# Negacyclic codes over the local ring $\mathbb{Z}_4[v]/\langle v^2+2v\rangle$ of oddly even length and their Gray images

## TL;DR

This paper classifies negacyclic codes over a specific local ring of odd length, analyzes their structure, duals, and Gray images, and constructs new self-dual codes with good parameters.

## Contribution

It provides a complete classification of negacyclic codes over the ring, including self-dual codes, and constructs new optimal self-dual codes via Gray images.

## Key findings

- Classification of negacyclic codes over the ring for odd lengths.
- Explicit enumeration of self-dual negacyclic codes.
- Construction of new self-dual 2-quasi-twisted codes with good parameters.

## Abstract

Let $R=\mathbb{Z}_{4}[v]/\langle v^2+2v\rangle=\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ ($v^2=2v$) and $n$ be an odd positive integer. Then $R$ is a local non-principal ideal ring of $16$ elements and there is a $\mathbb{Z}_{4}$-linear Gray map from $R$ onto $\mathbb{Z}_{4}^2$ which preserves Lee distance and orthogonality. First, a canonical form decomposition and the structure for any negacyclic code over $R$ of length $2n$ are presented. From this decomposition, a complete classification of all these codes is obtained. Then the cardinality and the dual code for each of these codes are given, and self-dual negacyclic codes over $R$ of length $2n$ are presented. Moreover, all $23\cdot(4^p+5\cdot 2^p+9)^{\frac{2^{p}-2}{p}}$ negacyclic codes over $R$ of length $2M_p$ and all $3\cdot(4^p+5\cdot 2^p+9)^{\frac{2^{p-1}-1}{p}}$ self-dual codes among them are presented precisely, where $M_p=2^p-1$ is a Mersenne prime. Finally, $36$ new and good self-dual $2$-quasi-twisted linear codes over $\mathbb{Z}_4$ with basic parameters $(28,2^{28}, d_L=8,d_E=12)$ and of type $2^{14}4^7$ and basic parameters $(28,2^{28}, d_L=6,d_E=12)$ and of type $2^{16}4^6$ which are Gray images of self-dual negacyclic codes over $R$ of length $14$ are listed.