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

A characterization of a class of optimal three-weight cyclic codes of dimension 3 over any finite field was recently presented in [10]. Almost immediately thereafter, several classes of cyclic codes with either optimal three weights or a few weights, were given in [3], showing at the same time that one of these classes can be constructed as a generalization of the sufficient numerical conditions of the characterization given in [10]. The main purpose of this work is to show that these numerical conditions that were found in [3], are also necessary. That is, the main contribution in this work is to present an extended characterization of a class of optimal three-weight cyclic codes, in terms of a weight distribution table. Now, note that the kind of characterizations that are given in terms of a weight distribution table are, in general, very difficult to establish. On the other hand, 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.