Sequences of Primitive and Non-primitive BCH Codes

ABSTRACT

In this work, we introduce a method by which it is established that how a sequence of non-primitive BCH codes can be obtained by a given primitive BCH code. For this, we rush to the out of routine assembling technique of BCH codes and use the structure of monoid rings instead of polynomial rings. Accordingly, it is gotten that there is a sequence ${\left\{C_{b^{j}n}\right\}}_{1\le j\le m}$, where b ^j n is the length of $C_{b^{j}n}$, of non-primitive binary BCH codes against a given binary BCH code C _n of length n. Matlab based simulated algorithms for encoding and decoding for these type of codes are introduced. Matlab provides in routines for construction of a primitive BCH code, but impose several constraints, like degree s of primitive irreducible polynomial should be less than 16. This work focuses on non-primitive irreducible polynomials having degree bs, which go far more than 16.

Keywords:
Monoid ring; BCH codes; primitive polynomial; non-primitive polynomial

RESUMO

Neste trabalho, apresentamos um método que estabelece como uma sequência de códigos BCH não primitivos pode ser obtida através de um dado código BCH primitivo. Para isso, utilizamos uma técnica de construção diferente da técnica rotineira de códigos BCH e usamos a estrutura de anéis monoidais em vez de anéis de polinômios. Consequentemente, mostramos que existe uma sequência ${\left\{C_{b^{j}n}\right\}}_{1\le j\le m}$, onde b ^j n é o comprimento do código $C_{b^{j}n}$, de códigos BCH binários não primitivos em vez de um dado código binário BCH C _n de comprimento n. Algoritmos simulados via Mathlab para codificação e decodificação para este tipo de códigos são introduzidos. O algoritmo via o Matlab fornece rotinas para a construção de um código BCH primitivo, mas impõe várias restrições, como por exemplo, o grau s de um polinômio irredutível primitivo deve ser menor que 16. Este trabalho trata-se de polinômios não-primitivos irredutíveis com grau bs, que são maiores do que 16.

Palavras-chave:
Anel monoidal; códigos BCH; polinômio primitivo; polinômio não-primitivo

1 INTRODUCTION

Introducing more general algebraic structures lead to various gains in coding applications and the generality of the algebraic structures helps to find more efficient encoding and decoding algorithms for known codes. This has motivated special attention of many researchers in considering ideals of certain rings ^{1), (5), (6), (9} and ¹⁰. The extension of a BCH code embedded in a semigroup ring was considered by Cazaran in ⁴. More information regarding ring constructions and its corresponding polynomial codes were given by Kelarev ⁵. In ¹, the authors elaborated cyclic, BCH, Alternant, Goppa and Srivastava codes over finite rings, which are constructed through a polynomial ring in one indeterminate. Several classes of cyclic codes constructed using the monoid rings are discussed in ^{2), (14), (13), (12), (11} and ¹⁵. These constructions address the error correction and the code rate in a good way. In ¹⁶, Shah et al., showed the existence of a binary cyclic code of length (n + 1)n such that a binary BCH code of length n is embedded in it. Though they were not succeeded to show the existence of binary BCH code of length (n + 1)n corresponding to a given binary BCH code of length n. In ¹⁷, a construction method is given by which cyclic codes are ideals in 𝔽₂[x; aℤ₀]n, 𝔽₂[x]_an , ${𝔽}_2{\left[x;\frac{a}{b}{\mathbb{Z}}_0\right]}_{bn}$ and ${𝔽}_2{\left[x;\frac{1}{b}{\mathbb{Z}}_0\right]}_{abn}$, where ℤ₀ is the set of non-negative integers, a;b ∈ ℤ and a; b > 1. These codes are capable of correcting random as well as burst errors. Moreover, a link between all these codes are also been developed. Furthermore, in ³, the work of ¹⁶ is improved and an association between primitive and non-primitive binary BCH codes is obtained by using the monoid ring ${𝔽}_2\left[x;\frac{a}{b}{\mathbb{Z}}_0\right]$, where a; b > 1. It is noticed that the monoid ring ${𝔽}_2\left[x;\frac{a}{b}{\mathbb{Z}}_0\right]$ does not contain the polynomial ring 𝔽₂[x] for a > 1. To handle this situation, a BCH code in the monoid ring 𝔽₂[x; aℤ₀] is constructed and then the existence of a non-primitive BCH code in the monoid ring ${𝔽}_2\left[x;\frac{a}{b}{\mathbb{Z}}_0\right]$ is showed. The non-primitive BCH code C _bn in the monoid ring ${𝔽}_2\left[x;\frac{a}{b}{\mathbb{Z}}_0\right]$ is of length bn and generated by a generalized polynomial $g\left(x^{\frac{a}{b}}\right)\in {𝔽}_2\left[x;\frac{a}{b}{\mathbb{Z}}_0\right]$ of degree br. In this line, corresponding to a binary BCH code C _n of length n generated by a generalized polynomial g(x ^a ) ∈ 𝔽₂[x; aℤ₀] of degree r it is constructed a code C _bn such that C _n is embedded in C _bn .

This work extends the work of ³, where the monoid ring ${𝔽}_2\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]$, with b = a + i, 1 ≤ i, j ≤ m, where m is a positive integer, is used. Corresponding to the sequence ${\left\{{𝔽}_2\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]\right\}}_{j\ge 1}$ of monoid rings, we obtain a sequence of non-primitive binary BCH codes ${\left\{C_{b^{j}n}^j\right\}}_{j\ge 1}$, based on a primitive BCH code C _n of length n. The non-primitive BCH code $C_{b^{j}n}^j$ in the monoid ring ${𝔽}_2\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]$ is of length b ^j n and generated by a generalized polynomial $g\left(x^{\frac{a}{b^j}}\right)\in {𝔽}_2\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]$ of degree b ^j r. Similarly, corresponding to a given binary BCH code C _n of length n generated by a polynomial g(x ^a ) ∈ 𝔽₂[x; aℤ₀] of degree r it is constructed a code $C_{b^{j}n}^j$ such that C _n is embedded in $C_{b^{j}n}^j$, where the length of the binary BCH code $C_{b^{j}n}^j$ is well controlled with better error correction capability. Along with the construction of a sequence ${\left\{C_{b^{j}n}^j\right\}}_{j\ge 1}$ of non-primitive binary BCH codes our focus is on its simulation as well, where the simulation is carried out using Matlab. It provides built in routines only for primitive BCH codes with degree of primitive polynomial less than 16. Whereas in constructing non-primitive BCH codes, the degree of non-primitive polynomial goes beyond 16, where to overcome this situation Generic Algorithm is developed in Matlab. The whole method of the algorithm is carried out in two major steps: I) Generating nonprimitive polynomial and II) Error correction in received polynomial. In Table 5, some examples are listed. The non-primitive BCH codes with same code rate and error corrections are found to be interleaved codes. By interleaving a t random error correcting (n, k) code to degree β, we obtain a (βn, βk) code which is capable of correcting any combination of t bursts of length β or less [⁷, Section 9.4], where the non-primitive BCH codes $C_{b^{j}n}^j$ have burst error correction capability along with the random error correction.

2 BCH CODES IN ${𝔽}_2{\left[X;\frac{A}{B^J}{\mathbb{Z}}_0\right]}_{B^{J}N}$, WHERE 1 ≤ J ≤ M.

The set of all finitely nonzero functions f from a commutative monoid (S,*) into the binary field ≤ ₂ is denoted by 𝔽₂(S). This set 𝔽₂(S) is a ring with respect to binary operations addition and multiplication defined as: (f + g)(s) = f(s) + g(s) and (fg)(s) = Σ_{t * u = s} f(t)g(u), where the symbol Σ_{t * u = s} indicates that the sum is taken over all pairs (t, u) of elements of S such that t * u = s and it is understood that in the situation where s is not expressible in the form t * u for any t, u ∈ S, then (fg)(s) = 0. The ring 𝔽₂(S) is known as the monoid ring of S over 𝔽₂ and it is represented by 𝔽₂[x; S] whenever S is an additive monoid. A nonzero element f of 𝔽₂[x; S] is uniquely represented in the canonical form ${\displaystyle {\sum }_{i=1}^{n}f\left(s_i\right)}{x}^{s_i}={\displaystyle {\sum }_{i=1}^n{f}_i{x}^{s_i}}$ where f _i ≠ 0 and s _i ≠ s _j for i ≠ j. The monoid ring 𝔽₂[x; S] is a polynomial ring in one indeterminate if S = ℤ₀.

