Counterexamples, Constructions, and Nonexistence Results for Optimal Ternary Cyclic Codes

TL;DR

This paper advances the understanding of optimal ternary cyclic codes by providing counterexamples, constructing new optimal codes, and establishing nonexistence results for certain parameter sets, thereby addressing open problems in the field.

Contribution

It offers the first counterexamples to some open problems and constructs new families of optimal codes, partially solving previously unresolved questions.

Findings

Provided counterexamples to open problems 3 and 4.

Constructed two new families of optimal codes with specific parameters.

Established nonexistence results for certain code parameters.

Abstract

Cyclic codes are an important subclass of linear codes with wide applications in communication systems and data storage systems. In 2013, Ding and Helleseth presented nine open problems on optimal ternary cyclic codes $\mathcal{C}_{(1,e)}$. While the first two and the sixth problems have been fully solved, others remain open. In this paper, we advance the study of the third and fourth open problems by providing the first counterexamples to both and constructing two families of optimal codes under certain conditions, thereby partially solving the third problem. Furthermore, we investigate the cyclic codes $\mathcal{C}_{(1,e)}$ where $e(3^h\pm 1)\equiv\frac{3^m-a}{2}\pmod{3^m-1}$ and $a$ is odd. For $a\equiv 3\pmod{4}$, we present two new families of optimal codes with parameters $[3^m-1,3^m-1-2m,4]$, generalizing known constructions. For $a\equiv 1\pmod{4}$, we obtain several nonexistence results on optimal codes $\mathcal{C}_{(1,e)}$ with the aforementioned parameters revealing the constraints of such codes.