Sequences of Primitive and Non-primitive BCH Codes

ANSARI, A.S.; SHAH, T.; RAHMAN, ZIA-UR; ANDRADE, A.A.

doi:10.5540/tema.2018.019.02.0369

A.S. ANSARI

Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan. E-mail: asia ansari@hotmail.com; stariqshah@gmail.com; ziaagikian@gmail.com

T. SHAH

Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan. E-mail: asia ansari@hotmail.com; stariqshah@gmail.com; ziaagikian@gmail.com

ZIA-UR RAHMAN

Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan. E-mail: asia ansari@hotmail.com; stariqshah@gmail.com; ziaagikian@gmail.com

A.A. ANDRADE

Departamento de Matemática, Universidade Estadual Paulista - São José do Rio Preto, SP, Brazil. E-mail: andrade@ibilce.unesp.br

*Corresponding author: Antonio A. Andrade - E-mail: antonio.andrade@unesp.br

p(x^a)∈𝔽₂ [x;a ℤ₀ ]	$p (x^{\frac{a}{b}}) \in 𝔽_{2} [x; \frac{a}{b} ℤ_{0}]$	$p (x^{\frac{a}{b^{2}}}) \in 𝔽_{2} [x; \frac{a}{b} ℤ_{0}]$
1 + (x ^a ) + (x ^a )³	$1 + {(x^{\frac{a}{7}})}^{7} + {(x^{\frac{a}{7}})}^{21}$	$1 + {(x^{\frac{a}{49}})}^{49} + {(x^{\frac{a}{49}})}^{147}$

$1 + {(x^{\frac{a}{3}})}^{3} + {(x^{\frac{a}{3}})}^{12}$	$1 + {(x^{\frac{a}{9}})}^{9} + {(x^{\frac{a}{9}})}^{36}$
$1 + {(x^{\frac{a}{5}})}^{5} + {(x^{\frac{a}{5}})}^{20}$	$1 + {(x^{\frac{a}{25}})}^{25} + {(x^{\frac{a}{25}})}^{100}$

1 + (x ^a )³ + (x ^a )⁴	$1 + {(x^{\frac{a}{3}})}^{9} + {(x^{\frac{a}{3}})}^{12}$	$1 + {(x^{\frac{a}{9}})}^{27} + {(x^{\frac{a}{9}})}^{36}$
1 + (x ^a )³ + (x ^a )⁴	$1 + {(x^{\frac{a}{5}})}^{15} + {(x^{\frac{a}{5}})}^{20}$	$1 + {(x^{\frac{a}{25}})}^{75} + {(x^{\frac{a}{25}})}^{100}$

1 + (x ^a ) + (x ^a )⁶	$1 + {(x^{\frac{a}{3}})}^{3} + {(x^{\frac{a}{3}})}^{18}$	$1 + {(x^{\frac{a}{9}})}^{9} + {(x^{\frac{a}{9}})}^{54}$
1 + (x ^a ) + (x ^a )⁶	$1 + {(x^{\frac{a}{7}})}^{7} + {(x^{\frac{a}{7}})}^{42}$	$1 + {(x^{\frac{a}{49}})}^{49} + {(x^{\frac{a}{49}})}^{294}$
1 + (x ^a ) + (x ^a )³ + (x ^a )⁵ + (x ^a )⁸	$\begin{array}{l} 1 + {(x^{\frac{a}{3}})}^{3} + {(x^{\frac{a}{3}})}^{9} + \\ {(x^{\frac{a}{3}})}^{15} + {(x^{\frac{a}{3}})}^{24} \end{array}$	$\begin{array}{l} 1 + {(x^{\frac{a}{9}})}^{9} + {(x^{\frac{a}{9}})}^{27} + \\ {(x^{\frac{a}{9}})}^{45} + {(x^{\frac{a}{9}})}^{72} \end{array}$
1 + (x ^a ) + (x ^a )³ + (x ^a )⁵ + (x ^a )⁸	$\begin{array}{l} 1 + {(x^{\frac{a}{5}})}^{5} + {(x^{\frac{a}{5}})}^{15} + \\ {(x^{\frac{a}{5}})}^{25} + {(x^{\frac{a}{5}})}^{40} \end{array}$	$\begin{array}{l} 1 + {(x^{\frac{a}{25}})}^{25} + {(x^{\frac{a}{25}})}^{75} + \\ {(x^{\frac{a}{25}})}^{125} + {(x^{\frac{a}{25}})}^{200} \end{array}$

1 + (x ^a )⁴ + (x ^a )⁹	$1 + {(x^{\frac{a}{7}})}^{28} + {(x^{\frac{a}{7}})}^{63}$
1 + (x ^a )⁴ + (x ^a )⁹	$1 + {(x^{\frac{a}{49}})}^{196} + {(x^{\frac{a}{49}})}^{441}$
⋮	⋮	⋮

