A class of $p$-ary cyclic codes and their weight enumerators

Long  Yu; Hongwei Liu

doi:10.3934/amc.2016017

Article Contents

2016, Volume 10, Issue 2: 437-457. Doi: 10.3934/amc.2016017

This issue Previous Article Coherence of sensing matrices coming from algebraic-geometric codes Next Article Cyclic and BCH codes whose minimum distance equals their maximum BCH bound

A class of $p$-ary cyclic codes and their weight enumerators

Long Yu^1, and
Hongwei Liu^1,

1.
School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei 430079

Received: August 31, 2014

Revised: September 30, 2015

Published: April 2016

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Abstract

Let $\mathbb{F}_{p^m}$ be a finite field with $p^m$ elements, where $p$ is an odd prime, and $m$ is a positive integer. Let $h_1(x)$ and $h_2(x)$ be minimal polynomials of $-\pi^{-1}$ and $\pi^{-\frac{p^k+1}{2}}$ over $\mathbb{F}_p$, respectively, where $\pi $ is a primitive element of $\mathbb{F}_{p^m}$, and $k$ is a positive integer such that $\frac{m}{\gcd(m,k)}\geq 3$. In [23], Zhou et al. obtained the weight distribution of a class of cyclic codes over $\mathbb{F}_p$ with parity-check polynomial $h_1(x)h_2(x)$ in the following two cases:
• $k$ is even and $\gcd(m,k)$ is odd;
• $\frac{m}{\gcd(m,k)}$ and $\frac{k}{\gcd(m,k)}$ are both odd. In this paper, we further investigate this class of cyclic codes over $\mathbb{F}_p$ in other cases. We determine the weight distribution of this class of cyclic codes.

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Mathematics Subject Classification: Primary: 11T71; Secondary: 94B15.

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