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

Asma Shaheen Ansari,Tariq Shah

doi:10.1186/s13638-016-0526-3

Asma Shaheen Ansari, Tariq Shah

Open Access

https://doi.org/10.1186/s13638-016-0526-3

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Abstract

BCH codes are one of the most important classes of cyclic codes for error correction. In this study, we generalize BCH codes using monoid rings instead of a polynomial ring over the binary field F 2. We show the existence of a non-primitive binary BCH code C bn of length bn, corresponding to a given length n binary BCH code C n . The value of b is investigated for which the existence of the non-primitive BCH code C bn is assured. It is noticed that the code C n is embedded in the code C bn . Therefore, encoding and decoding of the codes C n and C bn can be done simultaneously. The data transmitted by C n can also be transmitted by C bn . The BCH code C bn has better error correction capability whereas the BCH code C n has better code rate, hence both gains can be achieved at the same time.

Highlights

1 Introduction The BCH codes form a large class of error-correcting cyclic codes that are constructed using finite fields
We show the existence of nonprimitive binary BCH code Cbn of length bn using an a irreducible polynomial p x b
As the binary BCH code Cn is embedded in the binary non-primitive BCH code Cbn, we only describe the decoding principal for the code Cbn

Summary

Introduction

The BCH codes form a large class of error-correcting cyclic codes that are constructed using finite fields. 1) for positive integers c , d , bn such that 2 ≤ d ≤ bn and bn is relatively prime to 2, there exists a non-primitive binary BCH code Cbn of length bn, where bn is order of an element α ∈ F2bs . We are in position to develop a link between a primitive (n, n − r) binary BCH code Cn and a nonprimitive (bn, bn − r ) binary BCH code Cbn, where r and r are, respectively, the degrees of their generating polynomials g (xa) and g a xb. Let Cn be a primitive binary BCH code of length n = 2s − 1 generated by r degree polynomial g(xa) in F2 x; aZ≥0 , : 1) There exists a bn length binary non-primitive BCH a code Cbn generated by br degree polynomial g x b in F2 x;. Proof is straightforward and follows from Examples 1 and 2

General decoding principle

Conclusions