Combinatorial Bounds for List Decoding of Subspace Codes (Extended Version)

Codes constructed as subsets of the projective geometry of a vector space over a finite field have recently been shown to have applications as unconditionally secure authentication codes and random network error correcting codes. If the dimension of each codeword is restricted to a fixed integer, the code forms a subset of a finite-field Grassmannian, or equivalently, a subset of the vertices of the corresponding Grassmannian graph. In this paper, we initiate the study of decoding subspace codes beyond half the minimum distance bound. Using random coding arguments, we derive lower and upper bounds on size of subspace codes for the first relaxation of bounded minimum distance decoding, i.e., when the worst-case list size is restricted to two. An important ingredient in establishing our results is generalization of sphere-packing (sphere-covering) conditions to volume-packing (volume-covering) conditions, which can be of independent interest.