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.