Cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$

TL;DR

This paper investigates the structure and properties of cyclic codes over the ring r=Z_4+uZ_4 with u^2=0, focusing on odd lengths, and provides conditions for codes to be free modules and their generator forms.

Contribution

It introduces new conditions and formulas for cyclic codes over r, especially for Z_4-free modules and principally generated codes, advancing understanding of their algebraic structure.

Findings

Provided a sufficient condition for cyclic codes over r to be Z_4-free modules.

Derived the general form of generators for cyclic codes over r.

Established necessary and sufficient conditions for codes to be R-free.

Abstract

In this paper, we have studied cyclic codes over the ring $R=\mathbb{Z}_4+u\mathbb{Z}_4$, $u^2=0$. We have considered cyclic codes of odd lengths. A sufficient condition for a cyclic code over $R$ to be a $\mathbb{Z}_4$-free module is presented. We have provided the general form of the generators of a cyclic code over $R$ and determined a formula for the ranks of such codes. In this paper we have mainly focused on principally generated cyclic codes of odd length over $R$. We have determined a necessary condition and a sufficient condition for cyclic codes of odd lengths over $R$ to be $R$-free.