On the non-existence of perfect codes in the sum-rank metric
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| Format: | Preprint |
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2025
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| _version_ | 1866916923706441728 |
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| author | Del Prete, Giuseppe Roccolano, Antonio Zullo, Ferdinando |
| author_facet | Del Prete, Giuseppe Roccolano, Antonio Zullo, Ferdinando |
| contents | We study perfect codes in the sum-rank metric, a generalization of both the Hamming and rank metrics relevant in multishot network coding and space-time coding. A perfect code attains equality in the sphere-packing bound, corresponding to a partition of the ambient space into disjoint metric balls. While perfect codes in the Hamming and rank metrics are completely classified, the existence of nontrivial perfect codes in the sum-rank metric remains largely open. In this paper, we investigate linear perfect codes in the sum-rank metric. We analyze the geometry of balls and derive bounds on their volumes, showing how the sphere-packing bound applies. For two-block spaces, we determine explicit parameter constraints for the existence of perfect codes. For multiple-block spaces, we establish non-existence results for various ranges of minimum distance, divisibility conditions, and code dimensions. We further provide computational evidence based on congruence conditions imposed by the volume of metric balls. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_20940 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the non-existence of perfect codes in the sum-rank metric Del Prete, Giuseppe Roccolano, Antonio Zullo, Ferdinando Information Theory Combinatorics We study perfect codes in the sum-rank metric, a generalization of both the Hamming and rank metrics relevant in multishot network coding and space-time coding. A perfect code attains equality in the sphere-packing bound, corresponding to a partition of the ambient space into disjoint metric balls. While perfect codes in the Hamming and rank metrics are completely classified, the existence of nontrivial perfect codes in the sum-rank metric remains largely open. In this paper, we investigate linear perfect codes in the sum-rank metric. We analyze the geometry of balls and derive bounds on their volumes, showing how the sphere-packing bound applies. For two-block spaces, we determine explicit parameter constraints for the existence of perfect codes. For multiple-block spaces, we establish non-existence results for various ranges of minimum distance, divisibility conditions, and code dimensions. We further provide computational evidence based on congruence conditions imposed by the volume of metric balls. |
| title | On the non-existence of perfect codes in the sum-rank metric |
| topic | Information Theory Combinatorics |
| url | https://arxiv.org/abs/2508.20940 |