New non-binary quantum codes from skew constacyclic codes over the ring $\mathbb{F}_{p^m}+v\mathbb{F}_{p^m}+v^2 \mathbb{F}_{p^m}$

TL;DR

This paper introduces a method to construct new non-binary quantum error-correcting codes using skew constacyclic codes over a specific finite ring, analyzing their structure and duals to enable quantum code development.

Contribution

It provides a novel construction of non-binary quantum codes from skew constacyclic codes over a non-chain ring, including conditions for dual containment and application of CSS and Gray map techniques.

Findings

Established necessary and sufficient conditions for dual-containing skew constacyclic codes over the ring.

Constructed numerous new non-binary quantum codes over finite fields.

Analyzed structural properties of skew constacyclic codes and their duals.

Abstract

In this article, we construct new non-binary quantum codes from skew constacyclic codes over finite commutative non-chain ring $\mathcal{R}= \mathbb{F}_{p^m}[v]/\langle v^3 =v \rangle$ where $p$ is an odd prime and $m \geq 1$. In order to obtain such quantum codes, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring $\mathcal{R}$. Then a necessary and sufficient condition for skew constacyclic codes over $\mathcal{R}$ to contain their Euclidean duals is established. Finally, with the help of CSS construction and using Gray map, many new non-binary quantum codes are obtained over $\mathbb{F}_{p^m}$.