New Quantum codes from constacyclic codes over a general non-chain ring

TL;DR

This paper introduces new quantum error-correcting codes derived from constacyclic codes over a complex non-chain ring, demonstrating improved code parameters and establishing a general construction method.

Contribution

It characterizes constacyclic codes over a broad class of non-chain rings and constructs new quantum codes with better parameters than existing ones, independent of polynomial choices.

Findings

New quantum codes with improved parameters are constructed.

Existence of Quantum MDS code [[n,n-2,2]]_q for certain n is proven.

Construction depends only on degrees of polynomials, not their specific form.

Abstract

Let $q$ be a prime power and let $\mathcal{R}=\mathbb{F}_{q}[u_1,u_2, \cdots, u_k]/\langle f_i(u_i),u_iu_j-u_ju_i\rangle$ be a finite non-chain ring, where $f_i(u_i), 1\leq i \leq k$ are polynomials, not all linear, which split into distinct linear factors over $\mathbb{F}_{q}$. We characterize constacyclic codes over the ring $\mathcal{R}$ and study quantum codes from these. As an application, some new and better quantum codes, as compared to the best known codes, are obtained. We also prove that the choice of the polynomials $f_i(u_i),$ $1 \leq i \leq k$ is irrelevant while constructing quantum codes from constacyclic codes over $\mathcal{R}$, it depends only on their degrees. It is shown that there always exists Quantum MDS code $[[n,n-2,2]]_q$ for any $n$ with $\gcd (n,q)\neq 1.$