Geometrical Aspects of Subspace Codes

Abstract

Subspace codes are codes whose codewords are equal to subspaces of a finite vector space V(n, q). Since the geometry of the subspaces of a finite vector space V(n, q) is equivalent to the geometry of the subspaces of a projective space PG\((n-1,q)\), problems on subspace codes can be investigated by using geometrical arguments. Here, we illustrate this approach by showing some recent results on subspace codes, obtained via geometrical arguments. We discuss upper bounds on the sizes of subspace codes, by showing the link between the Johnson bound and the size of partial spreads in finite projective spaces. We present geometrical constructions of subspace codes, and we also focus on subspace codes constructed from Maximum Rank Distance (MRD) codes. Here, we also present geometrical links of MRD codes to exterior sets of Segre varieties. Our aim is to motivate researchers on subspace codes to also consider geometrical arguments when investigating problems on subspace codes.