Rank-metric intersecting codes

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.