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It is well known that the problem of determining the weight distributions offamilies of cyclic codes is, in general, notoriously difficult. An even harderproblem is to find characterizations of families of cyclic codes in terms oftheir weight distributions. On the other hand, it is also well known thatcyclic codes with few weights have a great practical importance in codingtheory and cryptography. In particular, cyclic codes having three nonzeroweights have been studied by several authors, however, most of these effortsfocused on cyclic codes over a prime field. In this work we present acharacterization of a class of optimal three-weight cyclic codes of dimension 3over any finite field. The codes under this characterization are, indeed,optimal in the sense that their lengths reach the Griesmer lower bound forlinear codes. Consequently, these codes reach, simultaneously, the bestpossible coding capacity, and also the best possible capabilities of errordetection and correction for linear codes. But because they are cyclic innature, they also possess a rich algebraic structure that can be utilized in avariety of ways, particularly, in the design of very efficient coding anddecoding algorithms. What is also worth pointing out, is the simplicity of thenecessary and sufficient numerical conditions that characterize our class ofoptimal three-weight cyclic codes. As we already pointed out, it is a hardproblem to find this kind of characterizations. However, for this particularcase the fundamental tool that allowed us to find our characterization was thecharacterization for all two-weight irreducible cyclic codes that wasintroduced by B. Schmidt and C. White (2002). Lastly, another feature about thecodes in this class, is that their duals seem to have always the sameparameters as the best known linear codes.

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