  A class of optimal three-weight cyclic codes of dimension 3 over any finite
field was presented by Vega [Finite Fields Appl., 42 (2016) 23-38]. Shortly
thereafter, Heng and Yue [IEEE Trans. Inf. Theory, 62(8) (2016) 4501-4513]
generalized this result by presenting several classes of cyclic codes with
either optimal three weights or a few weights. On the other hand, a class of
optimal five-weight cyclic codes of dimension 4 over a prime field was recently
presented by Li, et al. [Adv. Math. Commun., 13(1) (2019) 137-156]. One of the
purposes of this work is to present a more general description for these
optimal five-weight cyclic codes, which gives place to an enlarged class of
optimal five-weight cyclic codes of dimension 4 over any finite field. As an
application of this enlarged class, we present the complete weight enumerator
of a subclass of the optimal three-weight cyclic codes over any finite field
that were studied by Vega [Finite Fields Appl., 42 (2016) 23-38]. In addition,
we study the dual codes in this enlarged class of optimal five-weight cyclic
codes, and show that they are cyclic codes of length $q^2-1$, dimension
$q^2-5$, and minimum Hamming distance 4. In fact, through several examples, we
see that those parameters are the best known parameters for linear codes.
