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<inf>q</inf> = 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 = 2t, m = 2m', (m', 2) = 1 and ordm'( 2ᵗ ) = 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<inf>m</inf>(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.