An extended characterization of a class of optimal three-weight cyclic codes over any finite field

TL;DR

This paper proves the necessity of certain numerical conditions for a class of optimal three-weight cyclic codes over finite fields, using novel methods to simplify and extend previous characterizations.

Contribution

It establishes the necessity of numerical conditions for these codes and introduces new, less complex methods for their characterization and analysis.

Findings

Numerical conditions are necessary for the code class.

Dual codes share parameters with the best known linear codes.

New methods simplify the proof of code characterization.

Abstract

A characterization of a class of optimal three-weight cyclic codes of dimension 3 over any finite field was recently presented in [10]. Shortly after this, a generalization for the sufficient numerical conditions of such characterization was given in [3]. The main purpose of this work is to show that the numerical conditions found in [3], are also necessary. As we will see later, an interesting feature of the present work, in clear contrast with these two preceding works, is that we use some new and non-conventional methods in order to achieve our goals. In fact, through these non-conventional methods, we not only were able to extend the characterization in [10], but also present a less complex proof of such extended characterization, which avoids the use of some of the sophisticated --but at the same time complex-- theorems, that are the key arguments of the proofs given in [10] and [3]. Furthermore, we also find the parameters for the dual code of any cyclic code in our extended characterization class. In fact, after the analysis of some examples, it seems that such dual codes always have the same parameters as the best known linear codes.