  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.
