Linear codes invariant under cyclic endomorphisms

Abstract

In this paper, we study linear codes invariant under a cyclic endomorphism [Formula: see text] called [Formula: see text]-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 [Formula: see text] that we call [Formula: see text]-cyclic codes, and when [Formula: see text] is the companion matrix [Formula: see text] of a given nonzero polynomial [Formula: see text] we call them [Formula: see text]-cyclic codes. The similarity relation between matrices helps us find connections between [Formula: see text]-cyclic codes and [Formula: see text]-cyclic codes, where [Formula: see text] is the minimal polynomial of [Formula: see text] The class of [Formula: see text]-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 [Formula: see text]-cyclic codes and ideals of the polynomial ring [Formula: see text] where [Formula: see text] is the minimal polynomial of [Formula: see text] 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 [Formula: see text]-dual of an [Formula: see text]-cyclic code is an [Formula: see text]-cyclic code, where [Formula: see text] is the adjoint matrix of [Formula: see text] with respect to [Formula: see text] 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.