On the Hamming Distance of Repeated-Root Cyclic Codes of Length 6ps

Hai Q Dinh,Xiaoqiang Wang,Paravee Maneejuk

doi:10.1109/access.2020.2974866

Abstract

Let $p$ be an odd prime, $s$ , $m$ be positive integers such that $p^{m}\equiv 2 \pmod 3$ . In this paper, using the relationship about Hamming distances between simple-root cyclic codes and repeated-root cyclic codes, the Hamming distance of all cyclic codes of length $6p^{s}$ over finite field $\mathbb F_{p^{m}}$ is obtained. All maximum distance separable (MDS) cyclic codes of length $6p^{s}$ are established.

Highlights

Cyclic codes over finite fields have been well studied since the late 1950s because of their rich algebraic structures and practical implementations
Cyclic codes of length n over Fpm are classified as the ideals g(x) of the quotient ring Fpm [x]/ xn − 1, where the generator polynomial g(x) is the unique monic polynomial of minimum degree in the code, which is a divisor of xn − 1
Using the similar way as we show the Hamming distance of C for the case 0 ≤ v ≤ u ≤ j ≤ i ≤ ps, we first determine the Hamming distance of Cz for the case 0 ≤ j ≤ i ≤ v ≤ u ≤ ps

Summary

INTRODUCTION

Cyclic codes over finite fields have been well studied since the late 1950s because of their rich algebraic structures and practical implementations. Over finite field Fpm of length ps, 2ps, 3ps, 4ps and 6ps Since these results have been extended to more general code lengths (see, for example, [2], [3], [11], [19].). In [4], Dinh determined the Hamming distance of cyclic codes of length ps over Fpm. Later, in [16] the authors computed the Hamming distance of cyclic codes of length 2ps by using the result of [1]. In this paper, we get all Hamming distance of cyclic codes of length 6ps over the finite field Fpm for the case pm ≡ 2 (mod 3).

PRELIMINARIES

CONCLUSION