Structure and Rank of Cyclic codes over a class of non-chain rings

TL;DR

This paper investigates the structure, generators, rank, and cardinality of cyclic codes over certain non-chain rings derived from Z4+νZ4, expanding understanding of code properties in non-chain ring contexts.

Contribution

It characterizes cyclic codes over non-chain rings Z4+νZ4 for specific ν^2 values, providing explicit generator forms and minimal spanning sets.

Findings

Established the structure of cyclic codes over non-chain rings.

Derived unique generator forms for these codes.

Determined rank and cardinality through minimal spanning sets.

Abstract

The rings $Z_{4}+\nu Z_{4}$ have been classified into chain rings and non-chain rings on the basis of the values of $\nu^{2} \in Z_{4}+\nu Z_{4}.$ In this paper, the structure of cyclic codes of arbitrary length over the rings $Z_{4}+\nu Z_{4}$ for those values of $\nu^{2}$ for which these are non-chain rings has been established. A unique form of generators of these codes has also been obtained. Further, rank and cardinality of these codes have been established by finding minimal spanning sets for these codes.