A characterization of a class of optimal three-weight cyclic codes of dimension 3 over any finite field

TL;DR

This paper characterizes a class of optimal three-weight cyclic codes of dimension 3 over any finite field, which are optimal in length, capacity, and error correction, with simple conditions for their identification.

Contribution

It provides a novel characterization of optimal three-weight cyclic codes of dimension 3 over any finite field, extending previous prime field results and utilizing a known two-weight code characterization.

Findings

Codes reach the Griesmer lower bound for length

Codes have optimal error detection and correction capabilities

Dual codes match parameters of best known linear codes

Abstract

It is well known that the problem of determining the weight distributions of families of cyclic codes is, in general, notoriously difficult. An even harder problem is to find characterizations of families of cyclic codes in terms of their weight distributions. On the other hand, it is also well known that cyclic codes with few weights have a great practical importance in coding theory and cryptography. In particular, cyclic codes having three nonzero weights have been studied by several authors, however, most of these efforts focused on cyclic codes over a prime field. In this work we present a characterization of a class of optimal three-weight cyclic codes of dimension 3 over any finite field. The codes under this characterization are, indeed, optimal in the sense that their lengths reach the Griesmer lower bound for linear codes. Consequently, these codes reach, simultaneously, the best possible coding capacity, and also the best possible capabilities of error detection and correction for linear codes. But because they are cyclic in nature, they also possess a rich algebraic structure that can be utilized in a variety of ways, particularly, in the design of very efficient coding and decoding algorithms. What is also worth pointing out, is the simplicity of the necessary and sufficient numerical conditions that characterize our class of optimal three-weight cyclic codes. As we already pointed out, it is a hard problem to find this kind of characterizations. However, for this particular case the fundamental tool that allowed us to find our characterization was the characterization for all two-weight irreducible cyclic codes that was introduced by B. Schmidt and C. White (2002). Lastly, another feature about the codes in this class, is that their duals seem to have always the same parameters as the best known linear codes.