α³⁶ = 1 + α⁹	α⁶² = α²⁶ + α³⁵	α⁸⁸ = α¹⁶ + α³⁴	α¹¹⁴ = α⁶ + α¹⁵ + α²⁴ + α³³
α³⁷ = α + α¹⁰	α⁶³ = 1 + α⁹ + α²⁷	α⁸⁹ = α¹⁷ + α³⁵	α¹¹⁵ = α⁷ + α¹⁶ + α²⁵ + α³⁴
α³⁸ = α² + α¹¹	α⁶⁴ = α + α¹⁰ + α²⁸	α⁹⁰ = 1 + α⁹ + α¹⁸	α¹¹⁶ = α⁸ + α¹⁷ + α²⁶ + α³⁵
α³⁹ = α³ + α¹²	α⁶⁵ = α² + α¹¹ + α²⁹	α⁹¹ = α + α¹⁰ + α¹⁹	α¹¹⁷ = 1 + α¹⁸ + α²⁷
α⁴⁰ = α⁴ + α¹³	α⁶⁶ = α³ + α¹² + α³⁰	α⁹² = α² + α¹¹ + α²⁰	α¹¹⁸ = α + α¹⁹ + α²⁸
α⁴¹ = α⁵ + α¹⁴	α⁶⁷ = α⁴ + α¹³ + α³¹	α⁹³ = α³ + α¹² + α²¹	α¹¹⁹ = α² + α²⁰ + α²⁹
α⁴² = α⁶ + α¹⁵	α⁶⁸ = α⁵ + α¹⁴ + α³²	α⁹⁴ = α⁴ + α¹³ + α²²	α¹²⁰ = α³ + α²¹ + α³⁰
α⁴³ = α⁷ + α¹⁶	α⁶⁹ = α⁶ + α¹⁵ + α³³	α⁹⁵ = α⁵ + α¹⁴ + α²³	α¹²¹ = α⁴ + α²² + α³¹
α⁴⁴ = α⁸ + α¹⁷	α⁷⁰ = α⁷ + α¹⁶ + α³⁴	α⁹⁶ = α⁶ + α¹⁵ + α²⁴	α¹²² = α⁵ + α²³ + α³²
α⁴⁵ = α⁹ + α¹⁸	α⁷¹ = α⁸ + α¹⁷ + α³⁵	α⁹⁷ = α⁷ + α¹⁶ + α²⁵	α¹²³ = α⁶ + α²⁴ + α³³
α⁴⁶ = α¹⁰ + α¹⁹	α⁷² = 1 + α¹⁸	α⁹⁸ = α⁸ + α¹⁷ + α²⁶	α¹²⁴ = α⁷ + α²⁵ + α³⁴
α⁴⁷ = α¹¹ + α²⁰	α⁷³ = α + α¹⁹	α⁹⁹ = α⁹ + α¹⁸ + α²⁷	α¹²⁵ = α⁸ + α²⁶ + α³⁵
α⁴⁸ = α¹² + α²¹	α⁷⁴ = α² + α²⁰	α¹⁰⁰ = α¹⁰ + α¹⁹ + α²⁸	α¹²⁶ = 1 + α²⁷
α⁴⁹ = α¹³ + α²²	α⁷⁵ = α³ + α²¹	α¹⁰¹ = α¹¹ + α²⁰ + α²⁹	α¹²⁷ = α + α²⁸
α⁵⁰ = α¹⁴ + α²³	α⁷⁶ = α⁴ + α²²	α¹⁰² = α¹² + α²¹ + α³⁰	α¹²⁸ = α² + α²⁹
α⁵¹ = α¹⁵ + α²⁴	α⁷⁷ = α⁵ + α²³	α¹⁰³ = α¹³ + α²² + α³¹	α¹²⁹ = α³ + α³⁰
α⁵² = α¹⁶ + α²⁵	α⁷⁸ = α⁶ + α²⁴	α¹⁰⁴ = α¹⁴ + α²³ + α³²	α¹³⁰ = α⁴ + α³¹
α⁵³ = α¹⁷ + α²⁶	α⁷⁹ = α⁷ + α²⁵	α¹⁰⁵ = α¹⁵ + α²⁴ + α³³	α¹³¹ = α⁵ + α³²
α⁵⁴ = α¹⁸ + α²⁷	α⁸⁰ = α⁸ + α²⁶	α¹⁰⁶ = α¹⁶ + α²⁵ + α³⁴	α¹³² = α⁶ + α³³
α⁵⁵ = α¹⁹ + α²⁸	α⁸¹ = α⁹ + α²⁷	α¹⁰⁷ = α¹⁷ + α²⁶ + α³⁵	α¹³³ = α⁷ + α³⁴
α⁵⁶ = α²⁰ + α²⁹	α⁸² = α¹⁰ + α²⁸	α¹⁰⁸ = 1 + α⁹ + α¹⁸ + α²⁷	α¹³⁴ = α⁸ + α³⁵
α⁵⁷ = α²¹ + α³⁰	α⁸³ = α¹¹ + α²⁹	α¹⁰⁹ = α + α¹⁰ + α¹⁹ + α²⁸	α¹³⁵ = 1
α⁵⁸ = α²² + α³¹	α⁸⁴ = α¹² + α³⁰	α¹¹⁰ = α² + α¹¹ + α²⁰ + α²⁹
α⁵⁹ = α²³ + α³²	α⁸⁵ = α¹³ + α³¹	α¹¹¹ = α³ + α¹² + α²¹ + α³⁰
α⁶⁰ = α²⁴ + α³³	α⁸⁶ = α¹⁴ + α³²	α¹¹² = α⁴ + α¹³ + α²² + α³¹
α⁶¹ = α²⁵ + α³⁴	α⁸⁷ = α¹⁵ + α³³	α¹¹³ = α⁵ + α¹⁴ + α²³ + α³²

