Partitions of finite vector spaces into subspaces |
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Authors: | S I El‐Zanati G F Seelinger P A Sissokho L E Spence C Vanden Eynden |
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Institution: | Mathematics Department, Illinois State University, Campus Box 4520, Stevenson Hall 313, Normal, IL 61790‐4520 |
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Abstract: | Let Vn(q) denote a vector space of dimension n over the field with q elements. A set of subspaces of Vn(q) is a partition of Vn(q) if every nonzero element of Vn(q) is contained in exactly one element of . Suppose there exists a partition of Vn(q) into xi subspaces of dimension ni, 1 ≤ i ≤ k. Then x1, …, xk satisfy the Diophantine equation . However, not every solution of the Diophantine equation corresponds to a partition of Vn(q). In this article, we show that there exists a partition of Vn(2) into x subspaces of dimension 3 and y subspaces of dimension 2 if and only if 7x + 3y = 2n ? 1 and y ≠ 1. In doing so, we introduce techniques useful in constructing further partitions. We also show that partitions of Vn(q) induce uniformly resolvable designs on qn points. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 329–341, 2008 |
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Keywords: | partitions of finite vector spaces partitions of finite abelian groups uniformly resolvable designs |
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