Four classes of optimal p-ary cyclic codes

TL;DR

This paper introduces four new classes of optimal p-ary cyclic codes with specific parameters, expanding the known families and providing infinite classes, based on analyzing solutions over finite fields and weakening code conditions.

Contribution

It presents four classes of optimal p-ary cyclic codes with parameters [p^m-1,p^m-2m-2,4], including three infinite classes, generalizing many known optimal quinary codes.

Findings

Four classes of optimal p-ary cyclic codes are constructed.

Three classes are infinite, providing broad families of codes.

Many known optimal quinary codes are special cases.

Abstract

Let p>3 be an odd prime and m be a positive integer. Little progress on the study of optimal p-ary cyclic codes with parameters [p^m-1,p^m-2m-2,4] has been made.In this paper, by weakening the necessary and sufficient conditions on cyclic codes to have codewords of Hamming weight 3 and analyzing the solutions of certain equations over finite fields, four classes of optimal p-ary cyclic codes deduced by p^m+1/2 with parameters [p^m-1,p^m-2m-2,4] are presented.Wherein three classes of optimal p-ary cyclic codes are infinite.Many classes of known optimal quinary cyclic codes with parameters [5^m-1,5^m-2m-2,4] are special cases of the codes constructed in this paper.