Several classes of optimal $p$-ary cyclic codes with minimal distance four

TL;DR

This paper constructs several classes of optimal p-ary cyclic codes with minimal distance four, expanding known results and analyzing polynomial solutions over finite fields to achieve optimal parameters.

Contribution

The paper introduces four new classes of optimal p-ary cyclic codes with specific parameters, generalizing previous quinary code results and analyzing polynomial factorizations for code optimality.

Findings

Four classes of optimal p-ary cyclic codes with parameters [p^m-1, p^m-2m-2, 4]

Special cases include previous quinary code results

Two classes of optimal quinary cyclic codes derived from polynomial analysis

Abstract

Cyclic codes are a subclass of linear codes and have wide applications in data storage systems, communication systems and consumer electronics due to their efficient encoding and decoding algorithms. Let $p\ge 5$ be an odd prime and $m$ be a positive integer. Let $\mathcal{C}_{(1,e,s)}$ denote the $p$-ary cyclic code with three nonzeros $\alpha$, $\alpha^e$, and $\alpha^s$, where $\alpha $ is a generator of ${\mathbb F}_{p^m}^*$, $s=\frac{p^m-1}{2}$, and $2\le e\le p^m-2$. In this paper, we present four classes of optimal $p$-ary cyclic codes $\mathcal{C}_{(1,e,s)}$ with parameters $[p^m-1,p^m-2m-2,4]$ by analyzing the solutions of certain polynomials over finite fields. Some previous results about optimal quinary cyclic codes with parameters $[5^m-1,5^m-2m-2,4]$ are special cases of our constructions. In addition, by analyzing the irreducible factors of certain polynomials over ${\mathbb F}_{5^m}$, we present two classes of optimal quinary cyclic codes $\mathcal{C}_{(1,e,s)}$.