	Module	Input	Output
a	Elements of Galois Field	b n*	α'[i]
b	Bchgenpoly	n, k	g[i]
c	Cyclotomic cosets	α'[i], b, k	c[i]
d	Design distance	c[i]	d
e	error	d	t

	Module	Input	Output
a	Syndrome Matrix	d, bn, bk, α', r[i]	S'[i]
b	Calculate D_matrix	t, S'[i]	D_matrix
c	Is D invertible	t, D_matrix	\|D\| ≠ 0
d	error_locator_poly	t, D_matrix, S'[i]	f[i]
e	error_position	f[i]	e[i]
f	error values	t, S[i], D_matrix, e[i], bn, pm[i]	eν[i]
h	Correct_recieved polynomial	t, bn, e[i], eν[i], r[i], α'	ν[i]

n	bn	k₁	t₁	R₁	b²n	k₂	t₂	R₂	b³n	k₃	t₃	R₃
15	45	33	1	0.733	135	99	1	0.733	405	297	1	0.733
		29	2	0.644		87	2	0.644		261	2	0.644
		23	3	0.511		69	3	0.511		207	3	0.511
		11	4	0.244		33	4	0.244		99	4	0.244
		7	7	0.155		29	7	0.214		87	7	0.214
		5	10	0.111		23	10	0.170		69	10	0.170
		1	22	0.022		11	13	0.081		33	13	0.081
						7	22	0.051		29	22	0.071
						5	31	0.037		23	31	0.056
						1	67	0.007		11	40	0.027
										7	67	0.017
										5	94	0.012
										1	202	0.002
63	189	171	1	0.904	567	513	1	0.904	1701	1539	1	0.904
		165	2	0.873		495	2	0.873		1485	2	0.873
		147	3	0.777		441	3	0.777		1323	3	0.777
		129	4	0.682		387	4	0.682		1161	4	0.682
		123	5	0.650		381	5	0.672		1143	5	0.671
		105	6	0.555		327	6	0.576		981	6	0.576
		87	7	0.460		273	7	0.481		819	7	0.481
		81	10	0.428		255	10	0.449		765	10	0.449
63	189	75	11	0.3968	567	237	11	0.4179	1701	711	11	0.417
		57	13	0.3015		183	13	0.3227		549	13	0.322
		54	15	0.2857		177	15	0.3121		543	15	0.319
		36	16	0.1904		123	16	0.2169		381	16	0.224
		30	19	0.1587		105	19	0.1851		327	19	0.192
		24	22	0.1269		87	22	0.1534		273	22	0.160
		18	31	0.0952		81	31	0.1428		255	31	0.149
		16	34	0.0846		75	34	0.1322		237	34	0.1393
		10	40	0.0529		57	40	0.1005		183	40	0.1075
		7	46	0.0370		54	46	0.0952		177	46	0.1040
		1	94	0.0053		36	49	0.0634		123	49	0.0723
						30	58	0.0529		105	58	0.0617
						24	67	0.0423		87	67	0.0511
						18	94	0.0317		81	94	0.0476
						16	103	0.0282		75	103	0.0440
						10	121	0.0176		57	121	0.0335
						7	139	0.0123		54	139	0.0317
						1	283	0.0017		36	148	0.0211
										30	175	0.0176
										10	364	0.0058
										1	850	0.0005

C_{b^{j} n} = {ν (x^{\frac{a}{b^{j}}}) \in 𝔽_{2} {[x; \frac{a}{b^{j}} ℤ_{0}]}_{b^{j} n} : ν (α^{i}) = 0 \begin{matrix}  \end{matrix} f o r \begin{matrix}  \end{matrix} a l l \begin{matrix}  \end{matrix} i = c_{j}, c_{j} + 1,..., c_{j} + d_{j} - 2.

H = [\begin{matrix} 1 & α^{c_{j}} & α^{2 c_{j}} & \dots & α^{(b n - 1) c_{j}} \\ 1 & α^{c_{j} + 1} & α^{2 (c_{j} + 1)} & \dots & α^{(b n - 1) (c_{j} + 1)} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 1 & α^{c_{j} + d_{j} - 2} & α^{2 (c_{j} + d_{j} - 2)} & \dots & α^{(b n - 1) (c_{j} + d_{j} - 2)} \end{matrix}] .

\begin{array}{l} m_{5} (x^{\frac{2}{9}}) = {(x^{\frac{2}{9}})}^{18} + {(x^{\frac{2}{9}})}^{9} + 1, m_{7} (x^{\frac{2}{9}}) = {(x^{\frac{2}{9}})}^{36} + {(x^{\frac{2}{9}})}^{27} + 1 \\ m_{9} (x^{\frac{2}{9}}) = {(x^{\frac{2}{9}})}^{4} + (x^{\frac{2}{9}}) + 1, m_{15} (x^{\frac{2}{9}}) = {(x^{\frac{2}{9}})}^{6} + {(x^{\frac{2}{9}})}^{3} + 1 \\ m_{21} (x^{\frac{2}{9}}) = {(x^{\frac{2}{9}})}^{12} + {(x^{\frac{2}{9}})}^{9} + 1 \\ m_{27} (x^{\frac{2}{9}}) = {(x^{\frac{2}{9}})}^{4} + {(x^{\frac{2}{9}})}^{3} + {(x^{\frac{2}{9}})}^{2} + (x^{\frac{2}{9}}) + 1 \\ m_{45} (x^{\frac{2}{9}}) = {(x^{\frac{2}{9}})}^{2} + (x^{\frac{2}{9}}) + 1, m_{63} (x^{\frac{2}{9}}) = {(x^{\frac{2}{9}})}^{4} + {(x^{\frac{2}{9}})}^{3} + 1. \end{array}

