	)	)	PUNCT
ejpam-3838	103	4	.	.	PUNCT
ejpam-3838	104	1	hence	hence	ADV
ejpam-3838	104	2	,	,	PUNCT
ejpam-3838	104	3	the	the	DET
ejpam-3838	104	4	number	number	NOUN
ejpam-3838	104	5	of	of	ADP
ejpam-3838	104	6	free	free	ADJ
ejpam-3838	104	7	euclidean	euclidean	ADJ
ejpam-3838	104	8	self	self	NOUN
ejpam-3838	104	9	-	-	PUNCT
ejpam-3838	104	10	orthogonal	orthogonal	ADJ
ejpam-3838	104	11	codes	code	NOUN
ejpam-3838	104	12	c	c	NOUN
ejpam-3838	104	13	with	with	ADP
ejpam-3838	104	14	residue	residue	NOUN
ejpam-3838	104	15	code	code	NOUN
ejpam-3838	104	16	c1	c1	PROPN
ejpam-3838	104	17	is∣∣{n	is∣∣{n	PROPN
ejpam-3838	104	18	∈mk0×(n−k0)|ik0	∈mk0×(n−k0)|ik0	ADJ
ejpam-3838	104	19	+	+	SYM
ejpam-3838	104	20	aat	aat	NOUN
ejpam-3838	104	21	+	+	CCONJ
ejpam-3838	104	22	u(ant	u(ant	PROPN
ejpam-3838	104	23	+	+	ADJ
ejpam-3838	104	24	nat	nat	NOUN
ejpam-3838	104	25	)	)	PUNCT
ejpam-3838	104	26	≡	≡	PROPN
ejpam-3838	104	27	0	0	PUNCT
ejpam-3838	105	1	(	(	PUNCT
ejpam-3838	105	2	u2	u2	NOUN
ejpam-3838	105	3	)	)	PUNCT
ejpam-3838	105	4	}	}	PUNCT
ejpam-3838	105	5	∣∣	∣∣	X
ejpam-3838	105	6	.	.	PUNCT
ejpam-3838	106	1	(	(	PUNCT
ejpam-3838	106	2	7	7	X
ejpam-3838	106	3	)	)	PUNCT
ejpam-3838	106	4	by	by	ADP
ejpam-3838	106	5	(	(	PUNCT
ejpam-3838	106	6	5	5	NUM
ejpam-3838	106	7	)	)	PUNCT
ejpam-3838	106	8	,	,	PUNCT
ejpam-3838	106	9	we	we	PRON
ejpam-3838	106	10	have	have	VERB
ejpam-3838	106	11	ant	ant	ADJ
ejpam-3838	107	1	+	+	ADP
ejpam-3838	107	2	nat	nat	NOUN
ejpam-3838	107	3	≡	≡	PROPN
ejpam-3838	107	4	0	0	PUNCT
ejpam-3838	107	5	(	(	PUNCT
ejpam-3838	107	6	u).therefore	u).therefore	PROPN
ejpam-3838	107	7	,	,	PUNCT
ejpam-3838	107	8	(	(	PUNCT
ejpam-3838	107	9	7	7	X
ejpam-3838	107	10	)	)	PUNCT
ejpam-3838	107	11	becomes	become	VERB
ejpam-3838	107	12	∣∣{n	∣∣{n	PROPN
ejpam-3838	107	13	∈mk0×(n−k0)|an	∈mk0×(n−k0)|an	X
ejpam-3838	107	14	t	t	NOUN
ejpam-3838	108	1	+	+	NOUN
ejpam-3838	108	2	nat	nat	PROPN
ejpam-3838	108	3	≡	≡	PROPN
ejpam-3838	108	4	0	0	PUNCT
ejpam-3838	108	5	(	(	PUNCT
ejpam-3838	108	6	u	u	NOUN
ejpam-3838	108	7	)	)	PUNCT
ejpam-3838	108	8	}	}	PUNCT
ejpam-3838	108	9	∣∣	∣∣	X
ejpam-3838	108	10	=	=	PUNCT
ejpam-3838	108	11	|ker	|ker	NOUN
ejpam-3838	108	12	ψa|	ψa|	NOUN
ejpam-3838	108	13	=	=	PUNCT
ejpam-3838	108	14	∣∣mk0×(n−k0	∣∣mk0×(n−k0	NOUN
ejpam-3838	108	15	)	)	PUNCT
ejpam-3838	108	16	∣∣	∣∣	NUM
ejpam-3838	108	17	|im	|im	NUM
ejpam-3838	108	18	ψa|	ψa|	PROPN
ejpam-3838	108	19	.	.	PUNCT
ejpam-3838	109	1	thus	thus	ADV
ejpam-3838	109	2	,	,	PUNCT
ejpam-3838	109	3	we	we	PRON
ejpam-3838	109	4	have	have	VERB
ejpam-3838	109	5	|ker	|ker	NOUN
ejpam-3838	109	6	ψa|	ψa|	NOUN
ejpam-3838	109	7	=	=	PUNCT
ejpam-3838	109	8	{	{	PUNCT
ejpam-3838	109	9	q	q	PROPN
ejpam-3838	109	10	k0(2n−3k0−1	k0(2n−3k0−1	PROPN
ejpam-3838	109	11	)	)	PUNCT
ejpam-3838	109	12	2	2	NUM
ejpam-3838	109	13	,	,	PUNCT
ejpam-3838	109	14	if	if	SCONJ
ejpam-3838	109	15	q	q	NOUN
ejpam-3838	109	16	is	be	AUX
ejpam-3838	109	17	odd	odd	ADJ
ejpam-3838	109	18	q	q	ADJ
ejpam-3838	109	19	k0(2n−3k0	k0(2n−3k0	X
ejpam-3838	109	20	+	+	PROPN
ejpam-3838	109	21	1	1	NUM
ejpam-3838	109	22	)	)	PUNCT
ejpam-3838	109	23	2	2	NUM
ejpam-3838	109	24	,	,	PUNCT
ejpam-3838	109	25	if	if	SCONJ
ejpam-3838	109	26	q	q	NOUN
ejpam-3838	109	27	is	be	AUX
ejpam-3838	109	28	even	even	ADV
ejpam-3838	109	29	,	,	PUNCT
ejpam-3838	109	30	by	by	ADP
ejpam-3838	109	31	lemma	lemma	PROPN
ejpam-3838	109	32	3	3	NUM
ejpam-3838	109	33	.	.	PUNCT
ejpam-3838	109	34	similarly	similarly	ADV
ejpam-3838	109	35	,	,	PUNCT
ejpam-3838	109	36	c	c	PROPN
ejpam-3838	109	37	is	be	AUX
ejpam-3838	109	38	hermitian	hermitian	ADJ
ejpam-3838	109	39	self	self	NOUN
ejpam-3838	109	40	-	-	PUNCT
ejpam-3838	109	41	orthogonal	orthogonal	ADJ
ejpam-3838	109	42	if	if	SCONJ
ejpam-3838	109	43	and	and	CCONJ
ejpam-3838	109	44	only	only	ADV
ejpam-3838	109	45	if	if	SCONJ
ejpam-3838	109	46	ik0	ik0	VERB
ejpam-3838	109	47	+	+	ADV
ejpam-3838	109	48	aat	aat	VERB
ejpam-3838	109	49	+	+	CCONJ
ejpam-3838	109	50	u(ant	u(ant	ADJ
ejpam-3838	109	51	−nat	−nat	NOUN
ejpam-3838	109	52	)	)	PUNCT
ejpam-3838	110	1	≡	≡	PROPN
ejpam-3838	110	2	0	0	PUNCT
ejpam-3838	110	3	(	(	PUNCT
ejpam-3838	110	4	u2	u2	NOUN
ejpam-3838	110	5	)	)	PUNCT
ejpam-3838	110	6	.	.	PUNCT
ejpam-3838	111	1	hence	hence	ADV
ejpam-3838	111	2	,	,	PUNCT
ejpam-3838	111	3	by	by	ADP
ejpam-3838	111	4	(	(	PUNCT
ejpam-3838	111	5	5	5	NUM
ejpam-3838	111	6	)	)	PUNCT
ejpam-3838	111	7	and	and	CCONJ
ejpam-3838	111	8	lemma	lemma	PROPN
ejpam-3838	111	9	3	3	NUM
ejpam-3838	111	10	,	,	PUNCT
ejpam-3838	111	11	the	the	DET
ejpam-3838	111	12	number	number	NOUN
ejpam-3838	111	13	of	of	ADP
ejpam-3838	111	14	free	free	ADJ
ejpam-3838	111	15	hermitian	hermitian	ADJ
ejpam-3838	111	16	self	self	NOUN
ejpam-3838	111	17	-	-	PUNCT
ejpam-3838	111	18	orthogonal	orthogonal	ADJ
ejpam-3838	111	19	codes	code	NOUN
ejpam-3838	111	20	c	c	NOUN
ejpam-3838	111	21	with	with	ADP
ejpam-3838	111	22	residue	residue	NOUN
ejpam-3838	111	23	code	code	NOUN
ejpam-3838	111	24	c1	c1	PROPN
ejpam-3838	111	25	is∣∣{n	is∣∣{n	PROPN
ejpam-3838	111	26	∈mk0×(n−k0)|an	∈mk0×(n−k0)|an	X
ejpam-3838	111	27	t	t	PROPN
ejpam-3838	111	28	−nat	−nat	NOUN
ejpam-3838	111	29	≡	≡	PROPN
ejpam-3838	111	30	0	0	PUNCT
ejpam-3838	111	31	(	(	PUNCT
ejpam-3838	111	32	u	u	NOUN
ejpam-3838	111	33	)	)	PUNCT
ejpam-3838	111	34	}	}	PUNCT
ejpam-3838	111	35	∣∣	∣∣	X
ejpam-3838	111	36	=	=	SYM
ejpam-3838	111	37	|ker	|ker	NOUN
ejpam-3838	111	38	φa|	φa|	NOUN
ejpam-3838	111	39	=	=	PUNCT
ejpam-3838	111	40	q	q	X
ejpam-3838	111	41	k0(2n−3k0	k0(2n−3k0	PROPN
ejpam-3838	111	42	+	+	PROPN
ejpam-3838	111	43	1	1	NUM
ejpam-3838	111	44	)	)	PUNCT
ejpam-3838	111	45	2	2	NUM
ejpam-3838	111	46	.	.	PUNCT
ejpam-3838	112	1	l.e	l.e	PROPN
ejpam-3838	112	2	.	.	PROPN
ejpam-3838	112	3	galvez	galvez	PROPN
ejpam-3838	112	4	,	,	PUNCT
ejpam-3838	112	5	r.a	r.a	PROPN
ejpam-3838	112	6	.	.	PROPN
ejpam-3838	112	7	betty	betty	PROPN
ejpam-3838	112	8	,	,	PUNCT
ejpam-3838	112	9	f.	f.	PROPN
ejpam-3838	112	10	nemenzo	nemenzo	PROPN
ejpam-3838	112	11	/	/	SYM
ejpam-3838	112	12	eur	eur	NOUN
ejpam-3838	112	13	.	.	PUNCT
ejpam-3838	113	1	j.	j.	PROPN
ejpam-3838	113	2	pure	pure	PROPN
ejpam-3838	113	3	appl	appl	PROPN
ejpam-3838	113	4	.	.	PROPN
ejpam-3838	113	5	math	math	PROPN
ejpam-3838	113	6	,	,	PUNCT
ejpam-3838	113	7	13	13	NUM
ejpam-3838	113	8	(	(	PUNCT
ejpam-3838	113	9	4	4	NUM
ejpam-3838	113	10	)	)	PUNCT
ejpam-3838	113	11	(	(	PUNCT
ejpam-3838	113	12	2020	2020	NUM
ejpam-3838	113	13	)	)	PUNCT
ejpam-3838	113	14	,	,	PUNCT
ejpam-3838	113	15	873	873	NUM
ejpam-3838	113	16	-	-	NUM
ejpam-3838	113	17	892	892	NUM
ejpam-3838	113	18	878	878	NUM
ejpam-3838	113	19	�	�	NOUN
ejpam-3838	113	20	define	define	VERB
ejpam-3838	113	21	the	the	DET
ejpam-3838	113	22	sets	set	NOUN
ejpam-3838	113	23	x	x	PUNCT
ejpam-3838	113	24	=	=	PUNCT
ejpam-3838	113	25	{	{	PUNCT
ejpam-3838	114	1	c	c	NOUN
ejpam-3838	114	2	|	|	ADV
ejpam-3838	114	3	c	c	NOUN
ejpam-3838	114	4	⊆	⊆	NUM
ejpam-3838	114	5	rn1	rn1	NOUN
ejpam-3838	114	6	,	,	PUNCT
ejpam-3838	114	7	type	type	NOUN
ejpam-3838	114	8	{	{	PUNCT
ejpam-3838	114	9	k0	k0	PROPN
ejpam-3838	114	10	,	,	PUNCT
ejpam-3838	114	11	0	0	NUM
ejpam-3838	114	12	}	}	PUNCT
ejpam-3838	114	13	,	,	PUNCT
ejpam-3838	114	14	c	c	PROPN
ejpam-3838	114	15	⊆	⊆	NUM
ejpam-3838	114	16	c⊥	c⊥	PROPN
ejpam-3838	114	17	,	,	PUNCT
ejpam-3838	114	18	res(c	res(c	ADJ
ejpam-3838	114	19	)	)	PUNCT
ejpam-3838	114	20	=	=	SYM
ejpam-3838	114	21	c1	c1	PROPN
ejpam-3838	114	22	}	}	PUNCT
ejpam-3838	114	23	and	and	CCONJ
ejpam-3838	114	24	x	x	X
ejpam-3838	114	25	′	′	NUM
ejpam-3838	115	1	=	=	PUNCT
ejpam-3838	115	2	{	{	PUNCT
ejpam-3838	115	3	c	c	NOUN
ejpam-3838	115	4	′	′	NUM
ejpam-3838	116	1	|	|	ADV
ejpam-3838	116	2	c	c	NOUN
ejpam-3838	116	3	′	′	NOUN
ejpam-3838	117	1	⊆	⊆	NUM
ejpam-3838	117	2	rn1	rn1	NOUN
ejpam-3838	117	3	,	,	PUNCT
ejpam-3838	117	4	c	c	NOUN
ejpam-3838	117	5	′	′	NUM
ejpam-3838	118	1	⊆	⊆	NUM
ejpam-3838	118	2	c	c	NOUN
ejpam-3838	118	3	′⊥	′⊥	PROPN
ejpam-3838	118	4	,	,	PUNCT
ejpam-3838	118	5	res(c	res(c	PROPN
ejpam-3838	118	6	′	′	NOUN
ejpam-3838	118	7	)	)	PUNCT
ejpam-3838	118	8	=	=	SYM
ejpam-3838	118	9	c1	c1	PROPN
ejpam-3838	118	10	,	,	PUNCT
ejpam-3838	118	11	tor(c	tor(c	PROPN
ejpam-3838	118	12	′	′	NOUN
ejpam-3838	118	13	)	)	PUNCT
ejpam-3838	118	14	=	=	SYM
ejpam-3838	118	15	c2	c2	PROPN
ejpam-3838	118	16	}	}	PUNCT
ejpam-3838	118	17	,	,	PUNCT
ejpam-3838	118	18	where	where	SCONJ
ejpam-3838	118	19	self	self	NOUN
ejpam-3838	118	20	-	-	PUNCT
ejpam-3838	118	21	orthogonality	orthogonality	NOUN
ejpam-3838	118	22	is	be	AUX
ejpam-3838	118	23	either	either	CCONJ
ejpam-3838	118	24	in	in	ADP
ejpam-3838	118	25	the	the	DET
ejpam-3838	118	26	euclidean	euclidean	ADJ
ejpam-3838	118	27	or	or	CCONJ
ejpam-3838	118	28	hermitian	hermitian	ADJ
ejpam-3838	118	29	sense	sense	NOUN
ejpam-3838	118	30	.	.	PUNCT
ejpam-3838	119	1	lemma	lemma	PROPN
ejpam-3838	119	2	5	5	NUM
ejpam-3838	119	3	.	.	PUNCT
ejpam-3838	120	1	if	if	SCONJ
ejpam-3838	120	2	c	c	NOUN
ejpam-3838	121	1	′	′	NOUN
ejpam-3838	122	1	∈	∈	NOUN
ejpam-3838	122	2	x	x	SYM
ejpam-3838	122	3	′	′	NOUN
ejpam-3838	122	4	,	,	PUNCT
ejpam-3838	122	5	then	then	ADV
ejpam-3838	122	6	|{c	|{c	ADP
ejpam-3838	122	7	∈	∈	PROPN
ejpam-3838	122	8	x|c	x|c	PUNCT
ejpam-3838	123	1	⊆	⊆	NUM
ejpam-3838	123	2	c	c	NOUN
ejpam-3838	123	3	′}|	′}|	NOUN
ejpam-3838	123	4	=	=	SYM
ejpam-3838	123	5	qk0k1	qk0k1	NOUN
ejpam-3838	123	6	.	.	PUNCT
ejpam-3838	124	1	proof	proof	NOUN
ejpam-3838	124	2	.	.	PUNCT
ejpam-3838	125	1	by	by	ADP
ejpam-3838	125	2	lemma	lemma	PROPN
ejpam-3838	125	3	2	2	NUM
ejpam-3838	125	4	,	,	PUNCT
ejpam-3838	125	5	c	c	NOUN
ejpam-3838	125	6	′	′	NOUN
ejpam-3838	125	7	has	have	VERB
ejpam-3838	125	8	a	a	DET
ejpam-3838	125	9	generator	generator	NOUN
ejpam-3838	125	10	matrix	matrix	NOUN
ejpam-3838	125	11	(	(	PUNCT
ejpam-3838	125	12	4	4	NUM
ejpam-3838	125	13	)	)	PUNCT
ejpam-3838	125	14	.	.	PUNCT
ejpam-3838	126	1	consider	consider	VERB
ejpam-3838	126	2	the	the	DET
ejpam-3838	126	3	map	map	NOUN
ejpam-3838	126	4	ψ	ψ	X
ejpam-3838	126	5	:	:	PUNCT
ejpam-3838	126	6	mk0×k1(fq	mk0×k1(fq	PROPN
ejpam-3838	126	7	)	)	PUNCT
ejpam-3838	126	8	−→	−→	NOUN
ejpam-3838	126	9	{	{	PUNCT
ejpam-3838	126	10	c	c	NOUN
ejpam-3838	126	11	∈	∈	PROPN
ejpam-3838	127	1	x	x	INTJ
ejpam-3838	127	2	|	|	ADV
ejpam-3838	127	3	c	c	NOUN
ejpam-3838	127	4	⊆	⊆	NUM
ejpam-3838	127	5	c	c	NOUN
ejpam-3838	127	6	′	′	NUM
ejpam-3838	127	7	}	}	PUNCT
ejpam-3838	127	8	m	m	VERB
ejpam-3838	127	9	7−→	7−→	NOUN
ejpam-3838	127	10	rk01	rk01	PROPN
ejpam-3838	128	1	[	[	X
ejpam-3838	128	2	i	i	PRON
ejpam-3838	128	3	a+	a+	PUNCT
ejpam-3838	128	4	u(n	u(n	PROPN
ejpam-3838	128	5	+	+	PROPN
ejpam-3838	128	6	md	md	PROPN
ejpam-3838	128	7	)	)	PUNCT
ejpam-3838	128	8	]	]	PUNCT
ejpam-3838	128	9	.	.	PUNCT
ejpam-3838	129	1	clearly	clearly	ADV
ejpam-3838	129	2	,	,	PUNCT
ejpam-3838	129	3	ψ	ψ	X
ejpam-3838	129	4	is	be	AUX
ejpam-3838	129	5	well	well	ADV
ejpam-3838	129	6	-	-	PUNCT
ejpam-3838	129	7	defined	define	VERB
ejpam-3838	129	8	.	.	PUNCT
ejpam-3838	130	1	we	we	PRON
ejpam-3838	130	2	will	will	AUX
ejpam-3838	130	3	show	show	VERB
ejpam-3838	130	4	that	that	SCONJ
ejpam-3838	130	5	ψ	ψ	NOUN
ejpam-3838	130	6	is	be	AUX
ejpam-3838	130	7	bijective	bijective	ADJ
ejpam-3838	130	8	.	.	PUNCT
ejpam-3838	131	1	if	if	SCONJ
ejpam-3838	131	2	m1,m2	m1,m2	PROPN
ejpam-3838	131	3	∈mk0×k1(fq	∈mk0×k1(fq	NOUN
ejpam-3838	131	4	)	)	PUNCT
ejpam-3838	131	5	such	such	ADJ
ejpam-3838	131	6	that	that	SCONJ
ejpam-3838	131	7	ψ(m1	ψ(m1	NOUN
ejpam-3838	131	8	)	)	PUNCT
ejpam-3838	131	9	=	=	SYM
ejpam-3838	131	10	ψ(m2	ψ(m2	NOUN
ejpam-3838	131	11	)	)	PUNCT
ejpam-3838	131	12	,	,	PUNCT
ejpam-3838	131	13	then	then	ADV
ejpam-3838	131	14	rk01	rk01	PROPN
ejpam-3838	131	15	[	[	X
ejpam-3838	131	16	ik0	ik0	ADV
ejpam-3838	131	17	a+	a+	PUNCT
ejpam-3838	131	18	u(n	u(n	PROPN
ejpam-3838	131	19	+	+	PROPN
ejpam-3838	131	20	m1d	m1d	X
ejpam-3838	131	21	)	)	PUNCT
ejpam-3838	131	22	]	]	PUNCT
ejpam-3838	132	1	=	=	PUNCT
ejpam-3838	132	2	rk01	rk01	PROPN
ejpam-3838	132	3	[	[	X
ejpam-3838	132	4	ik0	ik0	ADV
ejpam-3838	132	5	a+	a+	PUNCT
ejpam-3838	132	6	u(n	u(n	PROPN
ejpam-3838	132	7	+	+	NOUN
ejpam-3838	132	8	m2d	m2d	NOUN
ejpam-3838	132	9	)	)	PUNCT
ejpam-3838	132	10	]	]	PUNCT
ejpam-3838	132	11	which	which	PRON
ejpam-3838	132	12	means	mean	VERB
ejpam-3838	132	13	a	a	PRON
ejpam-3838	132	14	+	+	CCONJ
ejpam-3838	132	15	u(n	u(n	PROPN
ejpam-3838	132	16	+	+	ADJ
ejpam-3838	132	17	m1d	m1d	X
ejpam-3838	132	18	)	)	PUNCT
ejpam-3838	132	19	≡	≡	PROPN
ejpam-3838	132	20	a	a	DET
ejpam-3838	132	21	+	+	X
ejpam-3838	132	22	u(n	u(n	PROPN
ejpam-3838	132	23	+	+	CCONJ
ejpam-3838	132	24	m2d	m2d	NOUN
ejpam-3838	132	25	)	)	PUNCT
ejpam-3838	132	26	(	(	PUNCT
ejpam-3838	132	27	u2	u2	NOUN
ejpam-3838	132	28	)	)	PUNCT
ejpam-3838	132	29	.	.	PUNCT
ejpam-3838	133	1	therefore	therefore	ADV
ejpam-3838	133	2	n	n	PROPN
ejpam-3838	133	3	+	+	CCONJ
ejpam-3838	133	4	m1d	m1d	X
ejpam-3838	133	5	≡	≡	PROPN
ejpam-3838	133	6	n	n	PROPN
ejpam-3838	134	1	+	+	CCONJ
ejpam-3838	135	1	m2d	m2d	PROPN
ejpam-3838	135	2	(	(	PUNCT
ejpam-3838	135	3	u	u	NOUN
ejpam-3838	135	4	)	)	PUNCT
ejpam-3838	135	5	.	.	PUNCT
ejpam-3838	136	1	since	since	SCONJ
ejpam-3838	136	2	d	d	PROPN
ejpam-3838	136	3	is	be	AUX
ejpam-3838	136	4	of	of	ADP
ejpam-3838	136	5	full	full	ADJ
ejpam-3838	136	6	row	row	NOUN
ejpam-3838	136	7	rank	rank	NOUN
ejpam-3838	136	8	,	,	PUNCT
ejpam-3838	136	9	we	we	PRON
ejpam-3838	136	10	have	have	VERB
ejpam-3838	136	11	m1	m1	PROPN
ejpam-3838	136	12	≡m2	≡m2	X
ejpam-3838	136	13	(	(	PUNCT
ejpam-3838	136	14	u	u	NOUN
ejpam-3838	136	15	)	)	PUNCT
ejpam-3838	136	16	.	.	PUNCT
ejpam-3838	137	1	hence	hence	ADV
ejpam-3838	137	2	,	,	PUNCT
ejpam-3838	137	3	ψ	ψ	NOUN
ejpam-3838	137	4	is	be	AUX
ejpam-3838	137	5	injective	injective	ADJ
ejpam-3838	137	6	.	.	PUNCT
ejpam-3838	138	1	suppose	suppose	VERB
ejpam-3838	138	2	c	c	X
ejpam-3838	138	3	∈	∈	PROPN
ejpam-3838	138	4	x	x	X
ejpam-3838	138	5	and	and	CCONJ
ejpam-3838	138	6	c	c	PROPN
ejpam-3838	138	7	⊆	⊆	NUM
ejpam-3838	138	8	c	c	NOUN
ejpam-3838	138	9	′.	′.	NOUN
ejpam-3838	138	10	by	by	ADP
ejpam-3838	138	11	lemma	lemma	PROPN
ejpam-3838	138	12	2	2	NUM
ejpam-3838	138	13	,	,	PUNCT
ejpam-3838	138	14	c	c	X
ejpam-3838	138	15	=	=	SYM
ejpam-3838	139	1	rk01	rk01	PROPN
ejpam-3838	139	2	[	[	X
ejpam-3838	139	3	ik0	ik0	X
ejpam-3838	139	4	a+	a+	PUNCT
ejpam-3838	139	5	uf	uf	NOUN
ejpam-3838	139	6	]	]	PUNCT
ejpam-3838	139	7	,	,	PUNCT
ejpam-3838	139	8	for	for	ADP
ejpam-3838	139	9	some	some	DET
ejpam-3838	139	10	matrix	matrix	NOUN
ejpam-3838	139	11	f	f	NOUN
ejpam-3838	139	12	.	.	PUNCT
ejpam-3838	140	1	the	the	DET
ejpam-3838	140	2	inclusion	inclusion	NOUN
ejpam-3838	140	3	c	c	NOUN
ejpam-3838	140	4	⊆	⊆	NUM
ejpam-3838	140	5	c	c	NOUN
ejpam-3838	140	6	′	′	NOUN
ejpam-3838	140	7	implies	imply	VERB
ejpam-3838	140	8	that	that	SCONJ
ejpam-3838	140	9	a+	a+	PUNCT
ejpam-3838	140	10	uf	uf	PROPN
ejpam-3838	140	11	≡	≡	PROPN
ejpam-3838	140	12	a+	a+	PUNCT
ejpam-3838	141	1	u(n	u(n	PROPN
ejpam-3838	141	2	+	+	PROPN
ejpam-3838	141	3	md	md	PROPN
ejpam-3838	141	4	)	)	PUNCT
ejpam-3838	141	5	(	(	PUNCT
ejpam-3838	141	6	u2	u2	PROPN
ejpam-3838	141	7	)	)	PUNCT
ejpam-3838	141	8	for	for	ADP
ejpam-3838	141	9	some	some	DET
ejpam-3838	141	10	matrix	matrix	NOUN
ejpam-3838	141	11	m	m	NOUN
ejpam-3838	141	12	.	.	PUNCT
ejpam-3838	142	1	so	so	ADV
ejpam-3838	142	2	f	f	PROPN
ejpam-3838	142	3	≡	≡	PROPN
ejpam-3838	142	4	n	n	PROPN
ejpam-3838	142	5	+	+	PROPN
ejpam-3838	142	6	md	md	PROPN
ejpam-3838	142	7	(	(	PUNCT
ejpam-3838	142	8	u	u	NOUN
ejpam-3838	142	9	)	)	PUNCT
ejpam-3838	142	10	,	,	PUNCT
ejpam-3838	142	11	which	which	PRON
ejpam-3838	142	12	shows	show	VERB
ejpam-3838	142	13	that	that	SCONJ
ejpam-3838	142	14	ψ	ψ	NOUN
ejpam-3838	142	15	is	be	AUX
ejpam-3838	142	16	surjective	surjective	ADJ
ejpam-3838	142	17	,	,	PUNCT
ejpam-3838	142	18	and	and	CCONJ
ejpam-3838	142	19	hence	hence	ADV
ejpam-3838	142	20	,	,	PUNCT
ejpam-3838	142	21	bijective	bijective	ADJ
ejpam-3838	142	22	.	.	PUNCT
ejpam-3838	143	1	therefore	therefore	ADV
ejpam-3838	143	2	,	,	PUNCT
ejpam-3838	143	3	∣∣{c	∣∣{c	PROPN
ejpam-3838	143	4	∈	∈	PROPN
ejpam-3838	143	5	x|c	x|c	PUNCT
ejpam-3838	144	1	⊆	⊆	NUM
ejpam-3838	144	2	c	c	NOUN
ejpam-3838	144	3	′}∣∣	′}∣∣	NOUN
ejpam-3838	144	4	=	=	SYM
ejpam-3838	144	5	|mk0×k1(fq)|	|mk0×k1(fq)|	PUNCT
ejpam-3838	144	6	=	=	SYM
ejpam-3838	144	7	qk0k1	qk0k1	PROPN
ejpam-3838	144	8	.	.	PUNCT
ejpam-3838	145	1	�	�	PROPN
ejpam-3838	145	2	lemma	lemma	PROPN
ejpam-3838	145	3	6	6	NUM
ejpam-3838	145	4	.	.	PUNCT
ejpam-3838	146	1	if	if	SCONJ
ejpam-3838	146	2	c	c	PROPN
ejpam-3838	146	3	∈	∈	PROPN
ejpam-3838	146	4	x	x	NOUN
ejpam-3838	146	5	,	,	PUNCT
ejpam-3838	146	6	then	then	ADV
ejpam-3838	146	7	there	there	PRON
ejpam-3838	146	8	exists	exist	VERB
ejpam-3838	146	9	a	a	DET
ejpam-3838	146	10	unique	unique	ADJ
ejpam-3838	146	11	code	code	NOUN
ejpam-3838	146	12	c	c	NOUN
ejpam-3838	147	1	′	′	NOUN
ejpam-3838	147	2	∈	∈	NOUN
ejpam-3838	147	3	x	x	PUNCT
ejpam-3838	148	1	′	′	NUM
ejpam-3838	148	2	such	such	ADJ
ejpam-3838	148	3	that	that	SCONJ
ejpam-3838	148	4	c	c	PROPN
ejpam-3838	148	5	⊆	⊆	NUM
ejpam-3838	148	6	c	c	NOUN
ejpam-3838	148	7	′.	′.	NOUN
ejpam-3838	148	8	proof	proof	NOUN
ejpam-3838	148	9	.	.	PUNCT
ejpam-3838	149	1	since	since	SCONJ
ejpam-3838	149	2	c	c	PROPN
ejpam-3838	149	3	∈	∈	PROPN
ejpam-3838	149	4	x	x	X
ejpam-3838	149	5	,	,	PUNCT
ejpam-3838	149	6	c	c	PROPN
ejpam-3838	149	7	has	have	VERB
ejpam-3838	149	8	a	a	DET
ejpam-3838	149	9	generator	generator	NOUN
ejpam-3838	149	10	matrix	matrix	NOUN
ejpam-3838	150	1	[	[	X
ejpam-3838	150	2	i	i	PRON
ejpam-3838	150	3	a+	a+	X
ejpam-3838	150	4	un	un	PROPN
ejpam-3838	150	5	]	]	PUNCT
ejpam-3838	150	6	for	for	ADP
ejpam-3838	150	7	some	some	DET
ejpam-3838	150	8	unique	unique	ADJ
ejpam-3838	150	9	matrix	matrix	NOUN
ejpam-3838	150	10	n	n	NOUN
ejpam-3838	150	11	,	,	PUNCT
ejpam-3838	150	12	by	by	ADP
ejpam-3838	150	13	lemma	lemma	PROPN
ejpam-3838	150	14	2	2	NUM
ejpam-3838	150	15	.	.	PUNCT
ejpam-3838	151	1	let	let	VERB
ejpam-3838	151	2	c	c	NOUN
ejpam-3838	151	3	′0	′0	NOUN
ejpam-3838	151	4	be	be	AUX
ejpam-3838	151	5	a	a	DET
ejpam-3838	151	6	code	code	NOUN
ejpam-3838	151	7	with	with	ADP
ejpam-3838	151	8	generator	generator	NOUN
ejpam-3838	151	9	matrix	matrix	NOUN
ejpam-3838	151	10	[	[	PUNCT
ejpam-3838	151	11	ik0	ik0	NOUN
ejpam-3838	151	12	a+	a+	PUNCT
ejpam-3838	151	13	un	un	PROPN
ejpam-3838	151	14	0	0	PROPN
ejpam-3838	151	15	ud	ud	PROPN
ejpam-3838	151	16	]	]	PUNCT
ejpam-3838	151	17	.	.	PUNCT
ejpam-3838	152	1	the	the	DET
ejpam-3838	152	2	code	code	NOUN
ejpam-3838	152	3	c	c	NOUN
ejpam-3838	152	4	′0	′0	NOUN
ejpam-3838	152	5	satisfies	satisfy	VERB
ejpam-3838	152	6	res(c	res(c	ADJ
ejpam-3838	152	7	′0	′0	NOUN
ejpam-3838	152	8	)	)	PUNCT
ejpam-3838	152	9	=	=	SYM
ejpam-3838	152	10	c1	c1	NOUN
ejpam-3838	152	11	and	and	CCONJ
ejpam-3838	152	12	tor(c	tor(c	PROPN
ejpam-3838	152	13	′0	′0	NOUN
ejpam-3838	152	14	)	)	PUNCT
ejpam-3838	152	15	=	=	SYM
ejpam-3838	152	16	c2	c2	PROPN
ejpam-3838	152	17	.	.	PUNCT
ejpam-3838	153	1	clearly	clearly	ADV
ejpam-3838	153	2	,	,	PUNCT
ejpam-3838	153	3	c	c	PROPN
ejpam-3838	153	4	⊆	⊆	NUM
ejpam-3838	153	5	c	c	NOUN
ejpam-3838	153	6	′0	′0	NOUN
ejpam-3838	153	7	.	.	PUNCT
ejpam-3838	154	1	since	since	SCONJ
ejpam-3838	154	2	c	c	PROPN
ejpam-3838	154	3	∈	∈	PROPN
ejpam-3838	154	4	x	x	X
ejpam-3838	154	5	,	,	PUNCT
ejpam-3838	154	6	(	(	PUNCT
ejpam-3838	154	7	6	6	NUM
ejpam-3838	154	8	)	)	PUNCT
ejpam-3838	154	9	implies	imply	VERB
ejpam-3838	154	10	c	c	NOUN
ejpam-3838	154	11	′0	′0	NOUN
ejpam-3838	154	12	is	be	AUX
ejpam-3838	154	13	self	self	NOUN
ejpam-3838	154	14	-	-	PUNCT
ejpam-3838	154	15	orthogonal	orthogonal	ADJ
ejpam-3838	154	16	and	and	CCONJ
ejpam-3838	154	17	hence	hence	ADV
ejpam-3838	154	18	,	,	PUNCT
ejpam-3838	155	1	c	c	X
ejpam-3838	155	2	′0	′0	NOUN
ejpam-3838	155	3	∈	∈	NOUN
ejpam-3838	155	4	x	x	NOUN
ejpam-3838	155	5	′.	′.	NOUN
ejpam-3838	155	6	suppose	suppose	VERB
ejpam-3838	155	7	c	c	SYM
ejpam-3838	155	8	⊆	⊆	NUM
ejpam-3838	155	9	c	c	NOUN
ejpam-3838	155	10	′	′	NOUN
ejpam-3838	155	11	for	for	ADP
ejpam-3838	155	12	some	some	PRON
ejpam-3838	156	1	c	c	NOUN
ejpam-3838	156	2	′	′	NUM
ejpam-3838	157	1	∈	∈	NOUN
ejpam-3838	157	2	x	x	PUNCT
ejpam-3838	157	3	′.	′.	NOUN
ejpam-3838	157	4	because	because	SCONJ
ejpam-3838	157	5	c	c	PROPN
ejpam-3838	157	6	′	′	NOUN
ejpam-3838	157	7	has	have	VERB
ejpam-3838	157	8	torsion	torsion	NOUN
ejpam-3838	157	9	code	code	NOUN
ejpam-3838	157	10	c2	c2	PROPN
ejpam-3838	157	11	,	,	PUNCT
ejpam-3838	157	12	by	by	ADP
ejpam-3838	157	13	lemma	lemma	PROPN
ejpam-3838	157	14	2	2	NUM
ejpam-3838	157	15	,	,	PUNCT
ejpam-3838	157	16	rk11	rk11	PROPN
ejpam-3838	158	1	[	[	X
ejpam-3838	158	2	0	0	NUM
ejpam-3838	158	3	ud	ud	ADP
ejpam-3838	158	4	]	]	PUNCT
ejpam-3838	158	5	⊆	⊆	NUM
ejpam-3838	158	6	c	c	NOUN
ejpam-3838	158	7	′	′	NOUN
ejpam-3838	159	1	and	and	CCONJ
ejpam-3838	159	2	so	so	ADV
ejpam-3838	159	3	c	c	NOUN
ejpam-3838	159	4	′0	′0	NOUN
ejpam-3838	159	5	⊆	⊆	NUM
ejpam-3838	159	6	c	c	NOUN
ejpam-3838	159	7	′.	′.	NOUN
ejpam-3838	159	8	note	note	VERB
ejpam-3838	159	9	that	that	SCONJ
ejpam-3838	159	10	|c	|c	ADJ
ejpam-3838	159	11	′0|	′0|	NOUN
ejpam-3838	159	12	=	=	SYM
ejpam-3838	159	13	|c1|	|c1|	PROPN
ejpam-3838	159	14	|c2|	|c2|	NOUN
ejpam-3838	159	15	=	=	PUNCT
ejpam-3838	159	16	q2k0+k1	q2k0+k1	X
ejpam-3838	159	17	=	=	PUNCT
ejpam-3838	159	18	|c	|c	X
ejpam-3838	159	19	′|	′|	NUM
ejpam-3838	159	20	.	.	PUNCT
ejpam-3838	160	1	hence	hence	ADV
ejpam-3838	160	2	,	,	PUNCT
ejpam-3838	160	3	c	c	X
ejpam-3838	160	4	′0	′0	PROPN
ejpam-3838	160	5	=	=	SYM
ejpam-3838	160	6	c	c	NOUN
ejpam-3838	160	7	′.	′.	NOUN
ejpam-3838	160	8	�	�	PROPN
ejpam-3838	160	9	next	next	ADV
ejpam-3838	160	10	,	,	PUNCT
ejpam-3838	160	11	we	we	PRON
ejpam-3838	160	12	count	count	VERB
ejpam-3838	160	13	self	self	NOUN
ejpam-3838	160	14	-	-	PUNCT
ejpam-3838	160	15	orthogonal	orthogonal	ADJ
ejpam-3838	160	16	codes	code	NOUN
ejpam-3838	160	17	c	c	NOUN
ejpam-3838	160	18	with	with	ADP
ejpam-3838	160	19	given	give	VERB
ejpam-3838	160	20	residue	residue	NOUN
ejpam-3838	160	21	code	code	NOUN
ejpam-3838	160	22	and	and	CCONJ
ejpam-3838	160	23	torsion	torsion	NOUN
ejpam-3838	160	24	code	code	NOUN
ejpam-3838	160	25	.	.	PUNCT
ejpam-3838	161	1	l.e	l.e	PROPN
ejpam-3838	161	2	.	.	PROPN
ejpam-3838	161	3	galvez	galvez	PROPN
ejpam-3838	161	4	,	,	PUNCT
ejpam-3838	161	5	r.a	r.a	PROPN
ejpam-3838	161	6	.	.	PROPN
ejpam-3838	161	7	betty	betty	PROPN
ejpam-3838	161	8	,	,	PUNCT
ejpam-3838	161	9	f.	f.	PROPN
ejpam-3838	161	10	nemenzo	nemenzo	PROPN
ejpam-3838	161	11	/	/	SYM
ejpam-3838	161	12	eur	eur	NOUN
ejpam-3838	161	13	.	.	PUNCT
ejpam-3838	162	1	j.	j.	PROPN
ejpam-3838	162	2	pure	pure	PROPN
ejpam-3838	162	3	appl	appl	PROPN
ejpam-3838	162	4	.	.	PROPN
ejpam-3838	162	5	math	math	PROPN
ejpam-3838	162	6	,	,	PUNCT
ejpam-3838	162	7	13	13	NUM
ejpam-3838	162	8	(	(	PUNCT
ejpam-3838	162	9	4	4	NUM
ejpam-3838	162	10	)	)	PUNCT
ejpam-3838	162	11	(	(	PUNCT
ejpam-3838	162	12	2020	2020	NUM
ejpam-3838	162	13	)	)	PUNCT
ejpam-3838	162	14	,	,	PUNCT
ejpam-3838	162	15	873	873	NUM
ejpam-3838	162	16	-	-	SYM
ejpam-3838	162	17	892	892	NUM
ejpam-3838	162	18	879	879	NUM
ejpam-3838	162	19	theorem	theorem	NOUN
ejpam-3838	162	20	1	1	NUM
ejpam-3838	162	21	.	.	PUNCT
ejpam-3838	163	1	let	let	VERB
ejpam-3838	163	2	c1	c1	PROPN
ejpam-3838	163	3	and	and	CCONJ
ejpam-3838	163	4	c2	c2	PROPN
ejpam-3838	163	5	be	be	VERB
ejpam-3838	163	6	codes	code	NOUN
ejpam-3838	163	7	of	of	ADP
ejpam-3838	163	8	length	length	NOUN
ejpam-3838	163	9	n	n	PROPN
ejpam-3838	163	10	over	over	ADP
ejpam-3838	163	11	fq	fq	PROPN
ejpam-3838	163	12	where	where	SCONJ
ejpam-3838	163	13	c1	c1	PROPN
ejpam-3838	163	14	⊆	⊆	NUM
ejpam-3838	163	15	c2	c2	PROPN
ejpam-3838	163	16	⊆	⊆	NUM
ejpam-3838	163	17	c⊥1	c⊥1	NOUN
ejpam-3838	163	18	.	.	PUNCT
ejpam-3838	164	1	if	if	SCONJ
ejpam-3838	164	2	dim	dim	ADJ
ejpam-3838	164	3	c1	c1	NOUN
ejpam-3838	164	4	=	=	PROPN
ejpam-3838	164	5	k0	k0	PROPN
ejpam-3838	164	6	and	and	CCONJ
ejpam-3838	164	7	dim	dim	ADJ
ejpam-3838	164	8	c2	c2	PROPN
ejpam-3838	164	9	=	=	SYM
ejpam-3838	164	10	k0	k0	PROPN
ejpam-3838	164	11	+	+	CCONJ
ejpam-3838	164	12	k1	k1	PROPN
ejpam-3838	164	13	,	,	PUNCT
ejpam-3838	164	14	then	then	ADV
ejpam-3838	164	15	(	(	PUNCT
ejpam-3838	164	16	i	i	NOUN
ejpam-3838	164	17	)	)	PUNCT
ejpam-3838	164	18	the	the	DET
ejpam-3838	164	19	number	number	NOUN
ejpam-3838	164	20	of	of	ADP
ejpam-3838	164	21	euclidean	euclidean	ADJ
ejpam-3838	164	22	self	self	NOUN
ejpam-3838	164	23	-	-	PUNCT
ejpam-3838	164	24	orthogonal	orthogonal	ADJ
ejpam-3838	164	25	codes	code	NOUN
ejpam-3838	164	26	c	c	PROPN
ejpam-3838	164	27	of	of	ADP
ejpam-3838	164	28	length	length	NOUN
ejpam-3838	164	29	n	n	PROPN
ejpam-3838	164	30	over	over	ADP
ejpam-3838	164	31	fq	fq	PROPN
ejpam-3838	164	32	+	+	CCONJ
ejpam-3838	164	33	ufq	ufq	VERB
ejpam-3838	164	34	with	with	ADP
ejpam-3838	164	35	res(c	res(c	PROPN
ejpam-3838	164	36	)	)	PUNCT
ejpam-3838	164	37	=	=	SYM
ejpam-3838	164	38	c1	c1	PROPN
ejpam-3838	164	39	and	and	CCONJ
ejpam-3838	164	40	tor(c	tor(c	PROPN
ejpam-3838	164	41	)	)	PUNCT
ejpam-3838	164	42	=	=	SYM
ejpam-3838	164	43	c2	c2	PROPN
ejpam-3838	164	44	is	be	AUX
ejpam-3838	164	45	qk0(2n−3k0−2k1+ε)/2	qk0(2n−3k0−2k1+ε)/2	PROPN
ejpam-3838	164	46	,	,	PUNCT
ejpam-3838	164	47	where	where	SCONJ
ejpam-3838	164	48	ε	ε	PROPN
ejpam-3838	164	49	=	=	VERB
ejpam-3838	164	50	−1	−1	NOUN
ejpam-3838	164	51	if	if	SCONJ
ejpam-3838	164	52	q	q	NOUN
ejpam-3838	164	53	is	be	AUX
ejpam-3838	164	54	odd	odd	ADJ
ejpam-3838	164	55	and	and	CCONJ
ejpam-3838	164	56	ε	ε	PROPN
ejpam-3838	164	57	=	=	SYM
ejpam-3838	164	58	1	1	NUM
ejpam-3838	164	59	if	if	SCONJ
ejpam-3838	164	60	q	q	NOUN
ejpam-3838	164	61	is	be	AUX
ejpam-3838	164	62	even	even	ADV
ejpam-3838	164	63	,	,	PUNCT
ejpam-3838	164	64	and	and	CCONJ
ejpam-3838	164	65	(	(	PUNCT
ejpam-3838	164	66	ii	ii	NOUN
ejpam-3838	164	67	)	)	PUNCT
ejpam-3838	164	68	the	the	DET
ejpam-3838	164	69	number	number	NOUN
ejpam-3838	164	70	of	of	ADP
ejpam-3838	164	71	hermitian	hermitian	ADJ
ejpam-3838	164	72	self	self	NOUN
ejpam-3838	164	73	-	-	PUNCT
ejpam-3838	164	74	orthogonal	orthogonal	ADJ
ejpam-3838	164	75	codes	code	NOUN
ejpam-3838	164	76	c	c	PROPN
ejpam-3838	164	77	of	of	ADP
ejpam-3838	164	78	length	length	NOUN
ejpam-3838	164	79	n	n	PROPN
ejpam-3838	164	80	over	over	ADP
ejpam-3838	164	81	fq	fq	PROPN
ejpam-3838	164	82	+	+	CCONJ
ejpam-3838	164	83	ufq	ufq	VERB
ejpam-3838	164	84	with	with	ADP
ejpam-3838	164	85	res(c	res(c	PROPN
ejpam-3838	164	86	)	)	PUNCT
ejpam-3838	164	87	=	=	SYM
ejpam-3838	164	88	c1	c1	PROPN
ejpam-3838	164	89	and	and	CCONJ
ejpam-3838	164	90	tor(c	tor(c	PROPN
ejpam-3838	164	91	)	)	PUNCT
ejpam-3838	165	1	=	=	SYM
ejpam-3838	165	2	c2	c2	PROPN
ejpam-3838	165	3	is	be	AUX
ejpam-3838	165	4	qk0(2n−3k0−2k1	qk0(2n−3k0−2k1	VERB
ejpam-3838	165	5	+	+	PROPN
ejpam-3838	165	6	1)/2	1)/2	NOUN
ejpam-3838	165	7	.	.	PUNCT
ejpam-3838	166	1	proof	proof	NOUN
ejpam-3838	166	2	.	.	PUNCT
ejpam-3838	167	1	we	we	PRON
ejpam-3838	167	2	may	may	AUX
ejpam-3838	167	3	assume	assume	VERB
ejpam-3838	167	4	without	without	ADP
ejpam-3838	167	5	loss	loss	NOUN
ejpam-3838	167	6	of	of	ADP
ejpam-3838	167	7	generality	generality	NOUN
ejpam-3838	167	8	that	that	PRON
ejpam-3838	167	9	c1	c1	PROPN
ejpam-3838	167	10	and	and	CCONJ
ejpam-3838	167	11	c2	c2	PROPN
ejpam-3838	167	12	are	be	AUX
ejpam-3838	167	13	codes	code	NOUN
ejpam-3838	167	14	with	with	ADP
ejpam-3838	167	15	generator	generator	NOUN
ejpam-3838	167	16	matrices	matrix	NOUN
ejpam-3838	167	17	(	(	PUNCT
ejpam-3838	167	18	2	2	NUM
ejpam-3838	167	19	)	)	PUNCT
ejpam-3838	167	20	and	and	CCONJ
ejpam-3838	167	21	(	(	PUNCT
ejpam-3838	167	22	3	3	NUM
ejpam-3838	167	23	)	)	PUNCT
ejpam-3838	167	24	,	,	PUNCT
