Strong Singleton-Like Bounds, Quasi-Perfect Codes and Distance-Optimal Codes in the Sum-Rank Metric
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910025328361472 |
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| author | Liu, Chao Chen, Hao Ji, Qinqin Xie, Ziyan Zheng, Dabin Xia, Yongbo |
| author_facet | Liu, Chao Chen, Hao Ji, Qinqin Xie, Ziyan Zheng, Dabin Xia, Yongbo |
| contents | Codes in the sum-rank metric have received many attentions in recent years, since they have wide applications in the multishot network coding, the space-time coding and the distributed storage. In this paper, by constructing covering codes in the sum-rank metric from covering codes in the Hamming metric, we derive new upper bounds on sizes, the covering radii and the block length functions of codes in the sum-rank metric. As applications, we present several strong Singleton-like bounds that are tighter than the classical Singleton-like bound when block lengths are large. In addition, we give the explicit constructions of the distance-optimal sum-rank codes of matrix sizes $s\times s$ and $2\times 2$ with minimum sum-rank distance four respectively by using cyclic codes in the Hamming metric. More importantly, we present an infinite families of quasi-perfect $q$-ary sum-rank codes with matrix sizes $2\times m$. Furthermore, we construct almost MSRD codes with larger block lengths and demonstrate how the Plotkin sum can be used to give more distance-optimal sum-rank codes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_05581 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Strong Singleton-Like Bounds, Quasi-Perfect Codes and Distance-Optimal Codes in the Sum-Rank Metric Liu, Chao Chen, Hao Ji, Qinqin Xie, Ziyan Zheng, Dabin Xia, Yongbo Information Theory Codes in the sum-rank metric have received many attentions in recent years, since they have wide applications in the multishot network coding, the space-time coding and the distributed storage. In this paper, by constructing covering codes in the sum-rank metric from covering codes in the Hamming metric, we derive new upper bounds on sizes, the covering radii and the block length functions of codes in the sum-rank metric. As applications, we present several strong Singleton-like bounds that are tighter than the classical Singleton-like bound when block lengths are large. In addition, we give the explicit constructions of the distance-optimal sum-rank codes of matrix sizes $s\times s$ and $2\times 2$ with minimum sum-rank distance four respectively by using cyclic codes in the Hamming metric. More importantly, we present an infinite families of quasi-perfect $q$-ary sum-rank codes with matrix sizes $2\times m$. Furthermore, we construct almost MSRD codes with larger block lengths and demonstrate how the Plotkin sum can be used to give more distance-optimal sum-rank codes. |
| title | Strong Singleton-Like Bounds, Quasi-Perfect Codes and Distance-Optimal Codes in the Sum-Rank Metric |
| topic | Information Theory |
| url | https://arxiv.org/abs/2601.05581 |