The indeterminate of generalized polynomials in monoid rings 𝔽₂[x; aℤ₀] and ${𝔽}_2\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]$ are given by x ^a and $x^{\frac{a}{b^j}}$ for each 1 ≤ j ≤ m, where m is a fix positive integer, and they behave like an indeterminate x in 𝔽₂[x]. The arbitrary elements in 𝔽₂[x; aℤ₀] and ${𝔽}_2\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]$ are written, respectively, as f(x ^a ) = 1 + (x ^a ) + (x ^a )² + ... + (x ^a )ⁿ and $f\left(x^{\frac{a}{b^j}}\right)=1+\left(x^{\frac{a}{b^j}}\right)+{\left(x^{\frac{a}{b^j}}\right)}^n+\mathrm{...}{\left(x^{\frac{a}{b^j}}\right)}^n$. The monoids aℤ₀ and $\frac{a}{b^j}{\mathbb{Z}}_0$ are totally ordered, so degree and order of elements of 𝔽₂[x; aℤ₀] and ${𝔽}_2\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]$ are defined.

As 𝔽₂[x; aℤ₀] ⊂ 𝔽₂[x], it follows that the construction of BCH code in the factor ring 𝔽₂[x; aℤ₀]_n is similar to the construction of a BCH code in 𝔽₂[x]_n . Whereas the construction of BCH codes in ${𝔽}_2\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]/\left({\left(x^{\frac{a}{b^j}}\right)}^{b^{j}n}-1\right)\equiv {𝔽}_2{\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]}_{b^{j}n}$ is based on the values of b. BCH codes of length b ^j n in ${𝔽}_2{\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]}_{b^{j}n}$, corresponding to a BCH code C _n of length n don’t exist for all values of b, where b = a + i, such that 1 ≤ i ≤ m, and a, m are positive integers. For that, consider the following map $p\left(x^a\right)=p_0+p_1{x}^a+\mathrm{...}+p_{n-1}{\left(x^a\right)}^{n-1}\mapsto p_0+p_1{\left(x^{\frac{a}{b^j}}\right)}^{b^j}+\mathrm{...}+p_{n-1}{\left(x^{\frac{a}{b^j}}\right)}^{b^j\left(n-1\right)}=p\left(x^{\frac{a}{b^j}}\right)$, which convert a primitive polynomial p(x ^a ) of degree s in 𝔽₂[x; aℤ₀] to a polynomial $p\left(x^{\frac{a}{b^j}}\right)$ of degree b ^j s in ${𝔽}_2\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]$. This converted polynomial is found to be non-primitive, but is irreducible for some values of b. Therefore, for the construction of a non-primitive BCH code in ${𝔽}_2{\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]}_{b^{j}n}$, only this specific value of b is selected for which there is an irreducible polynomial $p\left(x^{\frac{a}{b^j}}\right)$ in ${𝔽}_2\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]$. Thus, for positive integers c _j , d _j and b ^j n such that 2 ≤ d _j ≤ b ^j n with bjn is relatively prime to 2, there exists a non-primitive binary BCH code $C_{b^{j}n}$ of length b ^j n, where bjn is order of an element $\alpha \in {𝔽}_{2^{b^{j}s}}$.

In Table 1, we present a list of irreducible polynomials of degree b ^j s in ${𝔽}_2\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]$ corresponding to primitive irreducible polynomial of degree s in 𝔽₂[x; aℤ₀], where for $p\left(x^a\right)\in {𝔽}_2[x;a{\mathbb{Z}}_0],p\left(x^{\frac{a}{b}}\right)\in {𝔽}_2\left[x\frac{a}{b}{\mathbb{Z}}_0\right],p\left(x^{\frac{a}{b^2}}\right)\in {𝔽}_2\left[x;\frac{a}{b}{\mathbb{Z}}_0\right]$, replace $x^a,x^{\frac{a}{b}}$ and $x^{\frac{a}{b^2}}$ by x, y and z, respectively.

Table 1:
Irreducible polynomials of degree bjs in ${𝔽}_2\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]$ corresponding to primitive irreducible polynomial of degree s in 𝔽₂[x;aℤ₀].

p(xa)∈𝔽2 [x;a ℤ0 ]	$p\left(x^{\frac{a}{b}}\right)\in {𝔽}_2\left[x;\frac{a}{b}{\mathbb{Z}}_0\right]$	$p\left(x^{\frac{a}{b^2}}\right)\in {𝔽}_2\left[x;\frac{a}{b}{\mathbb{Z}}_0\right]$
1 + (x a ) + (x a )3	$1+{\left(x^{\frac{a}{7}}\right)}^7+{\left(x^{\frac{a}{7}}\right)}^{21}$	$1+{\left(x^{\frac{a}{49}}\right)}^{49}+{\left(x^{\frac{a}{49}}\right)}^{147}$
$1+{\left(x^{\frac{a}{3}}\right)}^3+{\left(x^{\frac{a}{3}}\right)}^{12}$	$1+{\left(x^{\frac{a}{9}}\right)}^9+{\left(x^{\frac{a}{9}}\right)}^{36}$
$1+{\left(x^{\frac{a}{5}}\right)}^5+{\left(x^{\frac{a}{5}}\right)}^{20}$	$1+{\left(x^{\frac{a}{25}}\right)}^{25}+{\left(x^{\frac{a}{25}}\right)}^{100}$
1 + (x a )3 + (x a )4	$1+{\left(x^{\frac{a}{3}}\right)}^9+{\left(x^{\frac{a}{3}}\right)}^{12}$	$1+{\left(x^{\frac{a}{9}}\right)}^{27}+{\left(x^{\frac{a}{9}}\right)}^{36}$
	$1+{\left(x^{\frac{a}{5}}\right)}^{15}+{\left(x^{\frac{a}{5}}\right)}^{20}$	$1+{\left(x^{\frac{a}{25}}\right)}^{75}+{\left(x^{\frac{a}{25}}\right)}^{100}$
1 + (x a ) + (x a )6	$1+{\left(x^{\frac{a}{3}}\right)}^3+{\left(x^{\frac{a}{3}}\right)}^{18}$	$1+{\left(x^{\frac{a}{9}}\right)}^9+{\left(x^{\frac{a}{9}}\right)}^{54}$
	$1+{\left(x^{\frac{a}{7}}\right)}^7+{\left(x^{\frac{a}{7}}\right)}^{42}$	$1+{\left(x^{\frac{a}{49}}\right)}^{49}+{\left(x^{\frac{a}{49}}\right)}^{294}$
1 + (x a ) + (x a )3 + (x a )5 + (x a )8	$\begin{array}{l}1+{\left(x^{\frac{a}{3}}\right)}^3+{\left(x^{\frac{a}{3}}\right)}^9+\\ {\left(x^{\frac{a}{3}}\right)}^{15}+{\left(x^{\frac{a}{3}}\right)}^{24}\end{array}$	$\begin{array}{l}1+{\left(x^{\frac{a}{9}}\right)}^9+{\left(x^{\frac{a}{9}}\right)}^{27}+\\ {\left(x^{\frac{a}{9}}\right)}^{45}+{\left(x^{\frac{a}{9}}\right)}^{72}\end{array}$
	$\begin{array}{l}1+{\left(x^{\frac{a}{5}}\right)}^5+{\left(x^{\frac{a}{5}}\right)}^{15}+\\ {\left(x^{\frac{a}{5}}\right)}^{25}+{\left(x^{\frac{a}{5}}\right)}^{40}\end{array}$	$\begin{array}{l}1+{\left(x^{\frac{a}{25}}\right)}^{25}+{\left(x^{\frac{a}{25}}\right)}^{75}+\\ {\left(x^{\frac{a}{25}}\right)}^{125}+{\left(x^{\frac{a}{25}}\right)}^{200}\end{array}$
1 + (x a )4 + (x a )9	$1+{\left(x^{\frac{a}{7}}\right)}^{28}+{\left(x^{\frac{a}{7}}\right)}^{63}$
	$1+{\left(x^{\frac{a}{49}}\right)}^{196}+{\left(x^{\frac{a}{49}}\right)}^{441}$
⋮	⋮	⋮

Proposition 1.[³, Proposition 2] If p(x^a ) ∈ 𝔽₂[x; aℤ₀] is a primitive irreducible polynomial of degree s ∈ {2l, 3l, 4l, 6l}, where l ∈ ℤ₀, then the corresponding generalized polynomial$p\left(x^{\frac{a}{b^{}}}\right)$of degree b ^j s in${𝔽}_2\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]$is non-primitive irreducible for b ∈ {3, 7, {3, 5}, {3, 7}}, respectively.

Theorem 2.[³, Theorem 3] Let n = 2^s - 1 be the length of a primitive BCH code C _n , where p(x ^a ) ∈ 𝔽2[x; aℤ₀] is a primitive irreducible polynomial of degree s such that$p\left(x^{\frac{a}{b^j}}\right)\in {𝔽}_2\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]$is a non-primitive irreducible polynomial of degree b ^j s.

1. For positive integers c_j;d_j;b^jn such that 2 ≤ d _j ≤ b ^j n and b ^j n are relatively prime to 2, there exist a non-primitive binary BCH code$C_{b^{j}n}$of length b ^j n, where b ^j n is the order of an element$\alpha \in {𝔽}_{2^{b^{j}s}}$.