ejpam-3838	167	25	respectively	respectively	ADV
ejpam-3838	167	26	.	.	PUNCT
ejpam-3838	168	1	then	then	ADV
ejpam-3838	168	2	we	we	PRON
ejpam-3838	168	3	have	have	VERB
ejpam-3838	168	4	to	to	PART
ejpam-3838	168	5	compute	compute	VERB
ejpam-3838	168	6	|x	|x	NOUN
ejpam-3838	168	7	′|	′|	PROPN
ejpam-3838	168	8	.	.	PUNCT
ejpam-3838	169	1	by	by	ADP
ejpam-3838	169	2	lemma	lemma	PROPN
ejpam-3838	169	3	5	5	NUM
ejpam-3838	169	4	and	and	CCONJ
ejpam-3838	169	5	lemma	lemma	PROPN
ejpam-3838	169	6	6	6	NUM
ejpam-3838	169	7	,	,	PUNCT
ejpam-3838	169	8	we	we	PRON
ejpam-3838	169	9	have	have	VERB
ejpam-3838	169	10	qk0k1	qk0k1	NOUN
ejpam-3838	169	11	∣∣x	∣∣x	ADP
ejpam-3838	169	12	′∣∣	′∣∣	NOUN
ejpam-3838	169	13	=	=	PUNCT
ejpam-3838	169	14	∑	∑	PUNCT
ejpam-3838	169	15	c′∈x′	c′∈x′	NOUN
ejpam-3838	169	16	∣∣{c	∣∣{c	PROPN
ejpam-3838	169	17	∈	∈	PROPN
ejpam-3838	169	18	x|c	x|c	PUNCT
ejpam-3838	170	1	⊆	⊆	NUM
ejpam-3838	170	2	c	c	NOUN
ejpam-3838	170	3	′}∣∣	′}∣∣	NOUN
ejpam-3838	170	4	=	=	PUNCT
ejpam-3838	170	5	∑	∑	PUNCT
ejpam-3838	170	6	c∈x	c∈x	NOUN
ejpam-3838	170	7	∣∣{c	∣∣{c	PROPN
ejpam-3838	170	8	′	′	NUM
ejpam-3838	170	9	∈	∈	PROPN
ejpam-3838	170	10	x	x	X
ejpam-3838	170	11	′|c	′|c	NOUN
ejpam-3838	170	12	⊆	⊆	NUM
ejpam-3838	170	13	c	c	NOUN
ejpam-3838	170	14	′}∣∣	′}∣∣	NOUN
ejpam-3838	170	15	=	=	PUNCT
ejpam-3838	170	16	∑	∑	PUNCT
ejpam-3838	170	17	c∈x	c∈x	NOUN
ejpam-3838	170	18	1	1	NUM
ejpam-3838	170	19	=	=	SYM
ejpam-3838	170	20	|x|	|x|	PROPN
ejpam-3838	170	21	.	.	PUNCT
ejpam-3838	171	1	the	the	DET
ejpam-3838	171	2	results	result	NOUN
ejpam-3838	171	3	follow	follow	VERB
ejpam-3838	171	4	from	from	ADP
ejpam-3838	171	5	lemma	lemma	PROPN
ejpam-3838	171	6	4	4	NUM
ejpam-3838	171	7	.	.	PUNCT
ejpam-3838	171	8	�	�	PROPN
ejpam-3838	171	9	4	4	NUM
ejpam-3838	171	10	.	.	PUNCT
ejpam-3838	171	11	mass	mass	ADJ
ejpam-3838	171	12	formula	formula	NOUN
ejpam-3838	171	13	for	for	ADP
ejpam-3838	171	14	self	self	NOUN
ejpam-3838	171	15	-	-	PUNCT
ejpam-3838	171	16	orthogonal	orthogonal	ADJ
ejpam-3838	171	17	codes	code	NOUN
ejpam-3838	171	18	over	over	ADP
ejpam-3838	171	19	fq	fq	PROPN
ejpam-3838	171	20	+	+	CCONJ
ejpam-3838	171	21	ufq	ufq	NOUN
ejpam-3838	171	22	let	let	VERB
ejpam-3838	171	23	σq(n	σq(n	NOUN
ejpam-3838	171	24	,	,	PUNCT
ejpam-3838	171	25	k0	k0	PROPN
ejpam-3838	171	26	)	)	PUNCT
ejpam-3838	171	27	denote	denote	VERB
ejpam-3838	171	28	the	the	DET
ejpam-3838	171	29	number	number	NOUN
ejpam-3838	171	30	of	of	ADP
ejpam-3838	171	31	distinct	distinct	ADJ
ejpam-3838	171	32	self	self	NOUN
ejpam-3838	171	33	-	-	PUNCT
ejpam-3838	171	34	orthogonal	orthogonal	ADJ
ejpam-3838	171	35	codes	code	NOUN
ejpam-3838	171	36	over	over	ADP
ejpam-3838	171	37	fq	fq	PROPN
ejpam-3838	171	38	of	of	ADP
ejpam-3838	171	39	length	length	NOUN
ejpam-3838	171	40	n	n	PROPN
ejpam-3838	171	41	and	and	CCONJ
ejpam-3838	171	42	dimension	dimension	PROPN
ejpam-3838	171	43	k0	k0	PROPN
ejpam-3838	171	44	(	(	PUNCT
ejpam-3838	171	45	see	see	VERB
ejpam-3838	171	46	[	[	X
ejpam-3838	171	47	9	9	NUM
ejpam-3838	171	48	,	,	PUNCT
ejpam-3838	171	49	10	10	NUM
ejpam-3838	171	50	]	]	PUNCT
ejpam-3838	171	51	)	)	PUNCT
ejpam-3838	171	52	.	.	PUNCT
ejpam-3838	172	1	we	we	PRON
ejpam-3838	172	2	define	define	VERB
ejpam-3838	172	3	the	the	DET
ejpam-3838	172	4	gaussian	gaussian	ADJ
ejpam-3838	172	5	coefficient	coefficient	NOUN
ejpam-3838	172	6	[	[	PUNCT
ejpam-3838	172	7	n	n	X
ejpam-3838	172	8	k	k	X
ejpam-3838	172	9	]	]	X
ejpam-3838	172	10	q	q	X
ejpam-3838	172	11	for	for	ADP
ejpam-3838	172	12	k	k	PROPN
ejpam-3838	172	13	≤	≤	PROPN
ejpam-3838	172	14	n	n	PRON
ejpam-3838	172	15	as	as	ADP
ejpam-3838	172	16	[	[	PUNCT
ejpam-3838	172	17	n	n	X
ejpam-3838	172	18	k	k	NOUN
ejpam-3838	172	19	]	]	X
ejpam-3838	172	20	q	q	X
ejpam-3838	173	1	=	=	PUNCT
ejpam-3838	173	2	(	(	PUNCT
ejpam-3838	173	3	qn	qn	NOUN
ejpam-3838	173	4	−	−	PROPN
ejpam-3838	173	5	1)(qn	1)(qn	NUM
ejpam-3838	173	6	−	−	PROPN
ejpam-3838	173	7	q	q	NOUN
ejpam-3838	173	8	)	)	PUNCT
ejpam-3838	173	9	·	·	PUNCT
ejpam-3838	173	10	·	·	PUNCT
ejpam-3838	173	11	·	·	PUNCT
ejpam-3838	173	12	(	(	PUNCT
ejpam-3838	173	13	qn	qn	INTJ
ejpam-3838	173	14	−	−	PROPN
ejpam-3838	173	15	qk−1	qk−1	PROPN
ejpam-3838	173	16	)	)	PUNCT
ejpam-3838	173	17	(	(	PUNCT
ejpam-3838	173	18	qk	qk	ADP
ejpam-3838	173	19	−	−	PROPN
ejpam-3838	173	20	1)(qk	1)(qk	NUM
ejpam-3838	173	21	−	−	NOUN
ejpam-3838	173	22	q	q	NOUN
ejpam-3838	173	23	)	)	PUNCT
ejpam-3838	173	24	·	·	PUNCT
ejpam-3838	173	25	·	·	PUNCT
ejpam-3838	173	26	·	·	PUNCT
ejpam-3838	173	27	(	(	PUNCT
ejpam-3838	173	28	qk	qk	ADP
ejpam-3838	173	29	−	−	PROPN
ejpam-3838	173	30	qk−1	qk−1	PROPN
ejpam-3838	173	31	)	)	PUNCT
ejpam-3838	173	32	,	,	PUNCT
ejpam-3838	173	33	which	which	PRON
ejpam-3838	173	34	gives	give	VERB
ejpam-3838	173	35	the	the	DET
ejpam-3838	173	36	number	number	NOUN
ejpam-3838	173	37	of	of	ADP
ejpam-3838	173	38	subspaces	subspace	NOUN
ejpam-3838	173	39	of	of	ADP
ejpam-3838	173	40	dimension	dimension	NOUN
ejpam-3838	173	41	k	k	PROPN
ejpam-3838	173	42	contained	contain	VERB
ejpam-3838	173	43	in	in	ADP
ejpam-3838	173	44	an	an	DET
ejpam-3838	173	45	n	n	ADV
ejpam-3838	173	46	-	-	PUNCT
ejpam-3838	173	47	dimensional	dimensional	ADJ
ejpam-3838	173	48	vector	vector	NOUN
ejpam-3838	173	49	space	space	NOUN
ejpam-3838	173	50	over	over	ADP
ejpam-3838	173	51	fq	fq	PROPN
ejpam-3838	173	52	.	.	PUNCT
ejpam-3838	174	1	we	we	PRON
ejpam-3838	174	2	now	now	ADV
ejpam-3838	174	3	have	have	VERB
ejpam-3838	174	4	the	the	DET
ejpam-3838	174	5	following	follow	VERB
ejpam-3838	174	6	mass	mass	NOUN
ejpam-3838	174	7	formula	formula	NOUN
ejpam-3838	174	8	for	for	ADP
ejpam-3838	174	9	self	self	NOUN
ejpam-3838	174	10	-	-	PUNCT
ejpam-3838	174	11	orthogonal	orthogonal	ADJ
ejpam-3838	174	12	codes	code	NOUN
ejpam-3838	174	13	over	over	ADP
ejpam-3838	174	14	r1	r1	PROPN
ejpam-3838	174	15	.	.	PUNCT
ejpam-3838	175	1	l.e	l.e	PROPN
ejpam-3838	175	2	.	.	PROPN
ejpam-3838	175	3	galvez	galvez	PROPN
ejpam-3838	175	4	,	,	PUNCT
ejpam-3838	175	5	r.a	r.a	PROPN
ejpam-3838	175	6	.	.	PROPN
ejpam-3838	175	7	betty	betty	PROPN
ejpam-3838	175	8	,	,	PUNCT
ejpam-3838	175	9	f.	f.	PROPN
ejpam-3838	175	10	nemenzo	nemenzo	PROPN
ejpam-3838	175	11	/	/	SYM
ejpam-3838	175	12	eur	eur	NOUN
ejpam-3838	175	13	.	.	PUNCT
ejpam-3838	176	1	j.	j.	PROPN
ejpam-3838	176	2	pure	pure	PROPN
ejpam-3838	176	3	appl	appl	PROPN
ejpam-3838	176	4	.	.	PROPN
ejpam-3838	176	5	math	math	PROPN
ejpam-3838	176	6	,	,	PUNCT
ejpam-3838	176	7	13	13	NUM
ejpam-3838	176	8	(	(	PUNCT
ejpam-3838	176	9	4	4	NUM
ejpam-3838	176	10	)	)	PUNCT
ejpam-3838	176	11	(	(	PUNCT
ejpam-3838	176	12	2020	2020	NUM
ejpam-3838	176	13	)	)	PUNCT
ejpam-3838	176	14	,	,	PUNCT
ejpam-3838	176	15	873	873	NUM
ejpam-3838	176	16	-	-	SYM
ejpam-3838	176	17	892	892	NUM
ejpam-3838	176	18	880	880	NUM
ejpam-3838	176	19	theorem	theorem	NOUN
ejpam-3838	176	20	2	2	NUM
ejpam-3838	176	21	.	.	PUNCT
ejpam-3838	177	1	let	let	AUX
ejpam-3838	177	2	mq(n	mq(n	NOUN
ejpam-3838	177	3	,	,	PUNCT
ejpam-3838	177	4	k0	k0	PROPN
ejpam-3838	177	5	,	,	PUNCT
ejpam-3838	177	6	k1)e	k1)e	NOUN
ejpam-3838	177	7	and	and	CCONJ
ejpam-3838	177	8	mq(n	mq(n	PROPN
ejpam-3838	177	9	,	,	PUNCT
ejpam-3838	177	10	k0	k0	PROPN
ejpam-3838	177	11	,	,	PUNCT
ejpam-3838	177	12	k1)h	k1)h	PROPN
ejpam-3838	177	13	denote	denote	VERB
ejpam-3838	177	14	the	the	DET
ejpam-3838	177	15	number	number	NOUN
ejpam-3838	177	16	of	of	ADP
ejpam-3838	177	17	distinct	distinct	ADJ
ejpam-3838	177	18	euclidean	euclidean	NOUN
ejpam-3838	177	19	and	and	CCONJ
ejpam-3838	177	20	hermitian	hermitian	ADJ
ejpam-3838	177	21	self	self	NOUN
ejpam-3838	177	22	-	-	PUNCT
ejpam-3838	177	23	orthogonal	orthogonal	ADJ
ejpam-3838	177	24	codes	code	NOUN
ejpam-3838	177	25	of	of	ADP
ejpam-3838	177	26	length	length	NOUN
ejpam-3838	177	27	n	n	PROPN
ejpam-3838	177	28	over	over	ADP
ejpam-3838	177	29	fq	fq	PROPN
ejpam-3838	177	30	+	+	CCONJ
ejpam-3838	177	31	ufq	ufq	NOUN
ejpam-3838	177	32	of	of	ADP
ejpam-3838	177	33	type	type	NOUN
ejpam-3838	177	34	{	{	PUNCT
ejpam-3838	177	35	k0	k0	PROPN
ejpam-3838	177	36	,	,	PUNCT
ejpam-3838	177	37	k1	k1	NOUN
ejpam-3838	177	38	}	}	PUNCT
ejpam-3838	177	39	,	,	PUNCT
ejpam-3838	177	40	respectively	respectively	ADV
ejpam-3838	177	41	.	.	PUNCT
ejpam-3838	178	1	we	we	PRON
ejpam-3838	178	2	have	have	AUX
ejpam-3838	178	3	mq(n	mq(n	NOUN
ejpam-3838	178	4	,	,	PUNCT
ejpam-3838	178	5	k0	k0	PROPN
ejpam-3838	178	6	,	,	PUNCT
ejpam-3838	178	7	k1)e	k1)e	NOUN
ejpam-3838	178	8	=	=	PUNCT
ejpam-3838	178	9	σq(n	σq(n	X
ejpam-3838	178	10	,	,	PUNCT
ejpam-3838	178	11	k0	k0	PROPN
ejpam-3838	178	12	)	)	PUNCT
ejpam-3838	178	13	[	[	PUNCT
ejpam-3838	178	14	n−	n−	NOUN
ejpam-3838	178	15	2k0	2k0	NUM
ejpam-3838	178	16	k1	k1	NOUN
ejpam-3838	178	17	]	]	PUNCT
ejpam-3838	178	18	q	q	PUNCT
ejpam-3838	178	19	qk0(2n−3k0−2k1+ε)/2	qk0(2n−3k0−2k1+ε)/2	NOUN
ejpam-3838	178	20	where	where	SCONJ
ejpam-3838	178	21	ε	ε	PROPN
ejpam-3838	178	22	=	=	SYM
ejpam-3838	178	23	−1	−1	NOUN
ejpam-3838	178	24	if	if	SCONJ
ejpam-3838	178	25	q	q	NOUN
ejpam-3838	178	26	is	be	AUX
ejpam-3838	178	27	odd	odd	ADJ
ejpam-3838	178	28	and	and	CCONJ
ejpam-3838	178	29	ε	ε	PROPN
ejpam-3838	178	30	=	=	SYM
ejpam-3838	178	31	1	1	NUM
ejpam-3838	178	32	if	if	SCONJ
ejpam-3838	178	33	q	q	NOUN
ejpam-3838	178	34	is	be	AUX
ejpam-3838	178	35	even	even	ADV
ejpam-3838	178	36	,	,	PUNCT
ejpam-3838	178	37	and	and	CCONJ
ejpam-3838	178	38	mq(n	mq(n	PROPN
ejpam-3838	178	39	,	,	PUNCT
ejpam-3838	178	40	k0	k0	PROPN
ejpam-3838	178	41	,	,	PUNCT
ejpam-3838	178	42	k1)h	k1)h	NOUN
ejpam-3838	178	43	=	=	PUNCT
ejpam-3838	178	44	σq(n	σq(n	PROPN
ejpam-3838	178	45	,	,	PUNCT
ejpam-3838	178	46	k0	k0	PROPN
ejpam-3838	178	47	)	)	PUNCT
ejpam-3838	178	48	[	[	PUNCT
ejpam-3838	178	49	n−	n−	NOUN
ejpam-3838	178	50	2k0	2k0	NUM
ejpam-3838	178	51	k1	k1	X
ejpam-3838	178	52	]	]	PUNCT
ejpam-3838	178	53	q	q	X
ejpam-3838	178	54	qk0(2n−3k0−2k1	qk0(2n−3k0−2k1	PROPN
ejpam-3838	178	55	+	+	NOUN
ejpam-3838	178	56	1)/2	1)/2	NOUN
ejpam-3838	178	57	.	.	PUNCT
ejpam-3838	179	1	proof	proof	NOUN
ejpam-3838	179	2	.	.	PUNCT
ejpam-3838	180	1	if	if	SCONJ
ejpam-3838	180	2	c	c	PROPN
ejpam-3838	180	3	is	be	AUX
ejpam-3838	180	4	a	a	DET
ejpam-3838	180	5	self	self	NOUN
ejpam-3838	180	6	-	-	PUNCT
ejpam-3838	180	7	orthogonal	orthogonal	ADJ
ejpam-3838	180	8	code	code	NOUN
ejpam-3838	180	9	of	of	ADP
ejpam-3838	180	10	length	length	NOUN
ejpam-3838	180	11	n	n	PROPN
ejpam-3838	180	12	over	over	ADP
ejpam-3838	180	13	fq	fq	PROPN
ejpam-3838	180	14	+	+	PROPN
ejpam-3838	180	15	ufq	ufq	NOUN
ejpam-3838	180	16	of	of	ADP
ejpam-3838	180	17	type	type	NOUN
ejpam-3838	180	18	{	{	PUNCT
ejpam-3838	180	19	k0	k0	PROPN
ejpam-3838	180	20	,	,	PUNCT
ejpam-3838	180	21	k1	k1	PROPN
ejpam-3838	180	22	}	}	PUNCT
ejpam-3838	180	23	,	,	PUNCT
ejpam-3838	180	24	then	then	ADV
ejpam-3838	180	25	by	by	ADP
ejpam-3838	180	26	setting	set	VERB
ejpam-3838	180	27	c1	c1	NOUN
ejpam-3838	180	28	=	=	PUNCT
ejpam-3838	180	29	res(c	res(c	PROPN
ejpam-3838	180	30	)	)	PUNCT
ejpam-3838	180	31	and	and	CCONJ
ejpam-3838	180	32	c2	c2	PROPN
ejpam-3838	180	33	=	=	SYM
ejpam-3838	180	34	tor(c	tor(c	PROPN
ejpam-3838	180	35	)	)	PUNCT
ejpam-3838	180	36	,	,	PUNCT
ejpam-3838	180	37	we	we	PRON
ejpam-3838	180	38	see	see	VERB
ejpam-3838	180	39	that	that	PRON
ejpam-3838	180	40	c1	c1	PROPN
ejpam-3838	180	41	and	and	CCONJ
ejpam-3838	180	42	c2	c2	PROPN
ejpam-3838	180	43	satisfies	satisfy	VERB
ejpam-3838	180	44	lemma	lemma	PROPN
ejpam-3838	180	45	1	1	X
ejpam-3838	180	46	.	.	PUNCT
ejpam-3838	181	1	there	there	PRON
ejpam-3838	181	2	are	be	VERB
ejpam-3838	181	3	σq(n	σq(n	NOUN
ejpam-3838	181	4	,	,	PUNCT
ejpam-3838	181	5	k0	k0	PROPN
ejpam-3838	181	6	)	)	PUNCT
ejpam-3838	181	7	self	self	NOUN
ejpam-3838	181	8	-	-	PUNCT
ejpam-3838	181	9	orthogonal	orthogonal	ADJ
ejpam-3838	181	10	codes	code	NOUN
ejpam-3838	181	11	c1	c1	PROPN
ejpam-3838	181	12	of	of	ADP
ejpam-3838	181	13	length	length	NOUN
ejpam-3838	181	14	n	n	PROPN
ejpam-3838	181	15	over	over	ADP
ejpam-3838	181	16	fq	fq	PROPN
ejpam-3838	181	17	.	.	PROPN
ejpam-3838	182	1	given	give	VERB
ejpam-3838	182	2	c1	c1	PROPN
ejpam-3838	182	3	,	,	PUNCT
ejpam-3838	182	4	there	there	PRON
ejpam-3838	182	5	are	be	VERB
ejpam-3838	182	6	[	[	PUNCT
ejpam-3838	182	7	n−	n−	NOUN
ejpam-3838	182	8	2k0	2k0	NUM
ejpam-3838	182	9	k1	k1	NOUN
ejpam-3838	182	10	]	]	PUNCT
ejpam-3838	183	1	q	q	PROPN
ejpam-3838	183	2	codes	code	NOUN
ejpam-3838	183	3	c2	c2	VERB
ejpam-3838	183	4	such	such	ADJ
ejpam-3838	183	5	that	that	DET
ejpam-3838	183	6	c1	c1	PROPN
ejpam-3838	183	7	⊆	⊆	NUM
ejpam-3838	183	8	c2	c2	PROPN
ejpam-3838	183	9	⊆	⊆	NUM
ejpam-3838	183	10	c⊥1	c⊥1	NOUN
ejpam-3838	183	11	.	.	PUNCT
ejpam-3838	184	1	then	then	ADV
ejpam-3838	184	2	the	the	DET
ejpam-3838	184	3	result	result	NOUN
ejpam-3838	184	4	follows	follow	VERB
ejpam-3838	184	5	from	from	ADP
ejpam-3838	184	6	theorem	theorem	ADJ
ejpam-3838	184	7	1	1	NUM
ejpam-3838	184	8	.	.	X
ejpam-3838	184	9	�	�	PROPN
ejpam-3838	184	10	we	we	PRON
ejpam-3838	184	11	have	have	VERB
ejpam-3838	184	12	the	the	DET
ejpam-3838	184	13	following	follow	VERB
ejpam-3838	184	14	mass	mass	NOUN
ejpam-3838	184	15	formula	formula	NOUN
ejpam-3838	184	16	for	for	ADP
ejpam-3838	184	17	self	self	NOUN
ejpam-3838	184	18	-	-	PUNCT
ejpam-3838	184	19	dual	dual	ADJ
ejpam-3838	184	20	codes	code	NOUN
ejpam-3838	184	21	over	over	ADP
ejpam-3838	184	22	r1	r1	PROPN
ejpam-3838	184	23	as	as	ADP
ejpam-3838	184	24	a	a	DET
ejpam-3838	184	25	direct	direct	ADJ
ejpam-3838	184	26	consequence	consequence	NOUN
ejpam-3838	184	27	of	of	ADP
ejpam-3838	184	28	the	the	DET
ejpam-3838	184	29	previous	previous	ADJ
ejpam-3838	184	30	theorem	theorem	PROPN
ejpam-3838	184	31	.	.	PROPN
ejpam-3838	184	32	corollary	corollary	ADJ
ejpam-3838	184	33	1	1	NUM
ejpam-3838	184	34	.	.	PUNCT
ejpam-3838	185	1	the	the	DET
ejpam-3838	185	2	number	number	NOUN
ejpam-3838	185	3	of	of	ADP
ejpam-3838	185	4	distinct	distinct	ADJ
ejpam-3838	185	5	euclidean	euclidean	ADJ
ejpam-3838	185	6	self	self	NOUN
ejpam-3838	185	7	-	-	PUNCT
ejpam-3838	185	8	dual	dual	ADJ
ejpam-3838	185	9	codes	code	NOUN
ejpam-3838	185	10	of	of	ADP
ejpam-3838	185	11	length	length	NOUN
ejpam-3838	185	12	n	n	PROPN
ejpam-3838	185	13	over	over	ADP
ejpam-3838	185	14	fq	fq	PROPN
ejpam-3838	185	15	+	+	CCONJ
ejpam-3838	185	16	ufq	ufq	NOUN
ejpam-3838	185	17	is	be	AUX
ejpam-3838	185	18	given	give	VERB
ejpam-3838	185	19	by	by	ADP
ejpam-3838	185	20	∑	∑	PROPN
ejpam-3838	185	21	0≤k0≤bn2	0≤k0≤bn2	PROPN
ejpam-3838	185	22	c	c	NOUN
ejpam-3838	185	23	σq(n	σq(n	X
ejpam-3838	185	24	,	,	PUNCT
ejpam-3838	185	25	k0)q	k0)q	PROPN
ejpam-3838	185	26	k0(k0+ε)/2	k0(k0+ε)/2	PROPN
ejpam-3838	185	27	,	,	PUNCT
ejpam-3838	185	28	(	(	PUNCT
ejpam-3838	185	29	8)	8)	NUM
ejpam-3838	185	30	where	where	SCONJ
ejpam-3838	185	31	ε	ε	PROPN
ejpam-3838	185	32	=	=	SYM
ejpam-3838	185	33	−1	−1	NOUN
ejpam-3838	185	34	if	if	SCONJ
ejpam-3838	185	35	q	q	NOUN
ejpam-3838	185	36	is	be	AUX
ejpam-3838	185	37	odd	odd	ADJ
ejpam-3838	185	38	and	and	CCONJ
ejpam-3838	185	39	ε	ε	PROPN
ejpam-3838	185	40	=	=	SYM
ejpam-3838	185	41	1	1	NUM
ejpam-3838	185	42	if	if	SCONJ
ejpam-3838	185	43	q	q	NOUN
ejpam-3838	185	44	is	be	AUX
ejpam-3838	185	45	even	even	ADV
ejpam-3838	185	46	,	,	PUNCT
ejpam-3838	185	47	and	and	CCONJ
ejpam-3838	185	48	the	the	DET
ejpam-3838	185	49	number	number	NOUN
ejpam-3838	185	50	of	of	ADP
ejpam-3838	185	51	distinct	distinct	ADJ
ejpam-3838	185	52	hermitian	hermitian	ADJ
ejpam-3838	185	53	self	self	NOUN
ejpam-3838	185	54	-	-	PUNCT
ejpam-3838	185	55	dual	dual	ADJ
ejpam-3838	185	56	codes	code	NOUN
ejpam-3838	185	57	of	of	ADP
ejpam-3838	185	58	length	length	NOUN
ejpam-3838	185	59	n	n	PROPN
ejpam-3838	185	60	over	over	ADP
ejpam-3838	185	61	fq	fq	PROPN
ejpam-3838	185	62	+	+	CCONJ
ejpam-3838	185	63	ufq	ufq	NOUN
ejpam-3838	185	64	is	be	AUX
ejpam-3838	185	65	given	give	VERB
ejpam-3838	185	66	by∑	by∑	NOUN
ejpam-3838	185	67	0≤k0≤bn2	0≤k0≤bn2	PUNCT
ejpam-3838	186	1	c	c	NOUN
ejpam-3838	186	2	σq(n	σq(n	X
ejpam-3838	186	3	,	,	PUNCT
ejpam-3838	186	4	k0)q	k0)q	VERB
ejpam-3838	186	5	k0(k0	k0(k0	PROPN
ejpam-3838	186	6	+	+	PROPN
ejpam-3838	186	7	1)/2	1)/2	NUM
ejpam-3838	186	8	.	.	PUNCT
ejpam-3838	187	1	(	(	PUNCT
ejpam-3838	187	2	9	9	X
ejpam-3838	187	3	)	)	PUNCT
ejpam-3838	187	4	proof	proof	NOUN
ejpam-3838	187	5	.	.	PUNCT
ejpam-3838	188	1	note	note	VERB
ejpam-3838	188	2	that	that	SCONJ
ejpam-3838	188	3	the	the	DET
ejpam-3838	188	4	number	number	NOUN
ejpam-3838	188	5	of	of	ADP
ejpam-3838	188	6	distinct	distinct	ADJ
ejpam-3838	188	7	euclidean	euclidean	ADJ
ejpam-3838	188	8	self	self	NOUN
ejpam-3838	188	9	-	-	PUNCT
ejpam-3838	188	10	dual	dual	ADJ
ejpam-3838	188	11	codes	code	NOUN
ejpam-3838	188	12	and	and	CCONJ
ejpam-3838	188	13	the	the	DET
ejpam-3838	188	14	number	number	NOUN
ejpam-3838	188	15	of	of	ADP
ejpam-3838	188	16	distinct	distinct	ADJ
ejpam-3838	188	17	hermitian	hermitian	ADJ
ejpam-3838	188	18	self	self	NOUN
ejpam-3838	188	19	-	-	PUNCT
ejpam-3838	188	20	dual	dual	ADJ
ejpam-3838	188	21	codes	code	NOUN
ejpam-3838	188	22	of	of	ADP
ejpam-3838	188	23	length	length	NOUN
ejpam-3838	188	24	n	n	PROPN
ejpam-3838	188	25	over	over	ADP
ejpam-3838	188	26	fq	fq	PROPN
ejpam-3838	188	27	+	+	CCONJ
ejpam-3838	188	28	ufq	ufq	NOUN
ejpam-3838	188	29	are	be	AUX
ejpam-3838	188	30	given	give	VERB
ejpam-3838	188	31	by∑	by∑	NOUN
ejpam-3838	188	32	0≤k0≤bn2	0≤k0≤bn2	PUNCT
ejpam-3838	188	33	c	c	PROPN
ejpam-3838	188	34	mq(n	mq(n	PROPN
ejpam-3838	188	35	,	,	PUNCT
ejpam-3838	188	36	k0	k0	PROPN
ejpam-3838	188	37	,	,	PUNCT
ejpam-3838	188	38	n−	n−	NOUN
ejpam-3838	188	39	2k0)e	2k0)e	NUM
ejpam-3838	188	40	,	,	PUNCT
ejpam-3838	188	41	and	and	CCONJ
ejpam-3838	188	42	∑	∑	ADP
ejpam-3838	188	43	0≤k0≤bn2	0≤k0≤bn2	PROPN
ejpam-3838	188	44	c	c	PROPN
ejpam-3838	188	45	mq(n	mq(n	PROPN
ejpam-3838	188	46	,	,	PUNCT
ejpam-3838	188	47	k0	k0	PROPN
ejpam-3838	188	48	,	,	PUNCT
ejpam-3838	188	49	n−	n−	PROPN
ejpam-3838	188	50	2k0)h	2k0)h	NUM
ejpam-3838	188	51	,	,	PUNCT
ejpam-3838	188	52	respectively	respectively	ADV
ejpam-3838	188	53	.	.	PUNCT
ejpam-3838	189	1	�	�	PROPN
ejpam-3838	189	2	in	in	ADP
ejpam-3838	189	3	[	[	X
ejpam-3838	189	4	6	6	NUM
ejpam-3838	189	5	,	,	PUNCT
ejpam-3838	189	6	theorem	theorem	VERB
ejpam-3838	189	7	3	3	NUM
ejpam-3838	189	8	]	]	PUNCT
ejpam-3838	189	9	,	,	PUNCT
ejpam-3838	189	10	gaborit	gaborit	NOUN
ejpam-3838	189	11	establishes	establish	VERB
ejpam-3838	189	12	the	the	DET
ejpam-3838	189	13	mass	mass	ADJ
ejpam-3838	189	14	formula	formula	NOUN
ejpam-3838	189	15	for	for	ADP
ejpam-3838	189	16	hermitian	hermitian	ADJ
ejpam-3838	189	17	self	self	NOUN
ejpam-3838	189	18	-	-	PUNCT
ejpam-3838	189	19	dual	dual	ADJ
ejpam-3838	189	20	codes	code	NOUN
ejpam-3838	189	21	over	over	ADP
ejpam-3838	189	22	fq	fq	PROPN
ejpam-3838	189	23	+	+	CCONJ
ejpam-3838	189	24	ufq	ufq	PROPN
ejpam-3838	189	25	,	,	PUNCT
ejpam-3838	189	26	but	but	CCONJ
ejpam-3838	189	27	gives	give	VERB
ejpam-3838	189	28	the	the	DET
ejpam-3838	189	29	formula	formula	NOUN
ejpam-3838	189	30	for	for	ADP
ejpam-3838	189	31	euclidean	euclidean	ADJ
ejpam-3838	189	32	self	self	NOUN
ejpam-3838	189	33	-	-	PUNCT
ejpam-3838	189	34	dual	dual	ADJ
ejpam-3838	189	35	codes	code	NOUN
ejpam-3838	189	36	instead	instead	ADV
ejpam-3838	189	37	.	.	PUNCT
ejpam-3838	190	1	the	the	DET
ejpam-3838	190	2	formula	formula	NOUN
ejpam-3838	190	3	(	(	PUNCT
ejpam-3838	190	4	9	9	X
ejpam-3838	190	5	)	)	PUNCT
ejpam-3838	190	6	corrects	correct	NOUN
ejpam-3838	190	7	this	this	PRON
ejpam-3838	190	8	.	.	PUNCT
ejpam-3838	191	1	next	next	ADV
ejpam-3838	191	2	,	,	PUNCT
ejpam-3838	191	3	we	we	PRON
ejpam-3838	191	4	establish	establish	VERB
ejpam-3838	191	5	another	another	DET
ejpam-3838	191	6	formula	formula	NOUN
ejpam-3838	191	7	for	for	ADP
ejpam-3838	191	8	the	the	DET
ejpam-3838	191	9	number	number	NOUN
ejpam-3838	191	10	of	of	ADP
ejpam-3838	191	11	distinct	distinct	ADJ
ejpam-3838	191	12	euclidean	euclidean	ADJ
ejpam-3838	191	13	self	self	NOUN
ejpam-3838	191	14	-	-	PUNCT
ejpam-3838	191	15	orthogonal	orthogonal	ADJ
ejpam-3838	191	16	codes	code	NOUN
ejpam-3838	191	17	when	when	SCONJ
ejpam-3838	191	18	the	the	DET
ejpam-3838	191	19	given	give	VERB
ejpam-3838	191	20	torsion	torsion	NOUN
ejpam-3838	191	21	is	be	AUX
ejpam-3838	191	22	self	self	NOUN
ejpam-3838	191	23	-	-	PUNCT
ejpam-3838	191	24	orthogonal	orthogonal	NOUN
ejpam-3838	191	25	.	.	PUNCT
ejpam-3838	192	1	l.e	l.e	PROPN
ejpam-3838	192	2	.	.	PROPN
ejpam-3838	192	3	galvez	galvez	PROPN
ejpam-3838	192	4	,	,	PUNCT
ejpam-3838	192	5	r.a	r.a	PROPN
ejpam-3838	192	6	.	.	PROPN
ejpam-3838	192	7	betty	betty	PROPN
ejpam-3838	192	8	,	,	PUNCT
ejpam-3838	192	9	f.	f.	PROPN
ejpam-3838	192	10	nemenzo	nemenzo	PROPN
ejpam-3838	192	11	/	/	SYM
ejpam-3838	192	12	eur	eur	NOUN
ejpam-3838	192	13	.	.	PUNCT
ejpam-3838	193	1	j.	j.	PROPN
ejpam-3838	193	2	pure	pure	PROPN
ejpam-3838	193	3	appl	appl	PROPN
ejpam-3838	193	4	.	.	PROPN
ejpam-3838	193	5	math	math	PROPN
ejpam-3838	193	6	,	,	PUNCT
ejpam-3838	193	7	13	13	NUM
ejpam-3838	193	8	(	(	PUNCT
ejpam-3838	193	9	4	4	NUM
ejpam-3838	193	10	)	)	PUNCT
ejpam-3838	193	11	(	(	PUNCT
ejpam-3838	193	12	2020	2020	NUM
ejpam-3838	193	13	)	)	PUNCT
ejpam-3838	193	14	,	,	PUNCT
ejpam-3838	193	15	873	873	NUM
ejpam-3838	193	16	-	-	NUM
ejpam-3838	193	17	892	892	NUM
ejpam-3838	193	18	881	881	NUM
ejpam-3838	193	19	corollary	corollary	NOUN
ejpam-3838	193	20	2	2	NUM
ejpam-3838	193	21	.	.	PUNCT
ejpam-3838	193	22	suppose	suppose	VERB
ejpam-3838	193	23	q	q	NOUN
ejpam-3838	193	24	is	be	AUX
ejpam-3838	193	25	odd	odd	ADJ
ejpam-3838	193	26	.	.	PUNCT
ejpam-3838	194	1	the	the	DET
ejpam-3838	194	2	number	number	NOUN
ejpam-3838	194	3	of	of	ADP
ejpam-3838	194	4	distinct	distinct	ADJ
ejpam-3838	194	5	euclidean	euclidean	ADJ
ejpam-3838	194	6	self	self	NOUN
ejpam-3838	194	7	-	-	PUNCT
ejpam-3838	194	8	orthogonal	orthogonal	ADJ
ejpam-3838	194	9	codes	code	NOUN
ejpam-3838	194	10	of	of	ADP
ejpam-3838	194	11	length	length	NOUN
ejpam-3838	194	12	n	n	PROPN
ejpam-3838	194	13	over	over	ADP
ejpam-3838	194	14	fq	fq	PROPN
ejpam-3838	194	15	+	+	CCONJ
ejpam-3838	194	16	ufq	ufq	NOUN
ejpam-3838	194	17	of	of	ADP
ejpam-3838	194	18	type	type	NOUN
ejpam-3838	194	19	{	{	PUNCT
ejpam-3838	194	20	k0	k0	PROPN
ejpam-3838	194	21	,	,	PUNCT
ejpam-3838	194	22	k1	k1	PROPN
ejpam-3838	194	23	}	}	PUNCT
ejpam-3838	194	24	with	with	ADP
ejpam-3838	194	25	self	self	NOUN
ejpam-3838	194	26	-	-	PUNCT
ejpam-3838	194	27	orthogonal	orthogonal	ADJ
ejpam-3838	194	28	torsion	torsion	NOUN
ejpam-3838	194	29	is	be	AUX
ejpam-3838	194	30	m̃q(n	m̃q(n	PROPN
ejpam-3838	194	31	,	,	PUNCT
ejpam-3838	194	32	k0	k0	PROPN
ejpam-3838	194	33	,	,	PUNCT
ejpam-3838	194	34	k1)e	k1)e	NOUN
ejpam-3838	194	35	=	=	PUNCT
ejpam-3838	194	36	[	[	PUNCT
ejpam-3838	194	37	k0	k0	PROPN
ejpam-3838	194	38	+	+	CCONJ
ejpam-3838	194	39	k1	k1	PROPN
ejpam-3838	194	40	k0	k0	PROPN
ejpam-3838	194	41	]	]	PUNCT
ejpam-3838	194	42	q	q	X
ejpam-3838	194	43	σq(n	σq(n	X
ejpam-3838	194	44	,	,	PUNCT
ejpam-3838	194	45	k0	k0	PROPN
ejpam-3838	194	46	+	+	CCONJ
ejpam-3838	194	47	k1)q	k1)q	PROPN
ejpam-3838	194	48	k0(2n−3k0−2k1−1)/2	k0(2n−3k0−2k1−1)/2	PROPN
ejpam-3838	194	49	.	.	PUNCT
ejpam-3838	195	1	proof	proof	NOUN
ejpam-3838	195	2	.	.	PUNCT
ejpam-3838	196	1	let	let	VERB
ejpam-3838	196	2	c1	c1	PROPN
ejpam-3838	196	3	and	and	CCONJ
ejpam-3838	196	4	c2	c2	PROPN
ejpam-3838	196	5	be	be	VERB
ejpam-3838	196	6	self	self	NOUN
ejpam-3838	196	7	-	-	PUNCT
ejpam-3838	196	8	orthogonal	orthogonal	ADJ
ejpam-3838	196	9	codes	code	NOUN
ejpam-3838	196	10	where	where	SCONJ
ejpam-3838	196	11	dim	dim	ADJ
ejpam-3838	196	12	c1	c1	NOUN
ejpam-3838	196	13	=	=	PROPN
ejpam-3838	196	14	k0	k0	PROPN
ejpam-3838	196	15	,	,	PUNCT
ejpam-3838	196	16	dim	dim	ADJ
ejpam-3838	196	17	c2	c2	PROPN
ejpam-3838	196	18	=	=	SYM
ejpam-3838	196	19	k0	k0	PROPN
ejpam-3838	196	20	+	+	CCONJ
ejpam-3838	196	21	k1	k1	PROPN
ejpam-3838	196	22	and	and	CCONJ
ejpam-3838	196	23	c1	c1	PROPN
ejpam-3838	196	24	⊆	⊆	NUM
ejpam-3838	196	25	c2	c2	PROPN
ejpam-3838	196	26	.	.	PUNCT
ejpam-3838	197	1	by	by	ADP
ejpam-3838	197	2	theorem	theorem	NOUN
ejpam-3838	197	3	2	2	NUM
ejpam-3838	197	4	,	,	PUNCT
ejpam-3838	197	5	we	we	PRON
ejpam-3838	197	6	have	have	VERB
ejpam-3838	197	7	m̃q(n	m̃q(n	PROPN
ejpam-3838	197	8	,	,	PUNCT
ejpam-3838	197	9	k0	k0	PROPN
ejpam-3838	197	10	,	,	PUNCT
ejpam-3838	197	11	k1)e	k1)e	NOUN
ejpam-3838	197	12	q	q	NOUN
ejpam-3838	198	1	−k0(2n−3k0−2k1−1)/2	−k0(2n−3k0−2k1−1)/2	PROPN
ejpam-3838	198	2	=	=	PUNCT
ejpam-3838	198	3	∑	∑	PUNCT
ejpam-3838	198	4	c2⊆c⊥2	c2⊆c⊥2	NOUN
ejpam-3838	198	5	|{c1	|{c1	VERB
ejpam-3838	198	6	|	|	ADV
ejpam-3838	198	7	c1	c1	PROPN
ejpam-3838	198	8	⊆	⊆	NUM
ejpam-3838	198	9	c2}|	c2}|	PROPN
ejpam-3838	199	1	=	=	PUNCT
ejpam-3838	200	1	[	[	PUNCT
ejpam-3838	200	2	k0	k0	PROPN
ejpam-3838	200	3	+	+	CCONJ
ejpam-3838	200	4	k1	k1	PROPN
ejpam-3838	200	5	k0	k0	PROPN
ejpam-3838	200	6	]	]	PUNCT
ejpam-3838	200	7	q	q	X
ejpam-3838	200	8	∣∣∣{c2	∣∣∣{c2	PROPN
ejpam-3838	200	9	|	|	ADV
ejpam-3838	200	10	c2	c2	PROPN
ejpam-3838	200	11	⊆	⊆	NUM
ejpam-3838	200	12	c⊥2	c⊥2	PUNCT
ejpam-3838	200	13	}	}	PUNCT
ejpam-3838	200	14	∣∣∣	∣∣∣	NOUN
ejpam-3838	201	1	=	=	PUNCT
ejpam-3838	202	1	[	[	PUNCT
ejpam-3838	202	2	k0	k0	PROPN
ejpam-3838	202	3	+	+	CCONJ
ejpam-3838	202	4	k1	k1	PROPN
ejpam-3838	202	5	k0	k0	PROPN
ejpam-3838	202	6	]	]	PUNCT
ejpam-3838	202	7	q	q	X
ejpam-3838	202	8	σq(n	σq(n	X
ejpam-3838	202	9	,	,	PUNCT
ejpam-3838	202	10	k0	k0	PROPN
ejpam-3838	202	11	+	+	CCONJ
ejpam-3838	202	12	k1	k1	NOUN
ejpam-3838	202	13	)	)	PUNCT
ejpam-3838	202	14	.	.	PUNCT
ejpam-3838	203	1	�	�	PROPN
ejpam-3838	204	1	this	this	DET
ejpam-3838	204	2	corollary	corollary	NOUN
ejpam-3838	204	3	will	will	AUX
ejpam-3838	204	4	be	be	AUX
ejpam-3838	204	5	useful	useful	ADJ
ejpam-3838	204	6	in	in	ADP
ejpam-3838	204	7	our	our	PRON
ejpam-3838	204	8	mass	mass	ADJ
ejpam-3838	204	9	formula	formula	NOUN
ejpam-3838	204	10	computations	computation	NOUN
ejpam-3838	204	11	on	on	ADP
ejpam-3838	204	12	later	later	ADJ
ejpam-3838	204	13	chapters	chapter	NOUN
ejpam-3838	204	14	.	.	PUNCT
ejpam-3838	205	1	5	5	X
ejpam-3838	205	2	.	.	X
ejpam-3838	205	3	classification	classification	NOUN
ejpam-3838	205	4	of	of	ADP
ejpam-3838	205	5	self	self	NOUN
ejpam-3838	205	6	-	-	PUNCT
ejpam-3838	205	7	orthogonal	orthogonal	ADJ
ejpam-3838	205	8	codes	code	NOUN
ejpam-3838	205	9	over	over	ADP
ejpam-3838	205	10	fq	fq	PROPN
ejpam-3838	205	11	+	+	CCONJ
ejpam-3838	205	12	ufq	ufq	NOUN
ejpam-3838	205	13	using	use	VERB
ejpam-3838	205	14	theorem	theorem	NOUN
ejpam-3838	205	15	2	2	NUM
ejpam-3838	205	16	,	,	PUNCT
ejpam-3838	205	17	we	we	PRON
ejpam-3838	205	18	classify	classify	VERB
ejpam-3838	205	19	euclidean	euclidean	ADJ
ejpam-3838	205	20	and	and	CCONJ
ejpam-3838	205	21	hermitian	hermitian	ADJ
ejpam-3838	205	22	self	self	NOUN
ejpam-3838	205	23	-	-	PUNCT
ejpam-3838	205	24	orthogonal	orthogonal	ADJ
ejpam-3838	205	25	codes	code	NOUN
ejpam-3838	205	26	over	over	ADP
ejpam-3838	205	27	f2	f2	PROPN
ejpam-3838	205	28	+	+	CCONJ
ejpam-3838	205	29	uf2	uf2	NOUN
ejpam-3838	205	30	and	and	CCONJ
ejpam-3838	205	31	f3	f3	PROPN
ejpam-3838	205	32	+	+	CCONJ
ejpam-3838	205	33	uf3	uf3	ADJ
ejpam-3838	205	34	,	,	PUNCT
ejpam-3838	205	35	of	of	ADP
ejpam-3838	205	36	given	give	VERB
ejpam-3838	205	37	type	type	NOUN
ejpam-3838	205	38	for	for	ADP
ejpam-3838	205	39	small	small	ADJ
ejpam-3838	205	40	lengths	length	NOUN
ejpam-3838	205	41	.	.	PUNCT
ejpam-3838	206	1	note	note	VERB
ejpam-3838	206	2	that	that	SCONJ
ejpam-3838	206	3	two	two	NUM
ejpam-3838	206	4	codes	code	NOUN
ejpam-3838	206	5	over	over	ADP
ejpam-3838	206	6	f2	f2	PROPN
ejpam-3838	206	7	+	+	CCONJ
ejpam-3838	206	8	uf2	uf2	NOUN
ejpam-3838	206	9	are	be	AUX
ejpam-3838	206	10	equivalent	equivalent	ADJ
ejpam-3838	206	11	if	if	SCONJ
