| 1. |
- Heden, Olof, et al.
(författare)
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Extremal sizes of subspace partitions
- 2012
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Ingår i: Designs, Codes and Cryptography. - : Springer Science and Business Media LLC. - 0925-1022 .- 1573-7586. ; 64:3, s. 265-274
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Tidskriftsartikel (refereegranskat)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. |
- Heden, Olof, et al.
(författare)
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On the existence of a (2,3)-spread in V(7,2)
- 2016
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Ingår i: Ars combinatoria. - : Charles Babbage Research Centre. - 0381-7032. ; 124, s. 161-164
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Tidskriftsartikel (refereegranskat)abstract
- An (s, t)-spread in a finite vector space V = V (n, q) is a collection F of t-dimensional subspaces of V with the property that every s-dimensional subspace of V is contained in exactly one member of F. It is remarkable that no (s, t)-spreads has been found yet, except in the case s = 1. In this note, the concept a-point to a (2,3)-spread F in V = V(7, 2) is introduced. A classical result of Thomas, applied to the vector space V, states that all points of V cannot be alpha-points to a given (2, 3)-spread.F. in V. In this note, we strengthened this result by proving that every 6-dimensional subspace of V must contain at least one point that is not an a-point to a given (2, 3)-spread of V.
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