\begin{array}{l} g (x^{\frac{2}{9}}) = (m_{1} (x^{\frac{2}{9}})) (m_{3} (x^{\frac{2}{9}})) = ({(x^{\frac{2}{9}})}^{36} + {(x^{\frac{2}{9}})}^{9} + 1) ({(x^{\frac{2}{9}})}^{12} + {(x^{\frac{2}{9}})}^{3} + 1) \\ \begin{matrix}  \end{matrix} = {(x^{\frac{2}{9}})}^{48} + {(x^{\frac{2}{9}})}^{39} + {(x^{\frac{2}{9}})}^{36} + {(x^{\frac{2}{9}})}^{21} + {(x^{\frac{2}{9}})}^{9} + {(x^{\frac{2}{9}})}^{3} + 1. \end{array}

\begin{array}{l} g (x^{\frac{2}{9}}) = (m_{1} (x^{\frac{2}{9}})) (m_{3} (x^{\frac{2}{9}})) (m_{5} (x^{\frac{2}{9}})) \\ \begin{matrix}  \end{matrix} = {(x^{\frac{2}{9}})}^{66} + {(x^{\frac{2}{9}})}^{54} + {(x^{\frac{2}{9}})}^{45} + {(x^{\frac{2}{9}})}^{36} + {(x^{\frac{2}{9}})}^{30} + {(x^{\frac{2}{9}})}^{27} + {(x^{\frac{2}{9}})}^{12} + {(x^{\frac{2}{9}})}^{3} + 1. \\ g (x^{\frac{2}{9}}) = (m_{1} (x^{\frac{2}{9}})) (m_{3} (x^{\frac{2}{9}})) (m_{5} (x^{\frac{2}{9}})) (m_{7} (x^{\frac{2}{9}})) \\ \begin{matrix}  \end{matrix} = {(x^{\frac{2}{9}})}^{102} + {(x^{\frac{2}{9}})}^{93} + {(x^{\frac{2}{9}})}^{90} + {(x^{\frac{2}{9}})}^{57} + {(x^{\frac{2}{9}})}^{48} + {(x^{\frac{2}{9}})}^{45} + {(x^{\frac{2}{9}})}^{12} + {(x^{\frac{2}{9}})}^{3} + 1. \\ g (x^{\frac{2}{9}}) = (m_{1} (x^{\frac{2}{9}})) (m_{3} (x^{\frac{2}{9}})) (m_{5} (x^{\frac{2}{9}})) (m_{7} (x^{\frac{2}{9}})) (m_{9} (x^{\frac{2}{9}})) \\ \begin{matrix}  \end{matrix} = {(x^{\frac{2}{9}})}^{106} + {(x^{\frac{2}{9}})}^{103} + {(x^{\frac{2}{9}})}^{102} + {(x^{\frac{2}{9}})}^{97} + {(x^{\frac{2}{9}})}^{93} + {(x^{\frac{2}{9}})}^{91} + {(x^{\frac{2}{9}})}^{90} + {(x^{\frac{2}{9}})}^{61} \\ \begin{matrix}  \end{matrix} + {(x^{\frac{2}{9}})}^{58} + {(x^{\frac{2}{9}})}^{57} + {(x^{\frac{2}{9}})}^{52} + {(x^{\frac{2}{9}})}^{48} + {(x^{\frac{2}{9}})}^{46} + {(x^{\frac{2}{9}})}^{45} + {(x^{\frac{2}{9}})}^{16} + {(x^{\frac{2}{9}})}^{13} \\ \begin{matrix}  \end{matrix} + {(x^{\frac{2}{9}})}^{12} + {(x^{\frac{2}{9}})}^{7} + {(x^{\frac{2}{9}})}^{3} + (x^{\frac{2}{9}}) + 1. \\ g (x^{\frac{2}{9}}) = (m_{1} (x^{\frac{2}{9}})) (m_{3} (x^{\frac{2}{9}})) (m_{5} (x^{\frac{2}{9}})) (m_{7} (x^{\frac{2}{9}})) (m_{9} (x^{\frac{2}{9}})) (m_{15} (x^{\frac{2}{9}})) \\ \begin{matrix}  \end{matrix} = {(x^{\frac{2}{9}})}^{112} + {(x^{\frac{2}{9}})}^{108} + {(x^{\frac{2}{9}})}^{105} + {(x^{\frac{2}{9}})}^{102} + {(x^{\frac{2}{9}})}^{100} + {(x^{\frac{2}{9}})}^{99} + {(x^{\frac{2}{9}})}^{94} + {(x^{\frac{2}{9}})}^{91} \\ \begin{matrix}  \end{matrix} + {(x^{\frac{2}{9}})}^{90} + {(x^{\frac{2}{9}})}^{67} + {(x^{\frac{2}{9}})}^{63} + {(x^{\frac{2}{9}})}^{60} + {(x^{\frac{2}{9}})}^{57} + {(x^{\frac{2}{9}})}^{55} + {(x^{\frac{2}{9}})}^{54} + {(x^{\frac{2}{9}})}^{49} \\ \begin{matrix}  \end{matrix} + {(x^{\frac{2}{9}})}^{46} + {(x^{\frac{2}{9}})}^{45} + {(x^{\frac{2}{9}})}^{22} + {(x^{\frac{2}{9}})}^{18} + {(x^{\frac{2}{9}})}^{15} + {(x^{\frac{2}{9}})}^{12} + {(x^{\frac{2}{9}})}^{10} + {(x^{\frac{2}{9}})}^{9} \\ \begin{matrix}  \end{matrix} + {(x^{\frac{2}{9}})}^{4} + (x^{\frac{2}{9}}) + 1. \\ g (x^{\frac{2}{9}}) = (m_{1} (x^{\frac{2}{9}})) (m_{3} (x^{\frac{2}{9}})) (m_{5} (x^{\frac{2}{9}})) (m_{7} (x^{\frac{2}{9}})) (m_{9} (x^{\frac{2}{9}})) (m_{15} (x^{\frac{2}{9}})) (m_{21} (x^{\frac{2}{9}})) \\ \begin{matrix}  \end{matrix} = {(x^{\frac{2}{9}})}^{124} + {(x^{\frac{2}{9}})}^{121} + {(x^{\frac{2}{9}})}^{120} + {(x^{\frac{2}{9}})}^{109} + {(x^{\frac{2}{9}})}^{106} + {(x^{\frac{2}{9}})}^{105} + {(x^{\frac{2}{9}})}^{94} + {(x^{\frac{2}{9}})}^{91} \\ \begin{matrix}  \end{matrix} + {(x^{\frac{2}{9}})}^{90} + {(x^{\frac{2}{9}})}^{79} + {(x^{\frac{2}{9}})}^{76} + {(x^{\frac{2}{9}})}^{75} + {(x^{\frac{2}{9}})}^{64} + {(x^{\frac{2}{9}})}^{61} + {(x^{\frac{2}{9}})}^{60} + {(x^{\frac{2}{9}})}^{49} \\ \begin{matrix}  \end{matrix} + {(x^{\frac{2}{9}})}^{46} + {(x^{\frac{2}{9}})}^{45} + {(x^{\frac{2}{9}})}^{34} + {(x^{\frac{2}{9}})}^{31} + {(x^{\frac{2}{9}})}^{30} + {(x^{\frac{2}{9}})}^{19} + {(x^{\frac{2}{9}})}^{16} + {(x^{\frac{2}{9}})}^{15} \\ \begin{matrix}  \end{matrix} + {(x^{\frac{2}{9}})}^{4} + (x^{\frac{2}{9}}) + 1. \\ g (x^{\frac{2}{9}}) = (m_{1} (x^{\frac{2}{9}})) (m_{3} (x^{\frac{2}{9}})) (m_{5} (x^{\frac{2}{9}})) (m_{7} (x^{\frac{2}{9}})) (m_{9} (x^{\frac{2}{9}})) (m_{15} (x^{\frac{2}{9}})) (m_{21} (x^{\frac{2}{9}})) (m_{27} (x^{\frac{2}{9}})) \\ \begin{matrix}  \end{matrix} = {(x^{\frac{2}{9}})}^{128} + {(x^{\frac{2}{9}})}^{127} + {(x^{\frac{2}{9}})}^{126} + {(x^{\frac{2}{9}})}^{124} + {(x^{\frac{2}{9}})}^{120} + {(x^{\frac{2}{9}})}^{113} + {(x^{\frac{2}{9}})}^{112} + {(x^{\frac{2}{9}})}^{111} \\ \begin{matrix}  \end{matrix} + {(x^{\frac{2}{9}})}^{109} + {(x^{\frac{2}{9}})}^{105} + {(x^{\frac{2}{9}})}^{98} + {(x^{\frac{2}{9}})}^{97} + {(x^{\frac{2}{9}})}^{96} + {(x^{\frac{2}{9}})}^{94} + {(x^{\frac{2}{9}})}^{90} + {(x^{\frac{2}{9}})}^{83} + {(x^{\frac{2}{9}})}^{82} \\ \begin{matrix}  \end{matrix} + {(x^{\frac{2}{9}})}^{81} + {(x^{\frac{2}{9}})}^{79} + {(x^{\frac{2}{9}})}^{75} + {(x^{\frac{2}{9}})}^{68} + {(x^{\frac{2}{9}})}^{67} + {(x^{\frac{2}{9}})}^{66} + {(x^{\frac{2}{9}})}^{64} + {(x^{\frac{2}{9}})}^{60} + {(x^{\frac{2}{9}})}^{53} \\ \begin{matrix}  \end{matrix} + {(x^{\frac{2}{9}})}^{52} + {(x^{\frac{2}{9}})}^{51} + {(x^{\frac{2}{9}})}^{49} + {(x^{\frac{2}{9}})}^{45} + {(x^{\frac{2}{9}})}^{38} + {(x^{\frac{2}{9}})}^{37} + {(x^{\frac{2}{9}})}^{36} + {(x^{\frac{2}{9}})}^{34} + {(x^{\frac{2}{9}})}^{30} \\ \begin{matrix}  \end{matrix} + {(x^{\frac{2}{9}})}^{23} + {(x^{\frac{2}{9}})}^{22} + {(x^{\frac{2}{9}})}^{21} + {(x^{\frac{2}{9}})}^{19} + {(x^{\frac{2}{9}})}^{15} + {(x^{\frac{2}{9}})}^{8} + {(x^{\frac{2}{9}})}^{7} + {(x^{\frac{2}{9}})}^{6} + {(x^{\frac{2}{9}})}^{4} + 