# On a class of constacyclic codes over the non-principal ideal ring $\mathbb{Z}_{p^s}+u\mathbb{Z}_{p^s}$

## TL;DR

This paper investigates the structure, enumeration, and duality of $(1+pw)$-constacyclic codes over a non-principal ideal ring, providing new insights into their properties and self-dual variants, especially over rings with characteristic 2.

## Contribution

It characterizes the structure, counts, and dual codes of $(1+pw)$-constacyclic codes over $Z_{p^s}+uZ_{p^s}$, including self-dual codes over rings with characteristic 2.

## Key findings

- Explicit structure of these constacyclic codes is provided.
- Formulas for counting the number of codes and codewords are derived.
- Self-dual codes over rings with characteristic 2 are characterized.

## Abstract

$(1+pw)$-constacyclic codes of arbitrary length over the non-principal ideal ring $\mathbb{Z}_{p^s} +u\mathbb{Z}_{p^s}$ are studied, where $p$ is a prime, $w\in \mathbb{Z}_{p^s}^{\times}$ and $s$ an integer satisfying $s\geq 2$. First, the structure of any $(1+pw)$-constacyclic code over $\mathbb{Z}_{p^s} +u\mathbb{Z}_{p^s}$ are presented. Then enumerations for the number of all codes and the number of codewords in each code, and the structure of dual codes for these codes are given, respectively. Then self-dual $(1+2w)$-constacyclic codes over $\mathbb{Z}_{2^s} +u\mathbb{Z}_{2^s}$ are investigated, where $w=2^{s-2}-1$ or $2^{s-1}-1$ if $s\geq 3$, and $w=1$ if $s=2$.