Partitions of finite vector spaces into subspaces

Abstract

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.

On vector space partitions and uniformly resolvable designs

“…In [5], when q = 2 and n = dim(V ) ≤ 7, we found all solutions to the Diophantine equation (1) with n 1 > · · · > n k for which there exists a partition of V (Heden [8] had settled the n = 6 case in 1986). In [6], we showed that if n > 2, and x and y are nonnegative integers satisfying x(2 3 − 1) + y(2 2 − 1) = 2 n − 1, then there exists a partition of V n (2) …”

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

“…Heden [11] gave lower bounds for the number of subspaces of least dimension in a given partition of V (n, q). The 4 6 3 21 2 6 -partition of V (8,2) in Example 17 shows that the bound in Theorem 1 (iii) from [11] is tight.…”

“…(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).…”

“…. 1 a 1 -partitions of V (n, q) for n ≤ 7 and q = 2 (see [5]), and the 3 a 2 b -partitions of V (n, 2) for all n ≥ 2 (see [6]). …”

“…For instance, the partitions in this article establish the tightness of bounds, obtained by Heden [11] (see Remark 20). Moreover, the constructions of vector space partitions of V (n, q) for small n provide base cases for recursive constructions with larger values of n (see [6,9]). This is similar to the situation in t-(v, k, ) designs where the designs with small parameters provide the building blocks for recursive constructions (e.g.…”