Multi-orbit cyclic subspace codes and linear sets

TL;DR

This paper explores cyclic subspace codes with multiple orbits linked to Sidon spaces and uses linear set geometry to establish bounds and construct new codes with specific intersection properties and limited weights.

Contribution

It investigates cyclic subspace codes associated with multiple Sidon spaces and employs linear set geometry to derive bounds and construct codes with unique intersection and weight properties.

Findings

Constructed large classes of cyclic subspace codes with multiple orbits.

Provided bounds on parameters of cyclic subspace codes using linear set geometry.

Developed new linear sets with specific intersection patterns and limited weights.

Abstract

Cyclic subspace codes gained a lot of attention especially because they may be 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 were introduced by Bachoc, Serra and Z\'emor in 2017 in relation with the linear analogue of Vosper's Theorem. This connection allowed Roth, Raviv and Tamo to construct large classes of cyclic subspace codes with one or more orbits. In this paper we will investigate cyclic subspace codes associated to a set of Sidon 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 the parameters of a cyclic subspace code. Conversely, cyclic subspace codes are used to construct families of linear sets which extend a class of linear sets recently introduced by Napolitano, Santonastaso, Polverino and the author. This yields large classes of linear sets with a special pattern of intersection with the hyperplanes, defining rank metric and Hamming metric codes with only three distinct weights.