ejpam-3838	206	12	one	one	PRON
ejpam-3838	206	13	can	can	AUX
ejpam-3838	206	14	be	be	AUX
ejpam-3838	206	15	obtained	obtain	VERB
ejpam-3838	206	16	from	from	ADP
ejpam-3838	206	17	the	the	DET
ejpam-3838	206	18	other	other	ADJ
ejpam-3838	206	19	by	by	ADP
ejpam-3838	206	20	permuting	permute	VERB
ejpam-3838	206	21	the	the	DET
ejpam-3838	206	22	coordinates	coordinate	NOUN
ejpam-3838	206	23	and	and	CCONJ
ejpam-3838	206	24	(	(	PUNCT
ejpam-3838	206	25	if	if	SCONJ
ejpam-3838	206	26	necessary	necessary	ADJ
ejpam-3838	206	27	)	)	PUNCT
ejpam-3838	206	28	multiplying	multiply	VERB
ejpam-3838	206	29	certain	certain	ADJ
ejpam-3838	206	30	coordinates	coordinate	NOUN
ejpam-3838	206	31	by	by	ADP
ejpam-3838	206	32	1	1	NUM
ejpam-3838	206	33	+	+	CCONJ
ejpam-3838	206	34	u.	u.	NOUN
ejpam-3838	206	35	on	on	ADP
ejpam-3838	206	36	the	the	DET
ejpam-3838	206	37	other	other	ADJ
ejpam-3838	206	38	hand	hand	NOUN
ejpam-3838	206	39	,	,	PUNCT
ejpam-3838	206	40	two	two	NUM
ejpam-3838	206	41	euclidean	euclidean	ADJ
ejpam-3838	206	42	self	self	NOUN
ejpam-3838	206	43	-	-	PUNCT
ejpam-3838	206	44	orthogonal	orthogonal	ADJ
ejpam-3838	206	45	codes	code	NOUN
ejpam-3838	206	46	over	over	ADP
ejpam-3838	206	47	f3+uf3	f3+uf3	NOUN
ejpam-3838	206	48	are	be	AUX
ejpam-3838	206	49	equivalent	equivalent	ADJ
ejpam-3838	206	50	if	if	SCONJ
ejpam-3838	206	51	one	one	PRON
ejpam-3838	206	52	can	can	AUX
ejpam-3838	206	53	be	be	AUX
ejpam-3838	206	54	obtained	obtain	VERB
ejpam-3838	206	55	from	from	ADP
ejpam-3838	206	56	the	the	DET
ejpam-3838	206	57	other	other	ADJ
ejpam-3838	206	58	by	by	ADP
ejpam-3838	206	59	permuting	permute	VERB
ejpam-3838	206	60	the	the	DET
ejpam-3838	206	61	coordinates	coordinate	NOUN
ejpam-3838	206	62	and	and	CCONJ
ejpam-3838	206	63	(	(	PUNCT
ejpam-3838	206	64	if	if	SCONJ
ejpam-3838	206	65	necessary	necessary	ADJ
ejpam-3838	206	66	)	)	PUNCT
ejpam-3838	206	67	multiplying	multiply	VERB
ejpam-3838	206	68	certain	certain	ADJ
ejpam-3838	206	69	coordinates	coordinate	NOUN
ejpam-3838	206	70	by	by	ADP
ejpam-3838	206	71	2	2	NUM
ejpam-3838	206	72	,	,	PUNCT
ejpam-3838	206	73	and	and	CCONJ
ejpam-3838	206	74	two	two	NUM
ejpam-3838	206	75	hermitian	hermitian	ADJ
ejpam-3838	206	76	self	self	NOUN
ejpam-3838	206	77	-	-	PUNCT
ejpam-3838	206	78	orthogonal	orthogonal	ADJ
ejpam-3838	206	79	codes	code	NOUN
ejpam-3838	206	80	over	over	ADP
ejpam-3838	206	81	f3	f3	PROPN
ejpam-3838	206	82	+	+	CCONJ
ejpam-3838	206	83	uf3	uf3	NOUN
ejpam-3838	206	84	are	be	AUX
ejpam-3838	206	85	equivalent	equivalent	ADJ
ejpam-3838	206	86	if	if	SCONJ
ejpam-3838	206	87	one	one	PRON
ejpam-3838	206	88	can	can	AUX
ejpam-3838	206	89	be	be	AUX
ejpam-3838	206	90	obtained	obtain	VERB
ejpam-3838	206	91	from	from	ADP
ejpam-3838	206	92	the	the	DET
ejpam-3838	206	93	other	other	ADJ
ejpam-3838	206	94	by	by	ADP
ejpam-3838	206	95	permuting	permute	VERB
ejpam-3838	206	96	the	the	DET
ejpam-3838	206	97	coordinates	coordinate	NOUN
ejpam-3838	206	98	and	and	CCONJ
ejpam-3838	206	99	(	(	PUNCT
ejpam-3838	206	100	if	if	SCONJ
ejpam-3838	206	101	necessary	necessary	ADJ
ejpam-3838	206	102	)	)	PUNCT
ejpam-3838	206	103	multiplying	multiply	VERB
ejpam-3838	206	104	certain	certain	ADJ
ejpam-3838	206	105	coordinates	coordinate	NOUN
ejpam-3838	206	106	by	by	ADP
ejpam-3838	206	107	r	r	NOUN
ejpam-3838	206	108	,	,	PUNCT
ejpam-3838	206	109	where	where	SCONJ
ejpam-3838	206	110	r	r	NOUN
ejpam-3838	206	111	∈	∈	PROPN
ejpam-3838	206	112	{	{	PUNCT
ejpam-3838	206	113	2	2	NUM
ejpam-3838	206	114	,	,	PUNCT
ejpam-3838	206	115	1	1	NUM
ejpam-3838	206	116	+	+	NUM
ejpam-3838	206	117	u	u	NOUN
ejpam-3838	206	118	,	,	PUNCT
ejpam-3838	206	119	1	1	NUM
ejpam-3838	206	120	+	+	CCONJ
ejpam-3838	206	121	2u	2u	NOUN
ejpam-3838	206	122	,	,	PUNCT
ejpam-3838	206	123	2	2	NUM
ejpam-3838	206	124	+	+	SYM
ejpam-3838	206	125	u	u	NOUN
ejpam-3838	206	126	,	,	PUNCT
ejpam-3838	206	127	2	2	NUM
ejpam-3838	206	128	+	+	CCONJ
ejpam-3838	206	129	2u	2u	NOUN
ejpam-3838	206	130	}	}	PUNCT
ejpam-3838	206	131	.	.	PUNCT
ejpam-3838	207	1	to	to	PART
ejpam-3838	207	2	illustrate	illustrate	VERB
ejpam-3838	207	3	,	,	PUNCT
ejpam-3838	207	4	we	we	PRON
ejpam-3838	207	5	classify	classify	VERB
ejpam-3838	207	6	euclidean	euclidean	ADJ
ejpam-3838	207	7	self	self	NOUN
ejpam-3838	207	8	-	-	PUNCT
ejpam-3838	207	9	orthogonal	orthogonal	ADJ
ejpam-3838	207	10	codes	code	NOUN
ejpam-3838	207	11	over	over	ADP
ejpam-3838	207	12	f3	f3	PROPN
ejpam-3838	207	13	+	+	PROPN
ejpam-3838	207	14	uf3	uf3	NOUN
ejpam-3838	207	15	of	of	ADP
ejpam-3838	207	16	length	length	NOUN
ejpam-3838	207	17	4	4	NUM
ejpam-3838	207	18	and	and	CCONJ
ejpam-3838	207	19	type	type	NOUN
ejpam-3838	207	20	{	{	PUNCT
ejpam-3838	207	21	2	2	NUM
ejpam-3838	207	22	,	,	PUNCT
ejpam-3838	207	23	0	0	NUM
ejpam-3838	207	24	}	}	PUNCT
ejpam-3838	207	25	.	.	PUNCT
ejpam-3838	208	1	let	let	VERB
ejpam-3838	208	2	c1	c1	PROPN
ejpam-3838	208	3	and	and	CCONJ
ejpam-3838	208	4	c2	c2	PROPN
ejpam-3838	208	5	be	be	AUX
ejpam-3838	208	6	inequivalent	inequivalent	ADJ
ejpam-3838	208	7	euclidean	euclidean	ADJ
ejpam-3838	208	8	self	self	NOUN
ejpam-3838	208	9	-	-	PUNCT
ejpam-3838	208	10	orthogonal	orthogonal	ADJ
ejpam-3838	208	11	codes	code	NOUN
ejpam-3838	208	12	over	over	ADP
ejpam-3838	208	13	f3	f3	PROPN
ejpam-3838	208	14	+	+	CCONJ
ejpam-3838	208	15	uf3	uf3	NOUN
ejpam-3838	208	16	of	of	ADP
ejpam-3838	208	17	length	length	NOUN
ejpam-3838	208	18	4	4	NUM
ejpam-3838	208	19	and	and	CCONJ
ejpam-3838	208	20	type	type	NOUN
ejpam-3838	208	21	{	{	PUNCT
ejpam-3838	208	22	2	2	NUM
ejpam-3838	208	23	,	,	PUNCT
ejpam-3838	208	24	0	0	NUM
ejpam-3838	208	25	}	}	PUNCT
ejpam-3838	208	26	with	with	ADP
ejpam-3838	208	27	generator	generator	NOUN
ejpam-3838	208	28	matrices	matrix	NOUN
ejpam-3838	208	29	[	[	PUNCT
ejpam-3838	208	30	1	1	NUM
ejpam-3838	208	31	0	0	NUM
ejpam-3838	208	32	2	2	NUM
ejpam-3838	208	33	2	2	NUM
ejpam-3838	208	34	0	0	NUM
ejpam-3838	208	35	1	1	NUM
ejpam-3838	208	36	2	2	NUM
ejpam-3838	208	37	1	1	NUM
ejpam-3838	208	38	]	]	PUNCT
ejpam-3838	208	39	and	and	CCONJ
ejpam-3838	208	40	[	[	PUNCT
ejpam-3838	208	41	1	1	NUM
ejpam-3838	208	42	0	0	NUM
ejpam-3838	208	43	2	2	NUM
ejpam-3838	208	44	+	+	CCONJ
ejpam-3838	208	45	2u	2u	NOUN
ejpam-3838	208	46	2	2	NUM
ejpam-3838	208	47	+	+	CCONJ
ejpam-3838	208	48	u	u	NOUN
ejpam-3838	208	49	0	0	NUM
ejpam-3838	208	50	1	1	NUM
ejpam-3838	208	51	2	2	NUM
ejpam-3838	208	52	+	+	NUM
ejpam-3838	208	53	u	u	NOUN
ejpam-3838	208	54	1	1	NUM
ejpam-3838	208	55	+	+	NUM
ejpam-3838	208	56	u	u	NOUN
ejpam-3838	208	57	]	]	PUNCT
ejpam-3838	208	58	,	,	PUNCT
ejpam-3838	208	59	respectively	respectively	ADV
ejpam-3838	208	60	.	.	PUNCT
ejpam-3838	209	1	the	the	DET
ejpam-3838	209	2	order	order	NOUN
ejpam-3838	209	3	of	of	ADP
ejpam-3838	209	4	their	their	PRON
ejpam-3838	209	5	automorphism	automorphism	NOUN
ejpam-3838	209	6	groups	group	NOUN
ejpam-3838	209	7	are	be	AUX
ejpam-3838	209	8	48	48	NUM
ejpam-3838	209	9	and	and	CCONJ
ejpam-3838	209	10	24	24	NUM
ejpam-3838	209	11	,	,	PUNCT
ejpam-3838	209	12	respectively	respectively	ADV
ejpam-3838	209	13	.	.	PUNCT
ejpam-3838	210	1	hence	hence	ADV
ejpam-3838	210	2	,	,	PUNCT
ejpam-3838	210	3	2∑	2∑	X
ejpam-3838	210	4	j=1	j=1	PROPN
ejpam-3838	210	5	|e4|	|e4|	NOUN
ejpam-3838	211	1	|aut(cj)|	|aut(cj)|	X
ejpam-3838	212	1	=	=	SYM
ejpam-3838	212	2	24	24	NUM
ejpam-3838	212	3	·	·	SYM
ejpam-3838	212	4	4	4	NUM
ejpam-3838	212	5	!	!	X
ejpam-3838	212	6	48	48	NUM
ejpam-3838	213	1	+	+	SYM
ejpam-3838	213	2	24	24	NUM
ejpam-3838	213	3	·	·	SYM
ejpam-3838	213	4	4	4	NUM
ejpam-3838	213	5	!	!	SYM
ejpam-3838	213	6	24	24	NUM
ejpam-3838	213	7	=	=	SYM
ejpam-3838	213	8	8	8	NUM
ejpam-3838	213	9	+	+	NUM
ejpam-3838	213	10	16	16	NUM
ejpam-3838	213	11	=	=	SYM
ejpam-3838	213	12	24	24	NUM
ejpam-3838	213	13	.	.	PUNCT
ejpam-3838	213	14	from	from	ADP
ejpam-3838	213	15	theorem	theorem	ADJ
ejpam-3838	213	16	2	2	NUM
ejpam-3838	213	17	,	,	PUNCT
ejpam-3838	213	18	m3(4	m3(4	PROPN
ejpam-3838	213	19	,	,	PUNCT
ejpam-3838	213	20	2	2	NUM
ejpam-3838	213	21	,	,	PUNCT
ejpam-3838	213	22	0)e	0)e	NOUN
ejpam-3838	213	23	=	=	SYM
ejpam-3838	213	24	σ3(4	σ3(4	NOUN
ejpam-3838	213	25	,	,	PUNCT
ejpam-3838	213	26	2	2	NUM
ejpam-3838	213	27	)	)	PUNCT
ejpam-3838	213	28	[	[	PUNCT
ejpam-3838	213	29	4−	4−	NOUN
ejpam-3838	213	30	2	2	NUM
ejpam-3838	213	31	·	·	SYM
ejpam-3838	213	32	2	2	NUM
ejpam-3838	213	33	0	0	NUM
ejpam-3838	213	34	]	]	SYM
ejpam-3838	213	35	3	3	NUM
ejpam-3838	213	36	32(8−6−0−1)/2	32(8−6−0−1)/2	NUM
ejpam-3838	213	37	=	=	SYM
ejpam-3838	213	38	8	8	NUM
ejpam-3838	213	39	·	·	SYM
ejpam-3838	213	40	1	1	NUM
ejpam-3838	213	41	·	·	SYM
ejpam-3838	213	42	3	3	NUM
ejpam-3838	213	43	=	=	SYM
ejpam-3838	213	44	24	24	NUM
ejpam-3838	213	45	.	.	PUNCT
ejpam-3838	214	1	l.e	l.e	PROPN
ejpam-3838	214	2	.	.	PROPN
ejpam-3838	214	3	galvez	galvez	PROPN
ejpam-3838	214	4	,	,	PUNCT
ejpam-3838	214	5	r.a	r.a	PROPN
ejpam-3838	214	6	.	.	PROPN
ejpam-3838	214	7	betty	betty	PROPN
ejpam-3838	214	8	,	,	PUNCT
ejpam-3838	214	9	f.	f.	PROPN
ejpam-3838	214	10	nemenzo	nemenzo	PROPN
ejpam-3838	214	11	/	/	SYM
ejpam-3838	214	12	eur	eur	NOUN
ejpam-3838	214	13	.	.	PUNCT
ejpam-3838	215	1	j.	j.	PROPN
ejpam-3838	215	2	pure	pure	PROPN
ejpam-3838	215	3	appl	appl	PROPN
ejpam-3838	215	4	.	.	PROPN
ejpam-3838	215	5	math	math	PROPN
ejpam-3838	215	6	,	,	PUNCT
ejpam-3838	215	7	13	13	NUM
ejpam-3838	215	8	(	(	PUNCT
ejpam-3838	215	9	4	4	NUM
ejpam-3838	215	10	)	)	PUNCT
ejpam-3838	215	11	(	(	PUNCT
ejpam-3838	215	12	2020	2020	NUM
ejpam-3838	215	13	)	)	PUNCT
ejpam-3838	215	14	,	,	PUNCT
ejpam-3838	215	15	873	873	NUM
ejpam-3838	215	16	-	-	SYM
ejpam-3838	215	17	892	892	NUM
ejpam-3838	215	18	882	882	NUM
ejpam-3838	215	19	therefore	therefore	ADV
ejpam-3838	215	20	,	,	PUNCT
ejpam-3838	215	21	there	there	PRON
ejpam-3838	215	22	are	be	VERB
ejpam-3838	215	23	two	two	NUM
ejpam-3838	215	24	euclidean	euclidean	ADJ
ejpam-3838	215	25	self	self	NOUN
ejpam-3838	215	26	-	-	PUNCT
ejpam-3838	215	27	orthogonal	orthogonal	ADJ
ejpam-3838	215	28	codes	code	NOUN
ejpam-3838	215	29	of	of	ADP
ejpam-3838	215	30	length	length	NOUN
ejpam-3838	215	31	4	4	NUM
ejpam-3838	215	32	and	and	CCONJ
ejpam-3838	215	33	type	type	NOUN
ejpam-3838	215	34	{	{	PUNCT
ejpam-3838	215	35	2	2	NUM
ejpam-3838	215	36	,	,	PUNCT
ejpam-3838	215	37	0	0	NUM
ejpam-3838	215	38	}	}	PUNCT
ejpam-3838	215	39	over	over	ADP
ejpam-3838	215	40	f3	f3	PROPN
ejpam-3838	215	41	+	+	CCONJ
ejpam-3838	215	42	uf3	uf3	ADJ
ejpam-3838	215	43	,	,	PUNCT
ejpam-3838	215	44	up	up	ADP
ejpam-3838	215	45	to	to	ADP
ejpam-3838	215	46	equivalence	equivalence	NOUN
ejpam-3838	215	47	.	.	PUNCT
ejpam-3838	216	1	we	we	PRON
ejpam-3838	216	2	note	note	VERB
ejpam-3838	216	3	that	that	SCONJ
ejpam-3838	216	4	euclidean	euclidean	VERB
ejpam-3838	216	5	self	self	NOUN
ejpam-3838	216	6	-	-	PUNCT
ejpam-3838	216	7	orthogonal	orthogonal	ADJ
ejpam-3838	216	8	and	and	CCONJ
ejpam-3838	216	9	hermitian	hermitian	ADJ
ejpam-3838	216	10	self	self	NOUN
ejpam-3838	216	11	-	-	PUNCT
ejpam-3838	216	12	orthogonal	orthogonal	ADJ
ejpam-3838	216	13	codes	code	NOUN
ejpam-3838	216	14	coincide	coincide	VERB
ejpam-3838	216	15	over	over	ADP
ejpam-3838	216	16	f2	f2	PROPN
ejpam-3838	216	17	+	+	CCONJ
ejpam-3838	216	18	uf2	uf2	NOUN
ejpam-3838	216	19	,	,	PUNCT
ejpam-3838	216	20	as	as	ADV
ejpam-3838	216	21	well	well	ADV
ejpam-3838	216	22	as	as	ADP
ejpam-3838	216	23	in	in	ADP
ejpam-3838	216	24	codes	code	NOUN
ejpam-3838	216	25	over	over	ADP
ejpam-3838	216	26	f3	f3	PROPN
ejpam-3838	216	27	+	+	CCONJ
ejpam-3838	216	28	uf3	uf3	NOUN
ejpam-3838	216	29	of	of	ADP
ejpam-3838	216	30	type	type	NOUN
ejpam-3838	216	31	{	{	PUNCT
ejpam-3838	216	32	0	0	NUM
ejpam-3838	216	33	,	,	PUNCT
ejpam-3838	216	34	k1	k1	NOUN
ejpam-3838	216	35	}	}	PUNCT
ejpam-3838	216	36	.	.	PUNCT
ejpam-3838	217	1	table	table	NOUN
ejpam-3838	217	2	1	1	NUM
ejpam-3838	217	3	gives	give	VERB
ejpam-3838	217	4	the	the	DET
ejpam-3838	217	5	number	number	NOUN
ejpam-3838	217	6	of	of	ADP
ejpam-3838	217	7	inequivalent	inequivalent	NOUN
ejpam-3838	217	8	euclidean	euclidean	ADJ
ejpam-3838	217	9	self	self	NOUN
ejpam-3838	217	10	-	-	PUNCT
ejpam-3838	217	11	orthogonal	orthogonal	ADJ
ejpam-3838	217	12	codes	code	NOUN
ejpam-3838	217	13	over	over	ADP
ejpam-3838	217	14	f2	f2	PROPN
ejpam-3838	217	15	+	+	CCONJ
ejpam-3838	217	16	uf2	uf2	NOUN
ejpam-3838	217	17	of	of	ADP
ejpam-3838	217	18	lengths	length	NOUN
ejpam-3838	217	19	2	2	NUM
ejpam-3838	217	20	up	up	ADP
ejpam-3838	217	21	to	to	ADP
ejpam-3838	217	22	7	7	NUM
ejpam-3838	217	23	,	,	PUNCT
ejpam-3838	217	24	while	while	SCONJ
ejpam-3838	217	25	table	table	NOUN
ejpam-3838	217	26	2	2	NUM
ejpam-3838	217	27	gives	give	VERB
ejpam-3838	217	28	the	the	DET
ejpam-3838	217	29	number	number	NOUN
ejpam-3838	217	30	of	of	ADP
ejpam-3838	217	31	inequivalent	inequivalent	ADJ
ejpam-3838	217	32	euclidean	euclidean	NOUN
ejpam-3838	217	33	and	and	CCONJ
ejpam-3838	217	34	hermitian	hermitian	ADJ
ejpam-3838	217	35	self	self	NOUN
ejpam-3838	217	36	-	-	PUNCT
ejpam-3838	217	37	orthogonal	orthogonal	ADJ
ejpam-3838	217	38	codes	code	NOUN
ejpam-3838	217	39	over	over	ADP
ejpam-3838	217	40	f3+uf3	f3+uf3	NOUN
ejpam-3838	217	41	of	of	ADP
ejpam-3838	217	42	lengths	length	NOUN
ejpam-3838	217	43	2	2	NUM
ejpam-3838	217	44	up	up	ADP
ejpam-3838	217	45	to	to	PART
ejpam-3838	217	46	6	6	NUM
ejpam-3838	217	47	,	,	PUNCT
ejpam-3838	217	48	for	for	ADP
ejpam-3838	217	49	each	each	DET
ejpam-3838	217	50	type	type	NOUN
ejpam-3838	217	51	.	.	PUNCT
ejpam-3838	218	1	note	note	VERB
ejpam-3838	218	2	that	that	SCONJ
ejpam-3838	218	3	the	the	DET
ejpam-3838	218	4	code	code	NOUN
ejpam-3838	218	5	of	of	ADP
ejpam-3838	218	6	length	length	NOUN
ejpam-3838	218	7	1	1	NUM
ejpam-3838	218	8	with	with	ADP
ejpam-3838	218	9	generator	generator	NOUN
ejpam-3838	218	10	matrix	matrix	NOUN
ejpam-3838	218	11	[	[	X
ejpam-3838	218	12	u	u	X
ejpam-3838	218	13	]	]	X
ejpam-3838	218	14	is	be	AUX
ejpam-3838	218	15	a	a	DET
ejpam-3838	218	16	euclidean	euclidean	ADJ
ejpam-3838	218	17	self	self	NOUN
ejpam-3838	218	18	-	-	PUNCT
ejpam-3838	218	19	orthogonal	orthogonal	ADJ
ejpam-3838	218	20	and	and	CCONJ
ejpam-3838	218	21	hermitian	hermitian	ADJ
ejpam-3838	218	22	self	self	NOUN
ejpam-3838	218	23	-	-	PUNCT
ejpam-3838	218	24	orthogonal	orthogonal	ADJ
ejpam-3838	218	25	code	code	NOUN
ejpam-3838	218	26	over	over	ADP
ejpam-3838	218	27	fq+ufq	fq+ufq	PROPN
ejpam-3838	218	28	.	.	PUNCT
ejpam-3838	219	1	therefore	therefore	ADV
ejpam-3838	219	2	,	,	PUNCT
ejpam-3838	219	3	there	there	PRON
ejpam-3838	219	4	is	be	VERB
ejpam-3838	219	5	a	a	DET
ejpam-3838	219	6	self	self	NOUN
ejpam-3838	219	7	-	-	PUNCT
ejpam-3838	219	8	orthogonal	orthogonal	ADJ
ejpam-3838	219	9	code	code	NOUN
ejpam-3838	219	10	for	for	ADP
ejpam-3838	219	11	any	any	DET
ejpam-3838	219	12	length	length	NOUN
ejpam-3838	219	13	n	n	CCONJ
ejpam-3838	219	14	,	,	PUNCT
ejpam-3838	219	15	since	since	SCONJ
ejpam-3838	219	16	one	one	PRON
ejpam-3838	219	17	can	can	AUX
ejpam-3838	219	18	just	just	ADV
ejpam-3838	219	19	form	form	VERB
ejpam-3838	219	20	a	a	DET
ejpam-3838	219	21	direct	direct	ADJ
ejpam-3838	219	22	sum	sum	NOUN
ejpam-3838	219	23	of	of	ADP
ejpam-3838	219	24	this	this	DET
ejpam-3838	219	25	length	length	NOUN
ejpam-3838	219	26	1	1	NUM
ejpam-3838	219	27	code	code	NOUN
ejpam-3838	219	28	.	.	PUNCT
ejpam-3838	220	1	our	our	PRON
ejpam-3838	220	2	classification	classification	NOUN
ejpam-3838	220	3	of	of	ADP
ejpam-3838	220	4	selforthogonal	selforthogonal	ADJ
ejpam-3838	220	5	codes	code	NOUN
ejpam-3838	220	6	over	over	ADP
ejpam-3838	220	7	f2+uf2	f2+uf2	NOUN
ejpam-3838	220	8	agrees	agree	VERB
ejpam-3838	220	9	with	with	ADP
ejpam-3838	220	10	the	the	DET
ejpam-3838	220	11	enumeration	enumeration	NOUN
ejpam-3838	220	12	in	in	ADP
ejpam-3838	220	13	[	[	X
ejpam-3838	220	14	5	5	NUM
ejpam-3838	220	15	]	]	PUNCT
ejpam-3838	220	16	for	for	ADP
ejpam-3838	220	17	self	self	NOUN
ejpam-3838	220	18	-	-	PUNCT
ejpam-3838	220	19	dual	dual	ADJ
ejpam-3838	220	20	codes	code	NOUN
ejpam-3838	220	21	(	(	PUNCT
ejpam-3838	220	22	codes	code	NOUN
ejpam-3838	220	23	of	of	ADP
ejpam-3838	220	24	type	type	NOUN
ejpam-3838	220	25	{	{	PUNCT
ejpam-3838	220	26	k0	k0	PROPN
ejpam-3838	220	27	,	,	PUNCT
ejpam-3838	220	28	n−	n−	NOUN
ejpam-3838	220	29	2k0	2k0	NUM
ejpam-3838	220	30	}	}	PUNCT
ejpam-3838	220	31	)	)	PUNCT
ejpam-3838	220	32	up	up	ADP
ejpam-3838	220	33	to	to	PART
ejpam-3838	220	34	length	length	NOUN
ejpam-3838	220	35	n	n	NOUN
ejpam-3838	220	36	=	=	SYM
ejpam-3838	220	37	7	7	X
ejpam-3838	221	1	.	.	PUNCT
ejpam-3838	221	2	generators	generator	NOUN
ejpam-3838	221	3	and	and	CCONJ
ejpam-3838	221	4	the	the	DET
ejpam-3838	221	5	order	order	NOUN
ejpam-3838	221	6	of	of	ADP
ejpam-3838	221	7	the	the	DET
ejpam-3838	221	8	automorphism	automorphism	NOUN
ejpam-3838	221	9	group	group	NOUN
ejpam-3838	221	10	of	of	ADP
ejpam-3838	221	11	each	each	DET
ejpam-3838	221	12	code	code	NOUN
ejpam-3838	221	13	in	in	ADP
ejpam-3838	221	14	table	table	NOUN
ejpam-3838	221	15	1	1	NUM
ejpam-3838	221	16	and	and	CCONJ
ejpam-3838	221	17	table	table	NOUN
ejpam-3838	221	18	2	2	NUM
ejpam-3838	221	19	may	may	AUX
ejpam-3838	221	20	be	be	AUX
ejpam-3838	221	21	requested	request	VERB
ejpam-3838	221	22	by	by	ADP
ejpam-3838	221	23	the	the	DET
ejpam-3838	221	24	interested	interested	ADJ
ejpam-3838	221	25	reader	reader	NOUN
ejpam-3838	221	26	from	from	ADP
ejpam-3838	221	27	the	the	DET
ejpam-3838	221	28	authors	author	NOUN
ejpam-3838	221	29	.	.	PUNCT
ejpam-3838	222	1	all	all	DET
ejpam-3838	222	2	computer	computer	NOUN
ejpam-3838	222	3	calculations	calculation	NOUN
ejpam-3838	222	4	in	in	ADP
ejpam-3838	222	5	this	this	DET
ejpam-3838	222	6	paper	paper	NOUN
ejpam-3838	222	7	were	be	AUX
ejpam-3838	222	8	done	do	VERB
ejpam-3838	222	9	with	with	ADP
ejpam-3838	222	10	the	the	DET
ejpam-3838	222	11	help	help	NOUN
ejpam-3838	222	12	of	of	ADP
ejpam-3838	222	13	magma[4	magma[4	NOUN
ejpam-3838	222	14	]	]	PUNCT
ejpam-3838	222	15	.	.	PUNCT
ejpam-3838	223	1	table	table	NOUN
ejpam-3838	223	2	1	1	NUM
ejpam-3838	223	3	:	:	PUNCT
ejpam-3838	223	4	the	the	DET
ejpam-3838	223	5	number	number	NOUN
ejpam-3838	223	6	of	of	ADP
ejpam-3838	223	7	inequivalent	inequivalent	NOUN
ejpam-3838	223	8	self	self	NOUN
ejpam-3838	223	9	-	-	PUNCT
ejpam-3838	223	10	orthogonal	orthogonal	ADJ
ejpam-3838	223	11	codes	code	NOUN
ejpam-3838	223	12	of	of	ADP
ejpam-3838	223	13	lengths	length	NOUN
ejpam-3838	223	14	2	2	NUM
ejpam-3838	223	15	≤	≤	NOUN
ejpam-3838	223	16	n	n	PRON
ejpam-3838	223	17	≤	≤	NUM
ejpam-3838	223	18	7	7	NUM
ejpam-3838	223	19	over	over	ADP
ejpam-3838	223	20	f2	f2	PROPN
ejpam-3838	223	21	+	+	CCONJ
ejpam-3838	223	22	uf2	uf2	NOUN
ejpam-3838	223	23	{	{	PUNCT
ejpam-3838	223	24	n	n	X
ejpam-3838	223	25	,	,	PUNCT
ejpam-3838	223	26	k0	k0	PROPN
ejpam-3838	223	27	,	,	PUNCT
ejpam-3838	223	28	k1	k1	NOUN
ejpam-3838	223	29	}	}	PUNCT
ejpam-3838	223	30	number	number	NOUN
ejpam-3838	223	31	of	of	ADP
ejpam-3838	223	32	{	{	PUNCT
ejpam-3838	223	33	n	n	X
ejpam-3838	223	34	,	,	PUNCT
ejpam-3838	223	35	k0	k0	PROPN
ejpam-3838	223	36	,	,	PUNCT
ejpam-3838	223	37	k1	k1	NOUN
ejpam-3838	223	38	}	}	PUNCT
ejpam-3838	223	39	number	number	NOUN
ejpam-3838	223	40	of	of	ADP
ejpam-3838	223	41	{	{	PUNCT
ejpam-3838	223	42	n	n	X
ejpam-3838	223	43	,	,	PUNCT
ejpam-3838	223	44	k0	k0	PROPN
ejpam-3838	223	45	,	,	PUNCT
ejpam-3838	223	46	k1	k1	NOUN
ejpam-3838	223	47	}	}	PUNCT
ejpam-3838	223	48	number	number	NOUN
ejpam-3838	223	49	of	of	ADP
ejpam-3838	223	50	codes	code	NOUN
ejpam-3838	223	51	codes	code	NOUN
ejpam-3838	223	52	codes	code	NOUN
ejpam-3838	223	53	{	{	PUNCT
ejpam-3838	223	54	2	2	NUM
ejpam-3838	223	55	,	,	PUNCT
ejpam-3838	223	56	1	1	NUM
ejpam-3838	223	57	,	,	PUNCT
ejpam-3838	223	58	0	0	NUM
ejpam-3838	223	59	}	}	SYM
ejpam-3838	223	60	1	1	NUM
ejpam-3838	223	61	{	{	PUNCT
ejpam-3838	223	62	5	5	NUM
ejpam-3838	223	63	,	,	PUNCT
ejpam-3838	223	64	2	2	NUM
ejpam-3838	223	65	,	,	PUNCT
ejpam-3838	223	66	1	1	NUM
ejpam-3838	223	67	}	}	SYM
ejpam-3838	223	68	2	2	NUM
ejpam-3838	223	69	{	{	PUNCT
ejpam-3838	223	70	6	6	NUM
ejpam-3838	223	71	,	,	PUNCT
ejpam-3838	223	72	0	0	NUM
ejpam-3838	223	73	,	,	PUNCT
ejpam-3838	223	74	6	6	NUM
ejpam-3838	223	75	}	}	SYM
ejpam-3838	223	76	1	1	NUM
ejpam-3838	223	77	{	{	PUNCT
ejpam-3838	223	78	2	2	NUM
ejpam-3838	223	79	,	,	PUNCT
ejpam-3838	223	80	0	0	NUM
ejpam-3838	223	81	,	,	PUNCT
ejpam-3838	223	82	1	1	NUM
ejpam-3838	223	83	}	}	SYM
ejpam-3838	223	84	2	2	NUM
ejpam-3838	223	85	{	{	PUNCT
ejpam-3838	223	86	5	5	NUM
ejpam-3838	223	87	,	,	PUNCT
ejpam-3838	223	88	0	0	NUM
ejpam-3838	223	89	,	,	PUNCT
ejpam-3838	223	90	1	1	NUM
ejpam-3838	223	91	}	}	SYM
ejpam-3838	223	92	5	5	NUM
ejpam-3838	223	93	{	{	PUNCT
ejpam-3838	223	94	7	7	NUM
ejpam-3838	223	95	,	,	PUNCT
ejpam-3838	223	96	1	1	NUM
ejpam-3838	223	97	,	,	PUNCT
ejpam-3838	223	98	0	0	NUM
ejpam-3838	223	99	}	}	SYM
ejpam-3838	223	100	12	12	NUM
ejpam-3838	223	101	{	{	PUNCT
ejpam-3838	223	102	2	2	NUM
ejpam-3838	223	103	,	,	PUNCT
ejpam-3838	223	104	0	0	NUM
ejpam-3838	223	105	,	,	PUNCT
ejpam-3838	223	106	2	2	NUM
ejpam-3838	223	107	}	}	SYM
ejpam-3838	223	108	1	1	NUM
ejpam-3838	223	109	{	{	PUNCT
ejpam-3838	223	110	5	5	NUM
ejpam-3838	223	111	,	,	PUNCT
ejpam-3838	223	112	0	0	NUM
ejpam-3838	223	113	,	,	PUNCT
ejpam-3838	223	114	2	2	NUM
ejpam-3838	223	115	}	}	SYM
ejpam-3838	223	116	10	10	NUM
ejpam-3838	223	117	{	{	PUNCT
ejpam-3838	223	118	7	7	NUM
ejpam-3838	223	119	,	,	PUNCT
ejpam-3838	223	120	1	1	NUM
ejpam-3838	223	121	,	,	PUNCT
ejpam-3838	223	122	1	1	NUM
ejpam-3838	223	123	}	}	SYM
ejpam-3838	223	124	54	54	NUM
ejpam-3838	223	125	{	{	PUNCT
ejpam-3838	223	126	3	3	NUM
ejpam-3838	223	127	,	,	PUNCT
ejpam-3838	223	128	1	1	NUM
ejpam-3838	223	129	,	,	PUNCT
ejpam-3838	223	130	0	0	NUM
ejpam-3838	223	131	}	}	SYM
ejpam-3838	223	132	2	2	NUM
ejpam-3838	223	133	{	{	PUNCT
ejpam-3838	223	134	5	5	NUM
ejpam-3838	223	135	,	,	PUNCT
ejpam-3838	223	136	0	0	NUM
ejpam-3838	223	137	,	,	PUNCT
ejpam-3838	223	138	3	3	NUM
ejpam-3838	223	139	}	}	SYM
ejpam-3838	223	140	10	10	NUM
ejpam-3838	223	141	{	{	PUNCT
ejpam-3838	223	142	7	7	NUM
ejpam-3838	223	143	,	,	PUNCT
ejpam-3838	223	144	1	1	NUM
ejpam-3838	223	145	,	,	PUNCT
ejpam-3838	223	146	2	2	NUM
ejpam-3838	223	147	}	}	SYM
ejpam-3838	223	148	100	100	NUM
ejpam-3838	223	149	{	{	PUNCT
ejpam-3838	223	150	3	3	NUM
ejpam-3838	223	151	,	,	PUNCT
ejpam-3838	223	152	1	1	NUM
ejpam-3838	223	153	,	,	PUNCT
ejpam-3838	223	154	1	1	NUM
ejpam-3838	223	155	}	}	SYM
ejpam-3838	223	156	1	1	NUM
ejpam-3838	223	157	{	{	PUNCT
ejpam-3838	223	158	5	5	NUM
ejpam-3838	223	159	,	,	PUNCT
ejpam-3838	223	160	0	0	NUM
ejpam-3838	223	161	,	,	PUNCT
ejpam-3838	223	162	4	4	NUM
ejpam-3838	223	163	}	}	SYM
ejpam-3838	223	164	5	5	NUM
ejpam-3838	223	165	{	{	PUNCT
ejpam-3838	223	166	7	7	NUM
ejpam-3838	223	167	,	,	PUNCT
ejpam-3838	223	168	1	1	NUM
ejpam-3838	223	169	,	,	PUNCT
ejpam-3838	223	170	3	3	X
ejpam-3838	223	171	}	}	SYM
ejpam-3838	223	172	73	73	NUM
ejpam-3838	223	173	{	{	PUNCT
ejpam-3838	223	174	3	3	NUM
ejpam-3838	223	175	,	,	PUNCT
ejpam-3838	223	176	0	0	NUM
ejpam-3838	223	177	,	,	PUNCT
ejpam-3838	223	178	1	1	NUM
ejpam-3838	223	179	}	}	SYM
ejpam-3838	223	180	3	3	NUM
ejpam-3838	223	181	{	{	PUNCT
ejpam-3838	223	182	5	5	NUM
ejpam-3838	223	183	,	,	PUNCT
ejpam-3838	223	184	0	0	NUM
ejpam-3838	223	185	,	,	PUNCT
ejpam-3838	223	186	5	5	NUM
ejpam-3838	223	187	}	}	SYM
ejpam-3838	223	188	1	1	NUM
ejpam-3838	223	189	{	{	PUNCT
ejpam-3838	223	190	7	7	NUM
ejpam-3838	223	191	,	,	PUNCT
ejpam-3838	223	192	1	1	NUM
ejpam-3838	223	193	,	,	PUNCT
ejpam-3838	223	194	4	4	NUM
ejpam-3838	223	195	}	}	SYM
ejpam-3838	223	196	24	24	NUM
ejpam-3838	223	197	{	{	PUNCT
ejpam-3838	223	198	3	3	NUM
ejpam-3838	223	199	,	,	PUNCT
ejpam-3838	223	200	0	0	NUM
ejpam-3838	223	201	,	,	PUNCT
ejpam-3838	223	202	2	2	NUM
ejpam-3838	223	203	}	}	SYM
ejpam-3838	223	204	3	3	NUM
ejpam-3838	223	205	{	{	PUNCT
ejpam-3838	223	206	6	6	NUM
ejpam-3838	223	207	,	,	PUNCT
ejpam-3838	223	208	1	1	NUM
ejpam-3838	223	209	,	,	PUNCT
ejpam-3838	223	210	0	0	NUM
ejpam-3838	223	211	}	}	SYM
ejpam-3838	223	212	9	9	NUM
ejpam-3838	223	213	{	{	PUNCT
ejpam-3838	223	214	7	7	NUM
ejpam-3838	223	215	,	,	PUNCT
ejpam-3838	223	216	1	1	NUM
ejpam-3838	223	217	,	,	PUNCT
ejpam-3838	223	218	5	5	NUM
ejpam-3838	223	219	}	}	SYM
ejpam-3838	223	220	3	3	NUM
ejpam-3838	223	221	{	{	PUNCT
ejpam-3838	223	222	3	3	NUM
ejpam-3838	223	223	,	,	PUNCT
ejpam-3838	223	224	0	0	NUM
ejpam-3838	223	225	,	,	PUNCT
ejpam-3838	223	226	3	3	X
ejpam-3838	223	227	}	}	SYM
ejpam-3838	223	228	1	1	NUM
ejpam-3838	223	229	{	{	PUNCT
ejpam-3838	223	230	6	6	NUM
ejpam-3838	223	231	,	,	PUNCT
ejpam-3838	223	232	1	1	NUM
ejpam-3838	223	233	,	,	PUNCT
ejpam-3838	223	234	1	1	NUM
ejpam-3838	223	235	}	}	SYM
ejpam-3838	223	236	29	29	NUM
ejpam-3838	223	237	{	{	PUNCT
ejpam-3838	223	238	7	7	NUM
ejpam-3838	223	239	,	,	PUNCT
ejpam-3838	223	240	2	2	NUM
ejpam-3838	223	241	,	,	PUNCT
ejpam-3838	223	242	0	0	NUM
ejpam-3838	223	243	}	}	SYM
ejpam-3838	223	244	43	43	NUM
ejpam-3838	223	245	{	{	PUNCT
ejpam-3838	223	246	4	4	NUM
ejpam-3838	223	247	,	,	PUNCT
ejpam-3838	223	248	1	1	NUM
ejpam-3838	223	249	,	,	PUNCT
ejpam-3838	223	250	0	0	NUM
ejpam-3838	223	251	}	}	SYM
ejpam-3838	223	252	4	4	NUM
ejpam-3838	223	253	{	{	SYM
ejpam-3838	223	254	6	6	NUM
ejpam-3838	223	255	,	,	PUNCT
ejpam-3838	223	256	1	1	NUM
ejpam-3838	223	257	,	,	PUNCT
ejpam-3838	223	258	2	2	NUM
ejpam-3838	223	259	}	}	SYM
ejpam-3838	223	260	36	36	NUM
ejpam-3838	223	261	{	{	PUNCT
ejpam-3838	223	262	7	7	NUM
ejpam-3838	223	263	,	,	PUNCT
ejpam-3838	223	264	2	2	NUM
ejpam-3838	223	265	,	,	PUNCT
ejpam-3838	223	266	1	1	NUM
ejpam-3838	223	267	}	}	SYM
ejpam-3838	223	268	74	74	NUM
ejpam-3838	223	269	{	{	PUNCT
ejpam-3838	223	270	4	4	NUM
ejpam-3838	223	271	,	,	PUNCT
ejpam-3838	223	272	1	1	NUM
ejpam-3838	223	273	,	,	PUNCT
ejpam-3838	223	274	1	1	NUM
ejpam-3838	223	275	}	}	SYM
ejpam-3838	223	276	5	5	NUM
ejpam-3838	223	277	{	{	PUNCT
ejpam-3838	223	278	6	6	NUM
ejpam-3838	223	279	,	,	PUNCT
ejpam-3838	223	280	1	1	NUM
ejpam-3838	223	281	,	,	PUNCT
ejpam-3838	223	282	3	3	NUM
ejpam-3838	223	283	}	}	SYM
ejpam-3838	223	284	16	16	NUM
ejpam-3838	223	285	{	{	PUNCT
ejpam-3838	223	286	7	7	NUM
ejpam-3838	223	287	,	,	PUNCT
ejpam-3838	223	288	2	2	NUM
ejpam-3838	223	289	,	,	PUNCT
ejpam-3838	223	290	2	2	NUM
ejpam-3838	223	291	}	}	SYM
ejpam-3838	223	292	40	40	NUM
ejpam-3838	223	293	{	{	PUNCT
ejpam-3838	223	294	4	4	NUM
ejpam-3838	223	295	,	,	PUNCT
ejpam-3838	223	296	1	1	NUM
ejpam-3838	223	297	,	,	PUNCT
ejpam-3838	223	298	2	2	NUM
ejpam-3838	223	299	}	}	SYM
ejpam-3838	223	300	2	2	NUM
ejpam-3838	223	301	{	{	PUNCT
ejpam-3838	223	302	6	6	NUM
ejpam-3838	223	303	,	,	PUNCT
ejpam-3838	223	304	1	1	NUM
ejpam-3838	223	305	,	,	PUNCT
ejpam-3838	223	306	4	4	NUM
ejpam-3838	223	307	}	}	SYM
ejpam-3838	223	308	3	3	NUM
ejpam-3838	223	309	{	{	PUNCT
ejpam-3838	223	310	7	7	NUM
ejpam-3838	223	311	,	,	PUNCT
ejpam-3838	223	312	2	2	NUM
ejpam-3838	223	313	,	,	PUNCT
ejpam-3838	223	314	3	3	NUM
ejpam-3838	223	315	}	}	SYM
ejpam-3838	223	316	5	5	NUM
ejpam-3838	223	317	{	{	PUNCT
ejpam-3838	223	318	4	4	NUM
ejpam-3838	223	319	,	,	PUNCT
ejpam-3838	223	320	2	2	NUM
ejpam-3838	223	321	,	,	PUNCT
ejpam-3838	223	322	0	0	NUM
ejpam-3838	223	323	}	}	SYM
ejpam-3838	223	324	2	2	NUM
ejpam-3838	223	325	{	{	PUNCT
ejpam-3838	223	326	6	6	NUM
ejpam-3838	223	327	,	,	PUNCT
ejpam-3838	223	328	2	2	NUM
ejpam-3838	223	329	,	,	PUNCT
ejpam-3838	223	330	0	0	NUM
ejpam-3838	223	331	}	}	SYM
ejpam-3838	223	332	19	19	NUM
ejpam-3838	223	333	{	{	PUNCT
ejpam-3838	223	334	7	7	NUM
ejpam-3838	223	335	,	,	PUNCT
ejpam-3838	223	336	3	3	NUM
ejpam-3838	223	337	,	,	PUNCT
ejpam-3838	223	338	0	0	NUM
ejpam-3838	223	339	}	}	SYM
ejpam-3838	223	340	22	22	NUM
ejpam-3838	223	341	{	{	PUNCT
ejpam-3838	223	342	4	4	NUM