2. The non-primitive BCH code $C_{b^{j}n}$ of length bjn is defined as

$C_{b^{j}n}=\{\nu \left(x^{\frac{a}{b^j}}\right)\in {𝔽}_2{\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]}_{b^{j}n}:\nu \left({\alpha }^i\right)=0\begin{array}{c}\end{array}for\begin{array}{c}\end{array}all\begin{array}{c}\end{array}i=c_j,c_j+\mathrm{1,...,}{c}_j+d_j-2.$

Equivalently, $C_{b^{j}n}$ is the null space of the matrix

$H=\left[\begin{array}{ccccc}1& {\alpha }^{c_j}& {\alpha }^{2c_j}& \cdots & {\alpha }^{\left(bn-1\right)c_j}\\ 1& {\alpha }^{c_j+1}& {\alpha }^{2\left(c_j+1\right)}& \cdots & {\alpha }^{\left(bn-1\right)\left(c_j+1\right)}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& {\alpha }^{c_j+d_j-2}& {\alpha }^{2\left(c_j+d_j-2\right)}& \cdots & {\alpha }^{\left(bn-1\right)\left(c_j+d_j-2\right)}\end{array}\right].$

The following example illustrates the construction of a non-primitive BCH code of length 3² n through ${𝔽}_2\left[x;\frac{2}{3^2}{\mathbb{Z}}_0\right]$.

Example 2.1.Corresponding to a primitive polynomial p(x²) = 1 + (x²) + (x²)⁴in 𝔽₂[x;2ℤ₀] there is a non-primitive irreducible polynomial$p\left(x^{\frac{2}{3^2}}\right)=1+{\left(x^{\frac{2}{9}}\right)}^9+{\left(x^{\frac{2}{9}}\right)}^{36}$in${𝔽}_2\left[;\frac{2}{3^2}{\mathbb{Z}}_0\right]$(Table 1). Let α ∈ GF(2³⁶) such that α satisfies the relation α³⁶ + α⁹ + 1 = 0. Using this relation, in theTable 2, we obtain the distinct elements of GF(2³⁶).

Table 2:
Distinct elements of GF(2³⁶).

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

ByTable 2, it follows that b²n = 3² × 15 = 135. Now, to calculate the generating polynomial$g\left(x^{\frac{2}{9}}\right)$, first calculate the minimal polynomials. By [⁸, Theorem 4.4.2], the following roots

α, α², α⁴, α⁸, α¹⁶, α³², α⁶⁴, α¹²⁸, α¹²¹, α¹⁰⁷, α⁷⁹, α²³, α⁴⁶, α⁹², α⁴⁹, α⁹⁸, α⁶¹, α¹²²,

α¹⁰⁹, α⁸³, α³¹, α⁶², α¹²⁴, α¹¹³, α⁹¹, α⁴⁷, α⁹⁴, α⁵³, α¹⁰⁶, α⁷⁷, α¹⁹α³⁸, α⁷⁶, α¹⁷, α³⁴, α⁶⁸

have same minimal polynomial$m_1\left(x^{\frac{2}{9}}\right)=p\left(x^{\frac{2}{9}}\right)=1+{\left(x^{\frac{2}{9}}\right)}^9+{\left(x^{\frac{2}{9}}\right)}^{36}$. It$m_3\left(x^{\frac{2}{9}}\right)$is the minimal polynomial for α³, then α³, α⁶, α¹², α²⁴, α⁴⁸, α⁹⁶, α⁵⁷, α¹¹⁴, α⁹³, α⁵¹, α¹⁰² and α⁶⁹ all are roots for$m_3\left(x^{\frac{2}{9}}\right)$. Therefore, it follows that$m_3\left(x^{\frac{2}{9}}\right)={\left(x^{\frac{2}{9}}\right)}^{12}+{\left(x^{\frac{2}{9}}\right)}^3+1$. Similarly,

$\begin{array}{l}{m}_5\left(x^{\frac{2}{9}}\right)={\left(x^{\frac{2}{9}}\right)}^{18}+{\left(x^{\frac{2}{9}}\right)}^9+\mathrm{1,}{m}_7\left(x^{\frac{2}{9}}\right)={\left(x^{\frac{2}{9}}\right)}^{36}+{\left(x^{\frac{2}{9}}\right)}^{27}+1\\ m_9\left(x^{\frac{2}{9}}\right)={\left(x^{\frac{2}{9}}\right)}^4+\left(x^{\frac{2}{9}}\right)+\mathrm{1,}{m}_{15}\left(x^{\frac{2}{9}}\right)={\left(x^{\frac{2}{9}}\right)}^6+{\left(x^{\frac{2}{9}}\right)}^3+1\\ m_{21}\left(x^{\frac{2}{9}}\right)={\left(x^{\frac{2}{9}}\right)}^{12}+{\left(x^{\frac{2}{9}}\right)}^9+1\\ m_{27}\left(x^{\frac{2}{9}}\right)={\left(x^{\frac{2}{9}}\right)}^4+{\left(x^{\frac{2}{9}}\right)}^3+{\left(x^{\frac{2}{9}}\right)}^2+\left(x^{\frac{2}{9}}\right)+1\\ m_{45}\left(x^{\frac{2}{9}}\right)={\left(x^{\frac{2}{9}}\right)}^2+\left(x^{\frac{2}{9}}\right)+\mathrm{1,}{m}_{63}\left(x^{\frac{2}{9}}\right)={\left(x^{\frac{2}{9}}\right)}^4+{\left(x^{\frac{2}{9}}\right)}^3+1.\end{array}$

The BCH code with d₂ = 3 has generator polynomial$g\left(x^{\frac{2}{9}}\right)=1+{\left(x^{\frac{2}{9}}\right)}^9+{\left(x^{\frac{2}{9}}\right)}^{36}$. It corrects up to 1 error and its code rate is$\frac{99}{135}=0.733$. The BCH code with d ₂ = 5 has generator polynomial

$\begin{array}{l}g\left(x^{\frac{2}{9}}\right)=\left(m_1\left(x^{\frac{2}{9}}\right)\right)\left(m_3\left(x^{\frac{2}{9}}\right)\right)=\left({\left(x^{\frac{2}{9}}\right)}^{36}+{\left(x^{\frac{2}{9}}\right)}^9+1\right)\left({\left(x^{\frac{2}{9}}\right)}^{12}+{\left(x^{\frac{2}{9}}\right)}^3+1\right)\\ \begin{array}{ccc}& & \end{array}={\left(x^{\frac{2}{9}}\right)}^{48}+{\left(x^{\frac{2}{9}}\right)}^{39}+{\left(x^{\frac{2}{9}}\right)}^{36}+{\left(x^{\frac{2}{9}}\right)}^{21}+{\left(x^{\frac{2}{9}}\right)}^9+{\left(x^{\frac{2}{9}}\right)}^3+1.\end{array}$

It corrects up to 3 errors and its code rate is$\frac{87}{135}=0.644$. Similarly, the BCH codes with d ₂ =7, 9, 15, 21, 27, 45, 63 and 135 have generator polynomials

