Semidefinite and linear programming bounds for sum-rank-metric codes and non-existence results
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866913077098708992 |
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| author | Abiad, Aida Khramova, Antonina P. Polak, Sven C. Zullo, Ferdinando |
| author_facet | Abiad, Aida Khramova, Antonina P. Polak, Sven C. Zullo, Ferdinando |
| contents | The sum-rank metric provides a unifying framework that generalizes both the celebrated Hamming and rank metrics, and has found applications in areas such as network coding, distributed storage, and space-time coding. A central problem is to determine the maximum size of a code with prescribed minimum distance. In this paper, we derive new sharp upper bounds on the size of a sum-rank-metric code using spectral and optimization techniques, including a semidefinite programming (SDP) bound that can outperform the best existing bounds based on computational experiments. Furthermore, we compare the Delsarte linear programming (LP) bound and a recent eigenvalue LP bound, and show equivalences between them, with particular emphasis on extremal regimes of the sum-rank metric. Finally, we show how to use the several SDP, LP and eigenvalue bounds to prove non-existence results for certain optimal and perfect sum-rank metric codes. Our results suggest that the combination of spectral and optimization methods effectively captures the hybrid nature of the sum-rank metric, providing new techniques that overcome the limitations of classical coding-theoretic approaches. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_27909 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Semidefinite and linear programming bounds for sum-rank-metric codes and non-existence results Abiad, Aida Khramova, Antonina P. Polak, Sven C. Zullo, Ferdinando Information Theory Combinatorics The sum-rank metric provides a unifying framework that generalizes both the celebrated Hamming and rank metrics, and has found applications in areas such as network coding, distributed storage, and space-time coding. A central problem is to determine the maximum size of a code with prescribed minimum distance. In this paper, we derive new sharp upper bounds on the size of a sum-rank-metric code using spectral and optimization techniques, including a semidefinite programming (SDP) bound that can outperform the best existing bounds based on computational experiments. Furthermore, we compare the Delsarte linear programming (LP) bound and a recent eigenvalue LP bound, and show equivalences between them, with particular emphasis on extremal regimes of the sum-rank metric. Finally, we show how to use the several SDP, LP and eigenvalue bounds to prove non-existence results for certain optimal and perfect sum-rank metric codes. Our results suggest that the combination of spectral and optimization methods effectively captures the hybrid nature of the sum-rank metric, providing new techniques that overcome the limitations of classical coding-theoretic approaches. |
| title | Semidefinite and linear programming bounds for sum-rank-metric codes and non-existence results |
| topic | Information Theory Combinatorics |
| url | https://arxiv.org/abs/2604.27909 |