Subspace Packings

Abstract

The Grassmannian ${\mathcal G}_q(n,k)$ is the set of all $k$-dimensional subspaces of the vector space $\mathbb{F}_q^n$. It is well known that codes in the 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 \emph{subspace designs}. The application of subspace codes has motivated extensive work on the $q$-analogs of block 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 \emph{subspace packings} is the $q$-analog of packings. This family of designs was considered recently for network coding solution for a family of multicast networks called the generalized combination networks. A \emph{subspace packing} $t$-$(n,k,\lambda)^m_q$ 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.