Intersecting Codes and the Connectivity of $q$-Matroids
Fabrizio Conca, Benjamin Jany, Alberto Ravagnani
TL;DR
This work links intersecting error-correcting codes to matroid theory and its $q$-analogs by proving that the intersecting property is equivalent to maximal vertical connectivity, i.e., $ appa(M)= abla ext{dim}( C)$ for the code's associated matroid $M$. It provides structural bounds and density results for intersecting codes, clarifies distinctions from minimal codes (notably outside the binary case), and extends the framework to rank-metric codes via $q$-matroids, where a rank-metric code is intersecting exactly when the associated $q$-matroid has vertical connectivity equal to the code's dimension. The results unify combinatorial and algebraic perspectives, offering a matroid-theoretic lens on code design, secret-sharing implications, and geometric connections in the $q$-analog setting.
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.