Linear rank-metric intersecting codes

TL;DR

This paper introduces rank-metric intersecting codes, exploring their properties, bounds, and constructions, and relating them to other combinatorial and geometric structures in coding theory.

Contribution

It defines and investigates a new class of rank-metric codes, providing structural insights, bounds, and connections to existing codes and combinatorial systems.

Findings

Derived structural properties and bounds for rank-metric intersecting codes.

Established geometric characterizations using 2-spannable q-systems.

Presented constructions and identified unexplored parameter ranges.

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.