  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.