ejpam-3838	223	343	,	,	PUNCT
ejpam-3838	223	344	0	0	NUM
ejpam-3838	223	345	,	,	PUNCT
ejpam-3838	223	346	1	1	NUM
ejpam-3838	223	347	}	}	SYM
ejpam-3838	223	348	4	4	NUM
ejpam-3838	223	349	{	{	SYM
ejpam-3838	223	350	6	6	NUM
ejpam-3838	223	351	,	,	PUNCT
ejpam-3838	223	352	2	2	NUM
ejpam-3838	223	353	,	,	PUNCT
ejpam-3838	223	354	1	1	NUM
ejpam-3838	223	355	}	}	SYM
ejpam-3838	223	356	18	18	NUM
ejpam-3838	223	357	{	{	PUNCT
ejpam-3838	223	358	7	7	NUM
ejpam-3838	223	359	,	,	PUNCT
ejpam-3838	223	360	3	3	NUM
ejpam-3838	223	361	,	,	PUNCT
ejpam-3838	223	362	1	1	NUM
ejpam-3838	223	363	}	}	SYM
ejpam-3838	223	364	5	5	NUM
ejpam-3838	223	365	{	{	PUNCT
ejpam-3838	223	366	4	4	NUM
ejpam-3838	223	367	,	,	PUNCT
ejpam-3838	223	368	0	0	NUM
ejpam-3838	223	369	,	,	PUNCT
ejpam-3838	223	370	2	2	NUM
ejpam-3838	223	371	}	}	SYM
ejpam-3838	223	372	6	6	NUM
ejpam-3838	223	373	{	{	PUNCT
ejpam-3838	223	374	6	6	NUM
ejpam-3838	223	375	,	,	PUNCT
ejpam-3838	223	376	2	2	NUM
ejpam-3838	223	377	,	,	PUNCT
ejpam-3838	223	378	2	2	NUM
ejpam-3838	223	379	}	}	SYM
ejpam-3838	223	380	5	5	NUM
ejpam-3838	223	381	{	{	PUNCT
ejpam-3838	223	382	7	7	NUM
ejpam-3838	223	383	,	,	PUNCT
ejpam-3838	223	384	0	0	NUM
ejpam-3838	223	385	,	,	PUNCT
ejpam-3838	223	386	1	1	NUM
ejpam-3838	223	387	}	}	SYM
ejpam-3838	223	388	7	7	NUM
ejpam-3838	223	389	{	{	PUNCT
ejpam-3838	223	390	4	4	NUM
ejpam-3838	223	391	,	,	PUNCT
ejpam-3838	223	392	0	0	NUM
ejpam-3838	223	393	,	,	PUNCT
ejpam-3838	223	394	3	3	X
ejpam-3838	223	395	}	}	SYM
ejpam-3838	223	396	4	4	NUM
ejpam-3838	223	397	{	{	SYM
ejpam-3838	223	398	6	6	NUM
ejpam-3838	223	399	,	,	PUNCT
ejpam-3838	223	400	3	3	NUM
ejpam-3838	223	401	,	,	PUNCT
ejpam-3838	223	402	0	0	NUM
ejpam-3838	223	403	}	}	SYM
ejpam-3838	223	404	4	4	NUM
ejpam-3838	223	405	{	{	SYM
ejpam-3838	223	406	7	7	NUM
ejpam-3838	223	407	,	,	PUNCT
ejpam-3838	223	408	0	0	NUM
ejpam-3838	223	409	,	,	PUNCT
ejpam-3838	223	410	2	2	NUM
ejpam-3838	223	411	}	}	SYM
ejpam-3838	223	412	23	23	NUM
ejpam-3838	223	413	{	{	PUNCT
ejpam-3838	223	414	4	4	NUM
ejpam-3838	223	415	,	,	PUNCT
ejpam-3838	223	416	0	0	NUM
ejpam-3838	223	417	,	,	PUNCT
ejpam-3838	223	418	4	4	NUM
ejpam-3838	223	419	}	}	SYM
ejpam-3838	223	420	1	1	NUM
ejpam-3838	223	421	{	{	PUNCT
ejpam-3838	223	422	6	6	NUM
ejpam-3838	223	423	,	,	PUNCT
ejpam-3838	223	424	0	0	NUM
ejpam-3838	223	425	,	,	PUNCT
ejpam-3838	223	426	1	1	NUM
ejpam-3838	223	427	}	}	SYM
ejpam-3838	223	428	6	6	NUM
ejpam-3838	223	429	{	{	PUNCT
ejpam-3838	223	430	7	7	NUM
ejpam-3838	223	431	,	,	PUNCT
ejpam-3838	223	432	0	0	NUM
ejpam-3838	223	433	,	,	PUNCT
ejpam-3838	223	434	3	3	X
ejpam-3838	223	435	}	}	SYM
ejpam-3838	223	436	43	43	NUM
ejpam-3838	223	437	{	{	PUNCT
ejpam-3838	223	438	5	5	NUM
ejpam-3838	223	439	,	,	PUNCT
ejpam-3838	223	440	1	1	NUM
ejpam-3838	223	441	,	,	PUNCT
ejpam-3838	223	442	0	0	NUM
ejpam-3838	223	443	}	}	SYM
ejpam-3838	223	444	6	6	NUM
ejpam-3838	223	445	{	{	SYM
ejpam-3838	223	446	6	6	NUM
ejpam-3838	223	447	,	,	PUNCT
ejpam-3838	223	448	0	0	NUM
ejpam-3838	223	449	,	,	PUNCT
ejpam-3838	223	450	2	2	NUM
ejpam-3838	223	451	}	}	SYM
ejpam-3838	223	452	16	16	NUM
ejpam-3838	223	453	{	{	PUNCT
ejpam-3838	223	454	7	7	NUM
ejpam-3838	223	455	,	,	PUNCT
ejpam-3838	223	456	0	0	NUM
ejpam-3838	223	457	,	,	PUNCT
ejpam-3838	223	458	4	4	NUM
ejpam-3838	223	459	}	}	SYM
ejpam-3838	223	460	43	43	NUM
ejpam-3838	223	461	{	{	PUNCT
ejpam-3838	223	462	5	5	NUM
ejpam-3838	223	463	,	,	PUNCT
ejpam-3838	223	464	1	1	NUM
ejpam-3838	223	465	,	,	PUNCT
ejpam-3838	223	466	1	1	NUM
ejpam-3838	223	467	}	}	SYM
ejpam-3838	223	468	13	13	NUM
ejpam-3838	223	469	{	{	PUNCT
ejpam-3838	223	470	6	6	NUM
ejpam-3838	223	471	,	,	PUNCT
ejpam-3838	223	472	0	0	NUM
ejpam-3838	223	473	,	,	PUNCT
ejpam-3838	223	474	3	3	X
ejpam-3838	223	475	}	}	SYM
ejpam-3838	223	476	22	22	NUM
ejpam-3838	223	477	{	{	PUNCT
ejpam-3838	223	478	7	7	NUM
ejpam-3838	223	479	,	,	PUNCT
ejpam-3838	223	480	0	0	NUM
ejpam-3838	223	481	,	,	PUNCT
ejpam-3838	223	482	5	5	NUM
ejpam-3838	223	483	}	}	SYM
ejpam-3838	223	484	23	23	NUM
ejpam-3838	223	485	{	{	PUNCT
ejpam-3838	223	486	5	5	NUM
ejpam-3838	223	487	,	,	PUNCT
ejpam-3838	223	488	1	1	NUM
ejpam-3838	223	489	,	,	PUNCT
ejpam-3838	223	490	2	2	NUM
ejpam-3838	223	491	}	}	SYM
ejpam-3838	223	492	10	10	NUM
ejpam-3838	223	493	{	{	PUNCT
ejpam-3838	223	494	6	6	NUM
ejpam-3838	223	495	,	,	PUNCT
ejpam-3838	223	496	0	0	NUM
ejpam-3838	223	497	,	,	PUNCT
ejpam-3838	223	498	4	4	NUM
ejpam-3838	223	499	}	}	SYM
ejpam-3838	223	500	16	16	NUM
ejpam-3838	223	501	{	{	PUNCT
ejpam-3838	223	502	7	7	NUM
ejpam-3838	223	503	,	,	PUNCT
ejpam-3838	223	504	0	0	NUM
ejpam-3838	223	505	,	,	PUNCT
ejpam-3838	223	506	6	6	NUM
ejpam-3838	223	507	}	}	SYM
ejpam-3838	223	508	7	7	NUM
ejpam-3838	223	509	{	{	PUNCT
ejpam-3838	223	510	5	5	NUM
ejpam-3838	223	511	,	,	PUNCT
ejpam-3838	223	512	1	1	NUM
ejpam-3838	223	513	,	,	PUNCT
ejpam-3838	223	514	3	3	NUM
ejpam-3838	223	515	}	}	SYM
ejpam-3838	223	516	2	2	NUM
ejpam-3838	223	517	{	{	PUNCT
ejpam-3838	223	518	6	6	NUM
ejpam-3838	223	519	,	,	PUNCT
ejpam-3838	223	520	0	0	NUM
ejpam-3838	223	521	,	,	PUNCT
ejpam-3838	223	522	5	5	NUM
ejpam-3838	223	523	}	}	SYM
ejpam-3838	223	524	6	6	NUM
ejpam-3838	223	525	{	{	PUNCT
ejpam-3838	223	526	7	7	NUM
ejpam-3838	223	527	,	,	PUNCT
ejpam-3838	223	528	0	0	NUM
ejpam-3838	223	529	,	,	PUNCT
ejpam-3838	223	530	7	7	NUM
ejpam-3838	223	531	}	}	SYM
ejpam-3838	223	532	1	1	NUM
ejpam-3838	223	533	{	{	PUNCT
ejpam-3838	223	534	5	5	NUM
ejpam-3838	223	535	,	,	PUNCT
ejpam-3838	223	536	2	2	NUM
ejpam-3838	223	537	,	,	PUNCT
ejpam-3838	223	538	0	0	NUM
ejpam-3838	223	539	}	}	SYM
ejpam-3838	223	540	6	6	NUM
ejpam-3838	223	541	l.e	l.e	PROPN
ejpam-3838	223	542	.	.	PROPN
ejpam-3838	223	543	galvez	galvez	PROPN
ejpam-3838	223	544	,	,	PUNCT
ejpam-3838	223	545	r.a	r.a	PROPN
ejpam-3838	223	546	.	.	PROPN
ejpam-3838	223	547	betty	betty	PROPN
ejpam-3838	223	548	,	,	PUNCT
ejpam-3838	223	549	f.	f.	PROPN
ejpam-3838	223	550	nemenzo	nemenzo	PROPN
ejpam-3838	223	551	/	/	SYM
ejpam-3838	223	552	eur	eur	NOUN
ejpam-3838	223	553	.	.	PUNCT
ejpam-3838	224	1	j.	j.	PROPN
ejpam-3838	224	2	pure	pure	PROPN
ejpam-3838	224	3	appl	appl	PROPN
ejpam-3838	224	4	.	.	PROPN
ejpam-3838	224	5	math	math	PROPN
ejpam-3838	224	6	,	,	PUNCT
ejpam-3838	224	7	13	13	NUM
ejpam-3838	224	8	(	(	PUNCT
ejpam-3838	224	9	4	4	NUM
ejpam-3838	224	10	)	)	PUNCT
ejpam-3838	224	11	(	(	PUNCT
ejpam-3838	224	12	2020	2020	NUM
ejpam-3838	224	13	)	)	PUNCT
ejpam-3838	224	14	,	,	PUNCT
ejpam-3838	224	15	873	873	NUM
ejpam-3838	224	16	-	-	NUM
ejpam-3838	224	17	892	892	NUM
ejpam-3838	224	18	883	883	NUM
ejpam-3838	224	19	table	table	NOUN
ejpam-3838	224	20	2	2	NUM
ejpam-3838	224	21	:	:	PUNCT
ejpam-3838	224	22	the	the	DET
ejpam-3838	224	23	number	number	NOUN
ejpam-3838	224	24	of	of	ADP
ejpam-3838	224	25	inequivalent	inequivalent	ADJ
ejpam-3838	224	26	euclidean	euclidean	NOUN
ejpam-3838	224	27	and	and	CCONJ
ejpam-3838	224	28	hermitian	hermitian	ADJ
ejpam-3838	224	29	self	self	NOUN
ejpam-3838	224	30	-	-	PUNCT
ejpam-3838	224	31	orthogonal	orthogonal	ADJ
ejpam-3838	224	32	codes	code	NOUN
ejpam-3838	224	33	of	of	ADP
ejpam-3838	224	34	lengths	length	NOUN
ejpam-3838	224	35	2	2	NUM
ejpam-3838	224	36	≤	≤	NOUN
ejpam-3838	224	37	n	n	PRON
ejpam-3838	224	38	≤	≤	NUM
ejpam-3838	224	39	6	6	NUM
ejpam-3838	224	40	over	over	ADP
ejpam-3838	224	41	f3	f3	PROPN
ejpam-3838	224	42	+	+	CCONJ
ejpam-3838	224	43	uf3	uf3	PROPN
ejpam-3838	224	44	{	{	PUNCT
ejpam-3838	224	45	n	n	X
ejpam-3838	224	46	,	,	PUNCT
ejpam-3838	224	47	k0	k0	PROPN
ejpam-3838	224	48	,	,	PUNCT
ejpam-3838	224	49	k1	k1	NOUN
ejpam-3838	224	50	}	}	PUNCT
ejpam-3838	224	51	number	number	NOUN
ejpam-3838	224	52	of	of	ADP
ejpam-3838	224	53	codes	code	NOUN
ejpam-3838	224	54	{	{	PUNCT
ejpam-3838	224	55	n	n	X
ejpam-3838	224	56	,	,	PUNCT
ejpam-3838	224	57	k0	k0	PROPN
ejpam-3838	224	58	,	,	PUNCT
ejpam-3838	224	59	k1	k1	NOUN
ejpam-3838	224	60	}	}	PUNCT
ejpam-3838	224	61	number	number	NOUN
ejpam-3838	224	62	of	of	ADP
ejpam-3838	224	63	codes	code	NOUN
ejpam-3838	224	64	euclidean	euclidean	PROPN
ejpam-3838	224	65	hermitian	hermitian	PROPN
ejpam-3838	224	66	euclidean	euclidean	PROPN
ejpam-3838	224	67	hermitian	hermitian	PROPN
ejpam-3838	224	68	{	{	PUNCT
ejpam-3838	224	69	2	2	NUM
ejpam-3838	224	70	,	,	PUNCT
ejpam-3838	224	71	1	1	NUM
ejpam-3838	224	72	,	,	PUNCT
ejpam-3838	224	73	0	0	NUM
ejpam-3838	224	74	}	}	SYM
ejpam-3838	224	75	0	0	NUM
ejpam-3838	224	76	0	0	NUM
ejpam-3838	224	77	{	{	PUNCT
ejpam-3838	224	78	5	5	NUM
ejpam-3838	224	79	,	,	PUNCT
ejpam-3838	224	80	2	2	NUM
ejpam-3838	224	81	,	,	PUNCT
ejpam-3838	224	82	1	1	NUM
ejpam-3838	224	83	}	}	SYM
ejpam-3838	224	84	2	2	NUM
ejpam-3838	224	85	1	1	NUM
ejpam-3838	224	86	{	{	PUNCT
ejpam-3838	224	87	2	2	NUM
ejpam-3838	224	88	,	,	PUNCT
ejpam-3838	224	89	0	0	NUM
ejpam-3838	224	90	,	,	PUNCT
ejpam-3838	224	91	1	1	NUM
ejpam-3838	224	92	}	}	SYM
ejpam-3838	224	93	2	2	NUM
ejpam-3838	224	94	2	2	NUM
ejpam-3838	224	95	{	{	PUNCT
ejpam-3838	224	96	5	5	NUM
ejpam-3838	224	97	,	,	PUNCT
ejpam-3838	224	98	0	0	NUM
ejpam-3838	224	99	,	,	PUNCT
ejpam-3838	224	100	1	1	NUM
ejpam-3838	224	101	}	}	SYM
ejpam-3838	224	102	5	5	NUM
ejpam-3838	224	103	5	5	NUM
ejpam-3838	224	104	{	{	PUNCT
ejpam-3838	224	105	2	2	NUM
ejpam-3838	224	106	,	,	PUNCT
ejpam-3838	224	107	0	0	NUM
ejpam-3838	224	108	,	,	PUNCT
ejpam-3838	224	109	2	2	NUM
ejpam-3838	224	110	}	}	SYM
ejpam-3838	224	111	1	1	NUM
ejpam-3838	224	112	1	1	NUM
ejpam-3838	224	113	{	{	PUNCT
ejpam-3838	224	114	5	5	NUM
ejpam-3838	224	115	,	,	PUNCT
ejpam-3838	224	116	0	0	NUM
ejpam-3838	224	117	,	,	PUNCT
ejpam-3838	224	118	2	2	NUM
ejpam-3838	224	119	}	}	SYM
ejpam-3838	224	120	12	12	NUM
ejpam-3838	224	121	12	12	NUM
ejpam-3838	224	122	{	{	PUNCT
ejpam-3838	224	123	3	3	NUM
ejpam-3838	224	124	,	,	PUNCT
ejpam-3838	224	125	1	1	NUM
ejpam-3838	224	126	,	,	PUNCT
ejpam-3838	224	127	0	0	NUM
ejpam-3838	224	128	}	}	SYM
ejpam-3838	224	129	2	2	NUM
ejpam-3838	224	130	1	1	NUM
ejpam-3838	224	131	{	{	PUNCT
ejpam-3838	224	132	5	5	NUM
ejpam-3838	224	133	,	,	PUNCT
ejpam-3838	224	134	0	0	NUM
ejpam-3838	224	135	,	,	PUNCT
ejpam-3838	224	136	3	3	NUM
ejpam-3838	224	137	}	}	SYM
ejpam-3838	224	138	12	12	NUM
ejpam-3838	224	139	12	12	NUM
ejpam-3838	224	140	{	{	PUNCT
ejpam-3838	224	141	3	3	NUM
ejpam-3838	224	142	,	,	PUNCT
ejpam-3838	224	143	1	1	NUM
ejpam-3838	224	144	,	,	PUNCT
ejpam-3838	224	145	1	1	NUM
ejpam-3838	224	146	}	}	SYM
ejpam-3838	224	147	1	1	NUM
ejpam-3838	224	148	1	1	NUM
ejpam-3838	224	149	{	{	PUNCT
ejpam-3838	224	150	5	5	NUM
ejpam-3838	224	151	,	,	PUNCT
ejpam-3838	224	152	0	0	NUM
ejpam-3838	224	153	,	,	PUNCT
ejpam-3838	224	154	4	4	NUM
ejpam-3838	224	155	}	}	SYM
ejpam-3838	224	156	5	5	NUM
ejpam-3838	224	157	5	5	NUM
ejpam-3838	224	158	{	{	PUNCT
ejpam-3838	224	159	3	3	NUM
ejpam-3838	224	160	,	,	PUNCT
ejpam-3838	224	161	0	0	NUM
ejpam-3838	224	162	,	,	PUNCT
ejpam-3838	224	163	1	1	NUM
ejpam-3838	224	164	}	}	SYM
ejpam-3838	224	165	3	3	NUM
ejpam-3838	224	166	3	3	NUM
ejpam-3838	224	167	{	{	PUNCT
ejpam-3838	224	168	5	5	NUM
ejpam-3838	224	169	,	,	PUNCT
ejpam-3838	224	170	0	0	NUM
ejpam-3838	224	171	,	,	PUNCT
ejpam-3838	224	172	5	5	NUM
ejpam-3838	224	173	}	}	SYM
ejpam-3838	224	174	1	1	NUM
ejpam-3838	224	175	1	1	NUM
ejpam-3838	224	176	{	{	PUNCT
ejpam-3838	224	177	3	3	NUM
ejpam-3838	224	178	,	,	PUNCT
ejpam-3838	224	179	0	0	NUM
ejpam-3838	224	180	,	,	PUNCT
ejpam-3838	224	181	2	2	NUM
ejpam-3838	224	182	}	}	SYM
ejpam-3838	224	183	3	3	NUM
ejpam-3838	224	184	3	3	NUM
ejpam-3838	224	185	{	{	PUNCT
ejpam-3838	224	186	6	6	NUM
ejpam-3838	224	187	,	,	PUNCT
ejpam-3838	224	188	1	1	NUM
ejpam-3838	224	189	,	,	PUNCT
ejpam-3838	224	190	0	0	NUM
ejpam-3838	224	191	}	}	SYM
ejpam-3838	224	192	12	12	NUM
ejpam-3838	224	193	5	5	NUM
ejpam-3838	224	194	{	{	PUNCT
ejpam-3838	224	195	3	3	NUM
ejpam-3838	224	196	,	,	PUNCT
ejpam-3838	224	197	0	0	NUM
ejpam-3838	224	198	,	,	PUNCT
ejpam-3838	224	199	3	3	NUM
ejpam-3838	224	200	}	}	SYM
ejpam-3838	224	201	1	1	NUM
ejpam-3838	224	202	1	1	NUM
ejpam-3838	224	203	{	{	PUNCT
ejpam-3838	224	204	6	6	NUM
ejpam-3838	224	205	,	,	PUNCT
ejpam-3838	224	206	1	1	NUM
ejpam-3838	224	207	,	,	PUNCT
ejpam-3838	224	208	1	1	NUM
ejpam-3838	224	209	}	}	SYM
ejpam-3838	224	210	57	57	NUM
ejpam-3838	224	211	27	27	NUM
ejpam-3838	224	212	{	{	PUNCT
ejpam-3838	224	213	4	4	NUM
ejpam-3838	224	214	,	,	PUNCT
ejpam-3838	224	215	1	1	NUM
ejpam-3838	224	216	,	,	PUNCT
ejpam-3838	224	217	0	0	NUM
ejpam-3838	224	218	}	}	SYM
ejpam-3838	224	219	4	4	NUM
ejpam-3838	224	220	2	2	NUM
ejpam-3838	224	221	{	{	PUNCT
ejpam-3838	224	222	6	6	NUM
ejpam-3838	224	223	,	,	PUNCT
ejpam-3838	224	224	1	1	NUM
ejpam-3838	224	225	,	,	PUNCT
ejpam-3838	224	226	2	2	NUM
ejpam-3838	224	227	}	}	PUNCT
ejpam-3838	224	228	64	64	NUM
ejpam-3838	224	229	34	34	NUM
ejpam-3838	224	230	{	{	PUNCT
ejpam-3838	224	231	4	4	NUM
ejpam-3838	224	232	,	,	PUNCT
ejpam-3838	224	233	1	1	NUM
ejpam-3838	224	234	,	,	PUNCT
ejpam-3838	224	235	1	1	NUM
ejpam-3838	224	236	}	}	SYM
ejpam-3838	224	237	6	6	NUM
ejpam-3838	224	238	4	4	NUM
ejpam-3838	224	239	{	{	PUNCT
ejpam-3838	224	240	6	6	NUM
ejpam-3838	224	241	,	,	PUNCT
ejpam-3838	224	242	1	1	NUM
ejpam-3838	224	243	,	,	PUNCT
ejpam-3838	224	244	3	3	NUM
ejpam-3838	224	245	}	}	SYM
ejpam-3838	224	246	20	20	NUM
ejpam-3838	224	247	13	13	NUM
ejpam-3838	224	248	{	{	PUNCT
ejpam-3838	224	249	4	4	NUM
ejpam-3838	224	250	,	,	PUNCT
ejpam-3838	224	251	1	1	NUM
ejpam-3838	224	252	,	,	PUNCT
ejpam-3838	224	253	2	2	NUM
ejpam-3838	224	254	}	}	SYM
ejpam-3838	224	255	1	1	NUM
ejpam-3838	224	256	1	1	NUM
ejpam-3838	224	257	{	{	PUNCT
ejpam-3838	224	258	6	6	NUM
ejpam-3838	224	259	,	,	PUNCT
ejpam-3838	224	260	1	1	NUM
ejpam-3838	224	261	,	,	PUNCT
ejpam-3838	224	262	4	4	NUM
ejpam-3838	224	263	}	}	SYM
ejpam-3838	224	264	2	2	NUM
ejpam-3838	224	265	2	2	NUM
ejpam-3838	224	266	{	{	PUNCT
ejpam-3838	224	267	4	4	NUM
ejpam-3838	224	268	,	,	PUNCT
ejpam-3838	224	269	2	2	NUM
ejpam-3838	224	270	,	,	PUNCT
ejpam-3838	224	271	0	0	NUM
ejpam-3838	224	272	}	}	SYM
ejpam-3838	224	273	2	2	NUM
ejpam-3838	224	274	1	1	NUM
ejpam-3838	224	275	{	{	PUNCT
ejpam-3838	224	276	6	6	NUM
ejpam-3838	224	277	,	,	PUNCT
ejpam-3838	224	278	2	2	NUM
ejpam-3838	224	279	,	,	PUNCT
ejpam-3838	224	280	0	0	NUM
ejpam-3838	224	281	}	}	SYM
ejpam-3838	224	282	22	22	NUM
ejpam-3838	224	283	8	8	NUM
ejpam-3838	224	284	{	{	PUNCT
ejpam-3838	224	285	4	4	NUM
ejpam-3838	224	286	,	,	PUNCT
ejpam-3838	224	287	0	0	NUM
ejpam-3838	224	288	,	,	PUNCT
ejpam-3838	224	289	1	1	NUM
ejpam-3838	224	290	}	}	SYM
ejpam-3838	224	291	4	4	NUM
ejpam-3838	224	292	4	4	NUM
ejpam-3838	224	293	{	{	PUNCT
ejpam-3838	224	294	6	6	NUM
ejpam-3838	224	295	,	,	PUNCT
ejpam-3838	224	296	2	2	NUM
ejpam-3838	224	297	,	,	PUNCT
ejpam-3838	224	298	1	1	NUM
ejpam-3838	224	299	}	}	SYM
ejpam-3838	224	300	18	18	NUM
ejpam-3838	224	301	9	9	NUM
ejpam-3838	224	302	{	{	PUNCT
ejpam-3838	224	303	4	4	NUM
ejpam-3838	224	304	,	,	PUNCT
ejpam-3838	224	305	0	0	NUM
ejpam-3838	224	306	,	,	PUNCT
ejpam-3838	224	307	2	2	NUM
ejpam-3838	224	308	}	}	SYM
ejpam-3838	224	309	7	7	NUM
ejpam-3838	224	310	7	7	NUM
ejpam-3838	224	311	{	{	PUNCT
ejpam-3838	224	312	6	6	NUM
ejpam-3838	224	313	,	,	PUNCT
ejpam-3838	224	314	2	2	NUM
ejpam-3838	224	315	,	,	PUNCT
ejpam-3838	224	316	2	2	NUM
ejpam-3838	224	317	}	}	SYM
ejpam-3838	224	318	4	4	NUM
ejpam-3838	224	319	3	3	NUM
ejpam-3838	224	320	{	{	PUNCT
ejpam-3838	224	321	4	4	NUM
ejpam-3838	224	322	,	,	PUNCT
ejpam-3838	224	323	0	0	NUM
ejpam-3838	224	324	,	,	PUNCT
ejpam-3838	224	325	3	3	X
ejpam-3838	224	326	}	}	SYM
ejpam-3838	224	327	4	4	NUM
ejpam-3838	224	328	4	4	NUM
ejpam-3838	224	329	{	{	PUNCT
ejpam-3838	224	330	6	6	NUM
ejpam-3838	224	331	,	,	PUNCT
ejpam-3838	224	332	3	3	NUM
ejpam-3838	224	333	,	,	PUNCT
ejpam-3838	224	334	0	0	NUM
ejpam-3838	224	335	}	}	SYM
ejpam-3838	224	336	0	0	NUM
ejpam-3838	224	337	0	0	NUM
ejpam-3838	224	338	{	{	PUNCT
ejpam-3838	224	339	4	4	NUM
ejpam-3838	224	340	,	,	PUNCT
ejpam-3838	224	341	0	0	NUM
ejpam-3838	224	342	,	,	PUNCT
ejpam-3838	224	343	4	4	NUM
ejpam-3838	224	344	}	}	SYM
ejpam-3838	224	345	1	1	NUM
ejpam-3838	224	346	1	1	NUM
ejpam-3838	224	347	{	{	PUNCT
ejpam-3838	224	348	6	6	NUM
ejpam-3838	224	349	,	,	PUNCT
ejpam-3838	224	350	0	0	NUM
ejpam-3838	224	351	,	,	PUNCT
ejpam-3838	224	352	1	1	NUM
ejpam-3838	224	353	}	}	SYM
ejpam-3838	224	354	6	6	NUM
ejpam-3838	224	355	6	6	NUM
ejpam-3838	224	356	{	{	PUNCT
ejpam-3838	224	357	5	5	NUM
ejpam-3838	224	358	,	,	PUNCT
ejpam-3838	224	359	1	1	NUM
ejpam-3838	224	360	,	,	PUNCT
ejpam-3838	224	361	0	0	NUM
ejpam-3838	224	362	}	}	SYM
ejpam-3838	224	363	6	6	NUM
ejpam-3838	224	364	3	3	NUM
ejpam-3838	224	365	{	{	PUNCT
ejpam-3838	224	366	6	6	NUM
ejpam-3838	224	367	,	,	PUNCT
ejpam-3838	224	368	0	0	NUM
ejpam-3838	224	369	,	,	PUNCT
ejpam-3838	224	370	2	2	NUM
ejpam-3838	224	371	}	}	SYM
ejpam-3838	224	372	20	20	NUM
ejpam-3838	224	373	20	20	NUM
ejpam-3838	224	374	{	{	PUNCT
ejpam-3838	224	375	5	5	NUM
ejpam-3838	224	376	,	,	PUNCT
ejpam-3838	224	377	1	1	NUM
ejpam-3838	224	378	,	,	PUNCT
ejpam-3838	224	379	1	1	NUM
ejpam-3838	224	380	}	}	SYM
ejpam-3838	224	381	19	19	NUM
ejpam-3838	224	382	11	11	NUM
ejpam-3838	224	383	{	{	PUNCT
ejpam-3838	224	384	6	6	NUM
ejpam-3838	224	385	,	,	PUNCT
ejpam-3838	224	386	0	0	NUM
ejpam-3838	224	387	,	,	PUNCT
ejpam-3838	224	388	3	3	NUM
ejpam-3838	224	389	}	}	SYM
ejpam-3838	224	390	31	31	NUM
ejpam-3838	224	391	31	31	NUM
ejpam-3838	224	392	{	{	PUNCT
ejpam-3838	224	393	5	5	NUM
ejpam-3838	224	394	,	,	PUNCT
ejpam-3838	224	395	1	1	NUM
ejpam-3838	224	396	,	,	PUNCT
ejpam-3838	224	397	2	2	NUM
ejpam-3838	224	398	}	}	SYM
ejpam-3838	224	399	10	10	NUM
ejpam-3838	224	400	7	7	NUM
ejpam-3838	224	401	{	{	PUNCT
ejpam-3838	224	402	6	6	NUM
ejpam-3838	224	403	,	,	PUNCT
ejpam-3838	224	404	0	0	NUM
ejpam-3838	224	405	,	,	PUNCT
ejpam-3838	224	406	4	4	NUM
ejpam-3838	224	407	}	}	SYM
ejpam-3838	224	408	20	20	NUM
ejpam-3838	224	409	20	20	NUM
ejpam-3838	224	410	{	{	PUNCT
ejpam-3838	224	411	5	5	NUM
ejpam-3838	224	412	,	,	PUNCT
ejpam-3838	224	413	1	1	NUM
ejpam-3838	224	414	,	,	PUNCT
ejpam-3838	224	415	3	3	NUM
ejpam-3838	224	416	}	}	SYM
ejpam-3838	224	417	1	1	NUM
ejpam-3838	224	418	1	1	NUM
ejpam-3838	224	419	{	{	PUNCT
ejpam-3838	224	420	6	6	NUM
ejpam-3838	224	421	,	,	PUNCT
ejpam-3838	224	422	0	0	NUM
ejpam-3838	224	423	,	,	PUNCT
ejpam-3838	224	424	5	5	NUM
ejpam-3838	224	425	}	}	SYM
ejpam-3838	224	426	6	6	NUM
ejpam-3838	224	427	6	6	NUM
ejpam-3838	224	428	{	{	PUNCT
ejpam-3838	224	429	5	5	NUM
ejpam-3838	224	430	,	,	PUNCT
ejpam-3838	224	431	2	2	NUM
ejpam-3838	224	432	,	,	PUNCT
ejpam-3838	224	433	0	0	NUM
ejpam-3838	224	434	}	}	SYM
ejpam-3838	224	435	4	4	NUM
ejpam-3838	224	436	2	2	NUM
ejpam-3838	224	437	{	{	PUNCT
ejpam-3838	224	438	6	6	NUM
ejpam-3838	224	439	,	,	PUNCT
ejpam-3838	224	440	0	0	NUM
ejpam-3838	224	441	,	,	PUNCT
ejpam-3838	224	442	6	6	NUM
ejpam-3838	224	443	}	}	SYM
ejpam-3838	224	444	1	1	NUM
ejpam-3838	224	445	1	1	NUM
ejpam-3838	224	446	6	6	NUM
ejpam-3838	224	447	.	.	PUNCT
ejpam-3838	225	1	codes	code	NOUN
ejpam-3838	225	2	over	over	ADP
ejpam-3838	225	3	fq	fq	PROPN
ejpam-3838	225	4	+	+	CCONJ
ejpam-3838	225	5	ufq	ufq	PROPN
ejpam-3838	225	6	+	+	NUM
ejpam-3838	225	7	u2fq	u2fq	PROPN
ejpam-3838	225	8	,	,	PUNCT
ejpam-3838	225	9	where	where	SCONJ
ejpam-3838	225	10	q	q	NOUN
ejpam-3838	225	11	is	be	AUX
ejpam-3838	225	12	odd	odd	ADJ
ejpam-3838	225	13	for	for	ADP
ejpam-3838	225	14	the	the	DET
ejpam-3838	225	15	rest	rest	NOUN
ejpam-3838	225	16	of	of	ADP
ejpam-3838	225	17	this	this	DET
ejpam-3838	225	18	paper	paper	NOUN
ejpam-3838	225	19	,	,	PUNCT
ejpam-3838	225	20	let	let	VERB
ejpam-3838	225	21	r2	r2	PROPN
ejpam-3838	225	22	be	be	AUX
ejpam-3838	225	23	the	the	DET
ejpam-3838	225	24	commutative	commutative	ADJ
ejpam-3838	225	25	ring	ring	NOUN
ejpam-3838	225	26	fq[u]/(u3	fq[u]/(u3	NOUN
ejpam-3838	225	27	)	)	PUNCT
ejpam-3838	226	1	=	=	SYM
ejpam-3838	226	2	fq+ufq+u2fq	fq+ufq+u2fq	NOUN
ejpam-3838	226	3	,	,	PUNCT
ejpam-3838	226	4	where	where	SCONJ
ejpam-3838	226	5	u3	u3	NOUN
ejpam-3838	226	6	=	=	SYM
ejpam-3838	226	7	0	0	NUM
ejpam-3838	226	8	and	and	CCONJ
ejpam-3838	226	9	q	q	NOUN
ejpam-3838	226	10	is	be	AUX
ejpam-3838	226	11	odd	odd	ADJ
ejpam-3838	226	12	.	.	PUNCT
ejpam-3838	227	1	we	we	PRON
ejpam-3838	227	2	will	will	AUX
ejpam-3838	227	3	only	only	ADV
ejpam-3838	227	4	consider	consider	VERB
ejpam-3838	227	5	euclidean	euclidean	ADJ
ejpam-3838	227	6	inner	inner	ADJ
ejpam-3838	227	7	product	product	NOUN
ejpam-3838	227	8	.	.	PUNCT
ejpam-3838	228	1	a	a	DET
ejpam-3838	228	2	code	code	NOUN
ejpam-3838	228	3	c	c	NOUN
ejpam-3838	228	4	of	of	ADP
ejpam-3838	228	5	length	length	NOUN
ejpam-3838	228	6	n	n	CCONJ
ejpam-3838	228	7	over	over	ADP
ejpam-3838	228	8	r2	r2	PROPN
ejpam-3838	228	9	is	be	AUX
ejpam-3838	228	10	permutation	permutation	NOUN
ejpam-3838	228	11	-	-	PUNCT
ejpam-3838	228	12	equivalent	equivalent	ADJ
ejpam-3838	228	13	to	to	ADP
ejpam-3838	228	14	a	a	DET
ejpam-3838	228	15	code	code	NOUN
ejpam-3838	228	16	with	with	ADP
ejpam-3838	228	17	generator	generator	NOUN
ejpam-3838	228	18	matrix	matrix	NOUN
ejpam-3838	228	19			PROPN
ejpam-3838	228	20	ik0	ik0	VERB
ejpam-3838	228	21	a0	a0	NOUN
ejpam-3838	228	22	b0	b0	PROPN
ejpam-3838	228	23	+	+	CCONJ
ejpam-3838	228	24	ub1	ub1	PROPN
ejpam-3838	228	25	+	+	CCONJ
ejpam-3838	228	26	u2b2	u2b2	PROPN
ejpam-3838	228	27	0	0	NUM
ejpam-3838	228	28	uik1	uik1	NOUN
ejpam-3838	228	29	ud1	ud1	NOUN
ejpam-3838	228	30	+	+	CCONJ
ejpam-3838	228	31	u2d2	u2d2	PROPN
ejpam-3838	228	32	0	0	NUM
ejpam-3838	228	33	0	0	NUM
ejpam-3838	228	34	u2f2	u2f2	ADP
ejpam-3838	228	35			NOUN
ejpam-3838	228	36	(	(	PUNCT
ejpam-3838	228	37	10	10	NUM
ejpam-3838	228	38	)	)	PUNCT
ejpam-3838	228	39	where	where	SCONJ
ejpam-3838	228	40	f2	f2	PROPN
ejpam-3838	228	41	∈mk2×(n−k0−k1)(fq	∈mk2×(n−k0−k1)(fq	NOUN
ejpam-3838	228	42	)	)	PUNCT
ejpam-3838	228	43	and	and	CCONJ
ejpam-3838	228	44	a0	a0	PROPN
ejpam-3838	228	45	,	,	PUNCT
ejpam-3838	228	46	b0	b0	NOUN
ejpam-3838	228	47	,	,	PUNCT
ejpam-3838	228	48	b1	b1	NOUN
ejpam-3838	228	49	,	,	PUNCT
ejpam-3838	228	50	b2	b2	NOUN
ejpam-3838	228	51	,	,	PUNCT
ejpam-3838	228	52	d1	d1	PROPN
ejpam-3838	228	53	,	,	PUNCT
ejpam-3838	228	54	d2	d2	PROPN
ejpam-3838	228	55	are	be	AUX
ejpam-3838	228	56	matrices	matrix	NOUN
ejpam-3838	228	57	of	of	ADP
ejpam-3838	228	58	appropriate	appropriate	ADJ
ejpam-3838	228	59	sizes	size	NOUN
ejpam-3838	228	60	over	over	ADP
ejpam-3838	228	61	fq	fq	PROPN
ejpam-3838	228	62	.	.	PUNCT
ejpam-3838	229	1	we	we	PRON
ejpam-3838	229	2	define	define	VERB
ejpam-3838	229	3	the	the	DET
ejpam-3838	229	4	torsion	torsion	NOUN
ejpam-3838	229	5	codes	code	NOUN
ejpam-3838	229	6	of	of	ADP
ejpam-3838	229	7	c	c	PROPN
ejpam-3838	229	8	as	as	SCONJ
ejpam-3838	229	9	follows	follow	VERB
ejpam-3838	229	10	:	:	PUNCT
ejpam-3838	229	11	tor0(c	tor0(c	NUM
ejpam-3838	229	12	)	)	PUNCT
ejpam-3838	229	13	=	=	PRON
ejpam-3838	229	14	{	{	PUNCT
ejpam-3838	229	15	v	v	NUM
ejpam-3838	229	16	∈	∈	PRON
ejpam-3838	230	1	fnq	fnq	NOUN
ejpam-3838	230	2	|	|	PROPN
ejpam-3838	230	3	∃w	∃w	PROPN
ejpam-3838	230	4	,	,	PUNCT
ejpam-3838	230	5	z	z	PROPN
ejpam-3838	230	6	∈	∈	PROPN
ejpam-3838	231	1	fnq	fnq	PROPN
ejpam-3838	231	2	,	,	PUNCT
ejpam-3838	231	3	v	v	PROPN
ejpam-3838	231	4	+	+	CCONJ
ejpam-3838	231	5	uw	uw	PROPN
ejpam-3838	231	6	+	+	CCONJ
ejpam-3838	231	7	u2z	u2z	PROPN
ejpam-3838	231	8	∈	∈	PROPN
ejpam-3838	231	9	c	c	AUX
ejpam-3838	231	10	}	}	PUNCT
ejpam-3838	231	11	and	and	CCONJ
ejpam-3838	231	12	tori(c	tori(c	NOUN
ejpam-3838	231	13	)	)	PUNCT
ejpam-3838	231	14	=	=	PRON
ejpam-3838	231	15	{	{	PUNCT
ejpam-3838	231	16	v	v	NUM
ejpam-3838	231	17	∈	∈	PRON
ejpam-3838	231	18	fnq	fnq	NOUN
ejpam-3838	232	1	|	|	ADV
ejpam-3838	232	2	uiv	uiv	VERB
ejpam-3838	232	3	∈	∈	PROPN
ejpam-3838	232	4	c	c	X
ejpam-3838	232	5	}	}	PUNCT
ejpam-3838	232	6	,	,	PUNCT
ejpam-3838	232	7	for	for	ADP
ejpam-3838	232	8	i	i	PROPN
ejpam-3838	232	9	=	=	SYM
ejpam-3838	232	10	1	1	NUM
ejpam-3838	232	11	,	,	PUNCT
ejpam-3838	232	12	2	2	NUM
ejpam-3838	232	13	.	.	PUNCT
ejpam-3838	233	1	the	the	DET
ejpam-3838	233	2	code	code	PROPN
ejpam-3838	233	3	tor0(c	tor0(c	NOUN
ejpam-3838	233	4	)	)	PUNCT
ejpam-3838	233	5	is	be	AUX
ejpam-3838	233	6	also	also	ADV
ejpam-3838	233	7	called	call	VERB
ejpam-3838	233	8	the	the	DET
ejpam-3838	233	9	residue	residue	NOUN
ejpam-3838	233	10	code	code	NOUN
ejpam-3838	233	11	of	of	ADP
ejpam-3838	233	12	c.	c.	PROPN
ejpam-3838	233	13	observe	observe	VERB
ejpam-3838	233	14	that	that	SCONJ
ejpam-3838	233	15	tor0(c	tor0(c	NOUN
ejpam-3838	233	16	)	)	PUNCT
ejpam-3838	233	17	⊆	⊆	NUM
ejpam-3838	233	18	tor1(c	tor1(c	NUM
ejpam-3838	233	19	)	)	PUNCT
ejpam-3838	234	1	⊆	⊆	NUM
ejpam-3838	234	2	tor2(c	tor2(c	NOUN
ejpam-3838	234	3	)	)	PUNCT
ejpam-3838	235	1	.	.	PUNCT
ejpam-3838	236	1	if	if	SCONJ
ejpam-3838	236	2	c	c	PROPN
ejpam-3838	236	3	has	have	VERB
ejpam-3838	236	4	generator	generator	NOUN
ejpam-3838	236	5	matrix	matrix	NOUN
ejpam-3838	236	6	(	(	PUNCT
ejpam-3838	236	7	10	10	NUM
ejpam-3838	236	8	)	)	PUNCT
ejpam-3838	236	9	,	,	PUNCT
ejpam-3838	236	10	then	then	ADV
ejpam-3838	236	11	the	the	DET
ejpam-3838	236	12	residue	residue	NOUN
ejpam-3838	236	13	code	code	NOUN
ejpam-3838	236	14	tor0(c	tor0(c	NOUN
ejpam-3838	236	15	)	)	PUNCT
ejpam-3838	236	16	has	have	VERB
ejpam-3838	236	17	dimension	dimension	NOUN
ejpam-3838	236	18	k0	k0	PROPN
ejpam-3838	236	19	and	and	CCONJ
ejpam-3838	236	20	generator	generator	NOUN
ejpam-3838	236	21	matrix	matrix	NOUN
ejpam-3838	236	22	[	[	PUNCT
ejpam-3838	236	23	ik0	ik0	VERB
ejpam-3838	236	24	a0	a0	NOUN
ejpam-3838	236	25	b0	b0	PROPN
ejpam-3838	236	26	]	]	PUNCT
ejpam-3838	236	27	,	,	PUNCT
ejpam-3838	236	28	(	(	PUNCT
ejpam-3838	236	29	11	11	X
ejpam-3838	236	30	)	)	PUNCT
ejpam-3838	236	31	l.e	l.e	PROPN
ejpam-3838	236	32	.	.	PROPN
ejpam-3838	236	33	galvez	galvez	PROPN
ejpam-3838	236	34	,	,	PUNCT
ejpam-3838	236	35	r.a	r.a	PROPN
ejpam-3838	236	36	.	.	PROPN
ejpam-3838	236	37	betty	betty	PROPN
ejpam-3838	236	38	,	,	PUNCT
ejpam-3838	236	39	f.	f.	PROPN
ejpam-3838	236	40	nemenzo	nemenzo	PROPN
ejpam-3838	236	41	/	/	SYM
ejpam-3838	236	42	eur	eur	NOUN
ejpam-3838	236	43	.	.	PUNCT
ejpam-3838	237	1	j.	j.	PROPN
ejpam-3838	237	2	pure	pure	PROPN
ejpam-3838	237	3	appl	appl	PROPN
ejpam-3838	237	4	.	.	PROPN
ejpam-3838	237	5	math	math	PROPN
ejpam-3838	237	6	,	,	PUNCT
ejpam-3838	237	7	13	13	NUM
ejpam-3838	237	8	(	(	PUNCT
ejpam-3838	237	9	4	4	NUM
ejpam-3838	237	10	)	)	PUNCT
ejpam-3838	237	11	(	(	PUNCT
ejpam-3838	237	12	2020	2020	NUM
ejpam-3838	237	13	)	)	PUNCT
ejpam-3838	237	14	,	,	PUNCT
ejpam-3838	237	15	873	873	NUM
ejpam-3838	237	16	-	-	NUM
ejpam-3838	237	17	892	892	NUM
ejpam-3838	237	18	884	884	NUM
ejpam-3838	237	19	tor1(c	tor1(c	NUM
ejpam-3838	237	20	)	)	PUNCT
ejpam-3838	237	21	has	have	AUX
ejpam-3838	237	22	dimension	dimension	NOUN
ejpam-3838	237	23	k0	k0	PROPN
ejpam-3838	237	24	+	+	CCONJ
ejpam-3838	237	25	k1	k1	PROPN
ejpam-3838	237	26	and	and	CCONJ
ejpam-3838	237	27	generator	generator	NOUN
ejpam-3838	237	28	matrix	matrix	NOUN
ejpam-3838	237	29	[	[	PUNCT
ejpam-3838	237	30	ik0	ik0	VERB
ejpam-3838	237	31	a0	a0	NOUN
ejpam-3838	237	32	b0	b0	PROPN
ejpam-3838	237	33	0	0	PUNCT
ejpam-3838	237	34	ik1	ik1	ADJ
ejpam-3838	237	35	d1	d1	PROPN
ejpam-3838	237	36	]	]	PUNCT
ejpam-3838	237	37	(	(	PUNCT
ejpam-3838	237	38	12	12	NUM
ejpam-3838	237	39	)	)	PUNCT
ejpam-3838	237	40	and	and	CCONJ
ejpam-3838	237	41	tor2(c	tor2(c	NUM
ejpam-3838	237	42	)	)	PUNCT
ejpam-3838	237	43	has	have	AUX
ejpam-3838	237	44	dimension	dimension	NOUN
ejpam-3838	237	45	k0	k0	PROPN
ejpam-3838	237	46	+	+	CCONJ
ejpam-3838	237	47	k1	k1	PROPN
ejpam-3838	237	48	+	+	CCONJ
ejpam-3838	237	49	k2	k2	NOUN
ejpam-3838	237	50	and	and	CCONJ
ejpam-3838	237	51	generator	generator	NOUN
