Application of Constacyclic codes to Quantum MDS Codes

TL;DR

This paper studies the existence of dual-containing constacyclic codes to construct quantum MDS codes, resulting in new codes with improved parameters over existing ones.

Contribution

It provides a detailed analysis of conditions for dual-containing constacyclic codes and constructs several classes of such codes, leading to better quantum MDS codes.

Findings

Constructed new dual-containing MDS constacyclic codes

Derived quantum MDS codes with improved parameters

Enhanced understanding of existence conditions for dual-containing codes

Abstract

Quantum maximal-distance-separable (MDS) codes form an important class of quantum codes. To get $q$-ary quantum MDS codes, it suffices to find linear MDS codes $C$ over $\mathbb{F}_{q^2}$ satisfying $C^{\perp_H}\subseteq C$ by the Hermitian construction and the quantum Singleton bound. If $C^{\perp_{H}}\subseteq C$, we say that $C$ is a dual-containing code. Many new quantum MDS codes with relatively large minimum distance have been produced by constructing dual-containing constacyclic MDS codes (see \cite{Guardia11}, \cite{Kai13}, \cite{Kai14}). These works motivate us to make a careful study on the existence condition for nontrivial dual-containing constacyclic codes. This would help us to avoid unnecessary attempts and provide effective ideas in order to construct dual-containing codes. Several classes of dual-containing MDS constacyclic codes are constructed and their parameters are computed. Consequently, new quantum MDS codes are derived from these parameters. The quantum MDS codes exhibited here have parameters better than the ones available in the literature.