On vector space partitions and uniformly resolvable designs |
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Authors: | A D Blinco S I El-Zanati G F Seelinger P A Sissokho L E Spence C Vanden Eynden |
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Institution: | (1) 4520 Mathematics Department, Illinois State University, Normal, IL 61790–4520, USA |
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Abstract: | Let V
n
(q) denote a vector space of dimension n over the field with q elements. A set of subspaces of V
n
(q) is a partition of V
n
(q) if every nonzero vector in V
n
(q) is contained in exactly one subspace in . A uniformly resolvable design is a pairwise balanced design whose blocks can be resolved in such a way that all blocks in a given parallel class have the
same size. A partition of V
n
(q) containing a
i
subspaces of dimension n
i
for 1 ≤ i ≤ k induces a uniformly resolvable design on q
n
points with a
i
parallel classes with block size , 1 ≤ i ≤ k, and also corresponds to a factorization of the complete graph into -factors, 1 ≤ i ≤ k. We present some sufficient and some necessary conditions for the existence of certain vector space partitions. For the partitions
that are shown to exist, we give the corresponding uniformly resolvable designs. We also show that there exist uniformly resolvable
designs on q
n
points where corresponding partitions of V
n
(q) do not exist.
A. D. Blinco—Part of this research was done while the author was visiting Illinois State University. |
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Keywords: | Vector space partitions Uniformly resolvable designs |
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