New $q$-ary Quantum MDS Codes with Distances Bigger than $\frac{q}{2}$

TL;DR

This paper introduces new constructions of $q$-ary quantum MDS codes with larger minimum distances than $rac{q}{2}$, especially for lengths where previous results were incomplete, using a direct approach to Hermitian self-orthogonal codes.

Contribution

It provides novel methods to construct $q$-ary quantum MDS codes with larger minimum distances for certain lengths where no such codes were previously known.

Findings

Constructed new $q$-ary quantum MDS codes with $d > q/2$

Achieved codes with lengths of the form $rac{w(q^2-1)}{u}$

Extended the range of known quantum MDS codes for sparse lengths

Abstract

Constructions of quantum MDS codes have been studied by many authors. We refer to the table in page 1482 of [3] for known constructions. However there are only few $q$-ary quantum MDS $[[n,n-2d+2,d]]_q$ codes with minimum distances $d>\frac{q}{2}$ for sparse lengths $n>q+1$. In the case $n=\frac{q^2-1}{m}$ where $m|q+1$ or $m|q-1$ there are complete results. In the case $n=\frac{q^2-1}{m}$ where $m|q^2-1$ is not a factor of $q-1$ or $q+1$, there is no $q$-ary quantum MDS code with $d> \frac{q}{2}$ has been constructed. In this paper we propose a direct approch to construct Hermitian self-orthogonal codes over ${\bf F}_{q^2}$. Thus we give some new $q$-ary quantum codes in this case. Moreover we present many new $q$-ary quantum MDS codes with lengths of the form $\frac{w(q^2-1)}{u}$ and minimum distances $d > \frac{q}{2}$.