ejpam-3838	237	52	matrix	matrix	PROPN
ejpam-3838	237	53	ik0	ik0	VERB
ejpam-3838	237	54	a0	a0	PROPN
ejpam-3838	237	55	b0	b0	PROPN
ejpam-3838	237	56	0	0	PUNCT
ejpam-3838	237	57	ik1	ik1	VERB
ejpam-3838	237	58	d1	d1	PROPN
ejpam-3838	237	59	0	0	NUM
ejpam-3838	237	60	0	0	PUNCT
ejpam-3838	237	61	f2	f2	ADJ
ejpam-3838	237	62			NOUN
ejpam-3838	237	63	(	(	PUNCT
ejpam-3838	237	64	13	13	NUM
ejpam-3838	237	65	)	)	PUNCT
ejpam-3838	237	66	where	where	SCONJ
ejpam-3838	237	67	f2	f2	PROPN
ejpam-3838	237	68	is	be	AUX
ejpam-3838	237	69	of	of	ADP
ejpam-3838	237	70	full	full	ADJ
ejpam-3838	237	71	row	row	NOUN
ejpam-3838	237	72	rank	rank	NOUN
ejpam-3838	237	73	.	.	PUNCT
ejpam-3838	238	1	the	the	DET
ejpam-3838	238	2	code	code	NOUN
ejpam-3838	238	3	c	c	PROPN
ejpam-3838	238	4	is	be	AUX
ejpam-3838	238	5	of	of	ADP
ejpam-3838	238	6	type	type	NOUN
ejpam-3838	238	7	{	{	PUNCT
ejpam-3838	238	8	k0	k0	PROPN
ejpam-3838	238	9	,	,	PUNCT
ejpam-3838	238	10	k1	k1	PROPN
ejpam-3838	238	11	,	,	PUNCT
ejpam-3838	238	12	k2	k2	ADJ
ejpam-3838	238	13	}	}	PUNCT
ejpam-3838	238	14	and	and	CCONJ
ejpam-3838	238	15	|c|	|c|	PROPN
ejpam-3838	238	16	=	=	SYM
ejpam-3838	239	1	|tor0(c)|	|tor0(c)|	NOUN
ejpam-3838	239	2	|tor1(c)|	|tor1(c)|	VERB
ejpam-3838	239	3	|tor2(c)|	|tor2(c)|	NOUN
ejpam-3838	239	4	=	=	X
ejpam-3838	239	5	q3k0	q3k0	X
ejpam-3838	239	6	+	+	NOUN
ejpam-3838	239	7	2k1+k2	2k1+k2	NUM
ejpam-3838	239	8	.	.	PUNCT
ejpam-3838	240	1	suppose	suppose	VERB
ejpam-3838	240	2	c	c	NOUN
ejpam-3838	240	3	is	be	AUX
ejpam-3838	240	4	self	self	NOUN
ejpam-3838	240	5	-	-	PUNCT
ejpam-3838	240	6	orthogonal	orthogonal	NOUN
ejpam-3838	240	7	.	.	PUNCT
ejpam-3838	241	1	then	then	ADV
ejpam-3838	241	2	ik0	ik0	VERB
ejpam-3838	242	1	+	+	PROPN
ejpam-3838	242	2	a0a	a0a	PROPN
ejpam-3838	242	3	t	t	NOUN
ejpam-3838	242	4	0	0	PUNCT
ejpam-3838	243	1	+	+	NOUN
ejpam-3838	243	2	b0b	b0b	X
ejpam-3838	243	3	t	t	X
ejpam-3838	243	4	0	0	NUM
ejpam-3838	244	1	+	+	CCONJ
ejpam-3838	244	2	u(b0b	u(b0b	CCONJ
ejpam-3838	244	3	t	t	PROPN
ejpam-3838	244	4	1	1	NUM
ejpam-3838	244	5	+	+	NOUN
ejpam-3838	244	6	b1b	b1b	NOUN
ejpam-3838	244	7	t	t	NOUN
ejpam-3838	244	8	0	0	NUM
ejpam-3838	244	9	)	)	PUNCT
ejpam-3838	245	1	+	+	ADP
ejpam-3838	245	2	u2(b0b	u2(b0b	X
ejpam-3838	245	3	t	t	NOUN
ejpam-3838	245	4	2	2	NUM
ejpam-3838	245	5	+	+	NOUN
ejpam-3838	245	6	b1b	b1b	NOUN
ejpam-3838	245	7	t	t	PROPN
ejpam-3838	245	8	1	1	NUM
ejpam-3838	246	1	+	+	NOUN
ejpam-3838	246	2	b2b	b2b	PROPN
ejpam-3838	246	3	t	t	PROPN
ejpam-3838	246	4	0	0	NUM
ejpam-3838	246	5	)	)	PUNCT
ejpam-3838	246	6	≡	≡	PROPN
ejpam-3838	246	7	0	0	PUNCT
ejpam-3838	246	8	(	(	PUNCT
ejpam-3838	246	9	u3	u3	NOUN
ejpam-3838	246	10	)	)	PUNCT
ejpam-3838	246	11	u(a0	u(a0	NOUN
ejpam-3838	247	1	+	+	VERB
ejpam-3838	247	2	b0d	b0d	PROPN
ejpam-3838	247	3	t	t	AUX
ejpam-3838	247	4	1	1	NUM
ejpam-3838	247	5	)	)	PUNCT
ejpam-3838	248	1	+	+	CCONJ
ejpam-3838	248	2	u2(b1d	u2(b1d	PROPN
ejpam-3838	248	3	t	t	PROPN
ejpam-3838	248	4	1	1	NUM
ejpam-3838	249	1	+	+	NOUN
ejpam-3838	249	2	b0d	b0d	PROPN
ejpam-3838	249	3	t	t	AUX
ejpam-3838	249	4	2	2	NUM
ejpam-3838	249	5	)	)	PUNCT
ejpam-3838	249	6	≡	≡	PROPN
ejpam-3838	249	7	0	0	PUNCT
ejpam-3838	249	8	(	(	PUNCT
ejpam-3838	249	9	u3	u3	PROPN
ejpam-3838	249	10	)	)	PUNCT
ejpam-3838	249	11	u2(b0f	u2(b0f	PROPN
ejpam-3838	249	12	t	t	PROPN
ejpam-3838	249	13	2	2	X
ejpam-3838	249	14	)	)	PUNCT
ejpam-3838	249	15	≡	≡	PROPN
ejpam-3838	249	16	0	0	PUNCT
ejpam-3838	249	17	(	(	PUNCT
ejpam-3838	249	18	u3	u3	PROPN
ejpam-3838	249	19	)	)	PUNCT
ejpam-3838	249	20	u2(ik1	u2(ik1	VERB
ejpam-3838	249	21	+	+	NOUN
ejpam-3838	249	22	d1d	d1d	NOUN
ejpam-3838	249	23	t	t	NOUN
ejpam-3838	249	24	1	1	NUM
ejpam-3838	249	25	)	)	PUNCT
ejpam-3838	249	26	≡	≡	PROPN
ejpam-3838	249	27	0	0	PUNCT
ejpam-3838	249	28	(	(	PUNCT
ejpam-3838	249	29	u3	u3	PROPN
ejpam-3838	249	30	)	)	PUNCT
ejpam-3838	249	31	which	which	PRON
ejpam-3838	249	32	give	give	VERB
ejpam-3838	249	33	the	the	DET
ejpam-3838	249	34	following	following	NOUN
ejpam-3838	249	35	:	:	PUNCT
ejpam-3838	249	36	ik0	ik0	VERB
ejpam-3838	249	37	+	+	PROPN
ejpam-3838	249	38	a0a	a0a	PROPN
ejpam-3838	249	39	t	t	NOUN
ejpam-3838	249	40	0	0	PUNCT
ejpam-3838	250	1	+	+	NOUN
ejpam-3838	250	2	b0b	b0b	PROPN
ejpam-3838	250	3	t	t	X
ejpam-3838	250	4	0	0	NUM
ejpam-3838	250	5	≡	≡	PROPN
ejpam-3838	250	6	0	0	NUM
ejpam-3838	250	7	(	(	PUNCT
ejpam-3838	250	8	u	u	NOUN
ejpam-3838	250	9	)	)	PUNCT
ejpam-3838	250	10	(	(	PUNCT
ejpam-3838	250	11	14	14	NUM
ejpam-3838	250	12	)	)	PUNCT
ejpam-3838	250	13	b0b	b0b	PROPN
ejpam-3838	250	14	t	t	PROPN
ejpam-3838	250	15	1	1	NUM
ejpam-3838	251	1	+	+	NOUN
ejpam-3838	251	2	b1b	b1b	PROPN
ejpam-3838	251	3	t	t	PROPN
ejpam-3838	251	4	0	0	NUM
ejpam-3838	251	5	≡	≡	PROPN
ejpam-3838	251	6	0	0	NUM
ejpam-3838	251	7	(	(	PUNCT
ejpam-3838	251	8	u	u	NOUN
ejpam-3838	251	9	)	)	PUNCT
ejpam-3838	251	10	(	(	PUNCT
ejpam-3838	251	11	15	15	NUM
ejpam-3838	251	12	)	)	PUNCT
ejpam-3838	251	13	b0b	b0b	NOUN
ejpam-3838	251	14	t	t	PROPN
ejpam-3838	251	15	2	2	NUM
ejpam-3838	252	1	+	+	NOUN
ejpam-3838	252	2	b1b	b1b	NOUN
ejpam-3838	252	3	t	t	PROPN
ejpam-3838	252	4	1	1	NUM
ejpam-3838	253	1	+	+	NOUN
ejpam-3838	253	2	b2b	b2b	PROPN
ejpam-3838	253	3	t	t	NOUN
ejpam-3838	253	4	0	0	NUM
ejpam-3838	254	1	≡	≡	PROPN
ejpam-3838	254	2	0	0	NUM
ejpam-3838	254	3	(	(	PUNCT
ejpam-3838	254	4	u	u	NOUN
ejpam-3838	254	5	)	)	PUNCT
ejpam-3838	254	6	(	(	PUNCT
ejpam-3838	254	7	16	16	NUM
ejpam-3838	254	8	)	)	PUNCT
ejpam-3838	254	9	a0	a0	NOUN
ejpam-3838	255	1	+	+	PROPN
ejpam-3838	255	2	b0d	b0d	PROPN
ejpam-3838	255	3	t	t	PROPN
ejpam-3838	255	4	1	1	NUM
ejpam-3838	255	5	≡	≡	PROPN
ejpam-3838	255	6	0	0	NUM
ejpam-3838	255	7	(	(	PUNCT
ejpam-3838	255	8	u	u	NOUN
ejpam-3838	255	9	)	)	PUNCT
ejpam-3838	255	10	(	(	PUNCT
ejpam-3838	255	11	17	17	NUM
ejpam-3838	255	12	)	)	PUNCT
ejpam-3838	255	13	b1d	b1d	NOUN
ejpam-3838	255	14	t	t	NOUN
ejpam-3838	255	15	1	1	NUM
ejpam-3838	255	16	+	+	NOUN
ejpam-3838	255	17	b0d	b0d	PROPN
ejpam-3838	255	18	t	t	PROPN
ejpam-3838	255	19	2	2	NUM
ejpam-3838	255	20	≡	≡	PROPN
ejpam-3838	255	21	0	0	NUM
ejpam-3838	255	22	(	(	PUNCT
ejpam-3838	255	23	u	u	NOUN
ejpam-3838	255	24	)	)	PUNCT
ejpam-3838	255	25	(	(	PUNCT
ejpam-3838	255	26	18	18	NUM
ejpam-3838	255	27	)	)	PUNCT
ejpam-3838	255	28	f2b	f2b	PROPN
ejpam-3838	255	29	t	t	NOUN
ejpam-3838	255	30	0	0	NUM
ejpam-3838	255	31	≡	≡	PROPN
ejpam-3838	255	32	0	0	NUM
ejpam-3838	256	1	(	(	PUNCT
ejpam-3838	256	2	u	u	NOUN
ejpam-3838	256	3	)	)	PUNCT
ejpam-3838	256	4	(	(	PUNCT
ejpam-3838	256	5	19	19	NUM
ejpam-3838	256	6	)	)	PUNCT
ejpam-3838	256	7	ik1	ik1	VERB
ejpam-3838	257	1	+	+	PROPN
ejpam-3838	257	2	d1d	d1d	PROPN
ejpam-3838	257	3	t	t	NOUN
ejpam-3838	257	4	1	1	NUM
ejpam-3838	257	5	≡	≡	PROPN
ejpam-3838	257	6	0	0	NUM
ejpam-3838	257	7	(	(	PUNCT
ejpam-3838	257	8	u	u	NOUN
ejpam-3838	257	9	)	)	PUNCT
ejpam-3838	257	10	.	.	PUNCT
ejpam-3838	258	1	(	(	PUNCT
ejpam-3838	258	2	20	20	NUM
ejpam-3838	258	3	)	)	PUNCT
ejpam-3838	258	4	from	from	ADP
ejpam-3838	258	5	(	(	PUNCT
ejpam-3838	258	6	14	14	NUM
ejpam-3838	258	7	)	)	PUNCT
ejpam-3838	258	8	,	,	PUNCT
ejpam-3838	258	9	tor0(c	tor0(c	NUM
ejpam-3838	258	10	)	)	PUNCT
ejpam-3838	258	11	is	be	AUX
ejpam-3838	258	12	self	self	NOUN
ejpam-3838	258	13	-	-	PUNCT
ejpam-3838	258	14	orthogonal	orthogonal	ADJ
ejpam-3838	258	15	and	and	CCONJ
ejpam-3838	258	16	by	by	ADP
ejpam-3838	258	17	(	(	PUNCT
ejpam-3838	258	18	14	14	NUM
ejpam-3838	258	19	)	)	PUNCT
ejpam-3838	258	20	,	,	PUNCT
ejpam-3838	258	21	(	(	PUNCT
ejpam-3838	258	22	17	17	NUM
ejpam-3838	258	23	)	)	PUNCT
ejpam-3838	258	24	and	and	CCONJ
ejpam-3838	258	25	(	(	PUNCT
ejpam-3838	258	26	20	20	NUM
ejpam-3838	258	27	)	)	PUNCT
ejpam-3838	258	28	,	,	PUNCT
ejpam-3838	258	29	we	we	PRON
ejpam-3838	258	30	have	have	VERB
ejpam-3838	258	31	tor1(c	tor1(c	NUM
ejpam-3838	258	32	)	)	PUNCT
ejpam-3838	258	33	⊆	⊆	NUM
ejpam-3838	258	34	tor1(c)⊥	tor1(c)⊥	NOUN
ejpam-3838	258	35	,	,	PUNCT
ejpam-3838	258	36	that	that	ADV
ejpam-3838	258	37	is	is	ADV
ejpam-3838	258	38	,	,	PUNCT
ejpam-3838	258	39	tor1(c	tor1(c	NUM
ejpam-3838	258	40	)	)	PUNCT
ejpam-3838	258	41	is	be	AUX
ejpam-3838	258	42	self	self	NOUN
ejpam-3838	258	43	-	-	PUNCT
ejpam-3838	258	44	orthogonal	orthogonal	NOUN
ejpam-3838	258	45	.	.	PUNCT
ejpam-3838	259	1	moreover	moreover	ADV
ejpam-3838	259	2	,	,	PUNCT
ejpam-3838	259	3	by	by	ADP
ejpam-3838	259	4	(	(	PUNCT
ejpam-3838	259	5	14	14	NUM
ejpam-3838	259	6	)	)	PUNCT
ejpam-3838	259	7	,	,	PUNCT
ejpam-3838	259	8	(	(	PUNCT
ejpam-3838	259	9	17	17	NUM
ejpam-3838	259	10	)	)	PUNCT
ejpam-3838	259	11	and	and	CCONJ
ejpam-3838	259	12	(	(	PUNCT
ejpam-3838	259	13	19	19	NUM
ejpam-3838	259	14	)	)	PUNCT
ejpam-3838	259	15	we	we	PRON
ejpam-3838	259	16	have	have	VERB
ejpam-3838	259	17	tor0(c	tor0(c	NOUN
ejpam-3838	259	18	)	)	PUNCT
ejpam-3838	259	19	⊆	⊆	NUM
ejpam-3838	259	20	tor2(c)⊥.	tor2(c)⊥.	NOUN
ejpam-3838	259	21	we	we	PRON
ejpam-3838	259	22	will	will	AUX
ejpam-3838	259	23	introduce	introduce	VERB
ejpam-3838	259	24	another	another	DET
ejpam-3838	259	25	type	type	NOUN
ejpam-3838	259	26	of	of	ADP
ejpam-3838	259	27	residue	residue	NOUN
ejpam-3838	259	28	for	for	ADP
ejpam-3838	259	29	a	a	DET
ejpam-3838	259	30	code	code	NOUN
ejpam-3838	259	31	over	over	ADP
ejpam-3838	259	32	r2	r2	PROPN
ejpam-3838	259	33	.	.	PUNCT
ejpam-3838	260	1	definition	definition	NOUN
ejpam-3838	260	2	1	1	NUM
ejpam-3838	260	3	.	.	PUNCT
ejpam-3838	261	1	let	let	VERB
ejpam-3838	261	2	c	c	PRON
ejpam-3838	261	3	be	be	AUX
ejpam-3838	261	4	a	a	DET
ejpam-3838	261	5	code	code	NOUN
ejpam-3838	261	6	over	over	ADP
ejpam-3838	261	7	r2	r2	PROPN
ejpam-3838	261	8	.	.	PUNCT
ejpam-3838	262	1	the	the	DET
ejpam-3838	262	2	code	code	NOUN
ejpam-3838	262	3	over	over	ADP
ejpam-3838	262	4	fq	fq	PROPN
ejpam-3838	262	5	+	+	CCONJ
ejpam-3838	262	6	ufq	ufq	PROPN
ejpam-3838	262	7	obtained	obtain	VERB
ejpam-3838	262	8	from	from	ADP
ejpam-3838	262	9	c	c	NOUN
ejpam-3838	262	10	by	by	ADP
ejpam-3838	262	11	reduction	reduction	NOUN
ejpam-3838	262	12	modulo	modulo	NOUN
ejpam-3838	262	13	u2	u2	NOUN
ejpam-3838	262	14	is	be	AUX
ejpam-3838	262	15	called	call	VERB
ejpam-3838	262	16	the	the	DET
ejpam-3838	262	17	u2	u2	NOUN
ejpam-3838	262	18	-	-	PUNCT
ejpam-3838	262	19	residue	residue	NOUN
ejpam-3838	262	20	of	of	ADP
ejpam-3838	262	21	c	c	NOUN
ejpam-3838	262	22	and	and	CCONJ
ejpam-3838	262	23	will	will	AUX
ejpam-3838	262	24	be	be	AUX
ejpam-3838	262	25	denoted	denote	VERB
ejpam-3838	262	26	by	by	ADP
ejpam-3838	262	27	res(c	res(c	PROPN
ejpam-3838	262	28	)	)	PUNCT
ejpam-3838	262	29	.	.	PUNCT
ejpam-3838	263	1	l.e	l.e	PROPN
ejpam-3838	263	2	.	.	PROPN
ejpam-3838	263	3	galvez	galvez	PROPN
ejpam-3838	263	4	,	,	PUNCT
ejpam-3838	263	5	r.a	r.a	PROPN
ejpam-3838	263	6	.	.	PROPN
ejpam-3838	263	7	betty	betty	PROPN
ejpam-3838	263	8	,	,	PUNCT
ejpam-3838	263	9	f.	f.	PROPN
ejpam-3838	263	10	nemenzo	nemenzo	PROPN
ejpam-3838	263	11	/	/	SYM
ejpam-3838	263	12	eur	eur	NOUN
ejpam-3838	263	13	.	.	PUNCT
ejpam-3838	264	1	j.	j.	PROPN
ejpam-3838	264	2	pure	pure	PROPN
ejpam-3838	264	3	appl	appl	PROPN
ejpam-3838	264	4	.	.	PROPN
ejpam-3838	264	5	math	math	PROPN
ejpam-3838	264	6	,	,	PUNCT
ejpam-3838	264	7	13	13	NUM
ejpam-3838	264	8	(	(	PUNCT
ejpam-3838	264	9	4	4	NUM
ejpam-3838	264	10	)	)	PUNCT
ejpam-3838	264	11	(	(	PUNCT
ejpam-3838	264	12	2020	2020	NUM
ejpam-3838	264	13	)	)	PUNCT
ejpam-3838	264	14	,	,	PUNCT
ejpam-3838	264	15	873	873	NUM
ejpam-3838	264	16	-	-	SYM
ejpam-3838	264	17	892	892	NUM
ejpam-3838	264	18	885	885	NUM
ejpam-3838	264	19	it	it	PRON
ejpam-3838	264	20	is	be	AUX
ejpam-3838	264	21	easy	easy	ADJ
ejpam-3838	264	22	to	to	PART
ejpam-3838	264	23	see	see	VERB
ejpam-3838	264	24	that	that	SCONJ
ejpam-3838	264	25	a	a	DET
ejpam-3838	264	26	generator	generator	NOUN
ejpam-3838	264	27	matrix	matrix	NOUN
ejpam-3838	264	28	for	for	ADP
ejpam-3838	264	29	res(c	res(c	PROPN
ejpam-3838	264	30	)	)	PUNCT
ejpam-3838	264	31	is	be	AUX
ejpam-3838	264	32	[	[	PUNCT
ejpam-3838	264	33	ik0	ik0	PROPN
ejpam-3838	264	34	a0	a0	NOUN
ejpam-3838	264	35	b0	b0	PROPN
ejpam-3838	264	36	+	+	CCONJ
ejpam-3838	264	37	ub1	ub1	PROPN
ejpam-3838	264	38	0	0	NUM
ejpam-3838	264	39	uik1	uik1	PROPN
ejpam-3838	264	40	ud1	ud1	NOUN
ejpam-3838	264	41	]	]	PUNCT
ejpam-3838	264	42	.	.	PUNCT
ejpam-3838	265	1	it	it	PRON
ejpam-3838	265	2	is	be	AUX
ejpam-3838	265	3	also	also	ADV
ejpam-3838	265	4	clear	clear	ADJ
ejpam-3838	265	5	that	that	SCONJ
ejpam-3838	265	6	res(res(c	res(res(c	VERB
ejpam-3838	265	7	)	)	PUNCT
ejpam-3838	265	8	)	)	PUNCT
ejpam-3838	266	1	=	=	SYM
ejpam-3838	266	2	tor0(c	tor0(c	NOUN
ejpam-3838	266	3	)	)	PUNCT
ejpam-3838	266	4	,	,	PUNCT
ejpam-3838	266	5	tor(res(c	tor(res(c	NUM
ejpam-3838	266	6	)	)	PUNCT
ejpam-3838	266	7	)	)	PUNCT
ejpam-3838	267	1	=	=	PUNCT
ejpam-3838	267	2	tor1(c	tor1(c	NOUN
ejpam-3838	267	3	)	)	PUNCT
ejpam-3838	267	4	,	,	PUNCT
ejpam-3838	267	5	and	and	CCONJ
ejpam-3838	267	6	res(c	res(c	ADJ
ejpam-3838	267	7	)	)	PUNCT
ejpam-3838	267	8	is	be	AUX
ejpam-3838	267	9	of	of	ADP
ejpam-3838	267	10	type	type	NOUN
ejpam-3838	267	11	{	{	PUNCT
ejpam-3838	267	12	k0	k0	PROPN
ejpam-3838	267	13	,	,	PUNCT
ejpam-3838	267	14	k1	k1	NOUN
ejpam-3838	267	15	}	}	PUNCT
ejpam-3838	267	16	.	.	PUNCT
ejpam-3838	268	1	if	if	SCONJ
ejpam-3838	268	2	c	c	PROPN
ejpam-3838	268	3	is	be	AUX
ejpam-3838	268	4	self	self	NOUN
ejpam-3838	268	5	-	-	PUNCT
ejpam-3838	268	6	orthogonal	orthogonal	ADJ
ejpam-3838	268	7	,	,	PUNCT
ejpam-3838	268	8	by	by	ADP
ejpam-3838	268	9	(	(	PUNCT
ejpam-3838	268	10	14	14	NUM
ejpam-3838	268	11	)	)	PUNCT
ejpam-3838	268	12	,	,	PUNCT
ejpam-3838	268	13	(	(	PUNCT
ejpam-3838	268	14	15	15	NUM
ejpam-3838	268	15	)	)	PUNCT
ejpam-3838	268	16	and	and	CCONJ
ejpam-3838	268	17	(	(	PUNCT
ejpam-3838	268	18	17	17	NUM
ejpam-3838	268	19	)	)	PUNCT
ejpam-3838	268	20	we	we	PRON
ejpam-3838	268	21	have	have	AUX
ejpam-3838	268	22	res(c	res(c	ADJ
ejpam-3838	268	23	)	)	PUNCT
ejpam-3838	269	1	⊆	⊆	NUM
ejpam-3838	269	2	res(c)⊥	res(c)⊥	NOUN
ejpam-3838	269	3	,	,	PUNCT
ejpam-3838	269	4	that	that	ADV
ejpam-3838	269	5	is	is	ADV
ejpam-3838	269	6	,	,	PUNCT
ejpam-3838	269	7	res(c	res(c	ADJ
ejpam-3838	269	8	)	)	PUNCT
ejpam-3838	269	9	is	be	AUX
ejpam-3838	269	10	self	self	NOUN
ejpam-3838	269	11	-	-	PUNCT
ejpam-3838	269	12	orthogonal	orthogonal	NOUN
ejpam-3838	269	13	of	of	ADP
ejpam-3838	269	14	type	type	NOUN
ejpam-3838	269	15	{	{	PUNCT
ejpam-3838	269	16	k0	k0	PROPN
ejpam-3838	269	17	,	,	PUNCT
ejpam-3838	269	18	k1	k1	NOUN
ejpam-3838	269	19	}	}	PUNCT
ejpam-3838	269	20	.	.	PUNCT
ejpam-3838	270	1	also	also	ADV
ejpam-3838	270	2	,	,	PUNCT
ejpam-3838	270	3	since	since	SCONJ
ejpam-3838	270	4	tor(res(c	tor(res(c	NUM
ejpam-3838	270	5	)	)	PUNCT
ejpam-3838	270	6	)	)	PUNCT
ejpam-3838	270	7	=	=	SYM
ejpam-3838	270	8	tor1(c	tor1(c	NOUN
ejpam-3838	270	9	)	)	PUNCT
ejpam-3838	270	10	⊆	⊆	NUM
ejpam-3838	270	11	tor2(c	tor2(c	NOUN
ejpam-3838	270	12	)	)	PUNCT
ejpam-3838	270	13	and	and	CCONJ
ejpam-3838	270	14	tor2(c	tor2(c	NUM
ejpam-3838	270	15	)	)	PUNCT
ejpam-3838	270	16	⊆	⊆	NUM
ejpam-3838	270	17	tor0(c)⊥	tor0(c)⊥	NOUN
ejpam-3838	270	18	=	=	SYM
ejpam-3838	270	19	res(res(c))⊥	res(res(c))⊥	PROPN
ejpam-3838	270	20	,	,	PUNCT
ejpam-3838	270	21	we	we	PRON
ejpam-3838	270	22	have	have	VERB
ejpam-3838	270	23	tor(res(c	tor(res(c	VERB
ejpam-3838	270	24	)	)	PUNCT
ejpam-3838	270	25	)	)	PUNCT
ejpam-3838	271	1	⊆	⊆	NUM
ejpam-3838	271	2	tor2(c	tor2(c	NUM
ejpam-3838	271	3	)	)	PUNCT
ejpam-3838	271	4	⊆	⊆	NUM
ejpam-3838	271	5	res(res(c))⊥	res(res(c))⊥	NOUN
ejpam-3838	271	6	which	which	PRON
ejpam-3838	271	7	gives	give	VERB
ejpam-3838	271	8	the	the	DET
ejpam-3838	271	9	following	follow	VERB
ejpam-3838	271	10	lemma	lemma	PROPN
ejpam-3838	271	11	.	.	PUNCT
ejpam-3838	272	1	lemma	lemma	PROPN
ejpam-3838	272	2	7	7	X
ejpam-3838	272	3	.	.	PUNCT
ejpam-3838	273	1	let	let	VERB
ejpam-3838	273	2	c	c	PRON
ejpam-3838	273	3	be	be	AUX
ejpam-3838	273	4	a	a	DET
ejpam-3838	273	5	self	self	NOUN
ejpam-3838	273	6	-	-	PUNCT
ejpam-3838	273	7	orthogonal	orthogonal	ADJ
ejpam-3838	273	8	code	code	NOUN
ejpam-3838	273	9	over	over	ADP
ejpam-3838	273	10	r2	r2	PROPN
ejpam-3838	273	11	of	of	ADP
ejpam-3838	273	12	type	type	NOUN
ejpam-3838	273	13	{	{	PUNCT
ejpam-3838	273	14	k0	k0	PROPN
ejpam-3838	273	15	,	,	PUNCT
ejpam-3838	273	16	k1	k1	PROPN
ejpam-3838	273	17	,	,	PUNCT
ejpam-3838	273	18	k2	k2	NOUN
ejpam-3838	273	19	}	}	PUNCT
ejpam-3838	273	20	and	and	CCONJ
ejpam-3838	273	21	let	let	VERB
ejpam-3838	273	22	c1	c1	PROPN
ejpam-3838	273	23	=	=	SYM
ejpam-3838	273	24	res(c	res(c	PROPN
ejpam-3838	273	25	)	)	PUNCT
ejpam-3838	273	26	and	and	CCONJ
ejpam-3838	273	27	c2	c2	PROPN
ejpam-3838	273	28	=	=	PRON
ejpam-3838	273	29	tor2(c	tor2(c	NOUN
ejpam-3838	273	30	)	)	PUNCT
ejpam-3838	273	31	.	.	PUNCT
ejpam-3838	274	1	then	then	ADV
ejpam-3838	274	2	(	(	PUNCT
ejpam-3838	274	3	i	i	NOUN
ejpam-3838	274	4	)	)	PUNCT
ejpam-3838	274	5	c1	c1	PROPN
ejpam-3838	274	6	⊆	⊆	NUM
ejpam-3838	274	7	c⊥1	c⊥1	NOUN
ejpam-3838	274	8	,	,	PUNCT
ejpam-3838	274	9	(	(	PUNCT
ejpam-3838	274	10	ii	ii	NOUN
ejpam-3838	274	11	)	)	PUNCT
ejpam-3838	274	12	tor(c1	tor(c1	PROPN
ejpam-3838	274	13	)	)	PUNCT
ejpam-3838	274	14	⊆	⊆	NUM
ejpam-3838	274	15	tor(c1	tor(c1	NOUN
ejpam-3838	274	16	)	)	PUNCT
ejpam-3838	274	17	⊥	⊥	NOUN
ejpam-3838	274	18	,	,	PUNCT
ejpam-3838	274	19	and	and	CCONJ
ejpam-3838	274	20	(	(	PUNCT
ejpam-3838	274	21	iii	iii	NOUN
ejpam-3838	274	22	)	)	PUNCT
ejpam-3838	274	23	tor(c1	tor(c1	PROPN
ejpam-3838	274	24	)	)	PUNCT
ejpam-3838	274	25	⊆	⊆	NUM
ejpam-3838	274	26	c2	c2	PROPN
ejpam-3838	274	27	⊆	⊆	NUM
ejpam-3838	274	28	res(c1	res(c1	NOUN
ejpam-3838	274	29	)	)	PUNCT
ejpam-3838	274	30	⊥	⊥	NOUN
ejpam-3838	274	31	,	,	PUNCT
ejpam-3838	274	32	dim	dim	ADJ
ejpam-3838	274	33	c2	c2	PROPN
ejpam-3838	274	34	=	=	SYM
ejpam-3838	274	35	k0	k0	PROPN
ejpam-3838	274	36	+	+	CCONJ
ejpam-3838	274	37	k1	k1	NOUN
ejpam-3838	274	38	+	+	X
ejpam-3838	274	39	k2	k2	NOUN
ejpam-3838	274	40	.	.	PUNCT
ejpam-3838	275	1	7	7	X
ejpam-3838	275	2	.	.	X
ejpam-3838	275	3	codes	code	NOUN
ejpam-3838	275	4	over	over	ADP
ejpam-3838	275	5	fq	fq	PROPN
ejpam-3838	275	6	+	+	CCONJ
ejpam-3838	275	7	ufq	ufq	PROPN
ejpam-3838	275	8	+	+	PRON
ejpam-3838	275	9	u2fq	u2fq	PUNCT
ejpam-3838	275	10	with	with	ADP
ejpam-3838	275	11	prescribed	prescribe	VERB
ejpam-3838	275	12	u2	u2	NOUN
ejpam-3838	275	13	-	-	PUNCT
ejpam-3838	275	14	residue	residue	NOUN
ejpam-3838	275	15	and	and	CCONJ
ejpam-3838	275	16	torsion	torsion	NOUN
ejpam-3838	275	17	for	for	ADP
ejpam-3838	275	18	the	the	DET
ejpam-3838	275	19	rest	rest	NOUN
ejpam-3838	275	20	of	of	ADP
ejpam-3838	275	21	this	this	DET
ejpam-3838	275	22	chapter	chapter	NOUN
ejpam-3838	275	23	,	,	PUNCT
ejpam-3838	275	24	we	we	PRON
ejpam-3838	275	25	let	let	VERB
ejpam-3838	275	26	c1	c1	PROPN
ejpam-3838	275	27	be	be	AUX
ejpam-3838	275	28	a	a	DET
ejpam-3838	275	29	self	self	NOUN
ejpam-3838	275	30	-	-	PUNCT
ejpam-3838	275	31	orthogonal	orthogonal	ADJ
ejpam-3838	275	32	code	code	NOUN
ejpam-3838	275	33	over	over	ADP
ejpam-3838	275	34	r1	r1	PROPN
ejpam-3838	275	35	of	of	ADP
ejpam-3838	275	36	type	type	NOUN
ejpam-3838	275	37	{	{	PUNCT
ejpam-3838	275	38	k0	k0	PROPN
ejpam-3838	275	39	,	,	PUNCT
ejpam-3838	275	40	k1	k1	PROPN
ejpam-3838	275	41	}	}	PUNCT
ejpam-3838	275	42	such	such	ADJ
ejpam-3838	275	43	that	that	SCONJ
ejpam-3838	275	44	tor(c1	tor(c1	PROPN
ejpam-3838	275	45	)	)	PUNCT
ejpam-3838	275	46	is	be	AUX
ejpam-3838	275	47	self	self	NOUN
ejpam-3838	275	48	-	-	PUNCT
ejpam-3838	275	49	orthogonal	orthogonal	ADJ
ejpam-3838	275	50	.	.	PUNCT
ejpam-3838	276	1	we	we	PRON
ejpam-3838	276	2	assume	assume	VERB
ejpam-3838	276	3	without	without	ADP
ejpam-3838	276	4	loss	loss	NOUN
ejpam-3838	276	5	of	of	ADP
ejpam-3838	276	6	generality	generality	NOUN
ejpam-3838	276	7	that	that	PRON
ejpam-3838	276	8	c1	c1	PROPN
ejpam-3838	276	9	has	have	VERB
ejpam-3838	276	10	generator	generator	NOUN
ejpam-3838	276	11	matrix	matrix	NOUN
ejpam-3838	276	12	g1	g1	NOUN
ejpam-3838	276	13	=	=	PUNCT
ejpam-3838	277	1	[	[	PUNCT
ejpam-3838	277	2	ik0	ik0	VERB
ejpam-3838	277	3	a0	a0	NOUN
ejpam-3838	277	4	b0	b0	PROPN
ejpam-3838	277	5	+	+	CCONJ
ejpam-3838	277	6	ub1	ub1	PROPN
ejpam-3838	277	7	0	0	NUM
ejpam-3838	277	8	uik1	uik1	PROPN
ejpam-3838	277	9	ud1	ud1	NOUN
ejpam-3838	277	10	]	]	PUNCT
ejpam-3838	277	11	.	.	PUNCT
ejpam-3838	278	1	since	since	SCONJ
ejpam-3838	278	2	c1	c1	PROPN
ejpam-3838	278	3	is	be	AUX
ejpam-3838	278	4	self	self	NOUN
ejpam-3838	278	5	-	-	PUNCT
ejpam-3838	278	6	orthogonal	orthogonal	NOUN
ejpam-3838	278	7	,	,	PUNCT
ejpam-3838	278	8	we	we	PRON
ejpam-3838	278	9	have	have	AUX
ejpam-3838	278	10	ik0	ik0	VERB
ejpam-3838	279	1	+	+	PROPN
ejpam-3838	279	2	a0a	a0a	PROPN
ejpam-3838	279	3	t	t	NOUN
ejpam-3838	279	4	0	0	PUNCT
ejpam-3838	280	1	+	+	NOUN
ejpam-3838	280	2	b0b	b0b	X
ejpam-3838	280	3	t	t	X
ejpam-3838	280	4	0	0	PUNCT
ejpam-3838	281	1	+	+	CCONJ
ejpam-3838	281	2	u(a0a	u(a0a	PROPN
ejpam-3838	281	3	t	t	PROPN
ejpam-3838	281	4	1	1	NUM
ejpam-3838	281	5	+	+	PROPN
ejpam-3838	281	6	a1a	a1a	PROPN
ejpam-3838	281	7	t	t	NOUN
ejpam-3838	281	8	0	0	PUNCT
ejpam-3838	282	1	+	+	NOUN
ejpam-3838	282	2	b0b	b0b	PROPN
ejpam-3838	282	3	t	t	NOUN
ejpam-3838	282	4	1	1	NUM
ejpam-3838	282	5	+	+	NOUN
ejpam-3838	282	6	b1b	b1b	NOUN
ejpam-3838	282	7	t	t	PROPN
ejpam-3838	282	8	0	0	NUM
ejpam-3838	282	9	)	)	PUNCT
ejpam-3838	282	10	≡	≡	PROPN
ejpam-3838	282	11	0	0	PUNCT
ejpam-3838	282	12	(	(	PUNCT
ejpam-3838	282	13	u2	u2	NOUN
ejpam-3838	282	14	)	)	PUNCT
ejpam-3838	282	15	u(a0	u(a0	NOUN
ejpam-3838	283	1	+	+	VERB
ejpam-3838	283	2	b0d	b0d	PROPN
ejpam-3838	283	3	t	t	NOUN
ejpam-3838	283	4	1	1	NUM
ejpam-3838	283	5	)	)	PUNCT
ejpam-3838	283	6	≡	≡	PROPN
ejpam-3838	283	7	0	0	PUNCT
ejpam-3838	283	8	(	(	PUNCT
ejpam-3838	283	9	u2	u2	PROPN
ejpam-3838	283	10	)	)	PUNCT
ejpam-3838	283	11	which	which	PRON
ejpam-3838	283	12	are	be	AUX
ejpam-3838	283	13	equivalent	equivalent	ADJ
ejpam-3838	283	14	to	to	ADP
ejpam-3838	283	15	(	(	PUNCT
ejpam-3838	283	16	14	14	NUM
ejpam-3838	283	17	)	)	PUNCT
ejpam-3838	283	18	,	,	PUNCT
ejpam-3838	283	19	(	(	PUNCT
ejpam-3838	283	20	15	15	NUM
ejpam-3838	283	21	)	)	PUNCT
ejpam-3838	283	22	and	and	CCONJ
ejpam-3838	283	23	(	(	PUNCT
ejpam-3838	283	24	17	17	NUM
ejpam-3838	283	25	)	)	PUNCT
ejpam-3838	283	26	.	.	PUNCT
ejpam-3838	284	1	moreover	moreover	ADV
ejpam-3838	284	2	,	,	PUNCT
ejpam-3838	284	3	since	since	SCONJ
ejpam-3838	284	4	tor(c1	tor(c1	PROPN
ejpam-3838	284	5	)	)	PUNCT
ejpam-3838	284	6	is	be	AUX
ejpam-3838	284	7	self	self	NOUN
ejpam-3838	284	8	-	-	PUNCT
ejpam-3838	284	9	orthogonal	orthogonal	NOUN
ejpam-3838	284	10	,	,	PUNCT
ejpam-3838	284	11	we	we	PRON
ejpam-3838	284	12	have	have	AUX
ejpam-3838	284	13	ik0	ik0	VERB
ejpam-3838	285	1	+	+	PROPN
ejpam-3838	285	2	a0a	a0a	PROPN
ejpam-3838	285	3	t	t	NOUN
ejpam-3838	285	4	0	0	PUNCT
ejpam-3838	286	1	+	+	NOUN
ejpam-3838	286	2	b0b	b0b	PROPN
ejpam-3838	286	3	t	t	X
ejpam-3838	286	4	0	0	NUM
ejpam-3838	286	5	≡	≡	PROPN
ejpam-3838	286	6	0	0	NUM
ejpam-3838	286	7	(	(	PUNCT
ejpam-3838	286	8	u	u	NOUN
ejpam-3838	286	9	)	)	PUNCT
ejpam-3838	286	10	l.e	l.e	PROPN
ejpam-3838	286	11	.	.	PROPN
ejpam-3838	286	12	galvez	galvez	PROPN
ejpam-3838	286	13	,	,	PUNCT
ejpam-3838	286	14	r.a	r.a	PROPN
ejpam-3838	286	15	.	.	PROPN
ejpam-3838	286	16	betty	betty	PROPN
ejpam-3838	286	17	,	,	PUNCT
ejpam-3838	286	18	f.	f.	PROPN
ejpam-3838	286	19	nemenzo	nemenzo	PROPN
ejpam-3838	286	20	/	/	SYM
ejpam-3838	286	21	eur	eur	NOUN
ejpam-3838	286	22	.	.	PUNCT
ejpam-3838	287	1	j.	j.	PROPN
ejpam-3838	287	2	pure	pure	PROPN
ejpam-3838	287	3	appl	appl	PROPN
ejpam-3838	287	4	.	.	PROPN
ejpam-3838	287	5	math	math	PROPN
ejpam-3838	287	6	,	,	PUNCT
ejpam-3838	287	7	13	13	NUM
ejpam-3838	287	8	(	(	PUNCT
ejpam-3838	287	9	4	4	NUM
ejpam-3838	287	10	)	)	PUNCT
ejpam-3838	287	11	(	(	PUNCT
ejpam-3838	287	12	2020	2020	NUM
ejpam-3838	287	13	)	)	PUNCT
ejpam-3838	287	14	,	,	PUNCT
ejpam-3838	287	15	873	873	NUM
ejpam-3838	287	16	-	-	NUM
ejpam-3838	287	17	892	892	NUM
ejpam-3838	287	18	886	886	NUM
ejpam-3838	287	19	a0	a0	NOUN
ejpam-3838	287	20	+	+	PROPN
ejpam-3838	287	21	b0d	b0d	PROPN
ejpam-3838	287	22	t	t	PROPN
ejpam-3838	287	23	1	1	NUM
ejpam-3838	287	24	≡	≡	PROPN
ejpam-3838	287	25	0	0	NUM
ejpam-3838	287	26	(	(	PUNCT
ejpam-3838	287	27	u	u	NOUN
ejpam-3838	287	28	)	)	PUNCT
ejpam-3838	287	29	ik1	ik1	VERB
ejpam-3838	287	30	+	+	PROPN
ejpam-3838	287	31	d1d	d1d	PROPN
ejpam-3838	287	32	t	t	NOUN
ejpam-3838	287	33	1	1	NUM
ejpam-3838	287	34	≡	≡	PROPN
ejpam-3838	287	35	0	0	NUM
ejpam-3838	287	36	(	(	PUNCT
ejpam-3838	287	37	u	u	NOUN
ejpam-3838	287	38	)	)	PUNCT
ejpam-3838	287	39	which	which	PRON
ejpam-3838	287	40	are	be	AUX
ejpam-3838	287	41	equivalent	equivalent	ADJ
ejpam-3838	287	42	to	to	ADP
ejpam-3838	287	43	(	(	PUNCT
ejpam-3838	287	44	14	14	NUM
ejpam-3838	287	45	)	)	PUNCT
ejpam-3838	287	46	,	,	PUNCT
ejpam-3838	287	47	(	(	PUNCT
ejpam-3838	287	48	17	17	NUM
ejpam-3838	287	49	)	)	PUNCT
ejpam-3838	287	50	and	and	CCONJ
ejpam-3838	287	51	(	(	PUNCT
ejpam-3838	287	52	19	19	NUM
ejpam-3838	287	53	)	)	PUNCT
ejpam-3838	287	54	.	.	PUNCT
ejpam-3838	288	1	now	now	ADV
ejpam-3838	288	2	,	,	PUNCT
ejpam-3838	288	3	notice	notice	VERB
ejpam-3838	288	4	that	that	SCONJ
ejpam-3838	288	5	from	from	ADP
ejpam-3838	288	6	(	(	PUNCT
ejpam-3838	288	7	17	17	NUM
ejpam-3838	288	8	)	)	PUNCT
ejpam-3838	288	9	,	,	PUNCT
ejpam-3838	288	10	we	we	PRON
ejpam-3838	288	11	have	have	VERB
ejpam-3838	288	12	a0	a0	PROPN
ejpam-3838	288	13	≡	≡	PROPN
ejpam-3838	288	14	−b0d	−b0d	PROPN
ejpam-3838	289	1	t	t	PROPN
ejpam-3838	289	2	1	1	NUM
ejpam-3838	289	3	(	(	PUNCT
ejpam-3838	289	4	u	u	NOUN
ejpam-3838	289	5	)	)	PUNCT
ejpam-3838	289	6	.	.	PUNCT
ejpam-3838	290	1	by	by	ADP
ejpam-3838	290	2	(	(	PUNCT
ejpam-3838	290	3	14	14	NUM
ejpam-3838	290	4	)	)	PUNCT
ejpam-3838	290	5	,	,	PUNCT
ejpam-3838	290	6	we	we	PRON
ejpam-3838	290	7	have	have	AUX
ejpam-3838	290	8	ik0	ik0	VERB
ejpam-3838	290	9	+	+	ADV
ejpam-3838	290	10	b0d	b0d	PROPN
ejpam-3838	290	11	t	t	NOUN
ejpam-3838	290	12	1d1b	1d1b	NUM
ejpam-3838	290	13	t	t	NOUN
ejpam-3838	290	14	0	0	PUNCT
ejpam-3838	291	1	+	+	NOUN
ejpam-3838	291	2	b0b	b0b	PROPN
ejpam-3838	291	3	t	t	X
ejpam-3838	291	4	0	0	NUM
ejpam-3838	291	5	≡	≡	PROPN
ejpam-3838	291	6	0	0	NUM
ejpam-3838	291	7	(	(	PUNCT
ejpam-3838	291	8	u	u	NOUN
ejpam-3838	291	9	)	)	PUNCT
ejpam-3838	291	10	ik0	ik0	VERB
ejpam-3838	291	11	+	+	PROPN
ejpam-3838	291	12	b0	b0	NOUN
ejpam-3838	291	13	(	(	PUNCT
ejpam-3838	291	14	dt	dt	NOUN
ejpam-3838	291	15	1d1b	1d1b	NUM
ejpam-3838	291	16	+	+	CCONJ
ejpam-3838	291	17	ik0	ik0	NOUN
ejpam-3838	291	18	)	)	PUNCT
ejpam-3838	291	19	bt	bt	NOUN
ejpam-3838	291	20	0	0	NUM
ejpam-3838	291	21	≡	≡	PROPN
ejpam-3838	291	22	0	0	NUM
ejpam-3838	291	23	(	(	PUNCT
ejpam-3838	291	24	u	u	NOUN
ejpam-3838	291	25	)	)	PUNCT
ejpam-3838	291	26	which	which	PRON
ejpam-3838	291	27	implies	imply	VERB
ejpam-3838	291	28	b0	b0	NOUN
ejpam-3838	291	29	is	be	AUX
ejpam-3838	291	30	of	of	ADP
ejpam-3838	291	31	full	full	ADJ
ejpam-3838	291	32	row	row	NOUN
ejpam-3838	291	33	rank	rank	NOUN
ejpam-3838	291	34	.	.	PUNCT
ejpam-3838	292	1	we	we	PRON
ejpam-3838	292	2	start	start	VERB
ejpam-3838	292	3	by	by	ADP
ejpam-3838	292	4	counting	count	VERB
ejpam-3838	292	5	the	the	DET
ejpam-3838	292	6	number	number	NOUN
ejpam-3838	292	7	of	of	ADP
ejpam-3838	292	8	self	self	NOUN
ejpam-3838	292	9	-	-	PUNCT
ejpam-3838	292	10	orthogonal	orthogonal	ADJ
ejpam-3838	292	11	codes	code	NOUN
ejpam-3838	292	12	c	c	PROPN
ejpam-3838	292	13	of	of	ADP
ejpam-3838	292	14	type	type	NOUN
ejpam-3838	292	15	{	{	PUNCT
ejpam-3838	292	16	k0	k0	PROPN
ejpam-3838	292	17	,	,	PUNCT
ejpam-3838	292	18	k1	k1	NOUN
ejpam-3838	292	19	,	,	PUNCT
ejpam-3838	292	20	0	0	NUM
ejpam-3838	292	21	}	}	PUNCT
ejpam-3838	292	22	such	such	ADJ
ejpam-3838	292	23	that	that	DET
ejpam-3838	292	24	res(c	res(c	PROPN
