Two new classes of quantum MDS codes

TL;DR

This paper introduces two new classes of quantum MDS codes constructed via generalized Reed-Solomon codes, offering flexible parameters and improved minimum distances, thereby advancing quantum error correction capabilities.

Contribution

The paper presents novel quantum MDS codes using GRS and extended GRS codes, generalizing and improving upon previous constructions with flexible parameters and larger minimum distances.

Findings

Constructed quantum MDS codes with parameters [[tq, tq-2d+2, d]]_q and [[t(q+1)+2, t(q+1)-2d+4, d]]_q.

Codes have minimum distances exceeding q/2+1 when t > q/2.

The constructions extend and enhance earlier quantum code results.

Abstract

Let $p$ be a prime and let $q$ be a power of $p$. In this paper, by using generalized Reed-Solomon (GRS for short) codes and extended GRS codes, we construct two new classes of quantum maximum-distance- separable (MDS) codes with parameters \[ [[tq, tq-2d+2, d]]_{q} \] for any $1 \leq t \leq q, 2 \leq d \leq \lfloor \frac{tq+q-1}{q+1}\rfloor+1$, and \[ [[t(q+1)+2, t(q+1)-2d+4, d]]_{q} \] for any $1 \leq t \leq q-1, 2 \leq d \leq t+2$ with $(p,t,d) \neq (2, q-1, q)$. Our quantum codes have flexible parameters, and have minimum distances larger than $\frac{q}{2}+1$ when $t > \frac{q}{2}$. Furthermore, it turns out that our constructions generalize and improve some previous results.