Subspace Packings -- Constructions and Bounds

Abstract

The Grassmannian ${\mathcal{G}}_q(n,k)$ is the set of all $k$-dimensional subspaces of the vector space ${\mathbb{F}}_q^n$. Kötter and Kschischang showed that codes in Grassmannian space can be used for error-correction in random network coding. On the other hand, these codes are $q$-analogs of codes in the Johnson scheme, i.e., constant dimension codes. These codes of the Grassmannian ${\mathcal{G}}_q(n,k)$ also form a family of $q$-analogs of block designs and they are called subspace designs. In this paper, we examine one of the last families of $q$-analogs of block designs which was not considered before. This family, called subspace packings, is the $q$-analog of packings, and was considered recently for network coding solution for a family of multicast networks called the generalized combination networks. A subspace packing $t$-$(n,k,\lambda {)}_q$ is a set $\mathcal{S}$ of $k$-subspaces from ${\mathcal{G}}_q(n,k)$ such that each $t$-subspace of ${\mathcal{G}}_q(n,t)$ is contained in at most $\lambda$ elements of $\mathcal{S}$. The goal of this work is to consider the largest size of such subspace packings. We derive a sequence of lower and upper bounds on the maximum size of such packings, analyse these bounds, and identify the important problems for further research in this area.

30 pages, 27 tables, continuation of arXiv:1811.04611, typos corrected