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

TL;DR

This paper explores the structure of $i$-clubs in higher-dimensional projective spaces, establishing bounds, constructions, and classifications, and connects these objects to three-weight rank-metric codes in finite geometry and coding theory.

Contribution

It extends the understanding of $i$-clubs beyond the projective line, providing bounds, explicit constructions, classifications, and links to three-weight rank-metric codes.

Findings

Upper bounds on the rank of $i$-clubs established.

Explicit constructions of $i$-clubs reaching maximum rank for certain parameters.

Full classification of the case $i = m-1$.

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.