Optimal quinary cyclic codes with minimum distance four

TL;DR

This paper introduces new classes of optimal quinary cyclic codes with minimum distance four, constructed using solutions of equations in finite fields and properties of monomials, expanding coding theory knowledge.

Contribution

It presents three new classes of optimal quinary cyclic codes with specific parameters and provides two theorems to facilitate their construction using monomials.

Findings

Three classes of optimal quinary cyclic codes with parameters [5^m-1, 5^m-2m-2, 4] are constructed.

Two theorems are established to aid in code construction.

Utilization of perfect nonlinear and almost perfect nonlinear monomials in code design.

Abstract

In this paper, by analyzing solutions of certain equations in the finite field $\mathbb{F}_{5^m}$, three classes of new optimal quinary cyclic codes with parameters $[5^m-1,5^m-2m-2,4]$ and two theorems are presented. With the help of the two theorems, perfect nonlinear monomials, almost perfect nonlinear monomials and a number of other monomials are used to construct more classes of new optimal quinary cyclic codes with the same parameters.