IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Construction of New q-ary quantum MDS codes with distance greater than $\frac{q}{2}$

Yu SHAO, Yindong Chen, Jingli WANG, Xianmang He

Published Jan 1, 2026• 0 citations

Abstract

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 $\frac{q}{2}$.

DOI: 10.1587/transfun.2026eal2010

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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Construction of New q-ary quantum MDS codes with distance greater than $\frac{q}{2}$ Construction of New q-ary quantum MDS codes with distance greater than $\frac{q}{2}$

Yu SHAO, Yindong Chen, Jingli WANG, Xianmang He

Published Jan 1, 2026• 0 citations

Publisher

Abstract

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}

DOI: 10.1587/transfun.2026eal2010

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Exploring new lengths for q-ary quantum MDS codes with larger distance

Xianmang He, Jingli Wang, Chunfang Huang +1 more

In the past decade, the construction of quantum maximum distance separable codes (MDS for short) has been extensively studied. For the length [Formula: see text], where m is an integer that divides either q + 1 or q - 1, a complete set of results has been available. In this paper, we dedicate to a previously unexplored cases where the length [Formula: see text], subject to the conditions that m is neither a divisor of q - 1 nor q + 1. Ultimately, this problem can be summarized as exploring the necessary and sufficient conditions for the existence of pairs [Formula: see text], where [Formula: see text] is an integer, with the additional requirement that the greatest common divisor ([Formula: see text]) of m with both m1 and m2, [Formula: see text] and [Formula: see text], and [Formula: see text]. The quantum MDS codes presented herein are novel and exhibit distance parameters exceeding [Formula: see text].In the past decade, the construction of quantum maximum distance separable codes (MDS for short) has been extensively studied. For the length [Formula: see text], where m is an integer that divides either q + 1 or q - 1, a complete set of results has been available. In this paper, we dedicate to a previously unexplored cases where the length [Formula: see text], subject to the conditions that m is neither a divisor of q - 1 nor q + 1. Ultimately, this problem can be summarized as exploring the necessary and sufficient conditions for the existence of pairs [Formula: see text], where [Formula: see text] is an integer, with the additional requirement that the greatest common divisor ([Formula: see text]) of m with both m1 and m2, [Formula: see text] and [Formula: see text], and [Formula: see text]. The quantum MDS codes presented herein are novel and exhibit distance parameters exceeding [Formula: see text].

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arXiv:1403.2499IEEE Transactions on Information TheoryJan 7, 2015191 citations

Some New Constructions of Quantum MDS Codes

Weijun Fang, Fang‐Wei Fu

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.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.

arXiv:1804.08213IEEE Transactions on Information TheorySep 3, 201958 citations

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}

\frac{q}{2}

Quantum MDS Codes with length

n\equiv 0,1(

mod

\,\frac{q\pm1}{2})

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

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

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

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 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.