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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:
 5.1e+15 Finite partial geometries (general), nets, partial spreads 5.1e+24 Spreads and packing problems in finite geometry 5.1e+11 Steiner systems in finite geometry
##### Keywords:
vector space partition; subspace partition; partition type
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##### References:
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