A decoding method of an n length binary BCH code through (n + 1)n length binary cyclic code

Abstract

For a given binary BCH code C n of length n = 2 s − 1 generated by a polynomial g(x) 2 F 2 [x] of degree r there is no binary BCH code of length (n + 1)n generated by a generalized polynomial g(Z ≥ 0] of degree 2r. However, it does exist a binary cyclic code C (n+1)n of length (n + 1)n such that the binary BCH code C n is embedded in C (n+1)n . Accordingly a high code rate is attained through a binary cyclic code C (n+1)n for a binary BCH code C n . Furthermore, an algorithm proposed facilitates in a decoding of a binary BCH code C n through the decoding of a binary cyclic code C (n+1)n , while the codes C n and C (n+1)n have the same minimum hamming distance.

Sequences of Primitive and Non-primitive BCH Codes

“…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 F 2 [x; aZ 0 ] n , F 2 [x] an , F 2 [x; a b Z 0 ] bn and F 2 [x; 1 b Z 0 ] abn , where Z 0 is the set of non-negative integers, a, b ∈ Z and a, b > 1.…”

“…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 [x; a b Z 0 ], where a, b > 1. It is noticed that the monoid ring F 2 [x; a b Z 0 ] does not contain the polynomial ring F 2 [x] for a > 1.…”

An association between primitive and non-primitive BCH codes using monoid rings

“…The purpose of these constructions is to address the error correction and the code rate trade off in a smart way. However, for a particular interest in [18], it is established that, there does not exist a binary BCH code of length (n + 1)n in the factor ring F 2 x; (n+1)n − 1 generated by generalized polynomial g x 1 2 F 2 [x] /(x n − 1) having generator polynomial g(x) ∈ F 2 [x] of degree r. But, there does exist a binary cyclic code of length (n + 1)n such that the length n binary BCH code is embedded in it. Besides this, the existence of a binary cyclic (n + 1) 3 k − 1, (n + 1) 3 k − 1 − 3 k r code, where k is a positive integer, corresponding to a binary cyclic (n, n − r) code is established in [15] by the use of monoid ring F 2 x;…”

“…In both papers [18] and [15], the authors cannot show the existence of binary BCH codes corresponding to the length n binary BCH code in F 2 [x] / (x n − 1) . In this study, we address this issue and construct a binary BCH code using monoid ring F 2 x; a b Z ≥0 , where a, b are integers such that a, b > 1.…”

“…In this way a link between a primitive and non-primitive BCH code is attained. The length of the binary BCH C bn is well controlled and has better error correction capability as compared to the length (n+1)n binary cyclic code C (n+1)n initiated in [18]. This paper is organized in the following way: Section 2 contains a brief introduction to the monoid rings and a description of binary BCH codes as ideals in the factor ring F 2 x; aZ ≥0 n = F 2 x; aZ ≥0 / ((x a ) n − 1), a case parallel to Ham(r, 2).…”