# On an open problem about a class of optimal ternary cyclic codes

## TL;DR

This paper resolves an open problem by proving the optimality of certain ternary cyclic codes with specific parameters, expanding understanding of their structure and applications in coding theory.

## Contribution

It provides a complete characterization of when the cyclic codes $C_{(1,e)}$ are optimal based on the parameters $m$ and $e$, settling a previously open question.

## Key findings

- Identifies conditions under which $C_{(1,e)}$ is optimal.
- Shows $C_{(1,e)}$ has parameters $[3^m-1,3^m-1-2m,4]$ under specific conditions.
- Provides explicit formulas for $e$ depending on $m$.

## Abstract

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In this paper, we settle an open problem about a class of optimal ternary cyclic codes which was proposed by Ding and Helleseth. Let $C_{(1,e)}$ be a cyclic code of length $3^m-1$ over GF(3) with two nonzeros $\alpha$ and $\alpha^e$, where $\alpha$ is a generator of $GF(3^m)^*$ and e is a given integer. It is shown that $C_{(1,e)}$ is optimal with parameters $[3^m-1,3^m-1-2m,4]$ if one of the following conditions is met. 1) $m\equiv0(\mathrm{mod}~ 4)$, $m\geq 4$, and $e=3^\frac{m}{2}+5$. 2) $m\equiv2(\mathrm{mod}~ 4)$, $m\geq 6$, and $e=3^\frac{m+2}{2}+5$.