Covering Grassmannian Codes: Bounds and Constructions

TL;DR

This paper studies covering Grassmannian codes, providing new bounds on their size, which are relevant for network coding, by extending previous theoretical limits and improving parameter ranges.

Contribution

It introduces improved upper and lower bounds on covering Grassmannian codes, expanding the known parameter ranges and advancing theoretical understanding.

Findings

New upper bounds on code size

New lower bounds on code size

Extended parameter ranges for bounds

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.