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Partitions of \(V(n, q)\) into 2- and s-dimensional subspaces. (English) Zbl 1267.51007
Let \(V(n,q)\) denote a vector space of dimension \(n\) over the field with \(q\) elements. A vector space partition of \(V(n,q)\) is a collection \(\mathcal P\) of subspaces of \(V(n,q)\) such that each non-zero vector is contained in precisely one element of \(\mathcal P\). A vector space partition is of type \(d_ 1^{x_ 1} \cdots d_ k^{x_ k}\) if it contains precisely \(x_ i\) subspaces of dimension \(d_i\). Let \(s\) and \(n\) be integers with \(s \geq 3\) and \(n \geq 2s\). The authors show that the existence of partitions of \(V(n,q)\) across a suitable range of types \(s^ x 2^ y\) implies the existence of partitions of \(V(n+j,q)\) of essentially all the types \(s^ x 2^ y\) for any integer \(j \geq 1\). They apply this result to construct partitions of \(V(n,2)\) of types \(5^ x 2^ y\) for all \(n \geq 14\).

MSC:
51E14 Finite partial geometries (general), nets, partial spreads
51E23 Spreads and packing problems in finite geometry
51E10 Steiner systems in finite geometry
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