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Multi-orbit cyclic subspace codes and linear sets

Ferdinando Zullo
Dec 2021
摘要
Cyclic subspace codes gained a lot of attention especially because they maybe used in random network coding for correction of errors and erasures. Roth,Raviv and Tamo in 2018 established a connection between cyclic subspace codes(with certain parameters) and Sidon spaces. These latter objects wereintroduced by Bachoc, Serra and Z\'emor in 2017 in relation with the linearanalogue of Vosper's Theorem. This connection allowed Roth, Raviv and Tamo toconstruct large classes of cyclic subspace codes with one or more orbits. Inthis paper we will investigate cyclic subspace codes associated to a set ofSidon spaces, that is cyclic subspace codes with more than one orbit. Moreover,we will also use the geometry of linear sets to provide some bounds on theparameters of a cyclic subspace code. Conversely, cyclic subspace codes areused to construct families of linear sets which extend a class of linear setsrecently introduced by Napolitano, Santonastaso, Polverino and the author. Thisyields large classes of linear sets with a special pattern of intersection withthe hyperplanes, defining rank metric and Hamming metric codes with only threedistinct weights.
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