1. \\ g (x^{\frac{2}{9}}) = (m_{1} (x^{\frac{2}{9}})) (m_{3} (x^{\frac{2}{9}})) (m_{5} (x^{\frac{2}{9}})) (m_{7} (x^{\frac{2}{9}})) (m_{9} (x^{\frac{2}{9}})) (m_{9} (x^{\frac{2}{9}})) (m_{15} (x^{\frac{2}{9}})) (m_{21} (x^{\frac{2}{9}})) \\ \begin{matrix}  \end{matrix} (m_{27} (x^{\frac{2}{9}})) (m_{45} (x^{\frac{2}{9}})) = {(x^{\frac{2}{9}})}^{130} + {(x^{\frac{2}{9}})}^{128} + {(x^{\frac{2}{9}})}^{125} + {(x^{\frac{2}{9}})}^{124} + {(x^{\frac{2}{9}})}^{122} \\ \begin{matrix}  \end{matrix} + {(x^{\frac{2}{9}})}^{121} + {(x^{\frac{2}{9}})}^{120} + {(x^{\frac{2}{9}})}^{115} + {(x^{\frac{2}{9}})}^{113} + {(x^{\frac{2}{9}})}^{110} + {(x^{\frac{2}{9}})}^{109} + {(x^{\frac{2}{9}})}^{107} \\ \begin{matrix}  \end{matrix} + {(x^{\frac{2}{9}})}^{106} + {(x^{\frac{2}{9}})}^{105} + {(x^{\frac{2}{9}})}^{100} + {(x^{\frac{2}{9}})}^{98} + {(x^{\frac{2}{9}})}^{95} + {(x^{\frac{2}{9}})}^{94} + {(x^{\frac{2}{9}})}^{92} \\ \begin{matrix}  \end{matrix} + {(x^{\frac{2}{9}})}^{91} + {(x^{\frac{2}{9}})}^{90} + {(x^{\frac{2}{9}})}^{85} + {(x^{\frac{2}{9}})}^{83} + {(x^{\frac{2}{9}})}^{80} + {(x^{\frac{2}{9}})}^{79} + {(x^{\frac{2}{9}})}^{77} + {(x^{\frac{2}{9}})}^{76} \\ \begin{matrix}  \end{matrix} + {(x^{\frac{2}{9}})}^{75} + {(x^{\frac{2}{9}})}^{70} + {(x^{\frac{2}{9}})}^{68} + {(x^{\frac{2}{9}})}^{65} + {(x^{\frac{2}{9}})}^{64} + {(x^{\frac{2}{9}})}^{62} + {(x^{\frac{2}{9}})}^{61} + {(x^{\frac{2}{9}})}^{60} \\ \begin{matrix}  \end{matrix} + {(x^{\frac{2}{9}})}^{55} + {(x^{\frac{2}{9}})}^{53} + {(x^{\frac{2}{9}})}^{50} + {(x^{\frac{2}{9}})}^{49} + {(x^{\frac{2}{9}})}^{7} + {(x^{\frac{2}{9}})}^{46} + {(x^{\frac{2}{9}})}^{45} + {(x^{\frac{2}{9}})}^{40} \\ \begin{matrix}  \end{matrix} + {(x^{\frac{2}{9}})}^{38} + {(x^{\frac{2}{9}})}^{35} + {(x^{\frac{2}{9}})}^{34} + {(x^{\frac{2}{9}})}^{32} + {(x^{\frac{2}{9}})}^{31} + {(x^{\frac{2}{9}})}^{30} + {(x^{\frac{2}{9}})}^{25} \\ \begin{matrix}  \end{matrix} + {(x^{\frac{2}{9}})}^{23} + {(x^{\frac{2}{9}})}^{20} + {(x^{\frac{2}{9}})}^{19} + {(x^{\frac{2}{9}})}^{17} + {(x^{\frac{2}{9}})}^{16} + {(x^{\frac{2}{9}})}^{5} + {(x^{\frac{2}{9}})}^{10} + {(x^{\frac{2}{9}})}^{8} \\ \begin{matrix}  \end{matrix} + {(x^{\frac{2}{9}})}^{5} + {(x^{\frac{2}{9}})}^{4} + {(x^{\frac{2}{9}})}^{2} + (x^{\frac{2}{9}}) + 1. \\ g (x^{\frac{2}{9}}) = (m_{1} (x^{\frac{2}{9}})) (m_{3} (x^{\frac{2}{9}})) (m_{5} (x^{\frac{2}{9}})) (m_{7} (x^{\frac{2}{9}})) (m_{9} (x^{\frac{2}{9}})) (m_{15} (x^{\frac{2}{9}})) \\ \begin{matrix}  \end{matrix} (m_{21} (x^{\frac{2}{9}})) (m_{27} (x^{\frac{2}{9}})) (m_{45} (x^{\frac{2}{9}})) (m_{63} (x^{\frac{2}{9}})) \\ \begin{matrix}  \end{matrix} = {(x^{\frac{2}{9}})}^{134} + {(x^{\frac{2}{9}})}^{133} + ... + {(x^{\frac{2}{9}})}^{2} + (x^{\frac{2}{9}}) + 1. \end{array}

