$ m $-QMDS codes over mixed alphabets via orthogonal arrays

Mengqian Chen; Shanqi Pang; Rong Yan; Yan Zhu; Mengqian Chen; Shanqi Pang; Rong Yan; Yan Zhu

doi:10.3934/math.20261390

AIMS Mathematics

2026, Volume 11, Issue 11: 35855-35886. doi: 10.3934/math.20261390

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$ m $-QMDS codes over mixed alphabets via orthogonal arrays

1.
School of Mathematics, South China University of Technology, Guangzhou 510641, China
2.
School of Mathematics and Statistics, Henan Normal University, Xinxiang 453007, China
3.
School of Mathematics and Statistics, Anyang Normal University, Anyang 455000, China

Received: 31 August 2026 Revised: 02 October 2026 Accepted: 10 October 2026 Published: 02 November 2026
MSC : 81P45, 05B15, 94B60

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 (MDS) codes are optimal because the minimum distance cannot be improved for given length, dimension, and alphabet sizes. Quantum MDS codes over mixed alphabets are scarcely studied, and even the existence and construction of QECCs over mixed alphabets with a minimum distance of at least three remain an open question. In this paper, we define an $ m $-QMDS code over mixed alphabets, which is a generalization of quantum MDS (QMDS) codes. We establish a relation between such codes and asymmetric orthogonal arrays with orthogonal partitions. This relation is used for the construction of $ m $-QMDS codes with varying local dimension, length, and minimum distance. As applications of this method, numerous infinite families of $ m $-QMDS codes over mixed alphabets can be constructed explicitly. The constructed codes not only have more flexibility in the choice of parameters than other models but also possess better error correction capability and encoding efficiency.
- quantum error-correcting code over mixed alphabets,
- orthogonal array,
- orthogonal partition,
- $ m $-QMDS code over mixed alphabets,
- Hamming distance
Citation: Mengqian Chen, Shanqi Pang, Rong Yan, Yan Zhu. $ m $-QMDS codes over mixed alphabets via orthogonal arrays[J]. AIMS Mathematics, 2026, 11(11): 35855-35886. doi: 10.3934/math.20261390

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Abstract

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 (MDS) codes are optimal because the minimum distance cannot be improved for given length, dimension, and alphabet sizes. Quantum MDS codes over mixed alphabets are scarcely studied, and even the existence and construction of QECCs over mixed alphabets with a minimum distance of at least three remain an open question. In this paper, we define an $ m $-QMDS code over mixed alphabets, which is a generalization of quantum MDS (QMDS) codes. We establish a relation between such codes and asymmetric orthogonal arrays with orthogonal partitions. This relation is used for the construction of $ m $-QMDS codes with varying local dimension, length, and minimum distance. As applications of this method, numerous infinite families of $ m $-QMDS codes over mixed alphabets can be constructed explicitly. The constructed codes not only have more flexibility in the choice of parameters than other models but also possess better error correction capability and encoding efficiency.

References

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