Subspace Packings – Constructions and Bounds
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The Grassmannian G_q(n,k) is the set of all k-dimensional subspaces of the vector space 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 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,λ)_q is a set S of k-subspaces from G_q(n,k) such that each t-subspace of G_q(n,t) is contained in at most λ elements of 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.
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