$\begin{array}{l}g\left(x^{\frac{2}{9}}\right)=\left(m_1\left(x^{\frac{2}{9}}\right)\right)\left(m_3\left(x^{\frac{2}{9}}\right)\right)\left(m_5\left(x^{\frac{2}{9}}\right)\right)\\ \begin{array}{ccc}& & \end{array}={\left(x^{\frac{2}{9}}\right)}^{66}+{\left(x^{\frac{2}{9}}\right)}^{54}+{\left(x^{\frac{2}{9}}\right)}^{45}+{\left(x^{\frac{2}{9}}\right)}^{36}+{\left(x^{\frac{2}{9}}\right)}^{30}+{\left(x^{\frac{2}{9}}\right)}^{27}+{\left(x^{\frac{2}{9}}\right)}^{12}+{\left(x^{\frac{2}{9}}\right)}^3+1.\\ g\left(x^{\frac{2}{9}}\right)=\left(m_1\left(x^{\frac{2}{9}}\right)\right)\left(m_3\left(x^{\frac{2}{9}}\right)\right)\left(m_5\left(x^{\frac{2}{9}}\right)\right)\left(m_7\left(x^{\frac{2}{9}}\right)\right)\\ \begin{array}{ccc}& & \end{array}={\left(x^{\frac{2}{9}}\right)}^{102}+{\left(x^{\frac{2}{9}}\right)}^{93}+{\left(x^{\frac{2}{9}}\right)}^{90}+{\left(x^{\frac{2}{9}}\right)}^{57}+{\left(x^{\frac{2}{9}}\right)}^{48}+{\left(x^{\frac{2}{9}}\right)}^{45}+{\left(x^{\frac{2}{9}}\right)}^{12}+{\left(x^{\frac{2}{9}}\right)}^3+1.\\ g\left(x^{\frac{2}{9}}\right)=\left(m_1\left(x^{\frac{2}{9}}\right)\right)\left(m_3\left(x^{\frac{2}{9}}\right)\right)\left(m_5\left(x^{\frac{2}{9}}\right)\right)\left(m_7\left(x^{\frac{2}{9}}\right)\right)\left(m_9\left(x^{\frac{2}{9}}\right)\right)\\ \begin{array}{ccc}& & \end{array}={\left(x^{\frac{2}{9}}\right)}^{106}+{\left(x^{\frac{2}{9}}\right)}^{103}+{\left(x^{\frac{2}{9}}\right)}^{102}+{\left(x^{\frac{2}{9}}\right)}^{97}+{\left(x^{\frac{2}{9}}\right)}^{93}+{\left(x^{\frac{2}{9}}\right)}^{91}+{\left(x^{\frac{2}{9}}\right)}^{90}+{\left(x^{\frac{2}{9}}\right)}^{61}\\ \begin{array}{cccc}& & & \end{array}+{\left(x^{\frac{2}{9}}\right)}^{58}+{\left(x^{\frac{2}{9}}\right)}^{57}+{\left(x^{\frac{2}{9}}\right)}^{52}+{\left(x^{\frac{2}{9}}\right)}^{48}+{\left(x^{\frac{2}{9}}\right)}^{46}+{\left(x^{\frac{2}{9}}\right)}^{45}+{\left(x^{\frac{2}{9}}\right)}^{16}+{\left(x^{\frac{2}{9}}\right)}^{13}\\ \begin{array}{cccc}& & & \end{array}+{\left(x^{\frac{2}{9}}\right)}^{12}+{\left(x^{\frac{2}{9}}\right)}^7+{\left(x^{\frac{2}{9}}\right)}^3+\left(x^{\frac{2}{9}}\right)+1.\\ g\left(x^{\frac{2}{9}}\right)=\left(m_1\left(x^{\frac{2}{9}}\right)\right)\left(m_3\left(x^{\frac{2}{9}}\right)\right)\left(m_5\left(x^{\frac{2}{9}}\right)\right)\left(m_7\left(x^{\frac{2}{9}}\right)\right)\left(m_9\left(x^{\frac{2}{9}}\right)\right)\left(m_{15}\left(x^{\frac{2}{9}}\right)\right)\\ \begin{array}{ccc}& & \end{array}={\left(x^{\frac{2}{9}}\right)}^{112}+{\left(x^{\frac{2}{9}}\right)}^{108}+{\left(x^{\frac{2}{9}}\right)}^{105}+{\left(x^{\frac{2}{9}}\right)}^{102}+{\left(x^{\frac{2}{9}}\right)}^{100}+{\left(x^{\frac{2}{9}}\right)}^{99}+{\left(x^{\frac{2}{9}}\right)}^{94}+{\left(x^{\frac{2}{9}}\right)}^{91}\\ \begin{array}{cccc}& & & \end{array}+{\left(x^{\frac{2}{9}}\right)}^{90}+{\left(x^{\frac{2}{9}}\right)}^{67}+{\left(x^{\frac{2}{9}}\right)}^{63}+{\left(x^{\frac{2}{9}}\right)}^{60}+{\left(x^{\frac{2}{9}}\right)}^{57}+{\left(x^{\frac{2}{9}}\right)}^{55}+{\left(x^{\frac{2}{9}}\right)}^{54}+{\left(x^{\frac{2}{9}}\right)}^{49}\\ \begin{array}{cccc}& & & \end{array}+{\left(x^{\frac{2}{9}}\right)}^{46}+{\left(x^{\frac{2}{9}}\right)}^{45}+{\left(x^{\frac{2}{9}}\right)}^{22}+{\left(x^{\frac{2}{9}}\right)}^{18}+{\left(x^{\frac{2}{9}}\right)}^{15}+{\left(x^{\frac{2}{9}}\right)}^{12}+{\left(x^{\frac{2}{9}}\right)}^{10}+{\left(x^{\frac{2}{9}}\right)}^9\\ \begin{array}{cccc}& & & \end{array}+{\left(x^{\frac{2}{9}}\right)}^4+\left(x^{\frac{2}{9}}\right)+1.\\ g\left(x^{\frac{2}{9}}\right)=\left(m_1\left(x^{\frac{2}{9}}\right)\right)\left(m_3\left(x^{\frac{2}{9}}\right)\right)\left(m_5\left(x^{\frac{2}{9}}\right)\right)\left(m_7\left(x^{\frac{2}{9}}\right)\right)\left(m_9\left(x^{\frac{2}{9}}\right)\right)\left(m_{15}\left(x^{\frac{2}{9}}\right)\right)\left(m_{21}\left(x^{\frac{2}{9}}\right)\right)\\ \begin{array}{ccc}& & \end{array}={\left(x^{\frac{2}{9}}\right)}^{124}+{\left(x^{\frac{2}{9}}\right)}^{121}+{\left(x^{\frac{2}{9}}\right)}^{120}+{\left(x^{\frac{2}{9}}\right)}^{109}+{\left(x^{\frac{2}{9}}\right)}^{106}+{\left(x^{\frac{2}{9}}\right)}^{105}+{\left(x^{\frac{2}{9}}\right)}^{94}+{\left(x^{\frac{2}{9}}\right)}^{91}\\ \begin{array}{ccc}& & \end{array}+{\left(x^{\frac{2}{9}}\right)}^{90}+{\left(x^{\frac{2}{9}}\right)}^{79}+{\left(x^{\frac{2}{9}}\right)}^{76}+{\left(x^{\frac{2}{9}}\right)}^{75}+{\left(x^{\frac{2}{9}}\right)}^{64}+{\left(x^{\frac{2}{9}}\right)}^{61}+{\left(x^{\frac{2}{9}}\right)}^{60}+{\left(x^{\frac{2}{9}}\right)}^{49}\\ \begin{array}{ccc}& & \end{array}+{\left(x^{\frac{2}{9}}\right)}^{46}+{\left(x^{\frac{2}{9}}\right)}^{45}+{\left(x^{\frac{2}{9}}\right)}^{34}+{\left(x^{\frac{2}{9}}\right)}^{31}+{\left(x^{\frac{2}{9}}\right)}^{30}+{\left(x^{\frac{2}{9}}\right)}^{19}+{\left(x^{\frac{2}{9}}\right)}^{16}+{\left(x^{\frac{2}{9}}\right)}^{15}\\ \begin{array}{ccc}& & \end{array}+{\left(x^{\frac{2}{9}}\right)}^4+\left(x^{\frac{2}{9}}\right)+1.\\ g\left(x^{\frac{2}{9}}\right)=\left(m_1\left(x^{\frac{2}{9}}\right)\right)\left(m_3\left(x^{\frac{2}{9}}\right)\right)\left(m_5\left(x^{\frac{2}{9}}\right)\right)\left(m_7\left(x^{\frac{2}{9}}\right)\right)\left(m_9\left(x^{\frac{2}{9}}\right)\right)\left(m_{15}\left(x^{\frac{2}{9}}\right)\right)\left(m_{21}\left(x^{\frac{2}{9}}\right)\right)\left(m_{27}\left(x^{\frac{2}{9}}\right)\right)\\ \begin{array}{ccc}& & \end{array}={\left(x^{\frac{2}{9}}\right)}^{128}+{\left(x^{\frac{2}{9}}\right)}^{127}+{\left(x^{\frac{2}{9}}\right)}^{126}+{\left(x^{\frac{2}{9}}\right)}^{124}+{\left(x^{\frac{2}{9}}\right)}^{120}+{\left(x^{\frac{2}{9}}\right)}^{113}+{\left(x^{\frac{2}{9}}\right)}^{112}+{\left(x^{\frac{2}{9}}\right)}^{111}\\ \begin{array}{cccc}& & & \end{array}+{\left(x^{\frac{2}{9}}\right)}^{109}+{\left(x^{\frac{2}{9}}\right)}^{105}+{\left(x^{\frac{2}{9}}\right)}^{98}+{\left(x^{\frac{2}{9}}\right)}^{97}+{\left(x^{\frac{2}{9}}\right)}^{96}+{\left(x^{\frac{2}{9}}\right)}^{94}+{\left(x^{\frac{2}{9}}\right)}^{90}+{\left(x^{\frac{2}{9}}\right)}^{83}+{\left(x^{\frac{2}{9}}\right)}^{82}\\ \begin{array}{cccc}& & & \end{array}+{\left(x^{\frac{2}{9}}\right)}^{81}+{\left(x^{\frac{2}{9}}\right)}^{79}+{\left(x^{\frac{2}{9}}\right)}^{75}+{\left(x^{\frac{2}{9}}\right)}^{68}+{\left(x^{\frac{2}{9}}\right)}^{67}+{\left(x^{\frac{2}{9}}\right)}^{66}+{\left(x^{\frac{2}{9}}\right)}^{64}+{\left(x^{\frac{2}{9}}\right)}^{60}+{\left(x^{\frac{2}{9}}\right)}^{53}\\ \begin{array}{cccc}& & & \end{array}+{\left(x^{\frac{2}{9}}\right)}^{52}+{\left(x^{\frac{2}{9}}\right)}^{51}+{\left(x^{\frac{2}{9}}\right)}^{49}+{\left(x^{\frac{2}{9}}\right)}^{45}+{\left(x^{\frac{2}{9}}\right)}^{38}+{\left(x^{\frac{2}{9}}\right)}^{37}+{\left(x^{\frac{2}{9}}\right)}^{36}+{\left(x^{\frac{2}{9}}\right)}^{34}+{\left(x^{\frac{2}{9}}\right)}^{30}\\ \begin{array}{cccc}& & & \end{array}+{\left(x^{\frac{2}{9}}\right)}^{23}+{\left(x^{\frac{2}{9}}\right)}^{22}+{\left(x^{\frac{2}{9}}\right)}^{21}+{\left(x^{\frac{2}{9}}\right)}^{19}+{\left(x^{\frac{2}{9}}\right)}^{15}+{\left(x^{\frac{2}{9}}\right)}^8+{\left(x^{\frac{2}{9}}\right)}^7+{\left(x^{\frac{2}{9}}\right)}^6+{\left(x^{\frac{2}{9}}\right)}^4+1.\\ g\left(x^{\frac{2}{9}}\right)=\left(m_1\left(x^{\frac{2}{9}}\right)\right)\left(m_3\left(x^{\frac{2}{9}}\right)\right)\left(m_5\left(x^{\frac{2}{9}}\right)\right)\left(m_7\left(x^{\frac{2}{9}}\right)\right)\left(m_9\left(x^{\frac{2}{9}}\right)\right)\left(m_9\left(x^{\frac{2}{9}}\right)\right)\left(m_{15}\left(x^{\frac{2}{9}}\right)\right)\left(m_{21}\left(x^{\frac{2}{9}}\right)\right)\\ \begin{array}{ccc}& & \end{array}\left(m_{27}\left(x^{\frac{2}{9}}\right)\right)\left(m_{45}\left(x^{\frac{2}{9}}\right)\right)={\left(x^{\frac{2}{9}}\right)}^{130}+{\left(x^{\frac{2}{9}}\right)}^{128}+{\left(x^{\frac{2}{9}}\right)}^{125}+{\left(x^{\frac{2}{9}}\right)}^{124}+{\left(x^{\frac{2}{9}}\right)}^{122}\\ \begin{array}{ccc}& & \end{array}+{\left(x^{\frac{2}{9}}\right)}^{121}+{\left(x^{\frac{2}{9}}\right)}^{120}+{\left(x^{\frac{2}{9}}\right)}^{115}+{\left(x^{\frac{2}{9}}\right)}^{113}+{\left(x^{\frac{2}{9}}\right)}^{110}+{\left(x^{\frac{2}{9}}\right)}^{109}+{\left(x^{\frac{2}{9}}\right)}^{107}\\ \begin{array}{ccc}& & \end{array}+{\left(x^{\frac{2}{9}}\right)}^{106}+{\left(x^{\frac{2}{9}}\right)}^{105}+{\left(x^{\frac{2}{9}}\right)}^{100}+{\left(x^{\frac{2}{9}}\right)}^{98}+{\left(x^{\frac{2}{9}}\right)}^{95}+{\left(x^{\frac{2}{9}}\right)}^{94}+{\left(x^{\frac{2}{9}}\right)}^{92}\\ \begin{array}{ccc}& & \end{array}+{\left(x^{\frac{2}{9}}\right)}^{91}+{\left(x^{\frac{2}{9}}\right)}^{90}+{\left(x^{\frac{2}{9}}\right)}^{85}+{\left(x^{\frac{2}{9}}\right)}^{83}+{\left(x^{\frac{2}{9}}\right)}^{80}+{\left(x^{\frac{2}{9}}\right)}^{79}+{\left(x^{\frac{2}{9}}\right)}^{77}+{\left(x^{\frac{2}{9}}\right)}^{76}\\ \begin{array}{ccc}& & \end{array}+{\left(x^{\frac{2}{9}}\right)}^{75}+{\left(x^{\frac{2}{9}}\right)}^{70}+{\left(x^{\frac{2}{9}}\right)}^{68}+{\left(x^{\frac{2}{9}}\right)}^{65}+{\left(x^{\frac{2}{9}}\right)}^{64}+{\left(x^{\frac{2}{9}}\right)}^{62}+{\left(x^{\frac{2}{9}}\right)}^{61}+{\left(x^{\frac{2}{9}}\right)}^{60}\\ \begin{array}{ccc}& & \end{array}+{\left(x^{\frac{2}{9}}\right)}^{55}+{\left(x^{\frac{2}{9}}\right)}^{53}+{\left(x^{\frac{2}{9}}\right)}^{50}+{\left(x^{\frac{2}{9}}\right)}^{49}+{\left(x^{\frac{2}{9}}\right)}^7+{\left(x^{\frac{2}{9}}\right)}^{46}+{\left(x^{\frac{2}{9}}\right)}^{45}+{\left(x^{\frac{2}{9}}\right)}^{40}\\ \begin{array}{ccc}& & \end{array}+{\left(x^{\frac{2}{9}}\right)}^{38}+{\left(x^{\frac{2}{9}}\right)}^{35}+{\left(x^{\frac{2}{9}}\right)}^{34}+{\left(x^{\frac{2}{9}}\right)}^{32}+{\left(x^{\frac{2}{9}}\right)}^{31}+{\left(x^{\frac{2}{9}}\right)}^{30}+{\left(x^{\frac{2}{9}}\right)}^{25}\\ \begin{array}{ccc}& & \end{array}+{\left(x^{\frac{2}{9}}\right)}^{23}+{\left(x^{\frac{2}{9}}\right)}^{20}+{\left(x^{\frac{2}{9}}\right)}^{19}+{\left(x^{\frac{2}{9}}\right)}^{17}+{\left(x^{\frac{2}{9}}\right)}^{16}+{\left(x^{\frac{2}{9}}\right)}^5+{\left(x^{\frac{2}{9}}\right)}^{10}+{\left(x^{\frac{2}{9}}\right)}^8\\ \begin{array}{ccc}& & \end{array}+{\left(x^{\frac{2}{9}}\right)}^5+{\left(x^{\frac{2}{9}}\right)}^4+{\left(x^{\frac{2}{9}}\right)}^2+\left(x^{\frac{2}{9}}\right)+1.\\ g\left(x^{\frac{2}{9}}\right)=\left(m_1\left(x^{\frac{2}{9}}\right)\right)\left(m_3\left(x^{\frac{2}{9}}\right)\right)\left(m_5\left(x^{\frac{2}{9}}\right)\right)\left(m_7\left(x^{\frac{2}{9}}\right)\right)\left(m_9\left(x^{\frac{2}{9}}\right)\right)\left(m_{15}\left(x^{\frac{2}{9}}\right)\right)\\ \begin{array}{ccc}& & \end{array}\left(m_{21}\left(x^{\frac{2}{9}}\right)\right)\left(m_{27}\left(x^{\frac{2}{9}}\right)\right)\left(m_{45}\left(x^{\frac{2}{9}}\right)\right)\left(m_{63}\left(x^{\frac{2}{9}}\right)\right)\\ \begin{array}{ccc}& & \end{array}={\left(x^{\frac{2}{9}}\right)}^{134}+{\left(x^{\frac{2}{9}}\right)}^{133}+\mathrm{...}+{\left(x^{\frac{2}{9}}\right)}^2+\left(x^{\frac{2}{9}}\right)+1.\end{array}$

