| 1. |
- Heden, Olof, et al.
(author)
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Extremal sizes of subspace partitions
- 2012
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In: Designs, Codes and Cryptography. - : Springer Science and Business Media LLC. - 0925-1022 .- 1573-7586. ; 64:3, s. 265-274
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Journal article (peer-reviewed)abstract
- A subspace partition I 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 I . The size of I is the number of its subspaces. Let sigma (q) (n, t) denote the minimum size of a subspace partition of V in which the largest subspace has dimension t, and let rho (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 sigma (q) (n, t) and rho (q) (n, t) for all positive integers n and t. Furthermore, we prove that if n a parts per thousand yen 2t, then the minimum size of a maximal partial t-spread in V(n + t -1, q) is sigma (q) (n, t).
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| 2. |
- Lehmann, Juliane, et al.
(author)
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Some necessary conditions for vector space partitions
- 2012
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In: Discrete Mathematics. - : Elsevier BV. - 0012-365X .- 1872-681X. ; 312:2, s. 351-361
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Journal article (peer-reviewed)abstract
- Some new necessary conditions for the existence of vector space partitions are derived. They are applied to the problem of finding the maximum number of spaces of dimension t in a vector space partition of V(2t, q) that contains m(d) spaces of dimension d, where t/2 < d < t, and also spaces of other dimensions. It is also discussed how this problem is related to maximal partial t-spreads in V (2t, q). We also give a lower bound for the number of spaces in a vector space partition and verify that this bound is tight.
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