Clubs and their applications

TL;DR

This paper studies clubs of rank n in finite projective spaces, providing classifications, polynomial descriptions, and applications to blocking sets, KM-arcs, and rank metric codes, advancing understanding of their structure and equivalences.

Contribution

It offers a classification of (n-2)-clubs, analyzes their equivalences, and introduces polynomial descriptions, leading to new constructions and insights in finite geometry and coding theory.

Findings

Classification of (n-2)-clubs in PG(1,q^n)

Polynomial descriptions of club families

New constructions in blocking sets, KM-arcs, and rank metric codes

Abstract

Clubs of rank k are well-celebrated objects in finite geometries introduced by Fancsali and Sziklai in 2006. After the connection with a special type of arcs known as KM-arcs, they renewed their interest. This paper aims to study clubs of rank n in PG$(1,q^n)$. We provide a classification result for (n-2)-clubs of rank n, we analyze the $\mathrm{\Gamma L}(2,q^n)$-equivalence of the known subspaces defining clubs, for some of them the problem is then translated in determining whether or not certain scattered spaces are equivalent. Then we find a polynomial description of the known families of clubs via some linearized polynomials. Then we apply our results to the theory of blocking sets, KM-arcs, polynomials and rank metric codes, obtaining new constructions and classification results.