Optimal Ternary Cyclic Codes with Minimum Distance Four and Five

TL;DR

This paper introduces two new families of optimal ternary cyclic codes with minimum distances four and five, expanding the known classes and including codes derived from perfect nonlinear functions.

Contribution

The paper presents two new families of optimal ternary cyclic codes with specific parameters, including conjectured and new classes, and codes from perfect nonlinear functions.

Findings

Two families of optimal ternary cyclic codes with parameters [3^m-1, 3^m-1-2m, 4] and [3^m-1, 3^m-2-2m, 5]

Contains conjectured and new classes of codes

Codes derived from perfect nonlinear functions

Abstract

Cyclic codes are an important subclass of linear codes and have wide applications in data storage systems, communication systems and consumer electronics. In this paper, two families of optimal ternary cyclic codes are presented. The first family of cyclic codes has parameters $[3^m-1, 3^m-1-2m, 4]$ and contains a class of conjectured cyclic codes and several new classes of optimal cyclic codes. The second family of cyclic codes has parameters $[3^m-1, 3^m-2-2m, 5]$ and contains a number of classes of cyclic codes that are obtained from perfect nonlinear functions over $\fthreem$, where $m>1$ and is a positive integer.