3,072 research outputs found
Higgledy-piggledy subspaces and uniform subspace designs
In this article, we investigate collections of `well-spread-out' projective
(and linear) subspaces. Projective -subspaces in
are in `higgledy-piggledy arrangement' if they meet each projective subspace of
co-dimension in a generator set of points. We prove that the set
of higgledy-piggledy -subspaces has to contain more than
elements. We
also prove that has to contain more than
elements if the field is algebraically closed.
An -uniform weak subspace design is a set of linear subspaces
each of rank such that each linear subspace
of rank meets at most among them. This subspace
design is an -uniform strong subspace design if
for of
rank . We prove that if then the dual ()
of an -uniform weak (strong) subspace design of parameter is an
-uniform weak (strong) subspace design of parameter . We show the
connection between uniform weak subspace designs and higgledy-piggledy
subspaces proving that
for
-uniform weak or strong subspace designs in .
We show that the -uniform strong subspace
design constructed by Guruswami and Kopprty (based on multiplicity codes) has
parameter if we consider it as a weak subspace design. We give
some similar constructions of weak and strong subspace designs (and
higgledy-piggledy subspaces) and prove that the lower bound
over algebraically closed field is tight.Comment: 27 pages. Submitted to Designs Codes and Cryptograph
Lines in higgledy-piggledy position
We examine sets of lines in PG(d,F) meeting each hyperplane in a generator
set of points. We prove that such a set has to contain at least 1.5d lines if
the field F has more than 1.5d elements, and at least 2d-1 lines if the field F
is algebraically closed. We show that suitable 2d-1 lines constitute such a set
(if |F| > or = 2d-1), proving that the lower bound is tight over algebraically
closed fields. At last, we will see that the strong (s,A) subspace designs
constructed by Guruswami and Kopparty have better (smaller) parameter A than
one would think at first sight.Comment: 17 page
Evolutionary trees: an integer multicommodity max-flow-min-cut theorem
In biomathematics, the extensions of a leaf-colouration of a binary tree to the whole vertex set with minimum number of colour-changing edges are extensively studied. Our paper generalizes the problem for trees; algorithms and a Menger-type theorem are presented. The LP dual of the problem is a multicommodity flow problem, for which a max-flow-min-cut theorem holds. The problem that we solve is an instance of the NP-hard multiway cut problem
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