Intersecting Codes and the Connectivity of $q$-Matroids

TL;DR

This paper explores the relationship between intersecting error-correcting codes and matroid theory, introducing the concept of vertical connectivity for $q$-matroids and characterizing intersecting codes through this property.

Contribution

It establishes a connection between intersecting codes and matroid vertical connectivity, and introduces vertical connectivity for $q$-matroids, linking it to rank-metric codes.

Findings

Intersecting codes are characterized by maximum vertical connectivity.

Properties and bounds for intersecting codes with Hamming metric are established.

Vertical connectivity for $q$-matroids is introduced and linked to intersecting codes.

Abstract

We investigate the structure of intersecting error-correcting codes, with a particular focus on their connection to matroid theory. We establish properties and bounds for intersecting codes with the Hamming metric and illustrate how these distinguish the subfamily of minimal codes within the family of intersecting codes. We prove that the property of a code being intersecting is characterized by the matroid-theoretic notion of vertical connectivity, showing that intersecting codes are precisely those achieving the highest possible value of this parameter. We then introduce the concept of vertical connectivity for $q$-matroids and link it to the theory of intersecting codes endowed with the rank metric.