Intersecting Codes and the Connectivity of $q$-Matroids

2026-02-13

Technology Category

• Knowledge Representation and Reasoning: Qualitative Reasoning
• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Security and Privacy: Applications of cryptography

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.

Problem

• intersecting codes
• q-matroids
• vertical connectivity
• error-correcting codes
• matroid theory

Innovation

• intersecting codes
• vertical connectivity
• q-matroids
• rank metric
• matroid theory