On the structure of quasi T-cyclic codes

In this paper, we study linear codes of length Formula: see text that are invariant under an endomorphism Formula: see text (Formula: see text copies of Formula: see text), where Formula: see text is a cyclic endomorphism on Formula: see text. As each endomorphism can be represented by a matrix, we restrict our study on linear codes that are under a matrix Formula: see text, where Formula: see text is an Formula: see text cyclic matrix, called quasi-Formula: see text-cyclic codes of index Formula: see text, and quasi-Formula: see text-cyclic codes when Formula: see text is the companion matrix of a polynomial Formula: see text. We prove a one-to-one correspondence between quasi-Formula: see text-cyclic codes of index Formula: see text and Formula: see text-submodules of Formula: see text, where Formula: see text and Formula: see text is the minimal polynomial of Formula: see text. We prove the BCH-like and Hartmann–Tzeng-like bounds for Formula: see text-generator quasi-Formula: see text-cyclic codes. In addition, we study the additive structure of quasi-Formula: see text-cyclic codes by mapping them to Formula: see text via an Formula: see text-module morphism. Finally, we provide examples of new quantum codes derived from quasi-Formula: see text-cyclic codes as an application of our results.