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arXiv · 2507.00569

Linear rank-metric intersecting codes

Daniele Bartoli·Martino Borello·Giuseppe Marino·Martin Scotti
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Abstract

In this paper we introduce and investigate rank-metric intersecting codes, a new class of linear codes in the rank-metric context, inspired by the well-studied notion of intersecting codes in the Hamming metric. A rank-metric code is said to be intersecting if any two nonzero codewords have supports intersecting non trivially. We explore this class from both a coding-theoretic and geometric perspective, highlighting its relationship with minimal codes, MRD codes, and Hamming-metric intersecting codes. We derive structural properties, sufficient conditions based on minimum distance, and geometric characterizations in terms of 2-spannable $q$-systems. We establish upper and lower bounds on code parameters and show some constructions, which leave a range of unexplored parameters. Finally, we connect rank-intersecting codes to other combinatorial structures such as $(2,1)$-separating systems and frameproof codes.

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Daniele Bartoli, Martino Borello, Giuseppe Marino, Martin Scotti. 2025-07-01. Linear rank-metric intersecting codes. https://arxiv.org/abs/2507.00569

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