Covering Grassmannian Codes: Bounds and Constructions

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 $α$-$(n,k,δ)_q^c$ covering Grassmannian code $\mathcal{C}$ is a subset of $\mathcal{G}_q(n,k)$ such that every set of $α$ codewords of $\mathcal{C}$ spans a subspace of dimension at least $δ+k$ 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.

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