A Novel Framework for Relating Quasi-Cyclic Codes and Quasi-Twisted Codes

In this paper, we aim to analyze the algebraic structure of repeated-root quasi-cyclic codes of length p(k) nl and index l over the finite field F-q, where k is a positive integer, q = p(alpha) and (n, p) = 1. For this purpose, a quasi-cyclic code over F-q is regarded as a linear code over an auxiliary ring. By introducing a ring isomorphism, we provide a one-to-one correspondence between this class of quasi-cyclic codes and non-repeated-root (1 - u)-quasi-twisted codes of length nl and index l over the chain ring F-q + uF(q) + center dot center dot center dot + (upk-1) F-q, where u(pk) = 0. Our approach enables us to extend the results regarding non-repeated-root quasi-twisted codes over rings to repeated-root quasi-cyclic codes over finite fields. To illustrate the effectiveness of our method, we provide examples that demonstrate how it simplifies the structure of this class of codes.