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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pᵏn</tex> and index <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"></tex> over the finite field <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Fq</tex>, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> is a positive integer, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q=pα</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(n,p)=1$</tex>. For this purpose, a quasi-cyclic code over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Fq</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(1-u)$</tex>. quasi-twisted codes of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> and index <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"></tex> over the chain ring <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Fq+uFq+⋯+u^pᵏ-1Fq</tex>, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u^pᵏ=0</tex>. 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.