Covering Grassmannian Codes: Bounds and Constructions

Abstract

Grassmannian ${\mathcal{G}}_q(n,k)$ is the set of all $k$-dimensional subspaces of the vector space ${\mathbb{F}}_q^n.$ Recently, Etzion and Zhang introduced a new notion called covering Grassmannian code which can be used in network coding solutions for generalized combination networks. An $\alpha$-$(n,k,\delta {)}_q^c$ covering Grassmannian code $\mathcal{C}$ is a subset of ${\mathcal{G}}_q(n,k)$ such that every set of $\alpha$ codewords of $\mathcal{C}$ spans a subspace of dimension at least $\delta +k$ in ${\mathbb{F}}_q^n.$ In this paper, we derive new upper and lower bounds on the size of covering Grassmannian codes. These bounds improve and extend the parameter range of known bounds.

17 pages