A Construction of New Quantum MDS Codes

TL;DR

This paper presents a novel construction method for quantum MDS codes with larger minimum distances than previously known, expanding the range of lengths and distances achievable using Hermitian self-orthogonal codes.

Contribution

It introduces a new approach to constructing quantum MDS codes with larger minimum distances via Hermitian self-orthogonal codes, extending known parameters.

Findings

Existence of $q$-ary quantum MDS codes with length $n=q^2+1$ and minimum distance up to $q+1$.

Construction of codes with length $(q^2+2)/3$ and minimum distance up to $(2q+2)/3$ under certain divisibility conditions.

Method based on solvability of systems of homogeneous equations over finite fields.

Abstract

It has been a great challenge to construct new quantum MDS codes. In particular, it is very hard to construct quantum MDS codes with relatively large minimum distance. So far, except for some sparse lengths, all known $q$-ary quantum MDS codes have minimum distance less than or equal to $q/2+1$. In the present paper, we provide a construction of quantum MDS codes with minimum distance bigger than $q/2+1$. In particular, we show existence of $q$-ary quantum MDS codes with length $n=q^2+1$ and minimum distance $d$ for any $d\le q-1$ and $d= q+1$(this result extends those given in \cite{Gu11,Jin1,KZ12}); and with length $(q^2+2)/3$ and minimum distance $d$ for any $d\le (2q+2)/3$ if $3|(q+1)$. Our method is through Hermitian self-orthogonal codes. The main idea of constructing Hermitian self-orthogonal codes is based on the solvability in $\F_q$ of a system of homogenous equations over $\F_{q^2}$.