Extremal sizes of subspace partitions |
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Authors: | Olof Heden Juliane Lehmann Esmeralda N?stase Papa Sissokho |
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Institution: | 1. Department of Mathematics, KTH, 100 44, Stockholm, Sweden 2. Department of Mathematics, Bremen University, Bibliothekstrasse 1-MZH, 28359, Bremen, Germany 3. Department of Mathematics and Computer Science, Xavier University, 3800 Victory Parkway, Cincinnati, OH, 45207, USA 4. Mathematics Department, Illinois State University, Normal, IL, 61790, USA
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Abstract: | A subspace partition Π of V?= V(n, q) is a collection of subspaces of V such that each 1-dimensional subspace of V is in exactly one subspace of Π. The size of Π is the number of its subspaces. Let σ q (n, t) denote the minimum size of a subspace partition of V in which the largest subspace has dimension t, and let ρ q (n, t) denote the maximum size of a subspace partition of V in which the smallest subspace has dimension t. In this article, we determine the values of σ q (n, t) and ρ q (n, t) for all positive integers n and t. Furthermore, we prove that if n ≥?2t, then the minimum size of a maximal partial t-spread in V(n +?t ?1, q) is σ q (n, t). |
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