Finally, errors like 3, 4, 7, 10, 13, 22, 31 and 67 are corrected.

From Example 2.1 and [³, Example 1], it follows that the code generated through ${𝔽}_2\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]$ corrects more errors and has better code rate than the code generated through 𝔽₂[x; aℤ₀]. Now, we are in position to develop a link between a primitive (n;n - r) binary BCH code C _n and a nonprimitive (b ^j n;b ^j n - r _j ) binary BCH code $C_{b^{j}n}$, where r and r _j are, respectively, the degrees of their generating polynomials g(x ^a ) and $g\left(x^{\frac{a}{b^j}}\right)$. From Theorem 2, it follows that the generalized polynomial $g\left(x^{\frac{a}{b^j}}\right)$ in ${𝔽}_2\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]$ divides ${\left(x^{\frac{a}{b^j}}\right)}^{b^{j}n}-1$ in ${𝔽}_2\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]$. So, there is a non-primitive BCH code $C_{b^{j}n}$ generated by $g\left(x^{\frac{a}{b^j}}\right)$ in ${𝔽}_2{\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]}_{b^{j}n}$. By the same argument, as b ^j n divides $n_j=2^{b^{j}s}-1$, it follows that ${\left(x^{\frac{a}{b^j}}\right)}^{b^{j}n}-1$ divides ${\left(x^{\frac{a}{b^j}}\right)}^{n_j}-1$ in ${𝔽}_2\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]$. Therefore, $\left({\left(x^{\frac{a}{b^j}}\right)}^{n_j}-1\right)\subset \left({\left(x^{\frac{a}{b^j}}\right)}^{b^{j}n}-1\right)$. Consequently, from third isomorphism theorem for rings, it follows that

