Two Classes of Optimal Ternary Cyclic Codes with Minimum Distance Four

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 C_{(t, e)} denote the cyclic code with two nonzero α^t and α^e, where α is a generator of 𝔽^*_3^m. In this letter, we investigate the ternary cyclic codes with parameters [3^m-1, 3^m-1-2m, 4] based on some results proposed by Ding and Helleseth in 2013. Two new classes of optimal ternary cyclic codes C_{(t, e)} are presented by choosing the proper t and e and determining the solutions of certain equations over 𝔽_3^m.