Linear codes invariant under a linear endomorphism

Abstract

This paper deals with linear codes invariant under an endomorphism $ T, $ that we call $ T $-codes. Since any endomorphism can be represented by a matrix when a basis is fixed, we shall, for simplicity, consider linear codes invariant under right multiplication by a square matrix $ M \in \mathbb{M}_{n}( \mathbb{F}_{_q}) $, and we call them $ M $-codes. In particular, we distinguish three fundamental types: $ M $-cyclic codes, quasi $ M $-cyclic codes, and $ (M_1, M_2, \dots, M_r) $-multi-cyclic codes. The approach developed here allows us to realize cyclic codes and many of their generalizations as special cases. It also helps us better understand the structure of linear codes invariant under some special matrices, such as a permutation matrix or a monomial matrix. Next, we study the duality of $ M $-codes, where we show that the $ b $-dual of an $ M $-code is an $ M^{^{*}} $-code, where $ M^{^{*}} $ represents the adjoint matrix of $ M $ for a non-degenerate bilinear form $ b $, and we explore some important results on the duality of these codes. In order to use polynomial rings in the study of $ M $-codes, we establish a one-to-one correspondence between $ M $-codes and $ \mathbb{F}_{_q}[x] $-submodules of $ \prod_{i = 1}^{r}R_{_{f_i}}, $ where $ R_{_{f_i}} : = \mathbb{F}_{_q}[x]/\langle f_{i}(x)\rangle $ and $ ( f_{_1}(x), f_{_{2}}(x), \ldots, f_{_r}(x) ) $ is the sequence of invariant factors of $ M $. Finally, we investigate the structure of 1-generator $ M $-codes and provide BCH-like and Hartmann-Tzeng-like bounds for them. We construct optimal linear and new quantum codes by applying the method developed in this article.