A Novel Construction Technique for Some Classes of Quasi-Cyclic Codes

Let R be an arbitrary finite commutative chain ring, γ be a fixed generator of the maximal ideal of R, and F
q
= R/〈γ〉. In this paper, using the Generalized Discrete Fourier Transform, we derive the generator matrix of repeated-root quasi-cyclic codes of length n = ℓm, where q = 2
^t
, m = 2m
^'
, (m
^'
, 2) = 1 and $or{d_{m'}}\left( {{2^t}} \right) = m' - 1$, over R. Then, we specialize this result to derive the generator matrix for simple-root quasi-cyclic codes of length n = ℓm where ord
m
(q) = m − 1. Our work presented in this paper can be used to find all classes of quasi-cyclic codes whose co-index satisfies these conditions. Finally, this method is used to list some binary quasi-cyclic codes with index 2 and find optimal and self-dual codes among them.