$\frac{{𝔽}_2\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]/\left({\left(x^{\frac{a}{b^j}}\right)}^{n_j}-1\right)}{\left({\left(x^{\frac{a}{b^j}}\right)}^{b^{j}n}-1\right)/\left({\left(x^{\frac{a}{b^j}}\right)}^{n_j}-1\right)}\simeq \frac{{𝔽}_2\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]}{\left({\left(x^{\frac{a}{b^j}}\right)}^{b^{j}n}-1\right)}\simeq \frac{{𝔽}_2\left[x;a{\mathbb{Z}}_0\right]}{\left({\left(x^a\right)}^n-1\right)}.$

Thus, there are embeddings $C_n\hookrightarrow C_{b^{j}n}\hookrightarrow C_{n_j}$ of codes, whereas $C_n,C_{b^{j}n},C_{n_j}$ are, respectively, primitive BCH, non-primitive BCH and primitive BCH codes. Whereas the embeddings $C_n\hookrightarrow C_{b^{j}n}$ are defined as $a\left(x^a\right)=a_0+a_1\left(x^a\right)+\mathrm{...}+a_{n-1}{\left(x^a\right)}^{n-1}\mapsto a_0+a_1{\left(x^{\frac{a}{b^j}}\right)}^{b^j}+\mathrm{...}+a_{n-1}{\left(x^{\frac{a}{b^j}}\right)}^{b^j\left(n-1\right)}=a\left(x^{\frac{a}{b^j}}\right)$, where a(x ^a ) ∈ C _n and $a\left(x^{\frac{a}{b^j}}\right)\in C_{b^{j}n}$. Also, if $g\left(x^{\frac{a}{b^j-1}}\right)$ is the generator polynomial of a binary non-primitive BCH code $C_{b^{j-1}n}^{j-1}$ in ${𝔽}_2{\left[x;\frac{a}{b^{j-1}}{\mathbb{Z}}_0\right]}_{b^{j-1}n}$, then $g\left(x^{\frac{a}{b^j}}\right)$ is the generator polynomial of a binary non-primitive BCH code $C_{b^{j}n}$ in the monoid ring ${𝔽}_2{\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]}_{b^{j}n}$. Thus, a non-primitive BCH code $C_{b^{j-1}n}$ is embedded in a non-primitive BCH code $C_{b^{j}n}$ under the monomorphism defined as $a\left(x^{\frac{a}{b^j-1}}\right)\mapsto a\left(x^{\frac{a}{b^j}}\right)$.

With the above discussion it follows the following theorem.

Theorem 3.[³, Theorem 6] Let C_nbe a primitive binary BCH code of length n = 2^is - 1 generated by a polynomial g(x ^a ) of degree r in 𝔽₂[x; aℤ₀].

• 1. There exists a binary non-primitive BCH code$C_{b^{j}n}$of length b^jn generated by a polynomial$g\left(x^{\frac{a}{b^j}}\right)$of degree b^jr in${𝔽}_2\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]$.

• 2. The binary primitive BCH code C_nis embedded in a binary non-primitive BCH code$C_{b^{j}n}$, for each j ≥ 1.

• 3. The binary BCH codes of the sequence ${\left\{C_{b^{j}n}\right\}}_{j\ge 1}$ have the following embedding

$C_{bn}\hookrightarrow C_{b^{2}n}\hookrightarrow \mathrm{...}\hookrightarrow C_{b^{j}n}\hookrightarrow \mathrm{...}$

Hence,

$\begin{array}{l}{𝔽}_2\left[x;a{\mathbb{Z}}_0\right]\subset {𝔽}_2\left[x;\frac{a}{b}{\mathbb{Z}}_0\right]\begin{array}{c}\end{array}\subset {𝔽}_2\left[x;\frac{a}{b^2}{\mathbb{Z}}_0\right]\begin{array}{c}\end{array}\subset \cdots {𝔽}_2\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]\\ \frac{{𝔽}_2\left[x;a{\mathbb{Z}}_0\right]}{\left({\left(x^a\right)}^n-1\right)}\underset{\_}{\backsim }\frac{{𝔽}_2\left[x;\frac{a}{b}{\mathbb{Z}}_0\right]}{\left({\left(x^{\frac{a}{b}}\right)}^{bn}-1\right)}\underset{\_}{\backsim }\frac{{𝔽}_2\left[x;\frac{a}{b^2}{\mathbb{Z}}_0\right]}{\left({\left(x^{\frac{a}{b^2}}\right)}^{b^{2}n}-1\right)}\underset{\_}{\backsim }\cdots \frac{{𝔽}_2\left[x;\frac{a}{b^j}{\mathbb{Z}}_0\right]}{\left({\left(x^{\frac{a}{b^j}}\right)}^{b^{j}n}-1\right)}\\ \cup \begin{array}{ccccc}& & & & \cup \begin{array}{ccccc}& & & & \cup \begin{array}{ccccc}& & & & \cdots \begin{array}{c}\end{array}\cup \end{array}\end{array}\end{array}\\ C_n\begin{array}{ccc}& & \end{array}\hookrightarrow \begin{array}{c}\end{array}{C}_{bn}\begin{array}{ccc}& & \end{array}\hookrightarrow \begin{array}{c}\end{array}{C}_{b^{2}n}\begin{array}{cccc}& & & \end{array}\hookrightarrow \cdots \begin{array}{c}\end{array}{C}_{b^{j}n}\end{array}$

and a non-primitive BCH code$C_{b^{j-1}n}$is embedded in a non-primitive BCH code$C_{b^{j}n}$under the monomorphism defined by$a\left(x^{\frac{a}{b^j-1}}\right)\mapsto a\left(x^{\frac{a}{b^j}}\right)$.

Remark 4.The polynomial$g\left(x^{\frac{a}{b}}\right)$can be obtained from$g\left(x^{\frac{a}{b^j}}\right)$by substituting$x^{\frac{a}{b^j}}=y$and replacing y by$y^{b^j-1}=x^{\frac{a}{b}}$.

Example 2.2.By [³, Example 2] and Example 2.1, it follows that the BCH codes with designed distance d = 2, 3, … ,9 have generator polynomials$g\left(x^{\frac{2}{3}}\right),g\left(x^{\frac{2}{9}}\right)$with the same minimum distance, error correction capability and code rate, where the differences are degree, data bits, code length and check sum of the code C ₁₃₅ which is three times that of code C ₄₅ and nine times of the code C ₁₅. Similarly, on letting$\left(x^{\frac{2}{9}}\right)=y$, that is$x^{\frac{2}{3}}=y^3$, we get

