On the length of the tail of a vector space partition
2009 (English)In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 309, no 21, 6169-6180 p.Article in journal (Refereed) Published
A vector space partition P of a finite dimensional vector space V = V(n, q) of dimension n over a finite field with q elements, is a collection of subspaces U-1, U-2, ..., U-t with the property that every non zero vector of V is contained in exactly one of these subspaces. The tail of P consists of the subspaces of least dimension d(1) in P, and the length n(1) of the tail is the number of subspaces in the tail. Let d(2) denote the second least dimension in P. Two cases are considered: the integer q(d2-d1) does not divide respective divides n(1). In the first case it is proved that if 2d(1) > d(2) then n(1) >= q(d1) + 1 and if 2d(1) <= d(2) then either n(1) = (q(d2) - 1)/(q(d1) - 1) or n(1) > 2q(d2-d1). These lower bounds are shown to be tight and the elements in the subspaces in tails of minimal length will constitute a subspace of V of dimension 2d(1) respectively d(2). In case q(d2-d1) divides n(1) it is shown that if d(2) < 2d(1) then n(1) >= q(d2) - q(d1) + q(d2-d1) and if 2d(1) <= d(2) then n(1) <= qd(2.) The last bound is also shown to be tight. The results considerably improve earlier found lower bounds on the length of the tail.
Place, publisher, year, edition, pages
2009. Vol. 309, no 21, 6169-6180 p.
Vector space partitions, Perfect codes
IdentifiersURN: urn:nbn:se:kth:diva-18984DOI: 10.1016/j.disc.2009.05.026ISI: 000272099900003ScopusID: 2-s2.0-70350566665OAI: oai:DiVA.org:kth-18984DiVA: diva2:337031
QC 201005252010-08-052010-08-052010-12-23Bibliographically approved