Linear rank-metric intersecting codes

2025-07-01

AI Summary

This paper introduces rank-metric intersecting codes—a new class of linear codes wherein any two nonzero codewords have column spaces (support spaces) with nontrivial intersection. To characterize them, we unify rank-metric coding theory, finite projective geometry, and *q*-analog combinatorial design, establishing a geometric equivalence to 2-spanning *q*-systems. We derive sufficient existence conditions in terms of the minimum rank distance and uncover deep connections to minimal codes, MRD codes, Hamming-intersecting codes, (2,1)-separating systems, and authentication codes. Tight upper and lower bounds on code length, dimension, and minimum rank distance are obtained; several explicit constructions are provided, confirming existence across key parameter regimes. Moreover, we precisely delineate the remaining open parameter regions. Collectively, these results lay a rigorous theoretical foundation and a systematic technical framework for future investigation of rank-intersecting codes.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

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

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.

Problem

• Introducing rank-metric intersecting codes in coding theory
• Exploring properties and bounds of rank-metric intersecting codes
• Connecting rank-intersecting codes to combinatorial structures

Innovation

• Introducing rank-metric intersecting codes
• Exploring coding-theoretic and geometric perspectives
• Connecting to combinatorial structures like separating systems