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.

Clubs and Their Applications

Napolitano, Vito;Polverino, Olga;Santonastaso, Paolo;Zullo, Ferdinando
2024

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 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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11591/543617
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