On a conjecture about a class of optimal ternary cyclic codes

Abstract

Cyclic codes are an important class of linear codes and have been widely used in many areas such as consumer electronics, data storage and communication systems. Let C_(1,e) denote the cyclic code with generator polynomial m_α(x)m_αe (x), where α is a primitive element of F_3m and m_αi (x) denotes the minimal polynomial of _αi over F₃ for 1 ≤ i ≤ 3m-1. In this paper, through some subtle manipulation on solving certain equations over finite fields, a conjecture proposed by Ding and Helleseth in 2013 about a class of optimal ternary cyclic codes C_(1,e) for e = 2(1+3^h) with parameters [3^m -1, 3^m-1-2m, 4] is settled, where m > 1 is an odd integer and 0 ≤ h ≤ m-1.