An infinite family of non-extendable MRD codes
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910092840927232 |
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| author | Bartoli, Daniele Giannoni, Alessandro Marino, Giuseppe Neri, Alessandro |
| author_facet | Bartoli, Daniele Giannoni, Alessandro Marino, Giuseppe Neri, Alessandro |
| contents | In the realm of rank-metric codes, Maximum Rank Distance (MRD) codes are optimal algebraic structures attaining the Singleton-like bound. A major open problem in this field is determining whether an MRD code can be extended to a longer one while preserving its optimality. This work investigates $\mathbb{F}_{q^m}$-linear MRD codes that are non-extendable but do not attain the maximum possible length. Geometrically, these correspond to scattered subspaces with respect to hyperplanes that are maximal with respect to inclusion but not of maximum dimension. By exploiting this geometric connection, we introduce the first infinite family of non-extendable $[4,2,3]_{q^5/q}$ MRD codes. Furthermore, we prove that these codes are self-dual up to equivalence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27748 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An infinite family of non-extendable MRD codes Bartoli, Daniele Giannoni, Alessandro Marino, Giuseppe Neri, Alessandro Information Theory Combinatorics 94B05, 51E20, 05B25 In the realm of rank-metric codes, Maximum Rank Distance (MRD) codes are optimal algebraic structures attaining the Singleton-like bound. A major open problem in this field is determining whether an MRD code can be extended to a longer one while preserving its optimality. This work investigates $\mathbb{F}_{q^m}$-linear MRD codes that are non-extendable but do not attain the maximum possible length. Geometrically, these correspond to scattered subspaces with respect to hyperplanes that are maximal with respect to inclusion but not of maximum dimension. By exploiting this geometric connection, we introduce the first infinite family of non-extendable $[4,2,3]_{q^5/q}$ MRD codes. Furthermore, we prove that these codes are self-dual up to equivalence. |
| title | An infinite family of non-extendable MRD codes |
| topic | Information Theory Combinatorics 94B05, 51E20, 05B25 |
| url | https://arxiv.org/abs/2603.27748 |