Sequences of Primitive and Non-primitive BCH Codes

A S Ansari,Antonio A Andrade,Zia Ur-Rahman,T Shah

doi:10.5540/tema.2018.019.02.369

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 $\{C_{b^{j}n}\}_{1\leq j\leq m}$, 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 built 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.

Highlights

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
The extension of a BCH code embedded in a semigroup ring was considered by Cazaran in [4]
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

Summary

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. For positive selected integers for cj, which d j and there is an irreducible polyb jn such that 2 ≤ d j ≤ b jn with b jn is relatively prime to 2, there exists a non-primitive binary BCH code Cbjn of length b jn, where b jn is order of an element α ∈ F2bjs. 1. For positive integers c j, d j, b jn such that 2 ≤ d j ≤ b jn and b jn are relatively prime to 2, there exist a non-primitive binary BCH code Cbjn of length b jn, where b jn is the order of an element α ∈ F2bjs.

The non-primitive BCH code Cbjn of length b jn is defined as

ALGORITHM

INPUT n and k

Initialize len coef array 0