$\begin{array}{l}g\left(x^{\frac{2}{9}}\right)={\left(x^{\frac{2}{9}}\right)}^{106}+{\left(x^{\frac{2}{9}}\right)}^{103}+{\left(x^{\frac{2}{9}}\right)}^{102}+{\left(x^{\frac{2}{9}}\right)}^{97}+{\left(x^{\frac{2}{9}}\right)}^{93}+{\left(x^{\frac{2}{9}}\right)}^{91}+{\left(x^{\frac{2}{9}}\right)}^{90}+\\ \begin{array}{ccc}& & \end{array}{\left(x^{\frac{2}{9}}\right)}^{61}+{\left(x^{\frac{2}{9}}\right)}^{58}+{\left(x^{\frac{2}{9}}\right)}^{57}+{\left(x^{\frac{2}{9}}\right)}^{52}+{\left(x^{\frac{2}{9}}\right)}^{48}+{\left(x^{\frac{2}{9}}\right)}^{46}+{\left(x^{\frac{2}{9}}\right)}^{45}+{\left(x^{\frac{2}{9}}\right)}^{16}+\\ \begin{array}{ccc}& & \end{array}{\left(x^{\frac{2}{9}}\right)}^{13}+{\left(x^{\frac{2}{9}}\right)}^{12}+{\left(x^{\frac{2}{9}}\right)}^7+{\left(x^{\frac{2}{9}}\right)}^3+\left(x^{\frac{2}{9}}\right)+\mathrm{1,}\\ g\left(y\right)={\left(y\right)}^{106}+{\left(y\right)}^{103}+{\left(y\right)}^{102}+{\left(y\right)}^{97}+{\left(y\right)}^{93}+{\left(y\right)}^{91}+{\left(y\right)}^{90}+{\left(y\right)}^{61}+{\left(y\right)}^{58}+\\ \begin{array}{ccc}& & \end{array}{\left(y\right)}^{57}+{\left(y\right)}^{52}+{\left(y\right)}^{48}+{\left(y\right)}^{46}+{\left(y\right)}^{45}+{\left(y\right)}^{16}+{\left(y\right)}^{13}+{\left(y\right)}^{12}+{\left(y\right)}^7+\\ \begin{array}{ccc}& & \end{array}{\left(y\right)}^3+y+\mathrm{1,}\\ g\left(y^3\right)={\left(y^3\right)}^{106}+{\left(y^3\right)}^{103}+{\left(y^3\right)}^{102}+{\left(y^3\right)}^{97}+{\left(y^3\right)}^{93}+{\left(y^3\right)}^{91}+{\left(y^3\right)}^{90}+{\left(y^3\right)}^{61}+\\ \begin{array}{ccc}& & \end{array}{\left(y^3\right)}^{58}+{\left(y^3\right)}^{57}+{\left(y^3\right)}^{52}+{\left(y^3\right)}^{48}+{\left(y^3\right)}^{46}+{\left(y^3\right)}^{45}+{\left(y^3\right)}^{16}+{\left(y^3\right)}^{13}+\\ \begin{array}{ccc}& & \end{array}{\left(y^3\right)}^{12}+{\left(y^3\right)}^7+{\left(y^3\right)}^3+\left(y^3\right)+\mathrm{1,}\\ g\left(x^{\frac{2}{3}}\right)={\left(x^{\frac{2}{3}}\right)}^{106}+{\left(x^{\frac{2}{3}}\right)}^{103}+{\left(x^{\frac{2}{3}}\right)}^{102}+{\left(x^{\frac{2}{3}}\right)}^{97}+{\left(x^{\frac{2}{3}}\right)}^{93}+{\left(x^{\frac{2}{3}}\right)}^{91}+{\left(x^{\frac{2}{3}}\right)}^{90}+{\left(x^{\frac{2}{3}}\right)}^{61}\\ \begin{array}{ccc}& & \end{array}+{\left(x^{\frac{2}{3}}\right)}^{58}+{\left(x^{\frac{2}{3}}\right)}^{57}+{\left(x^{\frac{2}{3}}\right)}^{52}+{\left(x^{\frac{2}{3}}\right)}^{48}+{\left(x^{\frac{2}{3}}\right)}^{46}+{\left(x^{\frac{2}{3}}\right)}^{45}+{\left(x^{\frac{2}{3}}\right)}^{16}+{\left(x^{\frac{2}{3}}\right)}^{13}\\ \begin{array}{ccc}& & \end{array}+{\left(x^{\frac{2}{3}}\right)}^{12}+{\left(x^{\frac{2}{3}}\right)}^7+{\left(x^{\frac{2}{3}}\right)}^3+\left(x^{\frac{2}{3}}\right)+1\\ \begin{array}{ccc}& & \end{array}={\left(x^{\frac{2}{3}}\right)}^{16}+{\left(x^{\frac{2}{3}}\right)}^{13}+{\left(x^{\frac{2}{3}}\right)}^7+{\left(x^{\frac{2}{3}}\right)}^3+\left(x^{\frac{2}{3}}\right)+1\in {𝔽}_2{\left[x;\frac{2}{3}{\mathbb{Z}}_0\right]}_{45}.\end{array}$

Similarly, for

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

it follows that

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

which is the generating polynomial of a BCH code (45, 23) having design distance d ₂ = 6, 7. In this way. we can obtain a non-primitive binary BCH code C ₁₅ from a non-primitive binary BCH code C ₄₅. On writing the corresponding code vectors of the generating polynomials

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

it follows that

$\begin{array}{l}{\nu }^1=\left(\text{11010001000011001}\right)\\ {\nu }^2=(\text{11010001000011001000000000000000000000000000011010001}\\ \begin{array}{cc}& \end{array}\text{000011001000000000000000000000000000011010001000011001})\end{array}$

The binary bits of ν¹ are properly overlapped on the bits of ν², in fact, they are repeated three times after a particular pattern. Hence, the generating matrix G ₂ of$g\left(x^{\frac{2}{9}}\right)$contains the generating matrix G ₁ of$g\left(x^{\frac{2}{3}}\right)$ such that $G_2={\oplus }_1^3{G}_1$.

3 ALGORITHM

In this section, we propose an algorithm to calculate a non-primitive BCH code of length b ^j n using a primitive BCH code of length n, where the encoding and decoding of the code are carried out in Matlab. The process of developing algorithm is divided into two major steps, i.e., encoding and decoding of a non-primitive BCH code of length b ^j n.

3.1 Encoding of a non-primitive BCH code of length b ^j n.

In encoding, we first calculate a primitive polynomial of degree s by invoking Matlab’s built in command “bchgenpoly”. After this operation, a non-primitive polynomial of degree bs is calculated. With the help of a root, say α’, the elements of the Galois field GF(2^bn ) are calculated such that ${\left(\alpha \text{'}\right)}^{b^{j}n}=1$ and after we obtain the minimal polynomials of these elements. Finally, we get a non-primitive polynomial with the help of these minimal polynomials. Thus, these design distances are calculated through which the number of errors that can be corrected in each BCH code are determined. Some methods are developed in order to achieve a specified result. In Table 3, we show a list of these methods and its description.

Table 3:
Encoding modulation description.

	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

The steps of the algorithm are explained as follows:

• a. Elements of Galois field (alpha array) α’[i]).

This step calculates all the elements of the Galois field GF(2^bn ) using a root, say α’, of a non-primitive polynomial of degree bs such that (α’)^bn = 1. The element α’ is called of alpha array and denoted by α’[i]. Now, denote the index array by A[index]. For a given input b * n, the non-primitive polynomial of degree bs gives the first element of the array α’[i]. By increasing its power, each element of the array α’0[i] is calculated in outer loop. Then in nested while loop their corresponding values are determined in such a way that if the value of any element in the array exceed bs, we take the remainder rem(index, bs) (step 9 of the algorithm). The loop breaks when the condition for identity is met.

Algorithm 1:

• b. BCH generator polynomial (g[i]).

The Matlab builds a function and its complete documentation is found under http://www.mathworks.com/help/comm/ref/bchgenpoly.html. In this module, we get the generator polynomial of primitive BCH code of length n.

1 INPUT n and k

2 OUTPUT g[i]

• c. Cyclotomic cosets (c[i]).

Given α’[i], dimension of code k and positive integer b, cyclotomic cosets c[i] are calculated. Length of c[i] is initialized to at max b * k, in short all elements should not exceed the max length. Loop start from 2 to max length and calculate unique values in given α’[i]. The process stops when we get sum of two elements = 2. Once cyclotomic cosets are calculated, we can calculate the minimal polynomials for BCH codes.

Algorithm 2:

• d. Non-primitive generating polynomialg’[i].

Given primitive polynomial p[i] and b. First find the highest degree of p[i] i.e., s and initial degree of non-primitive polynomial p’[i] i.e., bs. Initialize each element of p’[i] of length bs to 0. Iterate loop from 2 to bs in order to initialize coefficient array to 1. Finally, iterate each element if coeft array and modify the value if p’[i] at position i to the value of coeft array at i. Finally insert 1 at position 0 of p’[i], i.e., when its first element becomes 0. The output of this module play an integral role for calculating non-primitive BCH generating polynomial. It is denoted by g’ in our algorithm. We are interested in rows of obtained matrix. Using the matrix we obtained, the code for non-primitive generating polynomial of length b. Finally these values are printed out and saved in file for further usage.

Algorithm 3:

• e. Designed distance (d).

Here design distance is calculated from g’[i]. The length of coset array cl is determined and then iterate index from 2 to cl, calculate next index ni by increment current index. If ni ≤ cl, then next element of coset is calculated to the value of coset array at position of next index. Otherwise the bn is assigned to next coset. The whole process iterates to lencoset coset and finally stops at the last coset. Design distance d is calculated from the last value of coset array at position 1.

Algorithm 4:

• f. Error correction capability.

For given designed distance d, the error correction capability of a code t is calculated.

Algorithm 5:

3.2 Error correction in received polynomial (decoding)

In decoding step for a received polynomial, we first calculate the syndrome matrix S[i]. Then the D_matrix is calculated that should be invertible. After this error locator polynomial is determined whose roots give the exact position of errors in the received polynomial. Finally the received polynomial is corrected. To find the error vector and obtain the corrected codeword following scheme is used. Table 4 shows list of the following steps for error correction.

Table 4:
Encoding modulation description.

	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]

• a. Syndrome matrixS(‘[i]).

Given design distance d, bn, message length bk and received polynomial r[i], syndrome matrix S’[i] can be calculated. The length of S’[i] is initialize to the difference of bn and bk, i.e., bn - bk. Furthermore, S’[i] is initialized by element of Galois field at position i i.e. GF[i]. Nested loop are used to calculate S’[i]. Upper loop is limited to the length of bn + d - 2, where S’[i] equalizes to power of α’ and in nested loop S’[i] equalize to power of S’[i] and the iterator. Once the values of S’[i] are filled with the above values of S’[i], it follows that these S’[i] can be calculated as product of r[i] and (S’[i])^t .

Algorithm 6:

• b. D matrix (D[i]).

Given error t and S’[i], D matrix is calculated and then double loops operation on syndrome matrix is carried out and suitable values from syndrome matrix is scanned out. The process is as follows: calculate GF[i] of length t and D-matrix is initialized to that value. D[i] in nested loop for loops. Both loops iterates from 1 to t. D[i] values are S’[i] values at position i of sum of loops iterators to 1.

