Optimal Ternary Cyclic Codes from Monomials

TL;DR

This paper constructs optimal ternary cyclic codes using almost perfect nonlinear monomials and other monomials over GF(3^m), extending previous work on perfect nonlinear monomials and addressing open problems.

Contribution

It introduces new constructions of optimal ternary cyclic codes using a broader class of monomials, including almost perfect nonlinear ones, and presents nine open problems.

Findings

Constructed optimal ternary cyclic codes with parameters [3^m-1, 3^m-1-2m, 4].

Extended previous constructions from perfect to almost perfect nonlinear monomials.

Presented nine open problems related to these codes.

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. Perfect nonlinear monomials were employed to construct optimal ternary cyclic codes with parameters $[3^m-1, 3^m-1-2m, 4]$ by Carlet, Ding and Yuan in 2005. In this paper, almost perfect nonlinear monomials, and a number of other monomials over $\gf(3^m)$ are used to construct optimal ternary cyclic codes with the same parameters. Nine open problems on such codes are also presented.