On the Ding and Helleseth's 8th open problem about optimal ternary cyclic codes

TL;DR

This paper addresses Ding and Helleseth's 8th open problem on optimal ternary cyclic codes by providing counterexamples, sufficient conditions for optimality, and constructing new optimal codes that are inequivalent to known ones.

Contribution

It offers new insights into the 8th open problem by identifying counterexamples, establishing conditions for optimality, and constructing novel optimal ternary cyclic codes.

Findings

Counterexample for the 8th problem presented

Sufficient condition for the optimality of certain codes established

Constructed new optimal codes not equivalent to existing ones

Abstract

The cyclic code is a subclass of linear codes and has applications in consumer electronics, data storage systems and communication systems due to the efficient encoding and decoding algorithms. In 2013, Ding, et al. presented nine open problems about optimal ternary cyclic codes. Till now, the 1st, 2nd, 6th and 7th problems were completely solved, the 3rd, 8th and 9th problems were incompletely solved. In this manuscript, we focus on the 8th problem. By determining the root set of some special polynomials over finite fields, we present a counterexample and a sufficient condition for the ternary cyclic code $\mathcal{C}_{(1, e)}$ optimal. Furthermore, basing on the properties of finite fields, we construct a class of optimal ternary cyclic codes with respect to the Sphere Packing Bound, and show that these codes are not equivalent to any known codes.