Partitions of finite vector spaces into subspaces

✍ Scribed by S. I. El-Zanati; G. F. Seelinger; P. A. Sissokho; L. E. Spence; C. Vanden Eynden

• Publisher: John Wiley and Sons
• Year: 2008
• Tongue: English
• Weight: 143 KB
• Volume: 16
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.20167

✦ Synopsis

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