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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 \(n \geq 14\).

51E14 Finite partial geometries (general), nets, partial spreads
51E23 Spreads and packing problems in finite geometry
51E10 Steiner systems in finite geometry
Full Text: DOI
[1] André, Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe, Math Zh pp 156– (1954) · Zbl 0056.38503 · doi:10.1007/BF01187370
[2] Beutelspacher, Partial spreads in finite projective spaces and partial designs, Math Zh pp 211– (1975) · Zbl 0297.50018 · doi:10.1007/BF01215286
[3] Blinco, On vector space partitions and uniformly resolvable designs, Des Codes Cryptogr pp 69– (2008) · Zbl 1184.15002 · doi:10.1007/s10623-008-9199-1
[4] Bu, Partitions of a vector space, Discrete Math pp 79– (1980) · Zbl 0445.15004 · doi:10.1016/0012-365X(80)90174-0
[5] Drake, Partial t-spreads and group constructible (s,r,\(\mu\))-nets, J Geometry pp 211– (1979) · Zbl 0428.51004
[6] El-Zanati, Partitions of finite vector spaces into subspaces, J Comb Des pp 329– (2008) · Zbl 1176.05018 · doi:10.1002/jcd.20167
[7] El-Zanati, On partitions of finite vector spaces of low dimension over GF(2), Discrete Math pp 4727– (2009) · Zbl 1269.15001 · doi:10.1016/j.disc.2008.05.044
[8] El-Zanati, Partitions of the 8-dimensional vector subspace over GF(2), J Comb Des pp 462– (2010) · Zbl 1204.51012 · doi:10.1002/jcd.20247
[9] Heden, Partitions of finite abelian groups, Eur J Comb pp 11– (1986) · Zbl 0651.20035 · doi:10.1016/S0195-6698(86)80014-2
[10] Heden, On the length of the tail of a vector space partition, Discrete Math pp 6169– (2009) · Zbl 1235.51013 · doi:10.1016/j.disc.2009.05.026
[11] Heden, Necessary and sufficient conditions for the existence of a class of partitions of a finite vector space, Des Codes Cryptogr pp 69– (2009) · Zbl 1181.51005 · doi:10.1007/s10623-009-9292-0
[12] Heden, Some necessary conditions for vector space partitions, Discrete Math pp 351– (2012)
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