ejpam-3838	292	25	)	)	PUNCT
ejpam-3838	292	26	=	=	SYM
ejpam-3838	292	27	c1	c1	PROPN
ejpam-3838	292	28	.	.	PUNCT
ejpam-3838	293	1	similar	similar	ADJ
ejpam-3838	293	2	to	to	ADP
ejpam-3838	293	3	what	what	PRON
ejpam-3838	293	4	we	we	PRON
ejpam-3838	293	5	did	do	AUX
ejpam-3838	293	6	in	in	ADP
ejpam-3838	293	7	the	the	DET
ejpam-3838	293	8	previous	previous	ADJ
ejpam-3838	293	9	chapter	chapter	NOUN
ejpam-3838	293	10	,	,	PUNCT
ejpam-3838	293	11	we	we	PRON
ejpam-3838	293	12	first	first	ADV
ejpam-3838	293	13	exhibit	exhibit	VERB
ejpam-3838	293	14	the	the	DET
ejpam-3838	293	15	generator	generator	NOUN
ejpam-3838	293	16	matrix	matrix	NOUN
ejpam-3838	293	17	of	of	ADP
ejpam-3838	293	18	such	such	ADJ
ejpam-3838	293	19	code	code	PROPN
ejpam-3838	293	20	c.	c.	PROPN
ejpam-3838	293	21	lemma	lemma	PROPN
ejpam-3838	294	1	8	8	NUM
ejpam-3838	294	2	.	.	PUNCT
ejpam-3838	295	1	if	if	SCONJ
ejpam-3838	295	2	c	c	PROPN
ejpam-3838	295	3	is	be	AUX
ejpam-3838	295	4	a	a	DET
ejpam-3838	295	5	code	code	NOUN
ejpam-3838	295	6	over	over	ADP
ejpam-3838	295	7	r2	r2	PROPN
ejpam-3838	295	8	of	of	ADP
ejpam-3838	295	9	type	type	NOUN
ejpam-3838	295	10	{	{	PUNCT
ejpam-3838	295	11	k0	k0	PROPN
ejpam-3838	295	12	,	,	PUNCT
ejpam-3838	295	13	k1	k1	NOUN
ejpam-3838	295	14	,	,	PUNCT
ejpam-3838	295	15	0	0	NUM
ejpam-3838	295	16	}	}	PUNCT
ejpam-3838	295	17	and	and	CCONJ
ejpam-3838	295	18	res(c	res(c	ADJ
ejpam-3838	295	19	)	)	PUNCT
ejpam-3838	295	20	=	=	SYM
ejpam-3838	295	21	c1	c1	PROPN
ejpam-3838	295	22	,	,	PUNCT
ejpam-3838	295	23	then	then	ADV
ejpam-3838	295	24	there	there	PRON
ejpam-3838	295	25	exist	exist	VERB
ejpam-3838	295	26	matrices	matrix	NOUN
ejpam-3838	295	27	n0	n0	X
ejpam-3838	295	28	∈mk0×(n−k0−k1)(fq	∈mk0×(n−k0−k1)(fq	ADJ
ejpam-3838	295	29	)	)	PUNCT
ejpam-3838	295	30	and	and	CCONJ
ejpam-3838	295	31	n1	n1	PROPN
ejpam-3838	295	32	∈mk1×(n−k0−k1)(fq	∈mk1×(n−k0−k1)(fq	ADJ
ejpam-3838	295	33	)	)	PUNCT
ejpam-3838	295	34	such	such	ADJ
ejpam-3838	295	35	that	that	SCONJ
ejpam-3838	295	36	[	[	PUNCT
ejpam-3838	295	37	ik0	ik0	VERB
ejpam-3838	295	38	a0	a0	NOUN
ejpam-3838	295	39	b0	b0	PROPN
ejpam-3838	295	40	+	+	CCONJ
ejpam-3838	295	41	ub1	ub1	PROPN
ejpam-3838	295	42	+	+	CCONJ
ejpam-3838	295	43	u2n0	u2n0	ADJ
ejpam-3838	295	44	0	0	NUM
ejpam-3838	295	45	uik1	uik1	NOUN
ejpam-3838	295	46	ud1	ud1	NOUN
ejpam-3838	295	47	+	+	CCONJ
ejpam-3838	295	48	u2n1	u2n1	PROPN
ejpam-3838	295	49	]	]	PUNCT
ejpam-3838	295	50	(	(	PUNCT
ejpam-3838	295	51	21	21	NUM
ejpam-3838	295	52	)	)	PUNCT
ejpam-3838	295	53	is	be	AUX
ejpam-3838	295	54	a	a	DET
ejpam-3838	295	55	generator	generator	NOUN
ejpam-3838	295	56	matrix	matrix	NOUN
ejpam-3838	295	57	for	for	ADP
ejpam-3838	295	58	c.	c.	NOUN
ejpam-3838	295	59	the	the	DET
ejpam-3838	295	60	matrices	matrix	NOUN
ejpam-3838	295	61	n0	n0	NOUN
ejpam-3838	295	62	and	and	CCONJ
ejpam-3838	295	63	n1	n1	NOUN
ejpam-3838	295	64	are	be	AUX
ejpam-3838	295	65	unique	unique	ADJ
ejpam-3838	295	66	.	.	PUNCT
ejpam-3838	296	1	proof	proof	NOUN
ejpam-3838	296	2	.	.	PUNCT
ejpam-3838	297	1	if	if	SCONJ
ejpam-3838	297	2	c	c	PROPN
ejpam-3838	297	3	is	be	AUX
ejpam-3838	297	4	a	a	DET
ejpam-3838	297	5	code	code	NOUN
ejpam-3838	297	6	over	over	ADP
ejpam-3838	297	7	r2	r2	PROPN
ejpam-3838	297	8	of	of	ADP
ejpam-3838	297	9	type	type	NOUN
ejpam-3838	297	10	{	{	PUNCT
ejpam-3838	297	11	k0	k0	PROPN
ejpam-3838	297	12	,	,	PUNCT
ejpam-3838	297	13	k1	k1	NOUN
ejpam-3838	297	14	,	,	PUNCT
ejpam-3838	297	15	0	0	NUM
ejpam-3838	297	16	}	}	PUNCT
ejpam-3838	297	17	such	such	ADJ
ejpam-3838	297	18	that	that	DET
ejpam-3838	297	19	res(c	res(c	PROPN
ejpam-3838	297	20	)	)	PUNCT
ejpam-3838	297	21	=	=	SYM
ejpam-3838	297	22	c1	c1	PROPN
ejpam-3838	297	23	,	,	PUNCT
ejpam-3838	297	24	then	then	ADV
ejpam-3838	297	25	for	for	ADP
ejpam-3838	297	26	some	some	DET
ejpam-3838	297	27	matrices	matrix	NOUN
ejpam-3838	297	28	m1,m2,m3,m4	m1,m2,m3,m4	PROPN
ejpam-3838	297	29	and	and	CCONJ
ejpam-3838	297	30	m5	m5	PROPN
ejpam-3838	297	31	over	over	ADP
ejpam-3838	297	32	fq	fq	PROPN
ejpam-3838	297	33	of	of	ADP
ejpam-3838	297	34	appropriate	appropriate	ADJ
ejpam-3838	297	35	sizes	size	NOUN
ejpam-3838	297	36	,	,	PUNCT
ejpam-3838	297	37	rk0+k12	rk0+k12	NOUN
ejpam-3838	297	38	[	[	PUNCT
ejpam-3838	297	39	ik0	ik0	VERB
ejpam-3838	297	40	+	+	CCONJ
ejpam-3838	297	41	u2m1	u2m1	ADP
ejpam-3838	297	42	a0	a0	NOUN
ejpam-3838	297	43	+	+	CCONJ
ejpam-3838	297	44	u2m2	u2m2	X
ejpam-3838	297	45	b0	b0	NOUN
ejpam-3838	297	46	+	+	CCONJ
ejpam-3838	297	47	ub1	ub1	PROPN
ejpam-3838	298	1	+	+	CCONJ
ejpam-3838	298	2	u2m3	u2m3	PROPN
ejpam-3838	298	3	0	0	NUM
ejpam-3838	298	4	ik1	ik1	ADJ
ejpam-3838	298	5	+	+	CCONJ
ejpam-3838	298	6	u2m4	u2m4	X
ejpam-3838	298	7	d1	d1	PROPN
ejpam-3838	298	8	+	+	CCONJ
ejpam-3838	298	9	u2m5	u2m5	X
ejpam-3838	298	10	]	]	X
ejpam-3838	298	11	⊆	⊆	NUM
ejpam-3838	298	12	c.	c.	NOUN
ejpam-3838	298	13	applying	apply	VERB
ejpam-3838	298	14	elementary	elementary	ADJ
ejpam-3838	298	15	row	row	NOUN
ejpam-3838	298	16	operations	operation	NOUN
ejpam-3838	298	17	,	,	PUNCT
ejpam-3838	298	18	[	[	PUNCT
ejpam-3838	298	19	ik0	ik0	NOUN
ejpam-3838	298	20	−	−	PROPN
ejpam-3838	298	21	u2m1	u2m1	ADP
ejpam-3838	298	22	0	0	NUM
ejpam-3838	298	23	0	0	NUM
ejpam-3838	299	1	ik1	ik1	ADJ
ejpam-3838	299	2	−	−	PROPN
ejpam-3838	300	1	u2m4	u2m4	PUNCT
ejpam-3838	300	2	]	]	PUNCT
ejpam-3838	300	3	[	[	PUNCT
ejpam-3838	300	4	ik0	ik0	X
ejpam-3838	300	5	+	+	CCONJ
ejpam-3838	300	6	u2m1	u2m1	ADP
ejpam-3838	300	7	a0	a0	NOUN
ejpam-3838	300	8	+	+	CCONJ
ejpam-3838	300	9	u2m2	u2m2	X
ejpam-3838	300	10	b0	b0	NOUN
ejpam-3838	300	11	+	+	CCONJ
ejpam-3838	300	12	ub1	ub1	PROPN
ejpam-3838	301	1	+	+	CCONJ
ejpam-3838	302	1	u2m3	u2m3	PROPN
ejpam-3838	303	1	0	0	NUM
ejpam-3838	304	1	ik1	ik1	ADJ
ejpam-3838	304	2	+	+	CCONJ
ejpam-3838	304	3	u2m4	u2m4	X
ejpam-3838	304	4	d1	d1	PROPN
ejpam-3838	304	5	+	+	CCONJ
ejpam-3838	304	6	u2m5	u2m5	X
ejpam-3838	304	7	]	]	X
ejpam-3838	304	8	=	=	X
ejpam-3838	305	1	[	[	PUNCT
ejpam-3838	305	2	ik0	ik0	PROPN
ejpam-3838	305	3	a0	a0	NOUN
ejpam-3838	305	4	+	+	CCONJ
ejpam-3838	305	5	u2(m2	u2(m2	ADP
ejpam-3838	305	6	−m1a0	−m1a0	NOUN
ejpam-3838	305	7	)	)	PUNCT
ejpam-3838	305	8	b0	b0	NOUN
ejpam-3838	305	9	+	+	CCONJ
ejpam-3838	305	10	ub1	ub1	PROPN
ejpam-3838	305	11	+	+	CCONJ
ejpam-3838	305	12	u2(m3	u2(m3	ADJ
ejpam-3838	305	13	−m1b0	−m1b0	NOUN
ejpam-3838	305	14	)	)	PUNCT
ejpam-3838	305	15	0	0	PUNCT
ejpam-3838	306	1	ik1	ik1	VERB
ejpam-3838	306	2	d1	d1	PROPN
ejpam-3838	306	3	+	+	CCONJ
ejpam-3838	306	4	u2(m5	u2(m5	PROPN
ejpam-3838	306	5	−m4d1	−m4d1	NUM
ejpam-3838	306	6	)	)	PUNCT
ejpam-3838	306	7	]	]	PUNCT
ejpam-3838	306	8	and	and	CCONJ
ejpam-3838	306	9	[	[	PUNCT
ejpam-3838	306	10	ik0	ik0	NOUN
ejpam-3838	306	11	−u2(m2	−u2(m2	NOUN
ejpam-3838	306	12	−m1a0	−m1a0	NOUN
ejpam-3838	306	13	)	)	PUNCT
ejpam-3838	306	14	0	0	PUNCT
ejpam-3838	307	1	ik1	ik1	VERB
ejpam-3838	307	2	]	]	X
ejpam-3838	307	3	[	[	PUNCT
ejpam-3838	307	4	ik0	ik0	PROPN
ejpam-3838	307	5	a0	a0	NOUN
ejpam-3838	307	6	+	+	CCONJ
ejpam-3838	307	7	u2(m2	u2(m2	ADP
ejpam-3838	307	8	−m1a0	−m1a0	NOUN
ejpam-3838	307	9	)	)	PUNCT
ejpam-3838	307	10	b0	b0	NOUN
ejpam-3838	307	11	+	+	CCONJ
ejpam-3838	307	12	ub1	ub1	PROPN
ejpam-3838	307	13	+	+	CCONJ
ejpam-3838	307	14	u2(m3	u2(m3	ADJ
ejpam-3838	307	15	−m1b0	−m1b0	NOUN
ejpam-3838	307	16	)	)	PUNCT
ejpam-3838	307	17	0	0	PUNCT
ejpam-3838	308	1	ik1	ik1	VERB
ejpam-3838	308	2	d1	d1	PROPN
ejpam-3838	308	3	+	+	CCONJ
ejpam-3838	308	4	u2(m5	u2(m5	PROPN
ejpam-3838	308	5	−m4d1	−m4d1	NUM
ejpam-3838	308	6	)	)	PUNCT
ejpam-3838	308	7	]	]	PUNCT
ejpam-3838	309	1	=	=	PUNCT
ejpam-3838	309	2	[	[	PUNCT
ejpam-3838	309	3	ik0	ik0	VERB
ejpam-3838	309	4	a0	a0	NOUN
ejpam-3838	309	5	b0	b0	PROPN
ejpam-3838	309	6	+	+	CCONJ
ejpam-3838	309	7	ub1	ub1	PROPN
ejpam-3838	309	8	+	+	CCONJ
ejpam-3838	309	9	u2(m3	u2(m3	ADJ
ejpam-3838	309	10	−m1b0	−m1b0	NUM
ejpam-3838	309	11	−m2d1	−m2d1	NOUN
ejpam-3838	309	12	+	+	NOUN
ejpam-3838	309	13	m1a0d1	m1a0d1	NOUN
ejpam-3838	309	14	)	)	PUNCT
ejpam-3838	309	15	0	0	PUNCT
ejpam-3838	310	1	ik1	ik1	VERB
ejpam-3838	310	2	d1	d1	PROPN
ejpam-3838	310	3	+	+	CCONJ
ejpam-3838	310	4	u2(m5	u2(m5	PROPN
ejpam-3838	310	5	−m4d1	−m4d1	NUM
ejpam-3838	310	6	)	)	PUNCT
ejpam-3838	310	7	]	]	PUNCT
ejpam-3838	310	8	.	.	PUNCT
ejpam-3838	311	1	l.e	l.e	PROPN
ejpam-3838	311	2	.	.	PROPN
ejpam-3838	311	3	galvez	galvez	PROPN
ejpam-3838	311	4	,	,	PUNCT
ejpam-3838	311	5	r.a	r.a	PROPN
ejpam-3838	311	6	.	.	PROPN
ejpam-3838	311	7	betty	betty	PROPN
ejpam-3838	311	8	,	,	PUNCT
ejpam-3838	311	9	f.	f.	PROPN
ejpam-3838	311	10	nemenzo	nemenzo	PROPN
ejpam-3838	311	11	/	/	SYM
ejpam-3838	311	12	eur	eur	NOUN
ejpam-3838	311	13	.	.	PUNCT
ejpam-3838	312	1	j.	j.	PROPN
ejpam-3838	312	2	pure	pure	PROPN
ejpam-3838	312	3	appl	appl	PROPN
ejpam-3838	312	4	.	.	PROPN
ejpam-3838	312	5	math	math	PROPN
ejpam-3838	312	6	,	,	PUNCT
ejpam-3838	312	7	13	13	NUM
ejpam-3838	312	8	(	(	PUNCT
ejpam-3838	312	9	4	4	NUM
ejpam-3838	312	10	)	)	PUNCT
ejpam-3838	312	11	(	(	PUNCT
ejpam-3838	312	12	2020	2020	NUM
ejpam-3838	312	13	)	)	PUNCT
ejpam-3838	312	14	,	,	PUNCT
ejpam-3838	312	15	873	873	NUM
ejpam-3838	312	16	-	-	NUM
ejpam-3838	312	17	892	892	NUM
ejpam-3838	312	18	887	887	NUM
ejpam-3838	312	19	letting	let	VERB
ejpam-3838	312	20	n0	n0	X
ejpam-3838	312	21	=	=	SYM
ejpam-3838	312	22	m3	m3	PROPN
ejpam-3838	313	1	−m1b0	−m1b0	AUX
ejpam-3838	313	2	−m2d1	−m2d1	PROPN
ejpam-3838	313	3	+	+	NOUN
ejpam-3838	313	4	m1a0d1	m1a0d1	PROPN
ejpam-3838	313	5	n1	n1	NOUN
ejpam-3838	313	6	=	=	SYM
ejpam-3838	313	7	m5	m5	PROPN
ejpam-3838	313	8	−m4d1	−m4d1	PROPN
ejpam-3838	313	9	,	,	PUNCT
ejpam-3838	313	10	we	we	PRON
ejpam-3838	313	11	have	have	AUX
ejpam-3838	313	12	rk0+k12	rk0+k12	VERB
ejpam-3838	313	13	[	[	PUNCT
ejpam-3838	313	14	ik0	ik0	VERB
ejpam-3838	313	15	a0	a0	NOUN
ejpam-3838	313	16	b0	b0	PROPN
ejpam-3838	313	17	+	+	CCONJ
ejpam-3838	313	18	ub1	ub1	PROPN
ejpam-3838	313	19	+	+	CCONJ
ejpam-3838	313	20	u2n0	u2n0	ADJ
ejpam-3838	313	21	0	0	NUM
ejpam-3838	313	22	uik1	uik1	NOUN
ejpam-3838	313	23	ud1	ud1	NOUN
ejpam-3838	313	24	+	+	CCONJ
ejpam-3838	313	25	u2n1	u2n1	SYM
ejpam-3838	313	26	]	]	PUNCT
ejpam-3838	313	27	⊆	⊆	NUM
ejpam-3838	313	28	c.	c.	PROPN
ejpam-3838	313	29	therefore	therefore	ADV
ejpam-3838	313	30	,	,	PUNCT
ejpam-3838	313	31	|c|	|c|	PROPN
ejpam-3838	313	32	≥	≥	NOUN
ejpam-3838	313	33	∣∣∣∣rk0+k12	∣∣∣∣rk0+k12	NOUN
ejpam-3838	313	34	[	[	PUNCT
ejpam-3838	313	35	ik0	ik0	PROPN
ejpam-3838	313	36	a0	a0	NOUN
ejpam-3838	313	37	b0	b0	PROPN
ejpam-3838	313	38	+	+	CCONJ
ejpam-3838	313	39	ub1	ub1	PROPN
ejpam-3838	313	40	+	+	CCONJ
ejpam-3838	313	41	u2n0	u2n0	ADJ
ejpam-3838	313	42	0	0	NUM
ejpam-3838	313	43	uik1	uik1	NOUN
ejpam-3838	313	44	ud1	ud1	NOUN
ejpam-3838	313	45	+	+	CCONJ
ejpam-3838	313	46	u2n1	u2n1	NOUN
ejpam-3838	313	47	]	]	PUNCT
ejpam-3838	313	48	∣∣∣∣	∣∣∣∣	NOUN
ejpam-3838	313	49	=	=	SYM
ejpam-3838	313	50	qk0+k1q2k0+k1	qk0+k1q2k0+k1	PROPN
ejpam-3838	313	51	=	=	PUNCT
ejpam-3838	313	52	q3k0	q3k0	X
ejpam-3838	313	53	+	+	NOUN
ejpam-3838	313	54	2k1	2k1	NUM
ejpam-3838	313	55	=	=	NOUN
ejpam-3838	313	56	|c|	|c|	PROPN
ejpam-3838	313	57	and	and	CCONJ
ejpam-3838	313	58	hence	hence	ADV
ejpam-3838	313	59	,	,	PUNCT
ejpam-3838	313	60	(	(	PUNCT
ejpam-3838	313	61	21	21	NUM
ejpam-3838	313	62	)	)	PUNCT
ejpam-3838	313	63	is	be	AUX
ejpam-3838	313	64	a	a	DET
ejpam-3838	313	65	generator	generator	NOUN
ejpam-3838	313	66	matrix	matrix	NOUN
ejpam-3838	313	67	for	for	ADP
ejpam-3838	313	68	c.	c.	PROPN
ejpam-3838	313	69	next	next	ADV
ejpam-3838	313	70	,	,	PUNCT
ejpam-3838	313	71	we	we	PRON
ejpam-3838	313	72	show	show	VERB
ejpam-3838	313	73	uniqueness	uniqueness	NOUN
ejpam-3838	313	74	of	of	ADP
ejpam-3838	313	75	the	the	DET
ejpam-3838	313	76	matrices	matrix	NOUN
ejpam-3838	313	77	n0	n0	NOUN
ejpam-3838	313	78	and	and	CCONJ
ejpam-3838	313	79	n1	n1	PROPN
ejpam-3838	313	80	over	over	ADP
ejpam-3838	313	81	fq	fq	PROPN
ejpam-3838	313	82	.	.	PROPN
ejpam-3838	313	83	suppose	suppose	VERB
ejpam-3838	313	84	there	there	PRON
ejpam-3838	313	85	exist	exist	VERB
ejpam-3838	313	86	matrices	matrix	NOUN
ejpam-3838	313	87	n	n	PRON
ejpam-3838	313	88	′0	′0	NOUN
ejpam-3838	313	89	∈mk0×(n−k0−k1)(fq	∈mk0×(n−k0−k1)(fq	ADJ
ejpam-3838	313	90	)	)	PUNCT
ejpam-3838	313	91	and	and	CCONJ
ejpam-3838	313	92	n	n	CCONJ
ejpam-3838	313	93	′1	′1	ADP
ejpam-3838	313	94	∈mk1×(n−k0−k1)(fq	∈mk1×(n−k0−k1)(fq	ADJ
ejpam-3838	313	95	)	)	PUNCT
ejpam-3838	313	96	such	such	ADJ
ejpam-3838	313	97	that	that	SCONJ
ejpam-3838	313	98	rk0+k12	rk0+k12	NOUN
ejpam-3838	313	99	[	[	PUNCT
ejpam-3838	313	100	ik0	ik0	NOUN
ejpam-3838	313	101	a0	a0	NOUN
ejpam-3838	313	102	b0	b0	PROPN
ejpam-3838	313	103	+	+	CCONJ
ejpam-3838	313	104	ub1	ub1	PROPN
ejpam-3838	313	105	+	+	CCONJ
ejpam-3838	313	106	u2n	u2n	PROPN
ejpam-3838	313	107	′0	′0	NOUN
ejpam-3838	313	108	0	0	NUM
ejpam-3838	313	109	uik1	uik1	PROPN
ejpam-3838	313	110	ud1	ud1	NOUN
ejpam-3838	313	111	+	+	CCONJ
ejpam-3838	313	112	u2n	u2n	NOUN
ejpam-3838	313	113	′1	′1	NOUN
ejpam-3838	313	114	]	]	PUNCT
ejpam-3838	313	115	=	=	PUNCT
ejpam-3838	314	1	rk0+k12	rk0+k12	NOUN
ejpam-3838	314	2	[	[	PUNCT
ejpam-3838	314	3	ik0	ik0	NOUN
ejpam-3838	314	4	a0	a0	NOUN
ejpam-3838	314	5	b0	b0	PROPN
ejpam-3838	314	6	+	+	CCONJ
ejpam-3838	314	7	ub1	ub1	PROPN
ejpam-3838	314	8	+	+	CCONJ
ejpam-3838	314	9	u2n0	u2n0	ADJ
ejpam-3838	314	10	0	0	NUM
ejpam-3838	314	11	uik1	uik1	NOUN
ejpam-3838	314	12	ud1	ud1	NOUN
ejpam-3838	314	13	+	+	CCONJ
ejpam-3838	314	14	u2n1	u2n1	NOUN
ejpam-3838	314	15	.	.	NOUN
ejpam-3838	314	16	]	]	PUNCT
ejpam-3838	314	17	.	.	PUNCT
ejpam-3838	315	1	this	this	PRON
ejpam-3838	315	2	means	mean	VERB
ejpam-3838	315	3	that	that	SCONJ
ejpam-3838	315	4	b0	b0	NOUN
ejpam-3838	315	5	+	+	CCONJ
ejpam-3838	315	6	ub1	ub1	PROPN
ejpam-3838	315	7	+	+	CCONJ
ejpam-3838	315	8	u2n	u2n	PROPN
ejpam-3838	315	9	′0	′0	NOUN
ejpam-3838	315	10	≡	≡	PROPN
ejpam-3838	315	11	b0	b0	PROPN
ejpam-3838	315	12	+	+	CCONJ
ejpam-3838	315	13	ub1	ub1	PROPN
ejpam-3838	315	14	+	+	CCONJ
ejpam-3838	315	15	u2n0	u2n0	ADJ
ejpam-3838	315	16	(	(	PUNCT
ejpam-3838	315	17	u3	u3	NOUN
ejpam-3838	315	18	)	)	PUNCT
ejpam-3838	315	19	ud1	ud1	NOUN
ejpam-3838	315	20	+	+	CCONJ
ejpam-3838	315	21	u2n	u2n	NOUN
ejpam-3838	315	22	′1	′1	NOUN
ejpam-3838	315	23	≡	≡	PROPN
ejpam-3838	315	24	ud1	ud1	VERB
ejpam-3838	315	25	+	+	CCONJ
ejpam-3838	316	1	u2n1	u2n1	PROPN
ejpam-3838	316	2	(	(	PUNCT
ejpam-3838	316	3	u3	u3	PROPN
ejpam-3838	316	4	)	)	PUNCT
ejpam-3838	316	5	which	which	PRON
ejpam-3838	316	6	imply	imply	VERB
ejpam-3838	316	7	that	that	SCONJ
ejpam-3838	316	8	n	n	NUM
ejpam-3838	316	9	′0	′0	PROPN
ejpam-3838	316	10	≡	≡	PROPN
ejpam-3838	316	11	n0	n0	X
ejpam-3838	316	12	(	(	PUNCT
ejpam-3838	316	13	u	u	NOUN
ejpam-3838	316	14	)	)	PUNCT
ejpam-3838	316	15	n	n	CCONJ
ejpam-3838	316	16	′1	′1	PROPN
ejpam-3838	316	17	≡	≡	PROPN
ejpam-3838	316	18	n1	n1	PROPN
ejpam-3838	316	19	(	(	PUNCT
ejpam-3838	316	20	u	u	NOUN
ejpam-3838	316	21	)	)	PUNCT
ejpam-3838	316	22	and	and	CCONJ
ejpam-3838	316	23	hence	hence	ADV
ejpam-3838	316	24	,	,	PUNCT
ejpam-3838	316	25	n0	n0	ADJ
ejpam-3838	316	26	and	and	CCONJ
ejpam-3838	316	27	n1	n1	NOUN
ejpam-3838	316	28	are	be	AUX
ejpam-3838	316	29	unique	unique	ADJ
ejpam-3838	316	30	.	.	PUNCT
ejpam-3838	317	1	�	�	PROPN
ejpam-3838	317	2	this	this	PRON
ejpam-3838	317	3	shows	show	VERB
ejpam-3838	317	4	that	that	SCONJ
ejpam-3838	317	5	the	the	DET
ejpam-3838	317	6	number	number	NOUN
ejpam-3838	317	7	of	of	ADP
ejpam-3838	317	8	self	self	NOUN
ejpam-3838	317	9	-	-	PUNCT
ejpam-3838	317	10	orthogonal	orthogonal	ADJ
ejpam-3838	317	11	codes	code	NOUN
ejpam-3838	317	12	c	c	NOUN
ejpam-3838	317	13	over	over	ADP
ejpam-3838	317	14	r2	r2	PROPN
ejpam-3838	317	15	of	of	ADP
ejpam-3838	317	16	type	type	NOUN
ejpam-3838	317	17	{	{	PUNCT
ejpam-3838	317	18	k0	k0	PROPN
ejpam-3838	317	19	,	,	PUNCT
ejpam-3838	317	20	k1	k1	NOUN
ejpam-3838	317	21	,	,	PUNCT
ejpam-3838	317	22	0	0	NUM
ejpam-3838	317	23	}	}	PUNCT
ejpam-3838	317	24	with	with	ADP
ejpam-3838	317	25	res(c	res(c	ADJ
ejpam-3838	317	26	)	)	PUNCT
ejpam-3838	317	27	=	=	SYM
ejpam-3838	317	28	c1	c1	PROPN
ejpam-3838	317	29	is	be	AUX
ejpam-3838	317	30	determined	determine	VERB
ejpam-3838	317	31	by	by	ADP
ejpam-3838	317	32	the	the	DET
ejpam-3838	317	33	number	number	NOUN
ejpam-3838	317	34	of	of	ADP
ejpam-3838	317	35	such	such	ADJ
ejpam-3838	317	36	matrices	matrix	NOUN
ejpam-3838	317	37	n0	n0	ADJ
ejpam-3838	317	38	and	and	CCONJ
ejpam-3838	317	39	n1	n1	PROPN
ejpam-3838	317	40	,	,	PUNCT
ejpam-3838	317	41	which	which	PRON
ejpam-3838	317	42	will	will	AUX
ejpam-3838	317	43	be	be	AUX
ejpam-3838	317	44	given	give	VERB
ejpam-3838	317	45	in	in	ADP
ejpam-3838	317	46	the	the	DET
ejpam-3838	317	47	next	next	ADJ
ejpam-3838	317	48	lemma	lemma	PROPN
ejpam-3838	317	49	.	.	PUNCT
ejpam-3838	318	1	lemma	lemma	PROPN
ejpam-3838	318	2	9	9	NUM
ejpam-3838	318	3	.	.	PUNCT
ejpam-3838	319	1	the	the	DET
ejpam-3838	319	2	number	number	NOUN
ejpam-3838	319	3	of	of	ADP
ejpam-3838	319	4	self	self	NOUN
ejpam-3838	319	5	-	-	PUNCT
ejpam-3838	319	6	orthogonal	orthogonal	ADJ
ejpam-3838	319	7	codes	code	NOUN
ejpam-3838	319	8	c	c	PROPN
ejpam-3838	319	9	of	of	ADP
ejpam-3838	319	10	type	type	NOUN
ejpam-3838	319	11	{	{	PUNCT
ejpam-3838	319	12	k0	k0	PROPN
ejpam-3838	319	13	,	,	PUNCT
ejpam-3838	319	14	k1	k1	NOUN
ejpam-3838	319	15	,	,	PUNCT
ejpam-3838	319	16	0	0	NUM
ejpam-3838	319	17	}	}	PUNCT
ejpam-3838	319	18	over	over	ADP
ejpam-3838	319	19	r2	r2	NOUN
ejpam-3838	319	20	such	such	ADJ
ejpam-3838	319	21	that	that	PRON
ejpam-3838	319	22	res(c	res(c	PROPN
ejpam-3838	319	23	)	)	PUNCT
ejpam-3838	319	24	=	=	PROPN
ejpam-3838	319	25	c1	c1	PROPN
ejpam-3838	319	26	is	be	AUX
ejpam-3838	319	27	q(k0+k1)(n−k0−k1)−k0(k0	q(k0+k1)(n−k0−k1)−k0(k0	PROPN
ejpam-3838	319	28	+	+	PROPN
ejpam-3838	319	29	1)/2−k0k1	1)/2−k0k1	PROPN
ejpam-3838	319	30	.	.	PUNCT
ejpam-3838	320	1	l.e	l.e	PROPN
ejpam-3838	320	2	.	.	PROPN
ejpam-3838	320	3	galvez	galvez	PROPN
ejpam-3838	320	4	,	,	PUNCT
ejpam-3838	320	5	r.a	r.a	PROPN
ejpam-3838	320	6	.	.	PROPN
ejpam-3838	320	7	betty	betty	PROPN
ejpam-3838	320	8	,	,	PUNCT
ejpam-3838	320	9	f.	f.	PROPN
ejpam-3838	320	10	nemenzo	nemenzo	PROPN
ejpam-3838	320	11	/	/	SYM
ejpam-3838	320	12	eur	eur	NOUN
ejpam-3838	320	13	.	.	PUNCT
ejpam-3838	321	1	j.	j.	PROPN
ejpam-3838	321	2	pure	pure	PROPN
ejpam-3838	321	3	appl	appl	PROPN
ejpam-3838	321	4	.	.	PROPN
ejpam-3838	321	5	math	math	PROPN
ejpam-3838	321	6	,	,	PUNCT
ejpam-3838	321	7	13	13	NUM
ejpam-3838	321	8	(	(	PUNCT
ejpam-3838	321	9	4	4	NUM
ejpam-3838	321	10	)	)	PUNCT
ejpam-3838	321	11	(	(	PUNCT
ejpam-3838	321	12	2020	2020	NUM
ejpam-3838	321	13	)	)	PUNCT
ejpam-3838	321	14	,	,	PUNCT
ejpam-3838	321	15	873	873	NUM
ejpam-3838	321	16	-	-	NUM
ejpam-3838	321	17	892	892	NUM
ejpam-3838	321	18	888	888	NUM
ejpam-3838	321	19	proof	proof	NOUN
ejpam-3838	321	20	.	.	PUNCT
ejpam-3838	322	1	by	by	ADP
ejpam-3838	322	2	lemma	lemma	PROPN
ejpam-3838	322	3	8	8	NUM
ejpam-3838	322	4	,	,	PUNCT
ejpam-3838	322	5	c	c	PROPN
ejpam-3838	322	6	has	have	VERB
ejpam-3838	322	7	generator	generator	NOUN
ejpam-3838	322	8	matrix	matrix	NOUN
ejpam-3838	322	9	[	[	PUNCT
ejpam-3838	322	10	ik0	ik0	NOUN
ejpam-3838	322	11	a0	a0	NOUN
ejpam-3838	322	12	b0	b0	PROPN
ejpam-3838	322	13	+	+	CCONJ
ejpam-3838	322	14	ub1	ub1	PROPN
ejpam-3838	322	15	+	+	CCONJ
ejpam-3838	322	16	u2n0	u2n0	ADJ
ejpam-3838	322	17	0	0	NUM
ejpam-3838	322	18	uik1	uik1	NOUN
ejpam-3838	322	19	ud1	ud1	NOUN
ejpam-3838	322	20	+	+	CCONJ
ejpam-3838	322	21	u2n1	u2n1	ADP
ejpam-3838	322	22	]	]	PUNCT
ejpam-3838	322	23	for	for	ADP
ejpam-3838	322	24	some	some	DET
ejpam-3838	322	25	matrices	matrix	NOUN
ejpam-3838	322	26	n0	n0	NUM
ejpam-3838	322	27	,	,	PUNCT
ejpam-3838	322	28	n1	n1	PROPN
ejpam-3838	322	29	over	over	ADP
ejpam-3838	322	30	fq	fq	PROPN
ejpam-3838	322	31	.	.	PROPN
ejpam-3838	323	1	from	from	ADP
ejpam-3838	323	2	this	this	PRON
ejpam-3838	323	3	,	,	PUNCT
ejpam-3838	323	4	c	c	PROPN
ejpam-3838	323	5	is	be	AUX
ejpam-3838	323	6	self	self	NOUN
ejpam-3838	323	7	-	-	PUNCT
ejpam-3838	323	8	orthogonal	orthogonal	ADJ
ejpam-3838	323	9	if	if	SCONJ
ejpam-3838	323	10	and	and	CCONJ
ejpam-3838	323	11	only	only	ADV
ejpam-3838	323	12	if	if	SCONJ
ejpam-3838	323	13	ik0	ik0	VERB
ejpam-3838	323	14	+	+	ADV
ejpam-3838	323	15	aat0	aat0	PROPN
ejpam-3838	323	16	+	+	PROPN
ejpam-3838	323	17	bbt	bbt	PROPN
ejpam-3838	323	18	0	0	NUM
ejpam-3838	324	1	+	+	CCONJ
ejpam-3838	324	2	u(b0b	u(b0b	CCONJ
ejpam-3838	324	3	t	t	PROPN
ejpam-3838	324	4	1	1	NUM
ejpam-3838	324	5	+	+	NOUN
ejpam-3838	324	6	b1b	b1b	NOUN
ejpam-3838	324	7	t	t	NOUN
ejpam-3838	324	8	0	0	NUM
ejpam-3838	324	9	)	)	PUNCT
ejpam-3838	325	1	+	+	PROPN
ejpam-3838	325	2	u2(b1b	u2(b1b	NOUN
ejpam-3838	325	3	t	t	NOUN
ejpam-3838	325	4	1	1	NUM
ejpam-3838	325	5	+	+	ADP
ejpam-3838	325	6	b0n	b0n	X
ejpam-3838	325	7	t	t	NOUN
ejpam-3838	325	8	0	0	PUNCT
ejpam-3838	325	9	+	+	ADJ
ejpam-3838	325	10	n0b	n0b	PROPN
ejpam-3838	325	11	t	t	PROPN
ejpam-3838	325	12	0	0	NUM
ejpam-3838	325	13	)	)	PUNCT
ejpam-3838	325	14	≡	≡	PROPN
ejpam-3838	325	15	0	0	PUNCT
ejpam-3838	325	16	(	(	PUNCT
ejpam-3838	325	17	u3	u3	PROPN
ejpam-3838	325	18	)	)	PUNCT
ejpam-3838	325	19	(	(	PUNCT
ejpam-3838	325	20	22	22	NUM
ejpam-3838	325	21	)	)	PUNCT
ejpam-3838	325	22	u(a0	u(a0	VERB
ejpam-3838	326	1	+	+	VERB
ejpam-3838	326	2	b0d	b0d	PROPN
ejpam-3838	326	3	t	t	AUX
ejpam-3838	326	4	1	1	NUM
ejpam-3838	326	5	)	)	PUNCT
ejpam-3838	327	1	+	+	CCONJ
ejpam-3838	328	1	u2(b1d	u2(b1d	PROPN
ejpam-3838	328	2	t	t	PROPN
ejpam-3838	328	3	1	1	NUM
ejpam-3838	329	1	+	+	ADP
ejpam-3838	329	2	b0n	b0n	X
ejpam-3838	329	3	t	t	NOUN
ejpam-3838	329	4	1	1	NUM
ejpam-3838	329	5	)	)	PUNCT
ejpam-3838	329	6	≡	≡	PROPN
ejpam-3838	329	7	0	0	PUNCT
ejpam-3838	329	8	(	(	PUNCT
ejpam-3838	329	9	u3	u3	PROPN
ejpam-3838	329	10	)	)	PUNCT
ejpam-3838	329	11	.	.	PUNCT
ejpam-3838	330	1	(	(	PUNCT
ejpam-3838	330	2	23	23	NUM
ejpam-3838	330	3	)	)	PUNCT
ejpam-3838	330	4	we	we	PRON
ejpam-3838	330	5	want	want	VERB
ejpam-3838	330	6	to	to	PART
ejpam-3838	330	7	count	count	VERB
ejpam-3838	330	8	the	the	DET
ejpam-3838	330	9	number	number	NOUN
ejpam-3838	330	10	of	of	ADP
ejpam-3838	330	11	such	such	ADJ
ejpam-3838	330	12	matricesn0	matricesn0	NOUN
ejpam-3838	330	13	andn1	andn1	NOUN
ejpam-3838	330	14	satisfying	satisfy	VERB
ejpam-3838	330	15	the	the	DET
ejpam-3838	330	16	above	above	ADJ
ejpam-3838	330	17	equivalences	equivalence	NOUN
ejpam-3838	330	18	.	.	PUNCT
ejpam-3838	331	1	first	first	ADV
ejpam-3838	331	2	,	,	PUNCT
ejpam-3838	331	3	consider	consider	VERB
ejpam-3838	331	4	the	the	DET
ejpam-3838	331	5	map	map	NOUN
ejpam-3838	331	6	φb0	φb0	VERB
ejpam-3838	331	7	:	:	PUNCT
ejpam-3838	331	8	mk0×(n−k0−k1)(fq	mk0×(n−k0−k1)(fq	NUM
ejpam-3838	331	9	)	)	PUNCT
ejpam-3838	331	10	−→	−→	NOUN
ejpam-3838	331	11	mk0(fq	mk0(fq	NOUN
ejpam-3838	331	12	)	)	PUNCT
ejpam-3838	331	13	n0	n0	NOUN
ejpam-3838	331	14	7−→	7−→	PROPN
ejpam-3838	331	15	b0n	b0n	PROPN
ejpam-3838	331	16	t	t	NOUN
ejpam-3838	331	17	0	0	PUNCT
ejpam-3838	332	1	+	+	ADJ
ejpam-3838	332	2	n0b	n0b	PROPN
ejpam-3838	332	3	t	t	PROPN
ejpam-3838	332	4	0	0	PUNCT
ejpam-3838	332	5	as	as	SCONJ
ejpam-3838	332	6	defined	define	VERB
ejpam-3838	332	7	in	in	ADP
ejpam-3838	332	8	the	the	DET
ejpam-3838	332	9	previous	previous	ADJ
ejpam-3838	332	10	chapter	chapter	NOUN
ejpam-3838	332	11	.	.	PUNCT
ejpam-3838	333	1	by	by	ADP
ejpam-3838	333	2	(	(	PUNCT
ejpam-3838	333	3	14	14	NUM
ejpam-3838	333	4	)	)	PUNCT
ejpam-3838	333	5	and	and	CCONJ
ejpam-3838	333	6	(	(	PUNCT
ejpam-3838	333	7	15	15	NUM
ejpam-3838	333	8	)	)	PUNCT
ejpam-3838	333	9	,	,	PUNCT
ejpam-3838	333	10	(	(	PUNCT
ejpam-3838	333	11	22	22	NUM
ejpam-3838	333	12	)	)	PUNCT
ejpam-3838	333	13	becomes	become	VERB
ejpam-3838	333	14	b1b	b1b	NOUN
ejpam-3838	333	15	t	t	PROPN
ejpam-3838	333	16	1	1	NUM
ejpam-3838	334	1	+	+	ADP
ejpam-3838	334	2	b0n	b0n	ADJ
ejpam-3838	334	3	t	t	NOUN
ejpam-3838	334	4	0	0	PUNCT
ejpam-3838	335	1	+	+	ADJ
ejpam-3838	335	2	n0b	n0b	PROPN
ejpam-3838	335	3	t	t	PROPN
ejpam-3838	335	4	0	0	NUM
ejpam-3838	335	5	≡	≡	PROPN
ejpam-3838	335	6	0	0	PUNCT
ejpam-3838	335	7	(	(	PUNCT
ejpam-3838	335	8	u	u	NOUN
ejpam-3838	335	9	)	)	PUNCT
ejpam-3838	335	10	.	.	PUNCT
ejpam-3838	336	1	hence	hence	ADV
ejpam-3838	336	2	,	,	PUNCT
ejpam-3838	336	3	|{n0	|{n0	PROPN
ejpam-3838	336	4	∈mk0×(n−k0−k1)|n0	∈mk0×(n−k0−k1)|n0	NOUN
ejpam-3838	336	5	satisfies	satisfy	VERB
ejpam-3838	336	6	(	(	PUNCT
ejpam-3838	336	7	22)}|	22)}|	NUM
ejpam-3838	336	8	=	=	SYM
ejpam-3838	336	9	|{φ−1b0	|{φ−1b0	X
ejpam-3838	336	10	(	(	PUNCT
ejpam-3838	336	11	−b1b	−b1b	X
ejpam-3838	336	12	t	t	PROPN
ejpam-3838	336	13	1	1	NUM
ejpam-3838	336	14	)	)	PUNCT
ejpam-3838	336	15	}	}	PUNCT
ejpam-3838	336	16	|	|	NOUN
ejpam-3838	336	17	=	=	PUNCT
ejpam-3838	336	18	|ker	|ker	NOUN
ejpam-3838	336	19	φb0	φb0	NOUN
ejpam-3838	336	20	|	|	ADV
ejpam-3838	336	21	=	=	SYM
ejpam-3838	336	22	∣∣mk0×(n−k0−k1	∣∣mk0×(n−k0−k1	PROPN
ejpam-3838	336	23	)	)	PUNCT
ejpam-3838	336	24	∣∣∣∣symk0(fq	∣∣∣∣symk0(fq	PROPN
ejpam-3838	336	25	)	)	PUNCT
ejpam-3838	336	26	∣∣	∣∣	NUM
ejpam-3838	336	27	=	=	PUNCT
ejpam-3838	337	1	qk0(n−k0−k1)−k0(k0	qk0(n−k0−k1)−k0(k0	X
ejpam-3838	337	2	+	+	NOUN
ejpam-3838	337	3	1)/2	1)/2	NOUN
ejpam-3838	337	4	.	.	PUNCT
ejpam-3838	338	1	define	define	VERB
ejpam-3838	338	2	another	another	DET
ejpam-3838	338	3	map	map	NOUN
ejpam-3838	338	4	β	β	X
ejpam-3838	338	5	:	:	PUNCT
ejpam-3838	338	6	mk1×(n−k0−k1)(fq	mk1×(n−k0−k1)(fq	X
ejpam-3838	338	7	)	)	PUNCT
ejpam-3838	338	8	−→	−→	NOUN
ejpam-3838	338	9	mk0×k1(fq	mk0×k1(fq	PROPN
ejpam-3838	338	10	)	)	PUNCT
ejpam-3838	338	11	n1	n1	NOUN
ejpam-3838	338	12	7−→	7−→	PROPN
ejpam-3838	338	13	b0n	b0n	X
ejpam-3838	338	14	t	t	PROPN
ejpam-3838	338	15	1	1	NUM
ejpam-3838	338	16	.	.	PUNCT
ejpam-3838	339	1	this	this	DET
ejpam-3838	339	2	map	map	NOUN
ejpam-3838	339	3	is	be	AUX
ejpam-3838	339	4	surjective	surjective	ADJ
ejpam-3838	339	5	because	because	SCONJ
ejpam-3838	339	6	b0	b0	NOUN
ejpam-3838	339	7	is	be	AUX
ejpam-3838	339	8	of	of	ADP
ejpam-3838	339	9	full	full	ADJ
ejpam-3838	339	10	row	row	NOUN
ejpam-3838	339	11	rank	rank	NOUN
ejpam-3838	339	12	.	.	PUNCT
ejpam-3838	340	1	therefore	therefore	ADV
ejpam-3838	340	2	by	by	ADP
ejpam-3838	340	3	(	(	PUNCT
ejpam-3838	340	4	17	17	NUM
ejpam-3838	340	5	)	)	PUNCT
ejpam-3838	340	6	,	,	PUNCT
ejpam-3838	340	7	|{n1	|{n1	VERB
ejpam-3838	340	8	∈mk1×(n−k0−k1)|n1	∈mk1×(n−k0−k1)|n1	ADJ
ejpam-3838	340	9	satisfies	satisfie	NOUN
ejpam-3838	340	10	(	(	PUNCT
ejpam-3838	340	11	23)}|	23)}|	NUM
ejpam-3838	340	12	=	=	SYM
ejpam-3838	340	13	|{β−1(−b1d	|{β−1(−b1d	NUM
ejpam-3838	340	14	t	t	NOUN
ejpam-3838	340	15	1	1	NUM
ejpam-3838	340	16	)	)	PUNCT
ejpam-3838	340	17	}	}	PUNCT
ejpam-3838	340	18	|	|	NOUN
ejpam-3838	340	19	=	=	PUNCT
ejpam-3838	340	20	|ker	|ker	NOUN
ejpam-3838	340	21	β|	β|	ADP
ejpam-3838	340	22	=	=	SYM
ejpam-3838	340	23	∣∣mk1×(n−k0−k1)(fq	∣∣mk1×(n−k0−k1)(fq	NUM
ejpam-3838	340	24	)	)	PUNCT
ejpam-3838	340	25	∣∣	∣∣	X
ejpam-3838	340	26	|mk0×k1(fq)|	|mk0×k1(fq)|	PUNCT
ejpam-3838	340	27	=	=	SYM
ejpam-3838	340	28	qk1(n−k0−k1)−(k0k1	qk1(n−k0−k1)−(k0k1	NOUN
ejpam-3838	340	29	)	)	PUNCT
ejpam-3838	340	30	.	.	PUNCT
ejpam-3838	341	1	finally	finally	ADV
ejpam-3838	341	2	,	,	PUNCT
ejpam-3838	341	3	the	the	DET
ejpam-3838	341	4	number	number	NOUN
ejpam-3838	341	5	of	of	ADP
ejpam-3838	341	6	self	self	NOUN
ejpam-3838	341	7	-	-	PUNCT
ejpam-3838	341	8	orthogonal	orthogonal	ADJ
ejpam-3838	341	9	codes	code	NOUN
ejpam-3838	341	10	c	c	PROPN
ejpam-3838	341	11	of	of	ADP
ejpam-3838	341	12	type	type	NOUN
ejpam-3838	341	13	{	{	PUNCT
ejpam-3838	341	14	k0	k0	PROPN
ejpam-3838	341	15	,	,	PUNCT
ejpam-3838	341	16	k1	k1	NOUN
ejpam-3838	341	17	,	,	PUNCT
ejpam-3838	341	18	0	0	NUM
ejpam-3838	341	19	}	}	PUNCT
ejpam-3838	341	20	over	over	ADP
