Linear codes invariant under cyclic endomorphisms

In this paper, we study linear codes invariant under a cyclic endomorphism T, called T-cyclic codes. Since every cyclic endomorphism can be represented by a cyclic matrix with respect to a given basis, all of these matrices are similar. For simplicity we restrict ourselves to linear codes invariant under the right multiplication by a cyclic matrix M, that we call M-cyclic codes, and when M is the companion matrix C-f of a given nonzero polynomial f(x) we call them f-cyclic codes. The similarity relation between matrices helps us find connections between M-cyclic codes and f-cyclic codes, where f(x) is the minimal polynomial of M. The class of M-cyclic codes contains cyclic codes and their various generalizations such as constacyclic codes, right and left polycyclic codes, monomial codes, and others. As common in the study of cyclic codes and their generalizations, we make use of the one-to-one correspondence between M-cyclic codes and ideals of the polynomial ring R-f := F-q [x]/< f(x)>, where f(x) is the minimal polynomial of M. This correspondence leads to some basic characterizations of these codes such as generator and parity check polynomials among others. Next, we study the duality of these codes, where we show that the b-dual of an M-cyclic code is an M*-cyclic code, where M* is the adjoint matrix of M with respect to b, and we explore some important results on the duality of these codes. Finally, we give examples as applications of some of the results and we construct some optimal codes.