Negacyclic codes over Z4+uZ4

TL;DR

This paper thoroughly investigates negacyclic codes over the ring Z4+uZ4, providing their structure, conditions for freeness, duals, and self-dual codes for odd and power-of-two lengths.

Contribution

It offers a complete structural characterization of negacyclic codes over Z4+uZ4, including duals and self-dual codes, for odd and length 2^k cases.

Findings

Complete structure of negacyclic codes over R

Necessary and sufficient conditions for freeness

Characterization of dual and self-dual negacyclic codes

Abstract

In this paper, we study negacyclic codes of odd length and of length $2^k$ over the ring $R=\mathbb{Z}_4+u\mathbb{Z}_4$, $u^2=0$. We give the complete structure of negacyclic codes for both the cases. We have obtained a minimal spanning set for negacyclic codes of odd lengths over $R$. A necessary and sufficient condition for negacyclic codes of odd lengths to be free is presented. We have determined the cardinality of negacyclic codes in each case. We have obtained the structure of the duals of negacyclic codes of length $2^k$ over $R$ and also characterized self-dual negacyclic codes of length $2^k$ over $R$.