an:06137922
Zbl 1267.51007
Seelinger, G.; Sissokho, P.; Spence, L.; Vanden Eynden, C.
Partitions of \(V(n, q)\) into 2- and s-dimensional subspaces
EN
J. Comb. Des. 20, No. 11-12, 467-482 (2012).
00310870
2012
j
51E14 51E23 51E10
vector space partition; subspace partition; partition type
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 \(n \geq 14\).
Norbert Knarr (Gie??en)