On the Ding and Helleseth's 9th open problem about optimal ternary cyclic codes

TL;DR

This paper addresses Ding and Helleseth's 9th open problem on optimal ternary cyclic codes by analyzing polynomial roots over finite fields and constructing two classes of codes that meet the Sphere Packing Bound.

Contribution

It provides an incomplete solution to the 9th problem and introduces two new classes of optimal ternary cyclic codes based on special polynomial root sets.

Findings

Constructed two classes of optimal ternary cyclic codes

Provided an incomplete answer to Ding and Helleseth's 9th problem

Codes meet the Sphere Packing Bound

Abstract

The cyclic code is a subclass of linear codes and has applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In 2013, Ding, et al. presented nine open problems about optimal ternary cyclic codes. Till now, the 1st, 2nd and 6th problems were completely solved, and the 3rd, 7th, 8th and 9th problems were partially solved. In this manuscript, we focus on the 9th problem. By determining the root set of some special polynomials over finite fields, we give an incomplete answer for the 9th problem, and then we construct two classes of optimal ternary cyclic codes with respect to the Sphere Packing Bound basing on some special polynomials over finite fields