ejpam-3838	341	21	r2	r2	NOUN
ejpam-3838	341	22	such	such	ADJ
ejpam-3838	341	23	that	that	PRON
ejpam-3838	341	24	res(c	res(c	PROPN
ejpam-3838	341	25	)	)	PUNCT
ejpam-3838	341	26	=	=	PROPN
ejpam-3838	341	27	c1	c1	PROPN
ejpam-3838	341	28	is	be	AUX
ejpam-3838	341	29	the	the	DET
ejpam-3838	341	30	number	number	NOUN
ejpam-3838	341	31	of	of	ADP
ejpam-3838	341	32	such	such	ADJ
ejpam-3838	341	33	matrices	matrix	NOUN
ejpam-3838	341	34	n0	n0	X
ejpam-3838	341	35	satisfying	satisfying	ADJ
ejpam-3838	341	36	(	(	PUNCT
ejpam-3838	341	37	22	22	NUM
ejpam-3838	341	38	)	)	PUNCT
ejpam-3838	341	39	multiplied	multiply	VERB
ejpam-3838	341	40	to	to	ADP
ejpam-3838	341	41	the	the	DET
ejpam-3838	341	42	number	number	NOUN
ejpam-3838	341	43	of	of	ADP
ejpam-3838	341	44	such	such	ADJ
ejpam-3838	341	45	matrices	matrix	NOUN
ejpam-3838	341	46	n1	n1	NOUN
ejpam-3838	341	47	satisfying	satisfying	ADJ
ejpam-3838	341	48	(	(	PUNCT
ejpam-3838	341	49	23	23	NUM
ejpam-3838	341	50	)	)	PUNCT
ejpam-3838	341	51	which	which	PRON
ejpam-3838	341	52	is	be	AUX
ejpam-3838	341	53	qk0(n−k0−k1)−k0(k0	qk0(n−k0−k1)−k0(k0	X
ejpam-3838	341	54	+	+	PROPN
ejpam-3838	341	55	1)/2qk1(n−k0−k1)−k0k1	1)/2qk1(n−k0−k1)−k0k1	NUM
ejpam-3838	341	56	.	.	PUNCT
ejpam-3838	342	1	l.e	l.e	PROPN
ejpam-3838	342	2	.	.	PROPN
ejpam-3838	342	3	galvez	galvez	PROPN
ejpam-3838	342	4	,	,	PUNCT
ejpam-3838	342	5	r.a	r.a	PROPN
ejpam-3838	342	6	.	.	PROPN
ejpam-3838	342	7	betty	betty	PROPN
ejpam-3838	342	8	,	,	PUNCT
ejpam-3838	342	9	f.	f.	PROPN
ejpam-3838	342	10	nemenzo	nemenzo	PROPN
ejpam-3838	342	11	/	/	SYM
ejpam-3838	342	12	eur	eur	NOUN
ejpam-3838	342	13	.	.	PUNCT
ejpam-3838	343	1	j.	j.	PROPN
ejpam-3838	343	2	pure	pure	PROPN
ejpam-3838	343	3	appl	appl	PROPN
ejpam-3838	343	4	.	.	PROPN
ejpam-3838	343	5	math	math	PROPN
ejpam-3838	343	6	,	,	PUNCT
ejpam-3838	343	7	13	13	NUM
ejpam-3838	343	8	(	(	PUNCT
ejpam-3838	343	9	4	4	NUM
ejpam-3838	343	10	)	)	PUNCT
ejpam-3838	343	11	(	(	PUNCT
ejpam-3838	343	12	2020	2020	NUM
ejpam-3838	343	13	)	)	PUNCT
ejpam-3838	343	14	,	,	PUNCT
ejpam-3838	343	15	873	873	NUM
ejpam-3838	343	16	-	-	SYM
ejpam-3838	343	17	892	892	NUM
ejpam-3838	343	18	889	889	NUM
ejpam-3838	343	19	the	the	DET
ejpam-3838	343	20	result	result	NOUN
ejpam-3838	343	21	follows	follow	VERB
ejpam-3838	343	22	by	by	ADP
ejpam-3838	343	23	simplifying	simplify	VERB
ejpam-3838	343	24	the	the	DET
ejpam-3838	343	25	above	above	ADJ
ejpam-3838	343	26	expression	expression	NOUN
ejpam-3838	343	27	.	.	PUNCT
ejpam-3838	344	1	�	�	PROPN
ejpam-3838	344	2	for	for	ADP
ejpam-3838	344	3	the	the	DET
ejpam-3838	344	4	rest	rest	NOUN
ejpam-3838	344	5	of	of	ADP
ejpam-3838	344	6	this	this	DET
ejpam-3838	344	7	chapter	chapter	NOUN
ejpam-3838	344	8	,	,	PUNCT
ejpam-3838	344	9	let	let	VERB
ejpam-3838	344	10	c2	c2	PROPN
ejpam-3838	344	11	be	be	AUX
ejpam-3838	344	12	a	a	DET
ejpam-3838	344	13	code	code	NOUN
ejpam-3838	344	14	over	over	ADP
ejpam-3838	344	15	fq	fq	PROPN
ejpam-3838	344	16	with	with	ADP
ejpam-3838	344	17	dimension	dimension	NOUN
ejpam-3838	344	18	k0	k0	PROPN
ejpam-3838	344	19	+	+	CCONJ
ejpam-3838	344	20	k1	k1	PROPN
ejpam-3838	344	21	+	+	CCONJ
ejpam-3838	344	22	k2	k2	NOUN
ejpam-3838	344	23	and	and	CCONJ
ejpam-3838	344	24	has	have	VERB
ejpam-3838	344	25	a	a	DET
ejpam-3838	344	26	generator	generator	NOUN
ejpam-3838	344	27	matrix	matrix	NOUN
ejpam-3838	344	28	g2	g2	PROPN
ejpam-3838	344	29	=	=	PUNCT
ejpam-3838	344	30			PROPN
ejpam-3838	344	31	ik0	ik0	VERB
ejpam-3838	344	32	a0	a0	PROPN
ejpam-3838	344	33	b0	b0	PROPN
ejpam-3838	344	34	0	0	PUNCT
ejpam-3838	345	1	ik1	ik1	VERB
ejpam-3838	345	2	d1	d1	PROPN
ejpam-3838	345	3	0	0	NUM
ejpam-3838	345	4	0	0	PUNCT
ejpam-3838	345	5	f2	f2	ADJ
ejpam-3838	345	6			NOUN
ejpam-3838	345	7	where	where	SCONJ
ejpam-3838	345	8	f2	f2	PROPN
ejpam-3838	345	9	is	be	AUX
ejpam-3838	345	10	of	of	ADP
ejpam-3838	345	11	full	full	ADJ
ejpam-3838	345	12	row	row	NOUN
ejpam-3838	345	13	rank	rank	NOUN
ejpam-3838	345	14	.	.	PUNCT
ejpam-3838	346	1	we	we	PRON
ejpam-3838	346	2	assume	assume	VERB
ejpam-3838	346	3	that	that	SCONJ
ejpam-3838	346	4	tor(c1	tor(c1	PROPN
ejpam-3838	346	5	)	)	PUNCT
ejpam-3838	346	6	⊆	⊆	NUM
ejpam-3838	346	7	c2	c2	PROPN
ejpam-3838	346	8	⊆	⊆	NUM
ejpam-3838	346	9	res(c1	res(c1	PROPN
ejpam-3838	346	10	)	)	PUNCT
ejpam-3838	346	11	⊥.	⊥.	PROPN
ejpam-3838	346	12	hence	hence	ADV
ejpam-3838	346	13	,	,	PUNCT
ejpam-3838	346	14	ik0	ik0	VERB
ejpam-3838	346	15	+	+	PROPN
ejpam-3838	346	16	a0a	a0a	PROPN
ejpam-3838	346	17	t	t	NOUN
ejpam-3838	346	18	0	0	PUNCT
ejpam-3838	347	1	+	+	NOUN
ejpam-3838	347	2	b0b	b0b	PROPN
ejpam-3838	347	3	t	t	X
ejpam-3838	347	4	0	0	NUM
ejpam-3838	347	5	≡	≡	PROPN
ejpam-3838	347	6	0	0	NUM
ejpam-3838	347	7	(	(	PUNCT
ejpam-3838	347	8	u	u	NOUN
ejpam-3838	347	9	)	)	PUNCT
ejpam-3838	347	10	a0	a0	NOUN
ejpam-3838	347	11	+	+	PROPN
ejpam-3838	347	12	b0d	b0d	PROPN
ejpam-3838	347	13	t	t	PROPN
ejpam-3838	347	14	1	1	NUM
ejpam-3838	347	15	≡	≡	PROPN
ejpam-3838	347	16	0	0	NUM
ejpam-3838	347	17	(	(	PUNCT
ejpam-3838	347	18	u	u	NOUN
ejpam-3838	347	19	)	)	PUNCT
ejpam-3838	348	1	f2b	f2b	PROPN
ejpam-3838	348	2	t	t	NOUN
ejpam-3838	348	3	0	0	NUM
ejpam-3838	348	4	≡	≡	PROPN
ejpam-3838	348	5	0	0	NUM
ejpam-3838	348	6	(	(	PUNCT
ejpam-3838	348	7	u	u	NOUN
ejpam-3838	348	8	)	)	PUNCT
ejpam-3838	348	9	which	which	PRON
ejpam-3838	348	10	are	be	AUX
ejpam-3838	348	11	equivalent	equivalent	ADJ
ejpam-3838	348	12	to	to	ADP
ejpam-3838	348	13	(	(	PUNCT
ejpam-3838	348	14	14	14	NUM
ejpam-3838	348	15	)	)	PUNCT
ejpam-3838	348	16	,	,	PUNCT
ejpam-3838	348	17	(	(	PUNCT
ejpam-3838	348	18	17	17	NUM
ejpam-3838	348	19	)	)	PUNCT
ejpam-3838	348	20	and	and	CCONJ
ejpam-3838	348	21	(	(	PUNCT
ejpam-3838	348	22	19	19	NUM
ejpam-3838	348	23	)	)	PUNCT
ejpam-3838	348	24	,	,	PUNCT
ejpam-3838	348	25	respectively	respectively	ADV
ejpam-3838	348	26	.	.	PUNCT
ejpam-3838	349	1	consider	consider	VERB
ejpam-3838	349	2	the	the	DET
ejpam-3838	349	3	following	follow	VERB
ejpam-3838	349	4	sets	set	NOUN
ejpam-3838	349	5	of	of	ADP
ejpam-3838	349	6	codes	code	NOUN
ejpam-3838	349	7	over	over	ADP
ejpam-3838	349	8	r2	r2	PROPN
ejpam-3838	349	9	:	:	PUNCT
ejpam-3838	349	10	y	y	PROPN
ejpam-3838	349	11	=	=	PUNCT
ejpam-3838	349	12	{	{	PUNCT
ejpam-3838	349	13	c	c	NOUN
ejpam-3838	349	14	|	|	ADV
ejpam-3838	349	15	c	c	PROPN
ejpam-3838	349	16	is	be	AUX
ejpam-3838	349	17	self	self	NOUN
ejpam-3838	349	18	-	-	PUNCT
ejpam-3838	349	19	orthogonal	orthogonal	NOUN
ejpam-3838	349	20	of	of	ADP
ejpam-3838	349	21	type	type	NOUN
ejpam-3838	349	22	{	{	PUNCT
ejpam-3838	349	23	k0	k0	PROPN
ejpam-3838	349	24	,	,	PUNCT
ejpam-3838	349	25	k1	k1	NOUN
ejpam-3838	349	26	,	,	PUNCT
ejpam-3838	349	27	0},res(c	0},res(c	NOUN
ejpam-3838	349	28	)	)	PUNCT
ejpam-3838	349	29	=	=	SYM
ejpam-3838	349	30	c1	c1	PROPN
ejpam-3838	349	31	}	}	PUNCT
ejpam-3838	349	32	;	;	PUNCT
ejpam-3838	350	1	y	y	PROPN
ejpam-3838	350	2	′	′	NUM
ejpam-3838	350	3	=	=	PUNCT
ejpam-3838	351	1	{	{	PUNCT
ejpam-3838	351	2	c	c	NOUN
ejpam-3838	351	3	′	′	NUM
ejpam-3838	352	1	|	|	ADV
ejpam-3838	352	2	c	c	NOUN
ejpam-3838	352	3	′	′	NOUN
ejpam-3838	352	4	is	be	AUX
ejpam-3838	352	5	self	self	NOUN
ejpam-3838	352	6	-	-	PUNCT
ejpam-3838	352	7	orthogonal	orthogonal	ADJ
ejpam-3838	352	8	,	,	PUNCT
ejpam-3838	352	9	res(c	res(c	PROPN
ejpam-3838	352	10	′	′	NOUN
ejpam-3838	352	11	)	)	PUNCT
ejpam-3838	352	12	=	=	SYM
ejpam-3838	352	13	c1	c1	NOUN
ejpam-3838	352	14	,	,	PUNCT
ejpam-3838	352	15	tor2(c	tor2(c	NOUN
ejpam-3838	352	16	′	′	NOUN
ejpam-3838	352	17	)	)	PUNCT
ejpam-3838	352	18	=	=	SYM
ejpam-3838	352	19	c2	c2	PROPN
ejpam-3838	352	20	}	}	PUNCT
ejpam-3838	352	21	.	.	PUNCT
ejpam-3838	353	1	note	note	VERB
ejpam-3838	353	2	that	that	SCONJ
ejpam-3838	353	3	|y	|y	NOUN
ejpam-3838	353	4	|	|	ADV
ejpam-3838	353	5	is	be	AUX
ejpam-3838	353	6	already	already	ADV
ejpam-3838	353	7	given	give	VERB
ejpam-3838	353	8	in	in	ADP
ejpam-3838	353	9	lemma	lemma	PROPN
ejpam-3838	353	10	9	9	NUM
ejpam-3838	353	11	.	.	PUNCT
ejpam-3838	354	1	our	our	PRON
ejpam-3838	354	2	next	next	ADJ
ejpam-3838	354	3	goal	goal	NOUN
ejpam-3838	354	4	is	be	AUX
ejpam-3838	354	5	to	to	PART
ejpam-3838	354	6	compute	compute	VERB
ejpam-3838	354	7	for	for	ADP
ejpam-3838	354	8	|y	|y	NOUN
ejpam-3838	354	9	′|	′|	NOUN
ejpam-3838	354	10	.	.	PUNCT
ejpam-3838	355	1	this	this	PRON
ejpam-3838	355	2	will	will	AUX
ejpam-3838	355	3	be	be	AUX
ejpam-3838	355	4	done	do	VERB
ejpam-3838	355	5	in	in	ADP
ejpam-3838	355	6	the	the	DET
ejpam-3838	355	7	same	same	ADJ
ejpam-3838	355	8	way	way	NOUN
ejpam-3838	355	9	as	as	ADP
ejpam-3838	355	10	in	in	ADP
ejpam-3838	355	11	the	the	DET
ejpam-3838	355	12	previous	previous	ADJ
ejpam-3838	355	13	chapter	chapter	NOUN
ejpam-3838	355	14	.	.	PUNCT
ejpam-3838	356	1	lemma	lemma	PROPN
ejpam-3838	356	2	10	10	NUM
ejpam-3838	356	3	.	.	PUNCT
ejpam-3838	357	1	if	if	SCONJ
ejpam-3838	357	2	c	c	PROPN
ejpam-3838	357	3	∈	∈	PROPN
ejpam-3838	357	4	y	y	PROPN
ejpam-3838	357	5	,	,	PUNCT
ejpam-3838	357	6	then	then	ADV
ejpam-3838	357	7	there	there	PRON
ejpam-3838	357	8	exists	exist	VERB
ejpam-3838	357	9	a	a	DET
ejpam-3838	357	10	unique	unique	ADJ
ejpam-3838	357	11	c	c	NOUN
ejpam-3838	357	12	′	′	NUM
ejpam-3838	358	1	∈	∈	PROPN
ejpam-3838	358	2	y	y	NOUN
ejpam-3838	358	3	′	′	NUM
ejpam-3838	358	4	such	such	ADJ
ejpam-3838	358	5	that	that	SCONJ
ejpam-3838	358	6	c	c	PROPN
ejpam-3838	358	7	⊆	⊆	NUM
ejpam-3838	358	8	c	c	NOUN
ejpam-3838	358	9	′.	′.	NOUN
ejpam-3838	358	10	proof	proof	NOUN
ejpam-3838	358	11	.	.	PUNCT
ejpam-3838	359	1	since	since	SCONJ
ejpam-3838	359	2	c	c	PROPN
ejpam-3838	359	3	∈	∈	PROPN
ejpam-3838	359	4	y	y	PROPN
ejpam-3838	359	5	,	,	PUNCT
ejpam-3838	359	6	c	c	PROPN
ejpam-3838	359	7	has	have	VERB
ejpam-3838	359	8	generator	generator	NOUN
ejpam-3838	359	9	matrix	matrix	NOUN
ejpam-3838	359	10	(	(	PUNCT
ejpam-3838	359	11	21	21	NUM
ejpam-3838	359	12	)	)	PUNCT
ejpam-3838	359	13	for	for	ADP
ejpam-3838	359	14	some	some	DET
ejpam-3838	359	15	matrices	matrix	NOUN
ejpam-3838	359	16	n0	n0	NOUN
ejpam-3838	359	17	and	and	CCONJ
ejpam-3838	359	18	n1	n1	PROPN
ejpam-3838	359	19	.	.	PUNCT
ejpam-3838	360	1	suppose	suppose	VERB
ejpam-3838	360	2	c	c	SYM
ejpam-3838	360	3	⊆	⊆	NUM
ejpam-3838	360	4	c	c	NOUN
ejpam-3838	360	5	′	′	NOUN
ejpam-3838	360	6	for	for	ADP
ejpam-3838	360	7	some	some	PRON
ejpam-3838	360	8	c	c	NOUN
ejpam-3838	360	9	′	′	NUM
ejpam-3838	361	1	∈	∈	PROPN
ejpam-3838	361	2	y	y	NOUN
ejpam-3838	361	3	′	′	NOUN
ejpam-3838	362	1	and	and	CCONJ
ejpam-3838	362	2	there	there	PRON
ejpam-3838	362	3	exists	exist	VERB
ejpam-3838	362	4	a	a	DET
ejpam-3838	362	5	code	code	NOUN
ejpam-3838	362	6	c	c	X
ejpam-3838	363	1	′′	′′	PROPN
ejpam-3838	363	2	with	with	ADP
ejpam-3838	363	3	generator	generator	NOUN
ejpam-3838	363	4	matrix	matrix	PROPN
ejpam-3838	363	5	ik0	ik0	VERB
ejpam-3838	363	6	a0	a0	PROPN
ejpam-3838	363	7	b0	b0	PROPN
ejpam-3838	363	8	+	+	CCONJ
ejpam-3838	363	9	ub1	ub1	PROPN
ejpam-3838	363	10	+	+	CCONJ
ejpam-3838	363	11	u2n0	u2n0	ADJ
ejpam-3838	363	12	0	0	NUM
ejpam-3838	363	13	uik1	uik1	NOUN
ejpam-3838	363	14	ud1	ud1	NOUN
ejpam-3838	363	15	+	+	CCONJ
ejpam-3838	363	16	u2n1	u2n1	PROPN
ejpam-3838	363	17	0	0	NUM
ejpam-3838	363	18	0	0	NUM
ejpam-3838	363	19	u2f2	u2f2	ADP
ejpam-3838	363	20			NOUN
ejpam-3838	363	21	.	.	PUNCT
ejpam-3838	364	1	clearly	clearly	ADV
ejpam-3838	364	2	,	,	PUNCT
ejpam-3838	364	3	c	c	PROPN
ejpam-3838	364	4	⊆	⊆	NUM
ejpam-3838	364	5	c	c	NOUN
ejpam-3838	364	6	′′.	′′.	NOUN
ejpam-3838	364	7	note	note	VERB
ejpam-3838	364	8	that	that	SCONJ
ejpam-3838	364	9	c	c	X
ejpam-3838	364	10	′′	′′	PROPN
ejpam-3838	364	11	satisfies	satisfy	VERB
ejpam-3838	364	12	res(c	res(c	PROPN
ejpam-3838	364	13	′′	′′	PROPN
ejpam-3838	364	14	)	)	PUNCT
ejpam-3838	364	15	=	=	PROPN
ejpam-3838	364	16	c1	c1	NOUN
ejpam-3838	364	17	and	and	CCONJ
ejpam-3838	364	18	tor2(c	tor2(c	NOUN
ejpam-3838	364	19	′′	′′	PROPN
ejpam-3838	364	20	)	)	PUNCT
ejpam-3838	364	21	=	=	PROPN
ejpam-3838	364	22	c2	c2	PROPN
ejpam-3838	364	23	.	.	PUNCT
ejpam-3838	365	1	using	use	VERB
ejpam-3838	365	2	(	(	PUNCT
ejpam-3838	365	3	19	19	NUM
ejpam-3838	365	4	)	)	PUNCT
ejpam-3838	365	5	,	,	PUNCT
ejpam-3838	365	6	we	we	PRON
ejpam-3838	365	7	conclude	conclude	VERB
ejpam-3838	365	8	that	that	PRON
ejpam-3838	365	9	c	c	VERB
ejpam-3838	365	10	′′	′′	PROPN
ejpam-3838	365	11	is	be	AUX
ejpam-3838	365	12	self	self	NOUN
ejpam-3838	365	13	-	-	PUNCT
ejpam-3838	365	14	orthogonal	orthogonal	NOUN
ejpam-3838	365	15	.	.	PUNCT
ejpam-3838	366	1	hence	hence	ADV
ejpam-3838	366	2	,	,	PUNCT
ejpam-3838	366	3	c	c	X
ejpam-3838	366	4	′′	′′	PROPN
ejpam-3838	366	5	∈	∈	PROPN
ejpam-3838	366	6	y	y	PROPN
ejpam-3838	366	7	′.	′.	NOUN
ejpam-3838	366	8	next	next	ADV
ejpam-3838	366	9	,	,	PUNCT
ejpam-3838	366	10	notice	notice	VERB
ejpam-3838	366	11	that	that	SCONJ
ejpam-3838	367	1	rk22	rk22	PROPN
ejpam-3838	367	2	[	[	X
ejpam-3838	367	3	0	0	NUM
ejpam-3838	367	4	0	0	NUM
ejpam-3838	367	5	u2f2	u2f2	ADP
ejpam-3838	367	6	]	]	X
ejpam-3838	367	7	⊆	⊆	NUM
ejpam-3838	367	8	c	c	NOUN
ejpam-3838	367	9	′.	′.	NOUN
ejpam-3838	367	10	this	this	PRON
ejpam-3838	367	11	,	,	PUNCT
ejpam-3838	367	12	together	together	ADV
ejpam-3838	367	13	with	with	ADP
ejpam-3838	367	14	the	the	DET
ejpam-3838	367	15	fact	fact	NOUN
ejpam-3838	367	16	that	that	SCONJ
ejpam-3838	367	17	c	c	NOUN
ejpam-3838	367	18	⊆	⊆	NUM
ejpam-3838	367	19	c	c	NOUN
ejpam-3838	367	20	′	′	NUM
ejpam-3838	367	21	,	,	PUNCT
ejpam-3838	367	22	forces	force	NOUN
ejpam-3838	367	23	c	c	NOUN
ejpam-3838	367	24	′′	′′	PROPN
ejpam-3838	367	25	⊆	⊆	NUM
ejpam-3838	367	26	c	c	NOUN
ejpam-3838	367	27	′.	′.	NOUN
ejpam-3838	367	28	but	but	CCONJ
ejpam-3838	367	29	|c	|c	PROPN
ejpam-3838	367	30	′′|	′′|	VERB
ejpam-3838	367	31	=	=	SYM
ejpam-3838	367	32	|c1||c2|	|c1||c2|	NOUN
ejpam-3838	367	33	=	=	SYM
ejpam-3838	367	34	q2k0+k1qk0+k1+k2	q2k0+k1qk0+k1+k2	PROPN
ejpam-3838	367	35	=	=	PUNCT
ejpam-3838	367	36	q3k0	q3k0	X
ejpam-3838	367	37	+	+	NOUN
ejpam-3838	367	38	2k1+k2	2k1+k2	ADJ
ejpam-3838	367	39	=	=	SYM
ejpam-3838	367	40	|c	|c	VERB
ejpam-3838	367	41	′|	′|	NUM
ejpam-3838	367	42	and	and	CCONJ
ejpam-3838	367	43	therefore	therefore	ADV
ejpam-3838	367	44	,	,	PUNCT
ejpam-3838	367	45	c	c	NOUN
ejpam-3838	367	46	′	′	NOUN
ejpam-3838	367	47	=	=	PUNCT
ejpam-3838	367	48	c	c	NOUN
ejpam-3838	367	49	′′.	′′.	PROPN
ejpam-3838	367	50	�	�	PROPN
ejpam-3838	367	51	lemma	lemma	PROPN
ejpam-3838	367	52	11	11	NUM
ejpam-3838	367	53	.	.	PUNCT
ejpam-3838	368	1	let	let	VERB
ejpam-3838	368	2	c	c	NOUN
ejpam-3838	368	3	′	′	NOUN
ejpam-3838	368	4	∈	∈	PROPN
ejpam-3838	368	5	y	y	PROPN
ejpam-3838	368	6	′.	′.	NOUN
ejpam-3838	368	7	then	then	ADV
ejpam-3838	368	8	|	|	ADV
ejpam-3838	368	9	{	{	PUNCT
ejpam-3838	368	10	c	c	NOUN
ejpam-3838	368	11	∈	∈	PROPN
ejpam-3838	369	1	y	y	NOUN
ejpam-3838	369	2	|	|	ADV
ejpam-3838	369	3	c	c	VERB
ejpam-3838	369	4	⊆	⊆	NUM
ejpam-3838	369	5	c	c	NOUN
ejpam-3838	369	6	′	′	NOUN
ejpam-3838	369	7	}	}	PUNCT
ejpam-3838	370	1	|	|	ADV
ejpam-3838	370	2	=	=	SYM
ejpam-3838	370	3	q(k0+k1)k2	q(k0+k1)k2	PROPN
ejpam-3838	370	4	.	.	PUNCT
ejpam-3838	371	1	l.e	l.e	PROPN
ejpam-3838	371	2	.	.	PROPN
ejpam-3838	371	3	galvez	galvez	PROPN
ejpam-3838	371	4	,	,	PUNCT
ejpam-3838	371	5	r.a	r.a	PROPN
ejpam-3838	371	6	.	.	PROPN
ejpam-3838	371	7	betty	betty	PROPN
ejpam-3838	371	8	,	,	PUNCT
ejpam-3838	371	9	f.	f.	PROPN
ejpam-3838	371	10	nemenzo	nemenzo	PROPN
ejpam-3838	371	11	/	/	SYM
ejpam-3838	371	12	eur	eur	NOUN
ejpam-3838	371	13	.	.	PUNCT
ejpam-3838	372	1	j.	j.	PROPN
ejpam-3838	372	2	pure	pure	PROPN
ejpam-3838	372	3	appl	appl	PROPN
ejpam-3838	372	4	.	.	PROPN
ejpam-3838	372	5	math	math	PROPN
ejpam-3838	372	6	,	,	PUNCT
ejpam-3838	372	7	13	13	NUM
ejpam-3838	372	8	(	(	PUNCT
ejpam-3838	372	9	4	4	NUM
ejpam-3838	372	10	)	)	PUNCT
ejpam-3838	372	11	(	(	PUNCT
ejpam-3838	372	12	2020	2020	NUM
ejpam-3838	372	13	)	)	PUNCT
ejpam-3838	372	14	,	,	PUNCT
ejpam-3838	372	15	873	873	NUM
ejpam-3838	372	16	-	-	NUM
ejpam-3838	372	17	892	892	NUM
ejpam-3838	372	18	890	890	NUM
ejpam-3838	372	19	proof	proof	NOUN
ejpam-3838	372	20	.	.	PUNCT
ejpam-3838	373	1	let	let	VERB
ejpam-3838	374	1	c	c	NOUN
ejpam-3838	374	2	′	′	NOUN
ejpam-3838	375	1	∈	∈	PROPN
ejpam-3838	376	1	y	y	NOUN
ejpam-3838	376	2	′	′	NUM
ejpam-3838	376	3	whose	whose	DET
ejpam-3838	376	4	generator	generator	NOUN
ejpam-3838	376	5	matrix	matrix	NOUN
ejpam-3838	376	6	is	is	NOUN
ejpam-3838	376	7	ik0	ik0	VERB
ejpam-3838	376	8	a0	a0	PROPN
ejpam-3838	376	9	b0	b0	PROPN
ejpam-3838	376	10	+	+	CCONJ
ejpam-3838	376	11	ub1	ub1	PROPN
ejpam-3838	376	12	+	+	CCONJ
ejpam-3838	376	13	u2n0	u2n0	ADJ
ejpam-3838	376	14	0	0	NUM
ejpam-3838	376	15	uik1	uik1	NOUN
ejpam-3838	376	16	ud1	ud1	NOUN
ejpam-3838	376	17	+	+	CCONJ
ejpam-3838	376	18	u2n1	u2n1	PROPN
ejpam-3838	376	19	0	0	NUM
ejpam-3838	376	20	0	0	NUM
ejpam-3838	377	1	u2f2	u2f2	ADP
ejpam-3838	377	2			NOUN
ejpam-3838	377	3	.	.	PUNCT
ejpam-3838	378	1	define	define	VERB
ejpam-3838	378	2	the	the	DET
ejpam-3838	378	3	map	map	NOUN
ejpam-3838	378	4	ψ	ψ	X
ejpam-3838	378	5	:	:	PUNCT
ejpam-3838	378	6	mk0×k2(fq)×mk1×k2(fq	mk0×k2(fq)×mk1×k2(fq	NOUN
ejpam-3838	378	7	)	)	PUNCT
ejpam-3838	378	8	−→	−→	NOUN
ejpam-3838	378	9	{	{	PUNCT
ejpam-3838	378	10	c	c	NOUN
ejpam-3838	378	11	∈	∈	PROPN
ejpam-3838	378	12	y	y	NOUN
ejpam-3838	379	1	|	|	ADV
ejpam-3838	379	2	c	c	VERB
ejpam-3838	379	3	⊆	⊆	NUM
ejpam-3838	379	4	c	c	NOUN
ejpam-3838	379	5	′	′	NOUN
ejpam-3838	379	6	}	}	PUNCT
ejpam-3838	379	7	as	as	ADP
ejpam-3838	379	8	ψ	ψ	X
ejpam-3838	379	9	(	(	PUNCT
ejpam-3838	379	10	m	m	NOUN
ejpam-3838	379	11	′,m	′,m	NUM
ejpam-3838	379	12	′′	′′	NOUN
ejpam-3838	379	13	)	)	PUNCT
ejpam-3838	380	1	=	=	PUNCT
ejpam-3838	381	1	rk0+k12	rk0+k12	NOUN
ejpam-3838	381	2	[	[	PUNCT
ejpam-3838	381	3	ik0	ik0	NOUN
ejpam-3838	381	4	a0	a0	NOUN
ejpam-3838	381	5	b0	b0	PROPN
ejpam-3838	381	6	+	+	CCONJ
ejpam-3838	381	7	ub1	ub1	PROPN
ejpam-3838	381	8	+	+	CCONJ
ejpam-3838	381	9	u2(n0	u2(n0	PRON
ejpam-3838	381	10	+	+	PROPN
ejpam-3838	381	11	m	m	NOUN
ejpam-3838	381	12	′f2	′f2	ADJ
ejpam-3838	381	13	)	)	PUNCT
ejpam-3838	381	14	0	0	NUM
ejpam-3838	381	15	uik1	uik1	PROPN
ejpam-3838	381	16	ud1	ud1	NOUN
ejpam-3838	381	17	+	+	CCONJ
ejpam-3838	381	18	u2(n1	u2(n1	NUM
ejpam-3838	381	19	+	+	NOUN
ejpam-3838	381	20	m	m	NOUN
ejpam-3838	381	21	′′f2	′′f2	ADJ
ejpam-3838	381	22	)	)	PUNCT
ejpam-3838	381	23	]	]	PUNCT
ejpam-3838	381	24	and	and	CCONJ
ejpam-3838	381	25	claim	claim	VERB
ejpam-3838	381	26	that	that	SCONJ
ejpam-3838	381	27	this	this	DET
ejpam-3838	381	28	map	map	NOUN
ejpam-3838	381	29	is	be	AUX
ejpam-3838	381	30	bijective	bijective	ADJ
ejpam-3838	381	31	.	.	PUNCT
ejpam-3838	382	1	indeed	indeed	ADV
ejpam-3838	382	2	,	,	PUNCT
ejpam-3838	382	3	ψ	ψ	X
ejpam-3838	382	4	is	be	AUX
ejpam-3838	382	5	injective	injective	ADJ
ejpam-3838	382	6	because	because	SCONJ
ejpam-3838	382	7	f2	f2	PROPN
ejpam-3838	382	8	is	be	AUX
ejpam-3838	382	9	of	of	ADP
ejpam-3838	382	10	full	full	ADJ
ejpam-3838	382	11	row	row	NOUN
ejpam-3838	382	12	rank	rank	NOUN
ejpam-3838	382	13	.	.	PUNCT
ejpam-3838	383	1	now	now	ADV
ejpam-3838	383	2	,	,	PUNCT
ejpam-3838	383	3	suppose	suppose	VERB
ejpam-3838	383	4	c	c	PROPN
ejpam-3838	383	5	∈	∈	PROPN
ejpam-3838	383	6	y	y	PROPN
ejpam-3838	383	7	such	such	ADJ
ejpam-3838	383	8	that	that	SCONJ
ejpam-3838	383	9	c	c	PROPN
ejpam-3838	383	10	⊆	⊆	NUM
ejpam-3838	383	11	c	c	NOUN
ejpam-3838	383	12	′.	′.	NOUN
ejpam-3838	383	13	then	then	ADV
ejpam-3838	383	14	by	by	ADP
ejpam-3838	383	15	lemma	lemma	PROPN
ejpam-3838	383	16	8	8	NUM
ejpam-3838	383	17	,	,	PUNCT
ejpam-3838	383	18	c	c	PROPN
ejpam-3838	383	19	has	have	VERB
ejpam-3838	383	20	generator	generator	NOUN
ejpam-3838	383	21	matrix	matrix	NOUN
ejpam-3838	383	22	[	[	PUNCT
ejpam-3838	383	23	ik0	ik0	NOUN
ejpam-3838	383	24	a0	a0	NOUN
ejpam-3838	383	25	b0	b0	PROPN
ejpam-3838	383	26	+	+	CCONJ
ejpam-3838	383	27	ub1	ub1	PROPN
ejpam-3838	383	28	+	+	CCONJ
ejpam-3838	383	29	u2f	u2f	ADJ
ejpam-3838	383	30	′	′	NUM
ejpam-3838	383	31	0	0	NUM
ejpam-3838	383	32	uik1	uik1	NOUN
ejpam-3838	383	33	ud1	ud1	NOUN
ejpam-3838	383	34	+	+	CCONJ
ejpam-3838	383	35	u2f	u2f	PUNCT
ejpam-3838	383	36	′′	′′	PROPN
ejpam-3838	383	37	]	]	PUNCT
ejpam-3838	383	38	for	for	ADP
ejpam-3838	383	39	some	some	DET
ejpam-3838	383	40	matrices	matrix	NOUN
ejpam-3838	384	1	f	f	NOUN
ejpam-3838	384	2	′	′	NOUN
ejpam-3838	385	1	and	and	CCONJ
ejpam-3838	385	2	f	f	PROPN
ejpam-3838	385	3	′′.	′′.	PROPN
ejpam-3838	385	4	since	since	SCONJ
ejpam-3838	385	5	c	c	PROPN
ejpam-3838	385	6	⊆	⊆	NUM
ejpam-3838	385	7	c	c	NOUN
ejpam-3838	385	8	′	′	NOUN
ejpam-3838	385	9	,	,	PUNCT
ejpam-3838	385	10	there	there	PRON
ejpam-3838	385	11	exist	exist	VERB
ejpam-3838	385	12	matrices	matrix	NOUN
ejpam-3838	385	13	m	m	VERB
ejpam-3838	385	14	′	′	NUM
ejpam-3838	386	1	and	and	CCONJ
ejpam-3838	386	2	m	m	VERB
ejpam-3838	386	3	′′	′′	NOUN
ejpam-3838	386	4	such	such	ADJ
ejpam-3838	386	5	that	that	SCONJ
ejpam-3838	386	6	[	[	PUNCT
ejpam-3838	386	7	ik0	ik0	VERB
ejpam-3838	386	8	a0	a0	NOUN
ejpam-3838	386	9	b0	b0	PROPN
ejpam-3838	386	10	+	+	CCONJ
ejpam-3838	386	11	ub1	ub1	PROPN
ejpam-3838	386	12	+	+	CCONJ
ejpam-3838	386	13	u2f	u2f	ADJ
ejpam-3838	386	14	′	′	NUM
ejpam-3838	386	15	0	0	NUM
ejpam-3838	386	16	uik1	uik1	NOUN
ejpam-3838	386	17	ud1	ud1	NOUN
ejpam-3838	386	18	+	+	CCONJ
ejpam-3838	386	19	u2f	u2f	VERB
ejpam-3838	387	1	′′	′′	PROPN
ejpam-3838	387	2	]	]	PUNCT
ejpam-3838	387	3	≡	≡	PROPN
ejpam-3838	387	4	[	[	PUNCT
ejpam-3838	387	5	ik0	ik0	NOUN
ejpam-3838	387	6	0	0	NUM
ejpam-3838	387	7	m	m	VERB
ejpam-3838	387	8	′	′	NOUN
ejpam-3838	387	9	0	0	PUNCT
ejpam-3838	388	1	ik1	ik1	ADJ
ejpam-3838	388	2	m	m	VERB
ejpam-3838	388	3	′′	′′	NOUN
ejpam-3838	388	4	]	]	PUNCT
ejpam-3838	388	5			PROPN
ejpam-3838	388	6	ik0	ik0	VERB
ejpam-3838	388	7	a0	a0	NOUN
ejpam-3838	388	8	b0	b0	PROPN
ejpam-3838	388	9	+	+	CCONJ
ejpam-3838	388	10	ub1	ub1	PROPN
ejpam-3838	388	11	+	+	CCONJ
ejpam-3838	388	12	u2n0	u2n0	ADJ
ejpam-3838	388	13	0	0	NUM
ejpam-3838	388	14	uik1	uik1	NOUN
ejpam-3838	388	15	ud1	ud1	NOUN
ejpam-3838	388	16	+	+	CCONJ
ejpam-3838	388	17	u2n1	u2n1	PROPN
ejpam-3838	388	18	0	0	NUM
ejpam-3838	388	19	0	0	NUM
ejpam-3838	388	20	u2f2	u2f2	ADP
ejpam-3838	388	21			NOUN
ejpam-3838	388	22	(	(	PUNCT
ejpam-3838	388	23	u3	u3	PROPN
ejpam-3838	388	24	)	)	PUNCT
ejpam-3838	388	25	.	.	PUNCT
ejpam-3838	389	1	then	then	ADV
ejpam-3838	389	2	we	we	PRON
ejpam-3838	389	3	have	have	VERB
ejpam-3838	389	4	f	f	NOUN
ejpam-3838	389	5	′	′	NOUN
ejpam-3838	389	6	=	=	SYM
ejpam-3838	389	7	n0	n0	PROPN
ejpam-3838	389	8	+	+	CCONJ
ejpam-3838	389	9	m	m	VERB
ejpam-3838	389	10	′f2	′f2	NOUN
ejpam-3838	389	11	and	and	CCONJ
ejpam-3838	389	12	f2	f2	PROPN
ejpam-3838	389	13	=	=	SYM
ejpam-3838	389	14	n1	n1	PROPN
ejpam-3838	389	15	+	+	CCONJ
ejpam-3838	389	16	m	m	VERB
ejpam-3838	389	17	′′f2	′′f2	ADJ
ejpam-3838	389	18	,	,	PUNCT
ejpam-3838	389	19	so	so	SCONJ
ejpam-3838	389	20	ψ	ψ	NOUN
ejpam-3838	389	21	is	be	AUX
ejpam-3838	389	22	surjective	surjective	ADJ
ejpam-3838	389	23	and	and	CCONJ
ejpam-3838	389	24	hence	hence	ADV
ejpam-3838	389	25	,	,	PUNCT
ejpam-3838	389	26	bijective	bijective	ADJ
ejpam-3838	389	27	.	.	PUNCT
ejpam-3838	390	1	therefore	therefore	ADV
ejpam-3838	390	2	,	,	PUNCT
ejpam-3838	390	3	|	|	ADV
ejpam-3838	390	4	{	{	PUNCT
ejpam-3838	390	5	c	c	NOUN
ejpam-3838	390	6	∈	∈	PROPN
ejpam-3838	390	7	y	y	NOUN
ejpam-3838	391	1	|	|	ADV
ejpam-3838	391	2	c	c	VERB
ejpam-3838	391	3	⊆	⊆	NUM
ejpam-3838	391	4	c	c	NOUN
ejpam-3838	391	5	′	′	NOUN
ejpam-3838	391	6	}	}	PUNCT
ejpam-3838	392	1	|	|	ADV
ejpam-3838	392	2	=	=	SYM
ejpam-3838	392	3	|mk0×k2(fq)×mk1×k2(fq)|	|mk0×k2(fq)×mk1×k2(fq)|	NOUN
ejpam-3838	392	4	=	=	SYM
ejpam-3838	392	5	qk0k2qk1k2	qk0k2qk1k2	PROPN
ejpam-3838	392	6	=	=	NOUN
ejpam-3838	392	7	q(k0+k1)k2	q(k0+k1)k2	PROPN
ejpam-3838	392	8	.	.	PUNCT
ejpam-3838	393	1	�	�	PROPN
ejpam-3838	393	2	given	give	VERB
ejpam-3838	393	3	c1	c1	PROPN
ejpam-3838	393	4	and	and	CCONJ
ejpam-3838	393	5	c2	c2	PROPN
ejpam-3838	393	6	,	,	PUNCT
ejpam-3838	393	7	we	we	PRON
ejpam-3838	393	8	can	can	AUX
ejpam-3838	393	9	now	now	ADV
ejpam-3838	393	10	count	count	VERB
ejpam-3838	393	11	the	the	DET
ejpam-3838	393	12	number	number	NOUN
ejpam-3838	393	13	of	of	ADP
ejpam-3838	393	14	self	self	NOUN
ejpam-3838	393	15	-	-	PUNCT
ejpam-3838	393	16	orthogonal	orthogonal	ADJ
ejpam-3838	393	17	codes	code	NOUN
ejpam-3838	393	18	over	over	ADP
ejpam-3838	393	19	r2	r2	PROPN
ejpam-3838	393	20	having	have	VERB
ejpam-3838	393	21	u2	u2	NOUN
ejpam-3838	393	22	-	-	PUNCT
ejpam-3838	393	23	residue	residue	NOUN
ejpam-3838	393	24	c1	c1	NOUN
ejpam-3838	393	25	and	and	CCONJ
ejpam-3838	393	26	torsion	torsion	PROPN
ejpam-3838	393	27	c2	c2	PROPN
ejpam-3838	393	28	.	.	PUNCT
ejpam-3838	394	1	theorem	theorem	VERB
ejpam-3838	394	2	3	3	X
ejpam-3838	394	3	.	.	PUNCT
ejpam-3838	395	1	suppose	suppose	VERB
ejpam-3838	395	2	c1	c1	PROPN
ejpam-3838	395	3	is	be	AUX
ejpam-3838	395	4	a	a	DET
ejpam-3838	395	5	self	self	NOUN
ejpam-3838	395	6	-	-	PUNCT
ejpam-3838	395	7	orthogonal	orthogonal	ADJ
ejpam-3838	395	8	code	code	NOUN
ejpam-3838	395	9	over	over	ADP
ejpam-3838	395	10	fq	fq	PROPN
ejpam-3838	395	11	+	+	CCONJ
ejpam-3838	395	12	ufq	ufq	PROPN
ejpam-3838	395	13	,	,	PUNCT
ejpam-3838	395	14	where	where	SCONJ
ejpam-3838	395	15	q	q	NOUN
ejpam-3838	395	16	is	be	AUX
ejpam-3838	395	17	odd	odd	ADJ
ejpam-3838	395	18	,	,	PUNCT
ejpam-3838	395	19	of	of	ADP
ejpam-3838	395	20	type	type	NOUN
ejpam-3838	395	21	{	{	PUNCT
ejpam-3838	395	22	k0	k0	PROPN
ejpam-3838	395	23	,	,	PUNCT
ejpam-3838	395	24	k1	k1	PROPN
ejpam-3838	395	25	}	}	PUNCT
ejpam-3838	395	26	such	such	ADJ
ejpam-3838	395	27	that	that	SCONJ
ejpam-3838	395	28	tor(c1	tor(c1	PROPN
ejpam-3838	395	29	)	)	PUNCT
ejpam-3838	395	30	is	be	AUX
ejpam-3838	395	31	self	self	NOUN
ejpam-3838	395	32	-	-	PUNCT
ejpam-3838	395	33	orthogonal	orthogonal	ADJ
ejpam-3838	395	34	and	and	CCONJ
ejpam-3838	395	35	c2	c2	PROPN
ejpam-3838	395	36	is	be	AUX
ejpam-3838	395	37	a	a	DET
ejpam-3838	395	38	code	code	NOUN
ejpam-3838	395	39	over	over	ADP
ejpam-3838	395	40	fq	fq	PROPN
ejpam-3838	395	41	of	of	ADP
ejpam-3838	395	42	dimension	dimension	PROPN
ejpam-3838	395	43	k0	k0	PROPN
ejpam-3838	396	1	+	+	PROPN
ejpam-3838	396	2	k1	k1	PROPN
ejpam-3838	396	3	+	+	NOUN
ejpam-3838	396	4	k2	k2	NOUN
ejpam-3838	396	5	such	such	ADJ
ejpam-3838	396	6	that	that	SCONJ
ejpam-3838	396	7	tor(c1	tor(c1	PROPN
ejpam-3838	396	8	)	)	PUNCT
ejpam-3838	396	9	⊆	⊆	NUM
ejpam-3838	396	10	c2	c2	PROPN
ejpam-3838	396	11	⊆	⊆	NUM
ejpam-3838	396	12	res(c1	res(c1	PROPN
ejpam-3838	396	13	)	)	PUNCT
ejpam-3838	396	14	⊥.	⊥.	PROPN
ejpam-3838	396	15	then	then	ADV
ejpam-3838	396	16	the	the	DET
ejpam-3838	396	17	number	number	NOUN
ejpam-3838	396	18	of	of	ADP
ejpam-3838	396	19	self	self	NOUN
ejpam-3838	396	20	-	-	PUNCT
ejpam-3838	396	21	orthogonal	orthogonal	ADJ
ejpam-3838	396	22	codes	code	NOUN
ejpam-3838	396	23	c	c	NOUN
ejpam-3838	396	24	′	′	NOUN
ejpam-3838	396	25	of	of	ADP
ejpam-3838	396	26	length	length	NOUN
ejpam-3838	396	27	n	n	PROPN
ejpam-3838	396	28	over	over	ADP
ejpam-3838	396	29	fq	fq	PROPN
ejpam-3838	396	30	+	+	CCONJ
ejpam-3838	396	31	ufq	ufq	PROPN
ejpam-3838	396	32	+	+	NUM
ejpam-3838	396	33	u2fq	u2fq	X
ejpam-3838	396	34	such	such	ADJ
ejpam-3838	396	35	that	that	DET
ejpam-3838	396	36	res(c	res(c	ADJ
ejpam-3838	396	37	′	′	NOUN
ejpam-3838	396	38	)	)	PUNCT
ejpam-3838	396	39	=	=	SYM
ejpam-3838	396	40	c1	c1	NOUN
ejpam-3838	396	41	and	and	CCONJ
ejpam-3838	396	42	tor2(c	tor2(c	NOUN
ejpam-3838	396	43	′	′	NOUN
ejpam-3838	396	44	)	)	PUNCT
ejpam-3838	397	1	=	=	SYM
ejpam-3838	397	2	c2	c2	PROPN
ejpam-3838	397	3	is	be	AUX
ejpam-3838	397	4	qk0(2n−3k0−6k1−2k2−1)/2+k1(n−k1−k2	qk0(2n−3k0−6k1−2k2−1)/2+k1(n−k1−k2	ADJ
ejpam-3838	397	5	)	)	PUNCT
ejpam-3838	397	6	.	.	PUNCT
ejpam-3838	398	1	l.e	l.e	PROPN
ejpam-3838	398	2	.	.	PROPN
