Combinatorial lower bound for list decoding of codes on finite-field Grassmannian

Abstract

Codes constructed as subsets of the projective geometry of a vector space over a finite field have been shown to have applications as 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. These codes are referred to as codes on finite-field Grassmannian or more generally as subspace codes. In this paper, we study fundamental limits to list decoding codes on finite-field Grassmannian. By exploiting the algebraic properties of the Grassmannian graph, we derive a new lower bound on the code size for the first relaxation of bounded minimum distance decoding, that is, when the worst-case list size is restricted to two. We show that, even for small finite field size and code parameters, codes on finite-field Grassmannian admit significant improvements in code rate when compared to bounded minimum distance decoding.