One-weight codes in the sum-rank metric
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914167909253120 |
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| author | Mushrraf, Usman Zullo, Ferdinando |
| author_facet | Mushrraf, Usman Zullo, Ferdinando |
| contents | One-weight codes, in which all nonzero codewords share the same weight, form a highly structured class of linear codes with deep connections to finite geometry. While their classification is well understood in the Hamming and rank metrics - being equivalent to (direct sums of) simplex codes - the sum-rank metric presents a far more intricate landscape. In this work, we explore the geometry of one-weight sum-rank metric codes, focusing on three distinct classes. First, we introduce and classify \emph{constant rank-list} sum-rank codes, where each nonzero codeword has the same tuple of ranks, extending results from the rank-metric setting. Next, we investigate the more general \emph{constant rank-profile} codes, where, up to reordering, each nonzero codeword has the same tuple of ranks. Although a complete classification remains elusive, we present the first examples and partial structural results for this class. Finally, we consider one-weight codes that are also MSRD (Maximum Sum-Rank Distance) codes. For dimension two, constructions arise from partitions of scattered linear sets on projective lines. For dimension three, we connect their existence to that of special $2$-fold blocking sets in the projective plane, leading to new bounds and nonexistence results over certain fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_04262 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | One-weight codes in the sum-rank metric Mushrraf, Usman Zullo, Ferdinando Information Theory Combinatorics One-weight codes, in which all nonzero codewords share the same weight, form a highly structured class of linear codes with deep connections to finite geometry. While their classification is well understood in the Hamming and rank metrics - being equivalent to (direct sums of) simplex codes - the sum-rank metric presents a far more intricate landscape. In this work, we explore the geometry of one-weight sum-rank metric codes, focusing on three distinct classes. First, we introduce and classify \emph{constant rank-list} sum-rank codes, where each nonzero codeword has the same tuple of ranks, extending results from the rank-metric setting. Next, we investigate the more general \emph{constant rank-profile} codes, where, up to reordering, each nonzero codeword has the same tuple of ranks. Although a complete classification remains elusive, we present the first examples and partial structural results for this class. Finally, we consider one-weight codes that are also MSRD (Maximum Sum-Rank Distance) codes. For dimension two, constructions arise from partitions of scattered linear sets on projective lines. For dimension three, we connect their existence to that of special $2$-fold blocking sets in the projective plane, leading to new bounds and nonexistence results over certain fields. |
| title | One-weight codes in the sum-rank metric |
| topic | Information Theory Combinatorics |
| url | https://arxiv.org/abs/2508.04262 |