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

Xianmang He, Jingli Wang, Chunfang Huang, Yindong Chen

Published Jun 5, 2025• 0 citations
Publisher

Abstract

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].

DOI
10.1371/journal.pone.0325027

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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].
Construction of New q-ary quantum MDS codes with distance greater than q2\frac{q}{2}2q​
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=q2−1mn=\frac{q^2-1}{m}n=mq2−1​, 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=q2−1mn=\frac{q^2-1}{m}n=mq2−1​, 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=m1⋅m2m1+m2−1m=\frac{m_1\cdot m_2}{m_1+m_2-1}m=m1​+m2​−1m1​⋅m2​​ 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 q2\frac{q}{2}2q​.
In this paper, we construct two classes of new quantum maximum-distance-separable (MDS) codes with parameters [Formula: see text], where q is an odd prime power with q ≡ 3 (mod 4) and [Formula: see text]; [[8(q - 1), 8(q - 1) - 2d + 2, d]]q, where q is an odd prime power with the form q = 8t - 1 (t is an even positive integer) and [Formula: see text]. Comparing the parameters with all known quantum MDS codes, the quantum MDS codes exhibited here have minimum distances bigger than the ones available in the literature.
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 ofqqq-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 thanq2+1\frac {q}{2}+12q​+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) codes play an important role in the quantum codes. The previous quantum MDS codes had been constructed according to [Formula: see text] is odd or even. However, such classification omits to consider some special categories of quantum MDS codes. Because of this, we will discuss the other classifications of [Formula: see text]. In this paper, we construct some new [Formula: see text]-ary quantum MDS codes from generalized Reed–Solomon codes by using Hermitian construction, and prove that these quantum MDS codes have minimum distance greater than [Formula: see text], where [Formula: see text].
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.
Quantum maximum-distance-separable (MDS) codes form an important class of quantum codes. To get q-ary quantum MDS codes, one of the effective ways is to find linear MDS codes C over Fq2satisfying C⊥H⊆ C, where C⊥Hdenotes the Hermitian dual code of C. For a linear code C of length n over Fq2, we say that C is a dual-containing code if C⊥H⊆ C and C≠ Fq2n. Several classes of new quantum MDS codes with relatively large minimum distance have been produced through dual-containing constacyclic MDS codes. These works motivate us to make a careful study on the existence conditions for dual-containing constacyclic codes. We obtain necessary and sufficient conditions for the existence of dual-containing constacyclic codes. Four classes of dual-containing constacyclic MDS codes are constructed and their parameters are computed. Consequently, the quantum MDS codes are derived from these parameters. The quantum MDS codes exhibited here have minimum distance bigger than the ones available in the literature.