Clubs and their applications (Q6594415)

The authors introduce essential concepts and results pertinent to our study. This includes a review of linearized polynomials, linear sets, clubs, dual bases, and other relevant algebraic results. They identify a family of linear sets of rank \(h+2\) that contains a point of weight \(h\). They demonstrate that all \((n-2)\)-clubs belong to this family, up to equivalence. This leads to an algebraic characterization of these linear sets, allowing us to apply a generalized version of the linear analogue of Vosper's theorem under certain conditions. Next part delves into the \(\Gamma L(2,q^n)\)-equivalence of the known subspaces associated with i-clubs. Their analysis reveals a rich variety of examples. In particular, the authors show that the number of inequivalent subspaces constructed by \textit{A. Gács} and \textit{Z. Weiner} [Des. Codes Cryptography 29, No. 1--3, 131--139 (2003; Zbl 1029.51011)] correlates with the number of scattered subspaces in a smaller extension. Also, the authors identify additional inequivalent subspaces when considering \((n-2)\)-clubs. They present linearized polynomials that define the known families of clubs, providing a complete description for \((n-2)\)-clubs and addressing an open problem posed in [\textit{B. Csajbók} et al., Ars Math. Contemp. 16, No. 2, 585--608 (2019; Zbl 1461.11159)]. They investigate blocking sets of Rédei type that are inherently associated with iii-clubs and translation KM-arcs. Discussion yield classification results and non-equivalent constructions stemming from the results on i-clubs. The authors explore rank metric codes. By leveraging the connection established by \textit{T. H. Randrianarisoa} [Des. Codes Cryptography 88, No. 7, 1331--1348 (2020; Zbl 1450.94053)] between \(q\) systems and linear rank metric codes, classified and constructed linear rank metric codes with specific parameters are presented. This area is particularly intriguing due to the challenge of finding examples when the number of nonzero weights is restricted to three. Finally, this work not only provides a comprehensive classification of \((n-2)\)-clubs but also offers significant insights into blocking sets, KM-arcs, and rank metric codes. The results highlight the intricate relationships between these mathematical structures and pave the way for future research in this domain.