ejpam-3838	398	3	galvez	galvez	PROPN
ejpam-3838	398	4	,	,	PUNCT
ejpam-3838	398	5	r.a	r.a	PROPN
ejpam-3838	398	6	.	.	PROPN
ejpam-3838	398	7	betty	betty	PROPN
ejpam-3838	398	8	,	,	PUNCT
ejpam-3838	398	9	f.	f.	PROPN
ejpam-3838	398	10	nemenzo	nemenzo	PROPN
ejpam-3838	398	11	/	/	SYM
ejpam-3838	398	12	eur	eur	NOUN
ejpam-3838	398	13	.	.	PUNCT
ejpam-3838	399	1	j.	j.	PROPN
ejpam-3838	399	2	pure	pure	PROPN
ejpam-3838	399	3	appl	appl	PROPN
ejpam-3838	399	4	.	.	PROPN
ejpam-3838	399	5	math	math	PROPN
ejpam-3838	399	6	,	,	PUNCT
ejpam-3838	399	7	13	13	NUM
ejpam-3838	399	8	(	(	PUNCT
ejpam-3838	399	9	4	4	NUM
ejpam-3838	399	10	)	)	PUNCT
ejpam-3838	399	11	(	(	PUNCT
ejpam-3838	399	12	2020	2020	NUM
ejpam-3838	399	13	)	)	PUNCT
ejpam-3838	399	14	,	,	PUNCT
ejpam-3838	399	15	873	873	NUM
ejpam-3838	399	16	-	-	NUM
ejpam-3838	399	17	892	892	NUM
ejpam-3838	399	18	891	891	NUM
ejpam-3838	399	19	proof	proof	NOUN
ejpam-3838	399	20	.	.	PUNCT
ejpam-3838	400	1	without	without	ADP
ejpam-3838	400	2	loss	loss	NOUN
ejpam-3838	400	3	of	of	ADP
ejpam-3838	400	4	generality	generality	NOUN
ejpam-3838	400	5	,	,	PUNCT
ejpam-3838	400	6	we	we	PRON
ejpam-3838	400	7	assume	assume	VERB
ejpam-3838	400	8	that	that	SCONJ
ejpam-3838	400	9	c1	c1	PROPN
ejpam-3838	400	10	has	have	VERB
ejpam-3838	400	11	generator	generator	NOUN
ejpam-3838	400	12	matrix	matrix	NOUN
ejpam-3838	400	13	g1	g1	NOUN
ejpam-3838	400	14	and	and	CCONJ
ejpam-3838	400	15	c2	c2	PROPN
ejpam-3838	400	16	has	have	VERB
ejpam-3838	400	17	generator	generator	NOUN
ejpam-3838	400	18	matrix	matrix	NOUN
ejpam-3838	400	19	g2	g2	PROPN
ejpam-3838	400	20	.	.	PUNCT
ejpam-3838	401	1	then	then	ADV
ejpam-3838	401	2	we	we	PRON
ejpam-3838	401	3	compute	compute	VERB
ejpam-3838	401	4	for	for	ADP
ejpam-3838	401	5	|y	|y	NOUN
ejpam-3838	401	6	′|	′|	NOUN
ejpam-3838	401	7	.	.	PUNCT
ejpam-3838	401	8	by	by	ADP
ejpam-3838	401	9	lemma	lemma	PROPN
ejpam-3838	401	10	10	10	NUM
ejpam-3838	401	11	and	and	CCONJ
ejpam-3838	401	12	lemma	lemma	PROPN
ejpam-3838	401	13	11	11	NUM
ejpam-3838	401	14	,	,	PUNCT
ejpam-3838	401	15	we	we	PRON
ejpam-3838	401	16	have	have	VERB
ejpam-3838	401	17	q(k0+k1)k2	q(k0+k1)k2	NOUN
ejpam-3838	401	18	∣∣y	∣∣y	NOUN
ejpam-3838	401	19	′∣∣	′∣∣	NOUN
ejpam-3838	401	20	=	=	PUNCT
ejpam-3838	401	21	∑	∑	PUNCT
ejpam-3838	401	22	c′∈y	c′∈y	NOUN
ejpam-3838	401	23	′	′	NUM
ejpam-3838	401	24	∣∣{c	∣∣{c	PROPN
ejpam-3838	401	25	∈	∈	PROPN
ejpam-3838	401	26	x|c	x|c	PUNCT
ejpam-3838	402	1	⊆	⊆	NUM
ejpam-3838	402	2	c	c	NOUN
ejpam-3838	402	3	′}∣∣	′}∣∣	NOUN
ejpam-3838	402	4	=	=	SYM
ejpam-3838	402	5	∑	∑	PUNCT
ejpam-3838	402	6	c∈y	c∈y	NOUN
ejpam-3838	402	7	∣∣{c	∣∣{c	PROPN
ejpam-3838	402	8	′	′	NUM
ejpam-3838	402	9	∈	∈	PROPN
ejpam-3838	402	10	x	x	X
ejpam-3838	402	11	′|c	′|c	NOUN
ejpam-3838	402	12	⊆	⊆	NUM
ejpam-3838	402	13	c	c	NOUN
ejpam-3838	402	14	′}∣∣	′}∣∣	NOUN
ejpam-3838	402	15	=	=	SYM
ejpam-3838	402	16	∑	∑	PUNCT
ejpam-3838	402	17	c∈y	c∈y	NOUN
ejpam-3838	402	18	1	1	NUM
ejpam-3838	402	19	=	=	NOUN
ejpam-3838	402	20	|y	|y	NOUN
ejpam-3838	402	21	|	|	ADV
ejpam-3838	402	22	.	.	PUNCT
ejpam-3838	403	1	the	the	DET
ejpam-3838	403	2	results	result	NOUN
ejpam-3838	403	3	follow	follow	VERB
ejpam-3838	403	4	from	from	ADP
ejpam-3838	403	5	lemma	lemma	PROPN
ejpam-3838	403	6	9	9	NUM
ejpam-3838	403	7	.	.	PUNCT
ejpam-3838	403	8	�	�	PROPN
ejpam-3838	403	9	8	8	NUM
ejpam-3838	403	10	.	.	PUNCT
ejpam-3838	403	11	mass	mass	ADJ
ejpam-3838	403	12	formula	formula	NOUN
ejpam-3838	403	13	for	for	ADP
ejpam-3838	403	14	self	self	NOUN
ejpam-3838	403	15	-	-	PUNCT
ejpam-3838	403	16	orthogonal	orthogonal	ADJ
ejpam-3838	403	17	codes	code	NOUN
ejpam-3838	403	18	over	over	ADP
ejpam-3838	403	19	fq	fq	PROPN
ejpam-3838	403	20	+	+	CCONJ
ejpam-3838	403	21	ufq	ufq	PROPN
ejpam-3838	403	22	+	+	NUM
ejpam-3838	403	23	u2fq	u2fq	PROPN
ejpam-3838	403	24	,	,	PUNCT
ejpam-3838	403	25	where	where	SCONJ
ejpam-3838	403	26	q	q	NOUN
ejpam-3838	403	27	is	be	AUX
ejpam-3838	403	28	odd	odd	ADJ
ejpam-3838	403	29	we	we	PRON
ejpam-3838	403	30	now	now	ADV
ejpam-3838	403	31	have	have	VERB
ejpam-3838	403	32	the	the	DET
ejpam-3838	403	33	following	follow	VERB
ejpam-3838	403	34	theorem	theorem	VERB
ejpam-3838	403	35	.	.	PUNCT
ejpam-3838	404	1	theorem	theorem	NOUN
ejpam-3838	404	2	4	4	NUM
ejpam-3838	404	3	.	.	PUNCT
ejpam-3838	404	4	suppose	suppose	VERB
ejpam-3838	404	5	q	q	NOUN
ejpam-3838	404	6	is	be	AUX
ejpam-3838	404	7	odd	odd	ADJ
ejpam-3838	404	8	.	.	PUNCT
ejpam-3838	405	1	the	the	DET
ejpam-3838	405	2	number	number	NOUN
ejpam-3838	405	3	of	of	ADP
ejpam-3838	405	4	distinct	distinct	ADJ
ejpam-3838	405	5	self	self	NOUN
ejpam-3838	405	6	-	-	PUNCT
ejpam-3838	405	7	orthogonal	orthogonal	ADJ
ejpam-3838	405	8	codes	code	NOUN
ejpam-3838	405	9	over	over	ADP
ejpam-3838	405	10	fq	fq	PROPN
ejpam-3838	405	11	+	+	CCONJ
ejpam-3838	405	12	ufq	ufq	PROPN
ejpam-3838	405	13	+	+	NUM
ejpam-3838	405	14	u2fq	u2fq	X
ejpam-3838	405	15	of	of	ADP
ejpam-3838	405	16	length	length	NOUN
ejpam-3838	405	17	n	n	PROPN
ejpam-3838	405	18	and	and	CCONJ
ejpam-3838	405	19	type	type	NOUN
ejpam-3838	405	20	{	{	PUNCT
ejpam-3838	405	21	k0	k0	PROPN
ejpam-3838	405	22	,	,	PUNCT
ejpam-3838	405	23	k1	k1	NOUN
ejpam-3838	405	24	,	,	PUNCT
ejpam-3838	405	25	k2	k2	NOUN
ejpam-3838	405	26	}	}	PUNCT
ejpam-3838	405	27	,	,	PUNCT
ejpam-3838	405	28	denoted	denote	VERB
ejpam-3838	405	29	by	by	ADP
ejpam-3838	405	30	mr2(n	mr2(n	PROPN
ejpam-3838	405	31	,	,	PUNCT
ejpam-3838	405	32	k0	k0	PROPN
ejpam-3838	405	33	,	,	PUNCT
ejpam-3838	405	34	k1	k1	PROPN
ejpam-3838	405	35	,	,	PUNCT
ejpam-3838	405	36	k2	k2	NOUN
ejpam-3838	405	37	)	)	PUNCT
ejpam-3838	405	38	is	be	AUX
ejpam-3838	405	39	[	[	PUNCT
ejpam-3838	405	40	n−	n−	NOUN
ejpam-3838	405	41	2k0	2k0	NUM
ejpam-3838	405	42	−	−	PROPN
ejpam-3838	405	43	k1	k1	PROPN
ejpam-3838	405	44	k2	k2	PROPN
ejpam-3838	405	45	]	]	PUNCT
ejpam-3838	405	46	q	q	X
ejpam-3838	406	1	[	[	PUNCT
ejpam-3838	406	2	k0	k0	PROPN
ejpam-3838	406	3	+	+	CCONJ
ejpam-3838	406	4	k1	k1	PROPN
ejpam-3838	406	5	k0	k0	PROPN
ejpam-3838	406	6	]	]	PUNCT
ejpam-3838	406	7	q	q	X
ejpam-3838	406	8	σq(n	σq(n	X
ejpam-3838	406	9	,	,	PUNCT
ejpam-3838	406	10	k0	k0	PROPN
ejpam-3838	406	11	+	+	CCONJ
ejpam-3838	406	12	k1)q	k1)q	NOUN
ejpam-3838	406	13	k0(2n−3k0−4k1−k2−1)+k1(n−k1−k2	k0(2n−3k0−4k1−k2−1)+k1(n−k1−k2	NOUN
ejpam-3838	406	14	)	)	PUNCT
ejpam-3838	406	15	.	.	PUNCT
ejpam-3838	407	1	proof	proof	NOUN
ejpam-3838	407	2	.	.	PUNCT
ejpam-3838	408	1	if	if	SCONJ
ejpam-3838	408	2	c	c	PROPN
ejpam-3838	408	3	is	be	AUX
ejpam-3838	408	4	a	a	DET
ejpam-3838	408	5	self	self	NOUN
ejpam-3838	408	6	-	-	PUNCT
ejpam-3838	408	7	orthogonal	orthogonal	ADJ
ejpam-3838	408	8	code	code	NOUN
ejpam-3838	408	9	of	of	ADP
ejpam-3838	408	10	length	length	NOUN
ejpam-3838	408	11	n	n	CCONJ
ejpam-3838	408	12	over	over	ADP
ejpam-3838	408	13	fq+ufq+u2fq	fq+ufq+u2fq	PROPN
ejpam-3838	408	14	of	of	ADP
ejpam-3838	408	15	type	type	NOUN
ejpam-3838	408	16	{	{	PUNCT
ejpam-3838	408	17	k0	k0	PROPN
ejpam-3838	408	18	,	,	PUNCT
ejpam-3838	408	19	k1	k1	NOUN
ejpam-3838	408	20	,	,	PUNCT
ejpam-3838	408	21	k2	k2	NOUN
ejpam-3838	408	22	}	}	PUNCT
ejpam-3838	408	23	,	,	PUNCT
ejpam-3838	408	24	then	then	ADV
ejpam-3838	408	25	by	by	ADP
ejpam-3838	408	26	setting	set	VERB
ejpam-3838	408	27	c1	c1	NOUN
ejpam-3838	408	28	=	=	PUNCT
ejpam-3838	408	29	res(c	res(c	PROPN
ejpam-3838	408	30	)	)	PUNCT
ejpam-3838	408	31	and	and	CCONJ
ejpam-3838	408	32	c2	c2	PROPN
ejpam-3838	408	33	=	=	PUNCT
ejpam-3838	408	34	tor2(c	tor2(c	NUM
ejpam-3838	408	35	)	)	PUNCT
ejpam-3838	408	36	,	,	PUNCT
ejpam-3838	408	37	we	we	PRON
ejpam-3838	408	38	see	see	VERB
ejpam-3838	408	39	that	that	DET
ejpam-3838	408	40	c1	c1	PROPN
ejpam-3838	408	41	and	and	CCONJ
ejpam-3838	408	42	c2	c2	PROPN
ejpam-3838	408	43	satisfies	satisfie	NOUN
ejpam-3838	408	44	(	(	PUNCT
ejpam-3838	408	45	i)–(iii	i)–(iii	NOUN
ejpam-3838	408	46	)	)	PUNCT
ejpam-3838	408	47	of	of	ADP
ejpam-3838	408	48	lemma	lemma	PROPN
ejpam-3838	408	49	7	7	NUM
ejpam-3838	408	50	.	.	PUNCT
ejpam-3838	409	1	the	the	DET
ejpam-3838	409	2	number	number	NOUN
ejpam-3838	409	3	of	of	ADP
ejpam-3838	409	4	self	self	NOUN
ejpam-3838	409	5	-	-	PUNCT
ejpam-3838	409	6	orthogonal	orthogonal	ADJ
ejpam-3838	409	7	codes	code	NOUN
ejpam-3838	409	8	with	with	ADP
ejpam-3838	409	9	given	give	VERB
ejpam-3838	409	10	u2	u2	NOUN
ejpam-3838	409	11	-	-	PUNCT
ejpam-3838	409	12	residue	residue	NOUN
ejpam-3838	409	13	c1	c1	NOUN
ejpam-3838	409	14	and	and	CCONJ
ejpam-3838	409	15	torsion	torsion	NOUN
ejpam-3838	409	16	c2	c2	PROPN
ejpam-3838	409	17	is	be	AUX
ejpam-3838	409	18	given	give	VERB
ejpam-3838	409	19	in	in	ADP
ejpam-3838	409	20	theorem	theorem	ADJ
ejpam-3838	409	21	3	3	NUM
ejpam-3838	409	22	.	.	PUNCT
ejpam-3838	410	1	the	the	DET
ejpam-3838	410	2	number	number	NOUN
ejpam-3838	410	3	of	of	ADP
ejpam-3838	410	4	self	self	NOUN
ejpam-3838	410	5	-	-	PUNCT
ejpam-3838	410	6	orthogonal	orthogonal	ADJ
ejpam-3838	410	7	codes	code	NOUN
ejpam-3838	410	8	c1	c1	PROPN
ejpam-3838	410	9	over	over	ADP
ejpam-3838	410	10	fq+ufq	fq+ufq	PROPN
ejpam-3838	410	11	satisfying	satisfying	NOUN
ejpam-3838	410	12	(	(	PUNCT
ejpam-3838	410	13	i	i	NOUN
ejpam-3838	410	14	)	)	PUNCT
ejpam-3838	410	15	and	and	CCONJ
ejpam-3838	410	16	(	(	PUNCT
ejpam-3838	410	17	ii	ii	NOUN
ejpam-3838	410	18	)	)	PUNCT
ejpam-3838	410	19	is	be	AUX
ejpam-3838	410	20	given	give	VERB
ejpam-3838	410	21	in	in	ADP
ejpam-3838	410	22	corollary	corollary	ADJ
ejpam-3838	410	23	2	2	NUM
ejpam-3838	410	24	.	.	PUNCT
ejpam-3838	411	1	the	the	DET
ejpam-3838	411	2	number	number	NOUN
ejpam-3838	411	3	of	of	ADP
ejpam-3838	411	4	codes	code	NOUN
ejpam-3838	411	5	c2	c2	PROPN
ejpam-3838	411	6	satisfying	satisfying	ADJ
ejpam-3838	411	7	(	(	PUNCT
ejpam-3838	411	8	iii	iii	X
ejpam-3838	411	9	)	)	PUNCT
ejpam-3838	411	10	is	be	AUX
ejpam-3838	411	11	[	[	PUNCT
ejpam-3838	411	12	n−	n−	NOUN
ejpam-3838	411	13	2k0	2k0	NUM
ejpam-3838	411	14	−	−	PROPN
ejpam-3838	411	15	k1	k1	PROPN
ejpam-3838	411	16	k2	k2	PROPN
ejpam-3838	411	17	]	]	PUNCT
ejpam-3838	411	18	q	q	X
ejpam-3838	411	19	.	.	PUNCT
ejpam-3838	412	1	the	the	DET
ejpam-3838	412	2	value	value	NOUN
ejpam-3838	412	3	of	of	ADP
ejpam-3838	412	4	mr2(n	mr2(n	PROPN
ejpam-3838	412	5	,	,	PUNCT
ejpam-3838	412	6	k0	k0	PROPN
ejpam-3838	412	7	,	,	PUNCT
ejpam-3838	412	8	k1	k1	PROPN
ejpam-3838	412	9	,	,	PUNCT
ejpam-3838	412	10	k2	k2	NOUN
ejpam-3838	412	11	)	)	PUNCT
ejpam-3838	412	12	is	be	AUX
ejpam-3838	412	13	obtained	obtain	VERB
ejpam-3838	412	14	by	by	ADP
ejpam-3838	412	15	the	the	DET
ejpam-3838	412	16	product	product	NOUN
ejpam-3838	412	17	of	of	ADP
ejpam-3838	412	18	these	these	PRON
ejpam-3838	412	19	.	.	PUNCT
ejpam-3838	413	1	�	�	PROPN
ejpam-3838	413	2	we	we	PRON
ejpam-3838	413	3	now	now	ADV
ejpam-3838	413	4	have	have	VERB
ejpam-3838	413	5	the	the	DET
ejpam-3838	413	6	following	follow	VERB
ejpam-3838	413	7	mass	mass	NOUN
ejpam-3838	413	8	formula	formula	NOUN
ejpam-3838	413	9	for	for	ADP
ejpam-3838	413	10	self	self	NOUN
ejpam-3838	413	11	-	-	PUNCT
ejpam-3838	413	12	dual	dual	ADJ
ejpam-3838	413	13	codes	code	NOUN
ejpam-3838	413	14	over	over	ADP
ejpam-3838	413	15	r2	r2	PROPN
ejpam-3838	413	16	as	as	ADP
ejpam-3838	413	17	a	a	DET
ejpam-3838	413	18	direct	direct	ADJ
ejpam-3838	413	19	consequence	consequence	NOUN
ejpam-3838	413	20	of	of	ADP
ejpam-3838	413	21	theorem	theorem	ADJ
ejpam-3838	413	22	4	4	NUM
ejpam-3838	413	23	.	.	PUNCT
ejpam-3838	413	24	corollary	corollary	ADJ
ejpam-3838	413	25	3	3	NUM
ejpam-3838	413	26	.	.	PUNCT
ejpam-3838	413	27	suppose	suppose	VERB
ejpam-3838	413	28	q	q	NOUN
ejpam-3838	413	29	is	be	AUX
ejpam-3838	413	30	odd	odd	ADJ
ejpam-3838	413	31	.	.	PUNCT
ejpam-3838	414	1	the	the	DET
ejpam-3838	414	2	number	number	NOUN
ejpam-3838	414	3	of	of	ADP
ejpam-3838	414	4	distinct	distinct	ADJ
ejpam-3838	414	5	self	self	NOUN
ejpam-3838	414	6	-	-	PUNCT
ejpam-3838	414	7	dual	dual	ADJ
ejpam-3838	414	8	codes	code	NOUN
ejpam-3838	414	9	of	of	ADP
ejpam-3838	414	10	even	even	ADV
ejpam-3838	414	11	length	length	NOUN
ejpam-3838	414	12	n	n	PROPN
ejpam-3838	414	13	over	over	ADP
ejpam-3838	414	14	fq	fq	PROPN
ejpam-3838	414	15	+	+	CCONJ
ejpam-3838	414	16	ufq	ufq	PROPN
ejpam-3838	414	17	+	+	NUM
ejpam-3838	414	18	u2fq	u2fq	PUNCT
ejpam-3838	414	19	is	be	AUX
ejpam-3838	414	20	given	give	VERB
ejpam-3838	414	21	by	by	ADP
ejpam-3838	414	22	n	n	PRON
ejpam-3838	414	23	2∑	2∑	PROPN
ejpam-3838	414	24	k0=0	k0=0	PROPN
ejpam-3838	414	25	mr2(n	mr2(n	PROPN
ejpam-3838	414	26	,	,	PUNCT
ejpam-3838	414	27	k0	k0	PROPN
ejpam-3838	414	28	,	,	PUNCT
ejpam-3838	414	29	n	n	PRON
ejpam-3838	414	30	2	2	NUM
ejpam-3838	414	31	−	−	PROPN
ejpam-3838	414	32	k0	k0	PROPN
ejpam-3838	414	33	,	,	PUNCT
ejpam-3838	414	34	n	n	PRON
ejpam-3838	414	35	2	2	NUM
ejpam-3838	414	36	−	−	PROPN
ejpam-3838	414	37	k0	k0	PROPN
ejpam-3838	414	38	)	)	PUNCT
ejpam-3838	414	39	.	.	PUNCT
ejpam-3838	415	1	(	(	PUNCT
ejpam-3838	415	2	24	24	NUM
ejpam-3838	415	3	)	)	PUNCT
ejpam-3838	415	4	proof	proof	NOUN
ejpam-3838	415	5	.	.	PUNCT
ejpam-3838	416	1	by	by	ADP
ejpam-3838	416	2	[	[	X
ejpam-3838	416	3	3	3	NUM
ejpam-3838	416	4	]	]	PUNCT
ejpam-3838	416	5	,	,	PUNCT
ejpam-3838	416	6	we	we	PRON
ejpam-3838	416	7	have	have	VERB
ejpam-3838	416	8	k1	k1	NOUN
ejpam-3838	416	9	=	=	SYM
ejpam-3838	416	10	k2	k2	PROPN
ejpam-3838	416	11	and	and	CCONJ
ejpam-3838	416	12	n	n	NOUN
ejpam-3838	416	13	=	=	SYM
ejpam-3838	416	14	2(k0	2(k0	NUM
ejpam-3838	416	15	+	+	NUM
ejpam-3838	416	16	k1	k1	NOUN
ejpam-3838	416	17	)	)	PUNCT
ejpam-3838	416	18	.	.	PUNCT
ejpam-3838	417	1	the	the	DET
ejpam-3838	417	2	result	result	NOUN
ejpam-3838	417	3	follows	follow	VERB
ejpam-3838	417	4	from	from	ADP
ejpam-3838	417	5	theorem	theorem	ADJ
ejpam-3838	417	6	4	4	NUM
ejpam-3838	417	7	.	.	PUNCT
ejpam-3838	417	8	�	�	PROPN
ejpam-3838	417	9	the	the	DET
ejpam-3838	417	10	formula	formula	NOUN
ejpam-3838	417	11	(	(	PUNCT
ejpam-3838	417	12	24	24	NUM
ejpam-3838	417	13	)	)	PUNCT
ejpam-3838	417	14	agrees	agree	VERB
ejpam-3838	417	15	with	with	ADP
ejpam-3838	417	16	[	[	X
ejpam-3838	417	17	3	3	NUM
ejpam-3838	417	18	,	,	PUNCT
ejpam-3838	417	19	theorem	theorem	VERB
ejpam-3838	417	20	1	1	NUM
ejpam-3838	417	21	]	]	PUNCT
ejpam-3838	417	22	.	.	PUNCT
ejpam-3838	418	1	references	reference	NOUN
ejpam-3838	418	2	892	892	NUM
ejpam-3838	418	3	acknowledgements	acknowledgement	NOUN
ejpam-3838	418	4	the	the	DET
ejpam-3838	418	5	authors	author	NOUN
ejpam-3838	418	6	gratefully	gratefully	ADV
ejpam-3838	418	7	acknowledge	acknowledge	VERB
ejpam-3838	418	8	the	the	DET
ejpam-3838	418	9	support	support	NOUN
ejpam-3838	418	10	of	of	ADP
ejpam-3838	418	11	the	the	DET
ejpam-3838	418	12	university	university	NOUN
ejpam-3838	418	13	of	of	ADP
ejpam-3838	418	14	the	the	DET
ejpam-3838	418	15	philippines	philippine	NOUN
ejpam-3838	418	16	office	office	NOUN
ejpam-3838	418	17	of	of	ADP
ejpam-3838	418	18	the	the	DET
ejpam-3838	418	19	vice	vice	NOUN
ejpam-3838	418	20	president	president	NOUN
ejpam-3838	418	21	for	for	ADP
ejpam-3838	418	22	academic	academic	ADJ
ejpam-3838	418	23	affairs	affair	NOUN
ejpam-3838	418	24	for	for	ADP
ejpam-3838	418	25	this	this	DET
ejpam-3838	418	26	project	project	NOUN
ejpam-3838	418	27	.	.	PUNCT
ejpam-3838	419	1	the	the	DET
ejpam-3838	419	2	authors	author	NOUN
ejpam-3838	419	3	also	also	ADV
ejpam-3838	419	4	thank	thank	VERB
ejpam-3838	419	5	dr	dr	PROPN
ejpam-3838	419	6	.	.	PROPN
ejpam-3838	419	7	markus	markus	PROPN
ejpam-3838	419	8	grassl	grassl	VERB
ejpam-3838	419	9	for	for	ADP
ejpam-3838	419	10	useful	useful	ADJ
ejpam-3838	419	11	suggestions	suggestion	NOUN
ejpam-3838	419	12	and	and	CCONJ
ejpam-3838	419	13	the	the	DET
ejpam-3838	419	14	reviewers	reviewer	NOUN
ejpam-3838	419	15	for	for	ADP
ejpam-3838	419	16	their	their	PRON
ejpam-3838	419	17	valuable	valuable	ADJ
ejpam-3838	419	18	comments	comment	NOUN
ejpam-3838	419	19	.	.	PUNCT
ejpam-3838	420	1	references	reference	NOUN
ejpam-3838	420	2	[	[	X
ejpam-3838	420	3	1	1	NUM
ejpam-3838	420	4	]	]	X
ejpam-3838	420	5	koichi	koichi	PROPN
ejpam-3838	420	6	betsumiya	betsumiya	PROPN
ejpam-3838	420	7	,	,	PUNCT
ejpam-3838	420	8	rowena	rowena	PROPN
ejpam-3838	420	9	alma	alma	PROPN
ejpam-3838	420	10	betty	betty	PROPN
ejpam-3838	420	11	,	,	PUNCT
ejpam-3838	420	12	and	and	CCONJ
ejpam-3838	420	13	akihiro	akihiro	PROPN
ejpam-3838	420	14	munemasa	munemasa	PROPN
ejpam-3838	420	15	.	.	PUNCT
ejpam-3838	421	1	mass	mass	ADJ
ejpam-3838	421	2	formula	formula	NOUN
ejpam-3838	421	3	for	for	ADP
ejpam-3838	421	4	even	even	ADV
ejpam-3838	421	5	codes	code	NOUN
ejpam-3838	421	6	over	over	ADP
ejpam-3838	421	7	z8	z8	NOUN
ejpam-3838	421	8	.	.	PUNCT
ejpam-3838	422	1	lecture	lecture	NOUN
ejpam-3838	422	2	notes	note	NOUN
ejpam-3838	422	3	in	in	ADP
ejpam-3838	422	4	computer	computer	NOUN
ejpam-3838	422	5	science	science	NOUN
ejpam-3838	422	6	,	,	PUNCT
ejpam-3838	422	7	5921:65–77	5921:65–77	PROPN
ejpam-3838	422	8	,	,	PUNCT
ejpam-3838	422	9	2009	2009	NUM
ejpam-3838	422	10	.	.	PUNCT
ejpam-3838	423	1	[	[	X
ejpam-3838	423	2	2	2	X
ejpam-3838	423	3	]	]	X
ejpam-3838	423	4	rowena	rowena	PROPN
ejpam-3838	423	5	alma	alma	PROPN
ejpam-3838	423	6	betty	betty	PROPN
ejpam-3838	423	7	and	and	CCONJ
ejpam-3838	423	8	akihiro	akihiro	PROPN
ejpam-3838	423	9	munemasa	munemasa	PROPN
ejpam-3838	423	10	.	.	PUNCT
ejpam-3838	424	1	mass	mass	ADJ
ejpam-3838	424	2	formula	formula	NOUN
ejpam-3838	424	3	for	for	ADP
ejpam-3838	424	4	self	self	NOUN
ejpam-3838	424	5	-	-	PUNCT
ejpam-3838	424	6	orthogonal	orthogonal	ADJ
ejpam-3838	424	7	codes	code	NOUN
ejpam-3838	424	8	over	over	ADP
ejpam-3838	424	9	zp2	zp2	PROPN
ejpam-3838	424	10	.	.	PUNCT
ejpam-3838	425	1	journal	journal	PROPN
ejpam-3838	425	2	of	of	ADP
ejpam-3838	425	3	combinatorics	combinatoric	NOUN
ejpam-3838	425	4	,	,	PUNCT
ejpam-3838	425	5	information	information	NOUN
ejpam-3838	425	6	and	and	CCONJ
ejpam-3838	425	7	system	system	NOUN
ejpam-3838	425	8	sciences	science	NOUN
ejpam-3838	425	9	,	,	PUNCT
ejpam-3838	425	10	34:51–66	34:51–66	NUM
ejpam-3838	425	11	,	,	PUNCT
ejpam-3838	425	12	2009	2009	NUM
ejpam-3838	425	13	.	.	PUNCT
ejpam-3838	426	1	[	[	X
ejpam-3838	426	2	3	3	X
ejpam-3838	426	3	]	]	X
ejpam-3838	426	4	rowena	rowena	PROPN
ejpam-3838	426	5	alma	alma	PROPN
ejpam-3838	426	6	betty	betty	PROPN
ejpam-3838	426	7	,	,	PUNCT
ejpam-3838	426	8	trilbe	trilbe	PROPN
ejpam-3838	426	9	lizann	lizann	PROPN
ejpam-3838	426	10	vasquez	vasquez	PROPN
ejpam-3838	426	11	,	,	PUNCT
ejpam-3838	426	12	and	and	CCONJ
ejpam-3838	426	13	fidel	fidel	NOUN
ejpam-3838	426	14	nemenzo	nemenzo	NOUN
ejpam-3838	426	15	.	.	PUNCT
ejpam-3838	427	1	mass	mass	ADJ
ejpam-3838	427	2	formula	formula	NOUN
ejpam-3838	427	3	for	for	ADP
ejpam-3838	427	4	self	self	NOUN
ejpam-3838	427	5	-	-	PUNCT
ejpam-3838	427	6	dual	dual	ADJ
ejpam-3838	427	7	codes	code	NOUN
ejpam-3838	427	8	over	over	ADP
ejpam-3838	427	9	fq+ufq+u2fq	fq+ufq+u2fq	PROPN
ejpam-3838	427	10	.	.	PROPN
ejpam-3838	427	11	journal	journal	PROPN
ejpam-3838	427	12	of	of	ADP
ejpam-3838	427	13	applied	apply	VERB
ejpam-3838	427	14	mathematics	mathematic	NOUN
ejpam-3838	427	15	and	and	CCONJ
ejpam-3838	427	16	computing	computing	NOUN
ejpam-3838	427	17	,	,	PUNCT
ejpam-3838	427	18	57:523–546	57:523–546	NUM
ejpam-3838	427	19	,	,	PUNCT
ejpam-3838	427	20	2018	2018	NUM
ejpam-3838	427	21	.	.	PUNCT
ejpam-3838	428	1	[	[	X
ejpam-3838	428	2	4	4	NUM
ejpam-3838	428	3	]	]	X
ejpam-3838	428	4	wieb	wieb	NOUN
ejpam-3838	428	5	bosma	bosma	PROPN
ejpam-3838	428	6	,	,	PUNCT
ejpam-3838	428	7	john	john	PROPN
ejpam-3838	428	8	cannon	cannon	PROPN
ejpam-3838	428	9	,	,	PUNCT
ejpam-3838	428	10	and	and	CCONJ
ejpam-3838	428	11	catherine	catherine	PROPN
ejpam-3838	428	12	playoust	playoust	PROPN
ejpam-3838	428	13	.	.	PUNCT
ejpam-3838	429	1	the	the	DET
ejpam-3838	429	2	magma	magma	NOUN
ejpam-3838	429	3	algebra	algebra	NOUN
ejpam-3838	429	4	system	system	NOUN
ejpam-3838	430	1	i	i	PRON
ejpam-3838	430	2	:	:	PUNCT
ejpam-3838	430	3	the	the	DET
ejpam-3838	430	4	user	user	NOUN
ejpam-3838	430	5	language	language	PROPN
ejpam-3838	430	6	.	.	PUNCT
ejpam-3838	431	1	journal	journal	PROPN
ejpam-3838	431	2	of	of	ADP
ejpam-3838	431	3	symbolic	symbolic	ADJ
ejpam-3838	431	4	computation	computation	NOUN
ejpam-3838	431	5	,	,	PUNCT
ejpam-3838	431	6	24(3	24(3	NUM
ejpam-3838	431	7	-	-	SYM
ejpam-3838	431	8	4):235–265	4):235–265	NUM
ejpam-3838	431	9	,	,	PUNCT
ejpam-3838	431	10	1997	1997	NUM
ejpam-3838	431	11	.	.	PUNCT
ejpam-3838	432	1	[	[	X
ejpam-3838	432	2	5	5	NUM
ejpam-3838	432	3	]	]	X
ejpam-3838	432	4	steven	steven	PROPN
ejpam-3838	432	5	dougherty	dougherty	PROPN
ejpam-3838	432	6	,	,	PUNCT
ejpam-3838	432	7	philippe	philippe	PROPN
ejpam-3838	432	8	gaborit	gaborit	PROPN
ejpam-3838	432	9	,	,	PUNCT
ejpam-3838	432	10	masaaki	masaaki	PROPN
ejpam-3838	432	11	harada	harada	PROPN
ejpam-3838	432	12	,	,	PUNCT
ejpam-3838	432	13	and	and	CCONJ
ejpam-3838	432	14	patrick	patrick	PROPN
ejpam-3838	432	15	solé.	solé.	PROPN
ejpam-3838	432	16	type	type	NOUN
ejpam-3838	432	17	ii	ii	PROPN
ejpam-3838	432	18	codes	code	NOUN
ejpam-3838	432	19	over	over	ADP
ejpam-3838	432	20	f2	f2	PROPN
ejpam-3838	432	21	+	+	CCONJ
ejpam-3838	432	22	uf2	uf2	NOUN
ejpam-3838	432	23	.	.	PUNCT
ejpam-3838	433	1	ieee	ieee	NOUN
ejpam-3838	433	2	transactions	transaction	NOUN
ejpam-3838	433	3	on	on	ADP
ejpam-3838	433	4	information	information	NOUN
ejpam-3838	433	5	theory	theory	NOUN
ejpam-3838	433	6	,	,	PUNCT
ejpam-3838	433	7	45(1):32–45	45(1):32–45	NUM
ejpam-3838	433	8	,	,	PUNCT
ejpam-3838	433	9	1999	1999	NUM
ejpam-3838	433	10	.	.	PUNCT
ejpam-3838	434	1	[	[	X
ejpam-3838	434	2	6	6	NUM
ejpam-3838	434	3	]	]	PUNCT
ejpam-3838	434	4	philippe	philippe	PROPN
ejpam-3838	434	5	gaborit	gaborit	PROPN
ejpam-3838	434	6	.	.	PUNCT
ejpam-3838	435	1	mass	mass	ADJ
ejpam-3838	435	2	formulas	formula	NOUN
ejpam-3838	435	3	for	for	ADP
ejpam-3838	435	4	self	self	NOUN
ejpam-3838	435	5	-	-	PUNCT
ejpam-3838	435	6	dual	dual	ADJ
ejpam-3838	435	7	codes	code	NOUN
ejpam-3838	435	8	over	over	ADP
ejpam-3838	435	9	z4	z4	PROPN
ejpam-3838	435	10	and	and	CCONJ
ejpam-3838	435	11	fq+ufq	fq+ufq	NOUN
ejpam-3838	435	12	rings	ring	NOUN
ejpam-3838	435	13	.	.	PUNCT
ejpam-3838	436	1	ieee	ieee	NOUN
ejpam-3838	436	2	transactions	transaction	NOUN
ejpam-3838	436	3	on	on	ADP
ejpam-3838	436	4	information	information	NOUN
ejpam-3838	436	5	theory	theory	NOUN
ejpam-3838	436	6	,	,	PUNCT
ejpam-3838	436	7	42(4):1222–1228	42(4):1222–1228	NUM
ejpam-3838	436	8	,	,	PUNCT
ejpam-3838	436	9	1996	1996	NUM
ejpam-3838	436	10	.	.	PUNCT
ejpam-3838	437	1	[	[	X
ejpam-3838	437	2	7	7	X
ejpam-3838	437	3	]	]	X
ejpam-3838	437	4	a	a	DET
ejpam-3838	437	5	roger	roger	PROPN
ejpam-3838	437	6	hammons	hammon	NOUN
ejpam-3838	437	7	,	,	PUNCT
ejpam-3838	437	8	p	p	PROPN
ejpam-3838	437	9	vijay	vijay	NOUN
ejpam-3838	437	10	kumar	kumar	PROPN
ejpam-3838	437	11	,	,	PUNCT
ejpam-3838	437	12	a	a	DET
ejpam-3838	437	13	robert	robert	PROPN
ejpam-3838	437	14	calderbank	calderbank	PROPN
ejpam-3838	437	15	,	,	PUNCT
ejpam-3838	437	16	neil	neil	PROPN
ejpam-3838	437	17	ja	ja	PROPN
ejpam-3838	437	18	sloane	sloane	PROPN
ejpam-3838	437	19	,	,	PUNCT
ejpam-3838	437	20	and	and	CCONJ
ejpam-3838	437	21	patrick	patrick	PROPN
ejpam-3838	437	22	solé.	solé.	PROPN
ejpam-3838	437	23	the	the	DET
ejpam-3838	437	24	z4	z4	PROPN
ejpam-3838	437	25	-	-	PUNCT
ejpam-3838	437	26	linearity	linearity	NOUN
ejpam-3838	437	27	of	of	ADP
ejpam-3838	437	28	kerdock	kerdock	NOUN
ejpam-3838	437	29	,	,	PUNCT
ejpam-3838	437	30	preparata	preparata	NOUN
ejpam-3838	437	31	,	,	PUNCT
ejpam-3838	437	32	goethals	goethal	NOUN
ejpam-3838	437	33	,	,	PUNCT
ejpam-3838	437	34	and	and	CCONJ
ejpam-3838	437	35	related	related	ADJ
ejpam-3838	437	36	codes	code	NOUN
ejpam-3838	437	37	.	.	PUNCT
ejpam-3838	438	1	ieee	ieee	NOUN
ejpam-3838	438	2	transactions	transaction	NOUN
ejpam-3838	438	3	on	on	ADP
ejpam-3838	438	4	information	information	NOUN
ejpam-3838	438	5	theory	theory	NOUN
ejpam-3838	438	6	,	,	PUNCT
ejpam-3838	438	7	40(2):301–319	40(2):301–319	PROPN
ejpam-3838	438	8	,	,	PUNCT
ejpam-3838	438	9	1994	1994	NUM
ejpam-3838	438	10	.	.	PUNCT
ejpam-3838	439	1	[	[	X
ejpam-3838	439	2	8	8	NUM
ejpam-3838	439	3	]	]	X
ejpam-3838	439	4	w	w	PROPN
ejpam-3838	439	5	cary	cary	PROPN
ejpam-3838	439	6	huffman	huffman	PROPN
ejpam-3838	439	7	and	and	CCONJ
ejpam-3838	439	8	vera	vera	PROPN
ejpam-3838	439	9	pless	pless	PROPN
ejpam-3838	439	10	.	.	PUNCT
ejpam-3838	440	1	fundamentals	fundamental	NOUN
ejpam-3838	440	2	of	of	ADP
ejpam-3838	440	3	error	error	NOUN
ejpam-3838	440	4	-	-	PUNCT
ejpam-3838	440	5	correcting	correct	VERB
ejpam-3838	440	6	codes	code	NOUN
ejpam-3838	440	7	.	.	PUNCT
ejpam-3838	441	1	cambridge	cambridge	PROPN
ejpam-3838	441	2	university	university	PROPN
ejpam-3838	441	3	press	press	NOUN
ejpam-3838	441	4	,	,	PUNCT
ejpam-3838	441	5	2010	2010	NUM
ejpam-3838	441	6	.	.	PUNCT
ejpam-3838	442	1	[	[	X
ejpam-3838	442	2	9	9	NUM
ejpam-3838	442	3	]	]	X
ejpam-3838	442	4	vera	vera	NOUN
ejpam-3838	442	5	pless	pless	PROPN
ejpam-3838	442	6	.	.	PUNCT
ejpam-3838	443	1	number	number	NOUN
ejpam-3838	443	2	of	of	ADP
ejpam-3838	443	3	isotropic	isotropic	ADJ
ejpam-3838	443	4	subspaces	subspace	NOUN
ejpam-3838	443	5	in	in	ADP
ejpam-3838	443	6	a	a	DET
ejpam-3838	443	7	finite	finite	ADJ
ejpam-3838	443	8	geometry	geometry	NOUN
ejpam-3838	443	9	.	.	PUNCT
ejpam-3838	444	1	atti	atti	PROPN
ejpam-3838	444	2	della	della	PROPN
ejpam-3838	444	3	accademia	accademia	PROPN
ejpam-3838	444	4	nazionale	nazionale	PROPN
ejpam-3838	444	5	dei	dei	PROPN
ejpam-3838	444	6	lincei	lincei	PROPN
ejpam-3838	444	7	rendiconti	rendiconti	ADJ
ejpam-3838	444	8	-	-	PROPN
ejpam-3838	444	9	classe	classe	ADJ
ejpam-3838	444	10	di	di	PROPN
ejpam-3838	444	11	scienze	scienze	PROPN
ejpam-3838	444	12	fisiche	fisiche	PROPN
ejpam-3838	444	13	-	-	PUNCT
ejpam-3838	444	14	matematiche	matematiche	PROPN
ejpam-3838	444	15	&	&	CCONJ
ejpam-3838	444	16	naturali	naturali	PROPN
ejpam-3838	444	17	,	,	PUNCT
ejpam-3838	444	18	39(6):418	39(6):418	NUM
ejpam-3838	444	19	,	,	PUNCT
ejpam-3838	444	20	1965	1965	NUM
ejpam-3838	444	21	.	.	PUNCT
ejpam-3838	445	1	[	[	X
ejpam-3838	445	2	10	10	NUM
ejpam-3838	445	3	]	]	X
ejpam-3838	445	4	vera	vera	NOUN
ejpam-3838	445	5	pless	pless	PROPN
ejpam-3838	445	6	.	.	PUNCT
ejpam-3838	446	1	on	on	ADP
ejpam-3838	446	2	the	the	DET
ejpam-3838	446	3	uniqueness	uniqueness	NOUN
ejpam-3838	446	4	of	of	ADP
ejpam-3838	446	5	the	the	DET
ejpam-3838	446	6	golay	golay	NOUN
ejpam-3838	446	7	codes	code	NOUN
ejpam-3838	446	8	.	.	PUNCT
ejpam-3838	447	1	journal	journal	NOUN
ejpam-3838	447	2	of	of	ADP
ejpam-3838	447	3	combinatorial	combinatorial	ADJ
ejpam-3838	447	4	theory	theory	NOUN
ejpam-3838	447	5	,	,	PUNCT
ejpam-3838	447	6	5(3):215–228	5(3):215–228	NUM
ejpam-3838	447	7	,	,	PUNCT
ejpam-3838	447	8	1968	1968	NUM
ejpam-3838	447	9	.	.	PUNCT
ejpam-3838	448	1	[	[	X
ejpam-3838	448	2	11	11	NUM
ejpam-3838	448	3	]	]	X
ejpam-3838	448	4	eric	eric	PROPN
ejpam-3838	448	5	m	m	PROPN
ejpam-3838	448	6	rains	rain	NOUN
ejpam-3838	448	7	and	and	CCONJ
ejpam-3838	448	8	nja	nja	PROPN
ejpam-3838	448	9	sloane	sloane	NOUN
ejpam-3838	448	10	.	.	PUNCT
ejpam-3838	449	1	self	self	NOUN
ejpam-3838	449	2	-	-	PUNCT
ejpam-3838	449	3	dual	dual	ADJ
ejpam-3838	449	4	codes	code	NOUN
ejpam-3838	449	5	,	,	PUNCT
ejpam-3838	449	6	in	in	ADP
ejpam-3838	449	7	handbook	handbook	NOUN
ejpam-3838	449	8	of	of	ADP
ejpam-3838	449	9	coding	code	VERB
ejpam-3838	449	10	theory	theory	NOUN
ejpam-3838	449	11	.	.	PUNCT
ejpam-3838	450	1	elsevier	elsevier	PROPN
ejpam-3838	450	2	,	,	PUNCT
ejpam-3838	450	3	amsterdam	amsterdam	PROPN
ejpam-3838	450	4	,	,	PUNCT
ejpam-3838	450	5	pages	page	NOUN
ejpam-3838	450	6	177–294	177–294	NUM
ejpam-3838	450	7	,	,	PUNCT
ejpam-3838	450	8	1998	1998	NUM
ejpam-3838	450	9	.	.	PUNCT