Algorithm 7:

• c. Is D invertible.

This module check if D[i] is invertible. If D[i] is invertible, then the algorithm works, otherwise error t is decremented and D matrix is again calculated. These operations are carried out till error t becomes 0. If t becomes 0, then algorithm will exit i.e panic condition occurs as in step 10.

Algorithm 8:

• d. Error locator polynomial (f[i]).

Given input t, D[i] and S’[i], error locator polynomial f[i] is calculated. First, initialize product matrix pm[i] and f[i] to α’^t [i]. Then, iterate the loop from 1 to t, pm[i] is filled with the value of S’[i] at t + i. After the loops ends, the value of (S’[i] * pm[i])^-1 get equals to temporary matrix T”[i]. Once the temporary matrix T”[i] is achieved f[i] is taken as the transpose of that temporary matrix (T”[i])^t discuss in step 10.

Algorithm 9:

• e. Error position matrix (e[i]).

Based on f[i] we can determine error position. First, the roots of f[i] is calculated and then we take its inverse. The values we obtain are in matrix form and these manifest error position.

Algorithm 10:

• f. Error values (eν[i]).

Once error position e[i] is determined we can easily calculate their respective values. The nested for loops are used to determine error values. Both of the loops iterate from 1 to t, in the first loop values from D_matrix can be taken while in the next loop value of product matrix pm[i] can be taken along with S’[i]. Finally, eν[i] get equated to (D_matrix*pm[i])^-1 as discusses on line 9.

Algorithm 11:

• g. Correct received polynomialr[i].

Once we have calculated error positions e[i] and error values eν[i] the received polynomial r[i] can be corrected. Here input parameters are e[i], eν[i], r[i] and bn. First estimated codeword denoted by est_code is calculated using various operations i.e., taking loops to t, bn and power of Galois field. The received polynomial is corrected by subtracting error polynomial and we get the corrected codeword ν[i].

Algorithm 12:

Example 3.1.For the code of length 45 simulation is carried out as follows: in this case b = 3 and n = 15. Using n = 15 and k = 11, Matlab’s build in function genpoly is invoked in order to find primitive polynomial, i.e., p(i) = x ⁴ + x + 1, as explain inTable 3. With b = 3 and p(i) = x ⁴ + x + 1, non_primitive_poly function is invoked, as described inTable 3step 4, here non-primitive polynomial named as p’(i) is obtained. Output for p’(i) is x ¹² + x ⁹ + 1. Cyclotomic cosets, i.e., coset_array values are also calculated. First elements of Galois field in step 1 ofTable 1, is invoked to find the power of alpha till α⁴⁵ = 1. With coset_array in hand, the designed distance d can be calculated, which is the first element of next coset array. Last but not the least error t is calculated against the given designed distance d. Code rate R is also calculated against each k ₁ and bn but is not mentioned in previous section. The output are as follows: Cyclotomic cosets for (45, 33) = [1 2 4 8 16 32 19 38 31 17 34 23], t ₁ = 1 and R ₁ = (0.73333). Cyclotomic cosets for (45, 29) = [3 6 12 24], t ₁ = 2 and R ₁ = (0.64444). Cyclotomic cosets for (45, 23) = [5 10 20 40 35 25], t ₁ =3 and R ₁ = (0.51111). Cyclotomic cosets for (45, 11) = [7 14 28 11 22 44 43 41 37 29 13 26], t ₁ = 4 and R ₁ = (0.24444). Cyclotomic cosets for (45, 5) = [15 30], t ₁ = 10 and R ₁ = (0.11111). Now comes error correction in received polynomial. In this the code (45, 29) is taken under consideration, with designed distance d ₁ = 5 and t ₁ = 2. Let the received polynomial be

$x^{44}+x^{16}+x^{13}+x^{12}+x^{11}+x^7+x^3+x+1.$

With the given values d₁, k₁, and received polynomial, syndrome matrix is calculated. The output for syndromes are S₁ = α₂, S ₂ = α₄, S ₃ = α₃₀, S ₄ = α₈. Next, we arrange syndrome values in linear equation form that is Ax = B. Where A = [S ₁, S ₂; S ₂, S ₃] and B = [S ₃, S ₄]. Matrix A is named as t_matrix of t × t dimension. Thus, we find the whether the t_matrix is singular or not. If the determinant of t_matrix is nonzero then error locator polynomial is calculated. Next error position is calculated from sigma_matrix which is obtained from the coefficients of error locator polynomial. For the given values, error_positions are 44 and 11. Hence the error polynomial is x ⁴⁴ + x ¹¹. On subtracting error polynomial from received polynomial the following code polynomial

$x^{16}+x^{13}+x^{12}+x^7+x^3+x+1$

is obtained. All of the above equations are obtained by using Matlab symbolic toolbox.

With the help of the above discussed algorithm many examples on non-primitive BCH codes of length bn, b ² n, b ³ n are constructed corresponding to primitive BCH code of length n. The parameters for all binary non-primitive BCH codes of length bn, b ² n, b³ n, where n ≤ 2⁶ - 1 and b is either 3 or 7 are given in Table 5.

Table 5:
Parameters of binary non-primitive BCH codes.

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

Table 5 manifests the error and code rate values against some selected codes which we have obtained after simulating our algorithm. These codes are of length bn, b ² n and b³ n, where n ≤ 2⁶ - 1, and b = 3, 7. k ₁;k ₂ and k ₃ are dimensions of the codes C _bn , $C_{b^{2}n}$ and $C_{b^{3}n}$, respectively.

Interleaved Codes

From Table 5, it is observed that corresponding to a primitive (n, k) code there are (bn, bk), (b ² n, b ² k ₂), (b ³ n, b ³ k) codes with same error correction capability and code rate. These codes are found to be interleaved codes (Interleaving is a periodic and reversible reordering of codes of L transmitted bits) of depth b, b ² and b ³. Hence along with random error correction capability these codes can correct burst of error of length b, b ² and b ³, respectively. The term burst of error means that two or more bits in the received word has changed from 1 to 0 or from 0 to 1. The length of the burst is measured from the first corrupted bit to the last corrupted bit. Similarly, for the code (bn, bk) the codes (b ² n, b ² k) and (b ³ n, b ³ k) are interleaved codes of depth b ² and b ³, respectively. The code (49, 28) is interleaved code of depth 7, which is formed by interleaving the following 7 codewords from (7, 4) code that is (0000000), (1101000), (0000000), (0000000), (0000000), (0000000) and (0000000). on writing them column by column it gives

${\nu }^1=\left(01000000100000000000001000000000000000000000000000\right)\in C_{49}.$

In a similar way, codeword of (343, 196) and (2401, 1372) are obtained by writing column by column 7 codeword of C ₄₉ and C ₃₄₃. Therefore, for decoding a received polynomial in C ₃₄₃ one can easily reverse the process and correct errors in the codeword of either C ₄₉ or C ₇.

ACKNOWLEDMENT

Acknowledgment to FAPESP by financial support 2013/25977-7. The authors would like to thank the anonymous reviewers for their intuitive commentary that signicantly improved the worth of this work.

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• ¹¹
  T. Shah, Amanullah & A.A. de Andrade. A decoding procedure which improves code rate and error corrections. Journal of Advanced Research in Applied Mathematics, 4(4) (2012), 37-50.
• ¹²
  T. Shah, Amanullah & A.A. de Andrade. A method for improving the code rate and error correction capability of a cyclic code. Computational and Applied Mathematics, 32(2) (2013), 261-274.
• ¹³
  T. Shah & A.A. de Andrade. Cyclic codes through B[X], B[X; 1/k_pZ₀] and B[X; 1/p_kZ₀]: a comparison. Journal of Algebra and its Applications, 11(4) (2012), 1250078, 19.
• ¹⁴
  T. Shah & A.A. de Andrade. Cyclic codes through B[X; a/bℤ₀], with a/b ∈ ℚ+ and b = a + 1, and encoding. Discrete Mathematics, Algorithms and Applications, 4(4) (2012), 1250059, 14.
• ¹⁵
  T. Shah, A. Khan & A.A. Andrade. Encoding through generalized polynomial codes. Comp. Appl. Math., 30(2) (2011), 349-366.
• ¹⁶
  T. Shah, M. Khan & A.A. de Andrade. A decoding method of an n length binary BCH code through (n + 1)n length binary cyclic code. Anais da Academia Brasileira de Ciências, 85(3) (2013), 863-872.
• ¹⁷
  T. Shah & A. Shaheen. Cyclic codes as ideals in F₂[x;aN₀]_n, F₂[x]_an, and F₂[x; 1/bN₀]_abn: a linkage. U.P.B. Sci. Bull., Series A, 78(3) (2016), 205-220.

Publication Dates

• Publication in this collection
  May-Aug 2018

History

• Received
  27 Mar 2017
• Accepted
  21 Mar 2018