Covering Grassmannian Codes: Bounds and Constructions

07/19/2022

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by   Bingchen Qian, et al.

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Grassmannian 𝒢_q(n,k) is the set of all k-dimensional subspaces of the vector space 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 𝒞 is a subset of 𝒢_q(n,k) such that every set of α codewords of 𝒞 spans a subspace of dimension at least δ +k in 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.