Rank-metric intersecting codes
Abstract
Intersecting codes are codes in which any two nonzero codewords share a common nonzero coordinate, and they have been extensively studied due to their applications in communication theory, cryptography, and combinatorics. Recently, geometric interpretations have provided new insights into intersecting codes, both in the Hamming metric and, more recently, in the rank metric for linear codes. In this paper, we introduce and study matrix rank-metric intersecting codes. By defining the support of a matrix as its row space, we extend the notion of intersecting codes to the matrix rank-metric setting. Using the slice space associated with the generator tensor of a rank-metric code, we derive a geometric characterization that is equivalent to the intersecting property. This characterization generalizes the known results for linear rank-metric intersecting codes and highlights both analogies and differences with the linear case. In particular, it yields bounds on the parameters of intersecting codes, one of which is sharp and differs from the one in the linear case, together with first constructions. We then consider codes that are closed under multiplication, i.e. matrix algebras. In this setting, the intersecting property can be read off from the algebraic structure, namely the Jacobson radical and the socle. This provides a simple and flexible way to construct many intersecting codes, including codes that do not satisfy the classical sufficient condition on the minimum distance.
Related
- Explicit and asymptotically good constructions of Algebraic Geometry codes in the sum-rank metric
- A geometric approach to the density of rank-metric codes
- Puncturing-inclusion constructions of optimal Ferrers diagram rank-metric codes using semifield codes
- Optimal Ferrers Diagram Rank-Metric Codes: New Constructions, Diagram Combinations, and Applications to Constant-Dimension Subspace Codes
- Breaking linearity in the sum-rank metric: MSRD codes from switched σ-rational normal curves
BibTeX
@misc{borello2026rankmetric,
title = {{Rank-metric intersecting codes}},
author = {Martino Borello and Martin Scotti and Lune Vila},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05059},
eprint = {2611.05059},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05059},
}
RIS
TY - UNPB
TI - Rank-metric intersecting codes
AU - Martino Borello
AU - Martin Scotti
AU - Lune Vila
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05059
UR - https://arxiv.org/abs/2611.05059
AB - Intersecting codes are codes in which any two nonzero codewords share a common nonzero coordinate, and they have been extensively studied due to their applications in communication theory, cryptography, and combinatorics. Recently, geometric interpretations have provided new insights into intersecting codes, both in the Hamming metric and, more recently, in the rank metric for linear codes. In this paper, we introduce and study matrix rank-metric intersecting codes. By defining the support of a matrix as its row space, we extend the notion of intersecting codes to the matrix rank-metric setting. Using the slice space associated with the generator tensor of a rank-metric code, we derive a geometric characterization that is equivalent to the intersecting property. This characterization generalizes the known results for linear rank-metric intersecting codes and highlights both analogies and differences with the linear case. In particular, it yields bounds on the parameters of intersecting codes, one of which is sharp and differs from the one in the linear case, together with first constructions. We then consider codes that are closed under multiplication, i.e. matrix algebras. In this setting, the intersecting property can be read off from the algebraic structure, namely the Jacobson radical and the socle. This provides a simple and flexible way to construct many intersecting codes, including codes that do not satisfy the classical sufficient condition on the minimum distance.
ER -