Let n ≥ 3 be an integer, let V n (2) denote the vector space of dimension n over G F(2), and let c be the least residue of n modulo 3. We prove that the maximum number of 3-dimensional subspaces in V n (2) with pairwise intersection {0} is 2 n −2 c 7 − c for n ≥ 8 and c = 2. (The cases c = 0 and c = 1 have already been settled.) We then use our results to construct new optimal orthogonal arrays and (s, k, λ)-nets.

Let V n (q) denote a vector space of dimension n over the field with q elements. A set 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 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(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 7x + 3y = 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.

Let V n (q) denote a vector space of dimension n over the field with q elements. A set P 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 P. 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 q n i , 1 ≤ i ≤ k, and also corresponds to a factorization of the complete graph K q n into a i K q n i -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.

Let V = V(n,q) denote the vector space of dimension n over GF(q). A set of subspaces of V is called a partition of V if every nonzero vector in V is contained in exactly one subspace of V. Given a partition P of V with exactly a i subspaces of dimension i for 1 ≤ i ≤ n, we have n i=1 a i (q i −1) = q n −1, and we call the n-tuple (a n , a n-1 ,...,a 1 ) the type of P. In this article we identify all 8-tuples (a 8 , a 7 ,...,a 2 , 0) that are the types of partitions of V(8,2). q

Let V=V(n,q) denote a vector space of dimension n over the field with q elements. A set scriptP of subspaces of V is a (vector space) partition of V if every nonzero element of V is contained in exactly one subspace in scriptP. Suppose that scriptP is a partition of V with xi subspaces of dimension di for 1≤i≤k. Then we call d1x1...dkxk the type of the partition scriptP. Which possible types correspond to actual partitions is in general an open question. We prove that for any odd integer s≥3 and for any integer n≥2s, the existence of partitions of V(n,q) across a suitable range of types sx2y guarantees the existence of partitions of V(n+js,q) of essentially all the types sx2y for any integer j≥1. We then apply this result to construct new classes of partitions of V. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 467‐482, 2012