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arXiv:2308.00406 · Quantum Science and Technology

Near MDS and near quantum MDS codes via orthogonal arrays

Shanqi Pang, Chaomeng Zhang, Mengqian Chen, Miaomiao Zhang

Published Mar 12, 2024• 6 citations
arXivHTMLPDF

Abstract

Abstract Near maximum distance separable (NMDS) codes are closely related to interesting objects in finite geometry and have nice applications in combinatorics and cryptography. But there are many unsolved problems about construction of NMDS codes. In this paper, by using symmetrical orthogonal arrays (OAs), we construct a lot of NMDS, m-MDS and almost extremal NMDS codes. Quantum error-correcting codes (QECCs) play a central role in quantum information processing and can protect quantum information from various quantum noises. We present a general method for constructing QECCs over mixed alphabets through asymmetrical OAs. Since quantum maximum distance separable (QMDS) codes over mixed alphabets with the dimension equal to one have not been found in all the literature so far, the definition of a near QMDS code over mixed alphabets is proposed. By using asymmetrical OAs, we obtain many such codes.

DOI
10.1088/2058-9565/ad332b

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Abstract Near maximum distance separable (NMDS) codes are closely related to interesting objects in finite geometry and have nice applications in combinatorics and cryptography. But there are many unsolved problems about construction of NMDS codes. In this paper, by using symmetrical orthogonal arrays (OAs), we construct a lot of NMDS, m-MDS and almost extremal NMDS codes. Quantum error-correcting codes (QECCs) play a central role in quantum information processing and can protect quantum information from various quantum noises. We present a general method for constructing QECCs over mixed alphabets through asymmetrical OAs. Since quantum maximum distance separable (QMDS) codes over mixed alphabets with the dimension equal to one have not been found in all the literature so far, the definition of a near QMDS code over mixed alphabets is proposed. By using asymmetrical OAs, we obtain many such codes.
The construction of quantum error-correcting codes (QECCs) with good parameters is a hot topic in the area of quantum information and quantum computing. Quantum maximum distance separable (QMDS) codes are optimal because the minimum distance cannot be improved for a given length and code size. The QMDS codes over mixed alphabets are rarely known even if the existence and construction of QECCs over mixed alphabets with minimum distance more than or equal to three are still an open question. In this paper, we define an mmm-QMDS code over mixed alphabets, which is a generalization of QMDS codes. We establish a relation between mmm-QMDS codes over mixed alphabets and asymmetrical orthogonal arrays (OAs) with orthogonal partitions. Using this relation, we propose a general method to construct mmm-QMDS codes. As applications of this method, numerous infinite families of mmm-QMDS codes over mixed alphabets can be constructed explicitly. Compared with existing codes, the constructed codes have more flexibility in the choice of parameters, such as the alphabet sizes, length and dimension of the encoding state.
Abstract The construction of quantum Maximum Distance Separable (MDS) error-correcting codes is of great significance in practical applications, as it plays a vital role in enhancing the reliability and efficiency of quantum communication and quantum computing. As a result, it remains an active research topic in the field of quantum information science. In this paper, based on generalized Reed-Solomon (GRS) codes and Hermitian construction, we construct two new classes of quantum MDS codes. Our constructed quantum MDS codes are novel in the sense that their parameters differ from all previously known codes. Furthermore, most of our codes exhibit significantly larger minimum distance compared to the existing ones of the same length.
Entanglement-assisted quantum error correcting codes (EAQECCs) play a significant role in protecting quantum information from decoherence and quantum noise. Recently, constructing entanglement-assisted quantum maximum distance separable (EAQMDS) codes with flexible parameters has received much attention. In this work, four families of EAQMDS codes with a more general length are presented. And the method of selecting defining set is different from others. Compared with all the previously known results, the EAQMDS codes we constructed have larger minimum distance. All of these EAQMDS codes are new in the sense that their parameters are not covered by the quantum codes available in the literature.
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.
Recently, considerable works of literature have grown up around the theme of quantum error-correcting codes. Quantum convolutional maximum distance separable (MDS for brevity) codes and MDS asymmetric quantum codes often have better properties in long-distance quantum communications and asymmetric quantum channels. Within this writing, the authors utilized constacyclic codes in constructing three new kinds of MDS quantum convolutional codes and a class of MDS asymmetric quantum codes. These results are new in comparison with those presented in the existing publications on this subject.
In this paper, by using the Hamming distance, we establish a relation between quantum error-correcting codes ((N,K,d+1))s and orthogonal arrays with orthogonal partitions. Therefore, this is a generalization of the relation between quantum error-correcting codes ((N,1,d+1))s and irredundant orthogonal arrays. This relation is used for the construction of pure quantum error-correcting codes. As applications of this method, numerous infinite families of optimal quantum codes can be constructed explicitly such as ((3,s,2))s for all si≥3, ((4,s2,2))s for all si≥5, ((5,s,3))s for all si≥4, ((6,s2,3))s for all si≥5, ((7,s3,3))s for all si≥7, ((8,s2,4))s for all si≥9, ((9,s3,4))s for all si≥11, ((9,s,5))s for all si≥9, ((10,s2,5))s for all si≥11, ((11,s,6))s for all si≥11, and ((12,s2,6))s for all si≥13, where s=s1⋯sn and s1,…,sn are all prime powers. The advantages of our approach over existing methods lie in the facts that these results are not just existence results, but constructive results, the codes constructed are pure, and each basis state of these codes has far less terms. Moreover, the above method developed can be extended to construction of quantum error-correcting codes over mixed alphabets.