# Subspace packings: constructions and bounds

2020-09-01
Etzion, Tuvi
Kurz, Sascha
Otal, Kamil
Özbudak, Ferruh
Grassmannian Gq (n, k) is the set of all k-dimensional subspaces of the vector space Fn q. Kotter 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 Gq (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,.) q is a set S of k-subspaces from Gq (n, k) such that each t-subspace of Gq (n, t) is contained in at most. 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.
DESIGNS CODES AND CRYPTOGRAPHY

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Citation Formats
T. Etzion, S. Kurz, K. Otal, and F. Özbudak, “Subspace packings: constructions and bounds,” DESIGNS CODES AND CRYPTOGRAPHY, pp. 1781–1810, 2020, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/34314.