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Partitions of finite vector spaces into subspaces
Author(s) -
ElZanati S. I.,
Seelinger G. F.,
Sissokho P. A.,
Spence L. E.,
Vanden Eynden C.
Publication year - 2008
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.20167
Subject(s) - linear subspace , mathematics , partition (number theory) , combinatorics , diophantine equation , dimension (graph theory) , vector space , finite field , discrete mathematics , pure mathematics
Abstract Let V n (q) denote a vector space of dimension n over the field with q elements. A set ${\cal P}$ of subspaces of V n (q) is a partition of V n (q) if every nonzero element of V n (q) is contained in exactly one element of ${\cal P}$ . Suppose there exists a partition of V n (q) into x i subspaces of dimension n i , 1 ≤ i ≤ k . Then x 1 , …, x k satisfy the Diophantine equation $\sum_{i=1}^{k}{(q^{n_i}-1)x_i}=q^n-1$ . However, not every solution of the Diophantine equation corresponds to a partition of V n (q). In this article, we show that there exists a partition of V n (2) into x subspaces of dimension 3 and y subspaces of dimension 2 if and only if 7 x + 3 y = 2 n − 1 and y ≠ 1. In doing so, we introduce techniques useful in constructing further partitions. We also show that partitions of V n (q) induce uniformly resolvable designs on q n points. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 329–341, 2008