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
$p^{k}{n}\ell$
and index
$\ell$
over the finite field
$\mathbb{F}_{q}$
, where
$k$
is a positive integer,
$q=p^{\alpha}$
and
$(n,p)=1$
. For this purpose, a quasi-cyclic code over
$\mathbb{F}_{q}$
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 nonrepeated-root
$(1-u)$
. quasi-twisted codes of length
$n\ell$
and index
$\ell$
over the chain ring
$\mathbb{F}_{q}+u\mathbb{F}_{q}+\cdots+u^{p^{k}-1}\mathbb{F}_{q}$
, where
$u^{p^{k}}=0$
. 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.