\frac{𝔽_{2} [x; \frac{a}{b^{j}} ℤ_{0}] / ({(x^{\frac{a}{b^{j}}})}^{n_{j}} - 1)}{({(x^{\frac{a}{b^{j}}})}^{b^{j} n} - 1) / ({(x^{\frac{a}{b^{j}}})}^{n_{j}} - 1)} ≃ \frac{𝔽_{2} [x; \frac{a}{b^{j}} ℤ_{0}]}{({(x^{\frac{a}{b^{j}}})}^{b^{j} n} - 1)} ≃ \frac{𝔽_{2} [x; a ℤ_{0}]}{({(x^{a})}^{n} - 1)} .

\begin{array}{l} 𝔽_{2} [x; a ℤ_{0}] \subset 𝔽_{2} [x; \frac{a}{b} ℤ_{0}] \begin{matrix}  \end{matrix} \subset 𝔽_{2} [x; \frac{a}{b^{2}} ℤ_{0}] \begin{matrix}  \end{matrix} \subset \dots 𝔽_{2} [x; \frac{a}{b^{j}} ℤ_{0}] \\ \frac{𝔽_{2} [x; a ℤ_{0}]}{({(x^{a})}^{n} - 1)} \underline{∽} \frac{𝔽_{2} [x; \frac{a}{b} ℤ_{0}]}{({(x^{\frac{a}{b}})}^{b n} - 1)} \underline{∽} \frac{𝔽_{2} [x; \frac{a}{b^{2}} ℤ_{0}]}{({(x^{\frac{a}{b^{2}}})}^{b^{2} n} - 1)} \underline{∽} \dots \frac{𝔽_{2} [x; \frac{a}{b^{j}} ℤ_{0}]}{({(x^{\frac{a}{b^{j}}})}^{b^{j} n} - 1)} \\ \cup \begin{matrix} \cup \begin{matrix} \cup \begin{matrix} \dots \begin{matrix}  \end{matrix} \cup \end{matrix} \end{matrix} \end{matrix} \\ C_{n} \begin{matrix}  \end{matrix} ↪ \begin{matrix}  \end{matrix} C_{b n} \begin{matrix}  \end{matrix} ↪ \begin{matrix}  \end{matrix} C_{b^{2} n} \begin{matrix}  \end{matrix} ↪ \dots \begin{matrix}  \end{matrix} C_{b^{j} n} \end{array}

\begin{array}{l} g (x^{\frac{2}{9}}) = {(x^{\frac{2}{9}})}^{106} + {(x^{\frac{2}{9}})}^{103} + {(x^{\frac{2}{9}})}^{102} + {(x^{\frac{2}{9}})}^{97} + {(x^{\frac{2}{9}})}^{93} + {(x^{\frac{2}{9}})}^{91} + {(x^{\frac{2}{9}})}^{90} + \\ \begin{matrix}  \end{matrix} {(x^{\frac{2}{9}})}^{61} + {(x^{\frac{2}{9}})}^{58} + {(x^{\frac{2}{9}})}^{57} + {(x^{\frac{2}{9}})}^{52} + {(x^{\frac{2}{9}})}^{48} + {(x^{\frac{2}{9}})}^{46} + {(x^{\frac{2}{9}})}^{45} + {(x^{\frac{2}{9}})}^{16} + \\ \begin{matrix}  \end{matrix} {(x^{\frac{2}{9}})}^{13} + {(x^{\frac{2}{9}})}^{12} + {(x^{\frac{2}{9}})}^{7} + {(x^{\frac{2}{9}})}^{3} + (x^{\frac{2}{9}}) + 1, \\ g (y) = {(y)}^{106} + {(y)}^{103} + {(y)}^{102} + {(y)}^{97} + {(y)}^{93} + {(y)}^{91} + {(y)}^{90} + {(y)}^{61} + {(y)}^{58} + \\ \begin{matrix}  \end{matrix} {(y)}^{57} + {(y)}^{52} + {(y)}^{48} + {(y)}^{46} + {(y)}^{45} + {(y)}^{16} + {(y)}^{13} + {(y)}^{12} + {(y)}^{7} + \\ \begin{matrix}  \end{matrix} {(y)}^{3} + y + 1, \\ g (y^{3}) = {(y^{3})}^{106} + {(y^{3})}^{103} + {(y^{3})}^{102} + {(y^{3})}^{97} + {(y^{3})}^{93} + {(y^{3})}^{91} + {(y^{3})}^{90} + {(y^{3})}^{61} + \\ \begin{matrix}  \end{matrix} {(y^{3})}^{58} + {(y^{3})}^{57} + {(y^{3})}^{52} + {(y^{3})}^{48} + {(y^{3})}^{46} + {(y^{3})}^{45} + {(y^{3})}^{16} + {(y^{3})}^{13} + \\ \begin{matrix}  \end{matrix} {(y^{3})}^{12} + {(y^{3})}^{7} + {(y^{3})}^{3} + (y^{3}) + 1, \\ g (x^{\frac{2}{3}}) = {(x^{\frac{2}{3}})}^{106} + {(x^{\frac{2}{3}})}^{103} + {(x^{\frac{2}{3}})}^{102} + {(x^{\frac{2}{3}})}^{97} + {(x^{\frac{2}{3}})}^{93} + {(x^{\frac{2}{3}})}^{91} + {(x^{\frac{2}{3}})}^{90} + {(x^{\frac{2}{3}})}^{61} \\ \begin{matrix}  \end{matrix} + {(x^{\frac{2}{3}})}^{58} + {(x^{\frac{2}{3}})}^{57} + {(x^{\frac{2}{3}})}^{52} + {(x^{\frac{2}{3}})}^{48} + {(x^{\frac{2}{3}})}^{46} + {(x^{\frac{2}{3}})}^{45} + {(x^{\frac{2}{3}})}^{16} + {(x^{\frac{2}{3}})}^{13} \\ \begin{matrix}  \end{matrix} + {(x^{\frac{2}{3}})}^{12} + {(x^{\frac{2}{3}})}^{7} + {(x^{\frac{2}{3}})}^{3} + (x^{\frac{2}{3}}) + 1 \\ \begin{matrix}  \end{matrix} = {(x^{\frac{2}{3}})}^{16} + {(x^{\frac{2}{3}})}^{13} + {(x^{\frac{2}{3}})}^{7} + {(x^{\frac{2}{3}})}^{3} + (x^{\frac{2}{3}}) + 1 \in 𝔽_{2} {[x; \frac{2}{3} ℤ_{0}]}_{45} . \end{array}

