Optimal quaternary Hermitian self-orthogonal $ [n, 5] $ codes of $ n \geq 492 $

Hao Song; Yuezhen Ren; Ruihu Li; Yang Liu; Hao Song; Yuezhen Ren; Ruihu Li; Yang Liu

doi:10.3934/math.2025430

AIMS Mathematics

2025, Volume 10, Issue 4: 9324-9331. doi: 10.3934/math.2025430

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Research article

Optimal quaternary Hermitian self-orthogonal $ [n, 5] $ codes of $ n \geq 492 $

Fundamentals Department, Air Force Engineering University, Xi'an 710051, China

Received: 27 January 2025 Revised: 19 March 2025 Accepted: 28 March 2025 Published: 23 April 2025
MSC : 94B05, 11T71

Self-orthogonal (SO) codes, including Hermitian self-orthogonal (HSO) codes, form an important class of linear codes, and such codes have close connections to other mathematical structures such as block designs, lattices, and sphere packings, which can also be used to construct quantum codes. Many scholars try to solve the problem of determining low-dimensional optimal HSO codes over small fields as it is done for optimal linear codes. Let $ d_{o}(n, k) $ be the minimum distance of an optimal quaternary $ [n, k] $ linear code, and $ d_{so}(n, k) $ be that of an optimal quaternary $ [n, k] $ HSO code. In this paper, we try to determine $ d_{so}(n, 5) $ for $ n \geq 492 $ by constructing quaternary $ [n, 5] $ HSO codes in detail. Some disjoint HSO blocks have been found from generator matrices of some special optimal HSO codes. These special optimal HSO codes are constructed from quaternary simplex codes and McDonald codes. Then, $ [n, 5] $ HSO codes have been constructed for $ n \geq 492 $, by removing those special blocks from the known optimal HSO codes. As a result, we could show $ [n, 5, d_{so}(n, 5)] $ $ = [n, 5, 2\lfloor \frac{d_{o}(n, 5)}{2}\rfloor] $ for $ n \geq 492 $.
- optimal code,
- Hermitian self-orthogonal (HSO) code,
- Griesmer bound,
- simplex code,
- McDonald code,
- HSO block
Citation: Hao Song, Yuezhen Ren, Ruihu Li, Yang Liu. Optimal quaternary Hermitian self-orthogonal $ [n, 5] $ codes of $ n \geq 492 $[J]. AIMS Mathematics, 2025, 10(4): 9324-9331. doi: 10.3934/math.2025430

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Abstract

Self-orthogonal (SO) codes, including Hermitian self-orthogonal (HSO) codes, form an important class of linear codes, and such codes have close connections to other mathematical structures such as block designs, lattices, and sphere packings, which can also be used to construct quantum codes. Many scholars try to solve the problem of determining low-dimensional optimal HSO codes over small fields as it is done for optimal linear codes. Let $ d_{o}(n, k) $ be the minimum distance of an optimal quaternary $ [n, k] $ linear code, and $ d_{so}(n, k) $ be that of an optimal quaternary $ [n, k] $ HSO code. In this paper, we try to determine $ d_{so}(n, 5) $ for $ n \geq 492 $ by constructing quaternary $ [n, 5] $ HSO codes in detail. Some disjoint HSO blocks have been found from generator matrices of some special optimal HSO codes. These special optimal HSO codes are constructed from quaternary simplex codes and McDonald codes. Then, $ [n, 5] $ HSO codes have been constructed for $ n \geq 492 $, by removing those special blocks from the known optimal HSO codes. As a result, we could show $ [n, 5, d_{so}(n, 5)] $ $ = [n, 5, 2\lfloor \frac{d_{o}(n, 5)}{2}\rfloor] $ for $ n \geq 492 $.

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