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

TL;DR

This paper investigates the weight distribution of a specific class of $p$-ary cyclic codes, extending previous results to new cases based on the parity of certain parameters, and provides explicit weight enumerators.

Contribution

It extends the analysis of cyclic codes' weight distribution to additional parameter cases, offering explicit formulas for their weight enumerators.

Findings

Determined weight distributions for new parameter cases.

Extended previous results to cover all cases of $rac{m}{ ext{gcd}(m,k)}$.

Provided explicit weight enumerators for the cyclic codes.

Abstract

Let $m$, $k$ be positive integers such that $\frac{m}{\gcd(m,k)}\geq 3$, $p$ be an odd prime and $\pi $ be a primitive element of $\mathbb{F}_{p^m}$. Let $h_1(x)$ and $h_2(x)$ be the minimal polynomials of $-\pi^{-1}$ and $\pi^{-\frac{p^k+1}{2}}$ over $\mathbb{F}_p$, respectively. In the case of odd $\frac{m}{\gcd(m,k)}$, when $k$ is even, $\gcd(m,k)$ is odd or when $\frac{k}{\gcd(m,k)}$ is odd, Zhou et~al. in \cite{zhou} obtained the weight distribution of a class of cyclic codes $\mathcal{C}$ over $\mathbb{F}_p$ with parity-check polynomial $h_1(x)h_2(x)$. In this paper, we further investigate this class of cyclic codes $\mathcal{C}$ over $\mathbb{F}_p$ in the rest case of odd $\frac{m}{\gcd(m,k)}$ and the case of even $\frac{m}{\gcd(m,k)}$. Moreover, we determine the weight distribution of cyclic codes $\mathcal{C}$.