\begin{array}{l} g (x^{\frac{2}{9}}) = {(x^{\frac{2}{9}})}^{112} + {(x^{\frac{2}{9}})}^{108} + {(x^{\frac{2}{9}})}^{105} + {(x^{\frac{2}{9}})}^{102} + {(x^{\frac{2}{9}})}^{100} + {(x^{\frac{2}{9}})}^{99} + {(x^{\frac{2}{9}})}^{94} + \\ \begin{matrix}  \end{matrix} {(x^{\frac{2}{9}})}^{91} + {(x^{\frac{2}{9}})}^{90} + {(x^{\frac{2}{9}})}^{67} + {(x^{\frac{2}{9}})}^{63} + {(x^{\frac{2}{9}})}^{60} + {(x^{\frac{2}{9}})}^{57} + {(x^{\frac{2}{9}})}^{55} + \\ \begin{matrix}  \end{matrix} {(x^{\frac{2}{9}})}^{54} + {(x^{\frac{2}{9}})}^{49} + {(x^{\frac{2}{9}})}^{46} + {(x^{\frac{2}{9}})}^{45} + {(x^{\frac{2}{9}})}^{22} + {(x^{\frac{2}{9}})}^{18} + {(x^{\frac{2}{9}})}^{15} + \\ \begin{matrix}  \end{matrix} {(x^{\frac{2}{9}})}^{12} + {(x^{\frac{2}{9}})}^{10} + {(x^{\frac{2}{9}})}^{9} + {(x^{\frac{2}{9}})}^{4} + (x^{\frac{2}{9}}) + 1, \end{array}

g (x^{\frac{2}{3}}) = {(x^{\frac{2}{3}})}^{22} + {(x^{\frac{2}{3}})}^{18} + {(x^{\frac{2}{3}})}^{15} + {(x^{\frac{2}{3}})}^{12} + {(x^{\frac{2}{3}})}^{10} + {(x^{\frac{2}{3}})}^{9} + {(x^{\frac{2}{3}})}^{4} + (x^{\frac{2}{3}}) + 1,

\begin{array}{l} g (x^{\frac{2}{3}}) = {(x^{\frac{2}{3}})}^{16} + {(x^{\frac{2}{3}})}^{13} + {(x^{\frac{2}{3}})}^{12} + {(x^{\frac{2}{3}})}^{7} + {(x^{\frac{2}{3}})}^{3} + (x^{\frac{2}{3}}) + 1 \\ g (x^{\frac{2}{9}}) = {(x^{\frac{2}{9}})}^{106} + {(x^{\frac{2}{9}})}^{103} + {(x^{\frac{2}{9}})}^{102} + {(x^{\frac{2}{9}})}^{97} + {(x^{\frac{2}{9}})}^{93} + {(x^{\frac{2}{9}})}^{91} + \\ \begin{matrix}  \end{matrix} {(x^{\frac{2}{9}})}^{90} + {(x^{\frac{2}{9}})}^{61} + {(x^{\frac{2}{9}})}^{58} + {(x^{\frac{2}{9}})}^{57} + {(x^{\frac{2}{9}})}^{52} + {(x^{\frac{2}{9}})}^{48} + {(x^{\frac{2}{9}})}^{46} + \\ \begin{matrix}  \end{matrix} {(x^{\frac{2}{9}})}^{45} + {(x^{\frac{2}{9}})}^{16} + {(x^{\frac{2}{9}})}^{13} + {(x^{\frac{2}{9}})}^{12} + {(x^{\frac{2}{9}})}^{7} + {(x^{\frac{2}{9}})}^{3} + (x^{\frac{2}{9}}) + 1, \end{array}

[1] *Corresponding author: Antonio A. Andrade - E-mail: antonio.andrade@unesp.br

Brasil

Brasil

Sequences of Primitive and Non-primitive BCH Codes

ABSTRACT