Partitions of a vector space

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.

Section: Another Necessary Conditionmentioning

confidence: 84%

Section: Lemma 12 (Bu [2]) Let N D Be Integers Such That 1 ≤ D ≤ Nmentioning

confidence: 93%

Section: Another Necessary Conditionmentioning

confidence: 94%

“…form a (set) partition of V 8 (2) in which X ⊕ U is a 5-dimensional subspace and each of the other 32 sets consists of 7 nonzero elements. Now define…”

Section: B Proof Of Theorem 61mentioning

confidence: 99%

See 2 more Smart Citations

Partitions of finite vector spaces into subspaces

El-Zanati

Seelinger

Sissokho

et al. 2007

“…(8,2) with (a, b, c) = (13,6,6). We then show in Section C that there is no 4 13 3 6 2 6 -partition of V (8,2).…”

Section: Introductionmentioning

confidence: 97%

Partitions of the 8‐dimensional vector space over GF(2)

El-Zanati

Heden

Seelinger

et al. 2010

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

Partitions of V(n, q) into 2‐ and s‐Dimensional Subspaces

Seelinger¹,

Sissokho²,

Spence³

et al. 2012

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