Clubs and Their Applications

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 arc known as a KM-arc, 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 n- 2)clubs of rank n , and we analyze the \Gamma L(2, , q n )-equivalence of the known subspaces defining clubs; for some of them the problem is then translated into 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.