Primitive to non-primitive BCH codes: An instantaneous path shifting scheme for data transmission

Abstract

In this paper, we present constructions of primitive and non-primitive BCH codes using monoid rings over the local ring Z(2m), with m >= 1. We show that there exist two sequences {C-bjn}(j >= 1). and {C'(bjn)}(j >= 1) of non-primitive BCH codes (over Z(2) and Z(2m) respectively) against primitive BCH codes C-n of length n and C'(n) (over Z(2) and Z(2m)), respectively. A technique is developed in an innovative way that enables the data path to shift instantaneously during transmission via the coding schemes of C-n, C'(n), {C-bjn}(j >= 1) and {C'(bjn)}(j >= 1). The selection of the schemes is subject to the choice of better code rate or better error-correction capability of the code. Finally, we present a decoding procedure for BCH codes over Galois rings, which is also used for the decoding of BCH codes over Galois fields, based on the modified Berlekarrip-Massey algorithm.