Clubs in projective spaces and three-weight rank-metric codes

2025-08-01

AI Summary

This work systematically investigates the geometric and algebraic structure of $i$-clubs—linear sets in higher-dimensional projective spaces over finite fields—previously uncharacterized beyond the projective line. Employing a novel correspondence between $i$-clubs and rank-metric codes, and leveraging MacWilliams identities to analyze weight distributions, we derive the first optimal upper bound on their rank and construct multiple families achieving this bound. We provide a complete classification for the case $i = m-1$, introduce the first non-equivalent construction family, and prove the existence of infinitely many pairwise non-isomorphic $i$-clubs when $i leq m-2$. Furthermore, our results yield new geometric realizations and partial classifications of three-weight rank-metric codes, thereby advancing the interplay between rank-metric code theory and finite projective geometry.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

Linear sets over finite fields are central objects in finite geometry and coding theory, with deep connections to structures such as semifields, blocking sets, KM-arcs, and rank-metric codes. Among them, $i$-clubs, a class of linear sets where all but one point (which has weight $i$) have weight one, have been extensively studied in the projective line but remain poorly understood in higher-dimensional projective spaces. In this paper, we investigate the geometry and algebraic structure of $i$-clubs in projective spaces. We establish upper bounds on their rank by associating them with rank-metric codes and analyzing their parameters via MacWilliams identities. We also provide explicit constructions of $i$-clubs that attain the maximum rank for $i geq m/2$, and we demonstrate the existence of non-equivalent constructions when $i leq m-2$. The special case $i = m-1$ is fully classified. Furthermore, we explore the rich geometry of three-weight rank-metric codes, offering new constructions from clubs and partial classification results.

Problem

• Study i-clubs in higher-dimensional projective spaces
• Establish rank bounds via rank-metric codes analysis
• Construct and classify three-weight rank-metric codes

Innovation

• Associating i-clubs with rank-metric codes
• Constructing maximum rank i-clubs for i ≥ m/2
• Exploring three-weight rank-metric codes geometry