A class of optimal ternary cyclic codes and their duals

TL;DR

This paper introduces a new class of optimal ternary cyclic codes with minimal distance four, matching the Sphere Packing bound, and analyzes their dual codes, advancing coding theory for data transmission.

Contribution

The paper constructs a novel class of cyclic codes over GF(3) with optimal minimal distance and characterizes their duals, based on properties of Welch-type exponents.

Findings

Cyclic codes $\\C_{(u,v)}$ have minimal distance four.

Codes achieve the Sphere Packing bound for given parameters.

Dual codes are explicitly characterized.

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. Let $m=2\ell+1$ for an integer $\ell\geq 1$ and $\pi$ be a generator of $\gf(3^m)^*$. In this paper, a class of cyclic codes $\C_{(u,v)}$ over $\gf(3)$ with two nonzeros $\pi^{u}$ and $\pi^{v}$ is studied, where $u=(3^m+1)/2$, and $v=2\cdot 3^{\ell}+1$ is the ternary Welch-type exponent. Based on a result on the non-existence of solutions to certain equation over $\gf(3^m)$, the cyclic code $\C_{(u,v)}$ is shown to have minimal distance four, which is the best minimal distance for any linear code over $\gf(3)$ with length $3^m-1$ and dimension $3^m-1-2m$ according to the Sphere Packing bound. The duals of this class of cyclic codes are also studied.