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arXiv:1507.08355

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

Xianmang He, Xu, Liqing, Hao Chen

Published Jul 30, 2015• 0 citations

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 $d > \frac{q}{2}$.

DOI: 10.48550/arxiv.1507.08355

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New

q

-ary Quantum MDS Codes with Distances Bigger than

\frac{q}{2}

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}

.

It has been a great challenge to construct new quantum maximum-distance-separable (MDS) codes. In particular, it is very hard to construct the 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 ≤q/2 + 1. In this paper, we provide a construction of the quantum MDS codes with minimum distance >q/2 + 1. In particular, we show the existence of the q-ary quantum MDS codes with length n = q2+ 1 and minimum distance d for any d q + 1 (this result extends those given in the works of Guardia (2011), Jin et al. (2010), and Kai an Zhu (2012)); and with length (q2+ 2)/3 and minimum distance d for any d (2q+2)/3 if 3|(q + 1). Our method is through Hermitian selforthogonal codes. The main idea of constructing the Hermitian self-orthogonal codes is based on the solvability in Fqof a system of homogenous equations over Fq2.

It is an important task to construct quantum maximum-distance-separable (MDS) codes with good parameters. In the present paper, we provide six new classes of

q

-ary quantum MDS codes by using generalized Reed–Solomon (GRS) codes and Hermitian construction. The minimum distances of our quantum MDS codes can be larger than

\frac {q}{2}+1

. Three of these six classes of quantum MDS codes have longer lengths than the ones constructed in[1]and[2], hence some of their results can be easily derived from ours via the propagation rule. Moreover, some known quantum MDS codes of specific lengths can be seen as special cases of ours and the minimum distances of some known quantum MDS codes are also improved as well.

Quantum maximum-distance-separable (MDS for short) codes are an important class of quantum codes. In this paper, by using Hermitian self-orthogonal generalized Reed-Solomon (GRS for short) codes, we construct five new classes of

q

-ary quantum MDS codes with minimum distance larger than

q/2+1

. Furthermore, the parameters of our quantum MDS code cannot be obtained from the previous constructions.

Construction of New q-ary quantum MDS codes with distance greater than

\frac{q}{2}

In the past decade, the construction of quantum maximum-distance separable (MDS) codes has been extensively and deeply explored. For lengths of the form

n=\frac{q^2-1}{m}

, where m is an integer such that m|q+1 or m|q-1, comprehensive results are available. In this paper, we investigate the case where the length

n=\frac{q^2-1}{m}

, yet m is neither a divisor of q-1 nor q+1. Ultimately, this problem can be reduced to identifying the necessary and sufficient conditions for the existence of pairs (m1, m2), where

m=\frac{m_1\cdot m_2}{m_1+m_2-1}

is an integer. In addition, the greatest common divisor of m with both m1 and m2, denoted as gcd(m, m1) and gcd(m, m2), must be greater than 1. The quantum MDS codes presented in this paper have not been previously investigated and their minimum distances exceed

\frac{q}{2}

.

The construction of quantum error-correcting codes has been an active field of quantum information theory since the publication of \cite{Shor1995Scheme,Steane1998Enlargement,Laflamme1996Perfect}. It is becoming more and more difficult to construct some new quantum MDS codes with large minimum distance. In this paper, based on the approach developed in the paper \cite{NewHeMDS2016}, we construct several new classes of quantum MDS codes. The quantum MDS codes exhibited here have not been constructed before and the distance parameters are bigger than q/2.