Quantum MDS Codes with length $n\equiv 0,1($mod$\,\frac{q\pm1}{2})$

TL;DR

This paper introduces new quantum MDS codes with novel lengths and large minimum distances, constructed via GRS codes and Hermitian methods, expanding the known parameters of quantum error correction.

Contribution

The paper presents new classes of quantum MDS codes with lengths congruent to 0 or 1 mod (q±1)/2, differing from prior codes, and with minimum distances exceeding q/2+1.

Findings

Constructed quantum MDS codes with new length parameters.

Achieved codes with minimum distances greater than q/2+1.

Expanded the known range of quantum MDS code parameters.

Abstract

An important family of quantum codes is the quantum maximum-distance-separable (MDS) codes. In this paper, we construct some new classes of quantum MDS codes by generalized Reed-Solomon (GRS) codes and Hermitian construction. In addition, the length $n$ of most of the quantum MDS codes we constructed satisfies $n\equiv 0,1($mod$\,\frac{q\pm1}{2})$, which is different from previously known code lengths. At the same time, the quantum MDS codes we construct have large minimum distances that are greater than $q/2+1$.