Do non-free LCD codes over finite commutative Frobenius rings exist?

Abstract

In this paper, we clarify some aspects of LCD codes in the literature. We first prove that non-free LCD codes do not exist over finite commutative Frobenius local rings. We then obtain a necessary and sufficient condition for the existence of LCD codes over a finite commutative Frobenius ring. We later show that a free constacyclic code over a finite chain ring is an LCD code if and only if it is reversible, and also provide a necessary and sufficient condition for a constacyclic code to be reversible. We illustrate the minimum Lee distance of LCD codes over some finite commutative chain rings with examples. We found some new optimal cyclic codes over $${\mathbb {Z}}_4$$ of different lengths which are LCD codes using computer algebra system MAGMA.