Subspace Packings : Constructions and Bounds
Tuvi Etzion, Sascha Kurz, Kamil Otal, Ferruh Özbudak · EPub Bayreuth (University of Bayreuth) · 2019
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 $${\mathbb {S}}$$ 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 $${\mathbb {S}}$$ . 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.