138 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
On the structure of the directions not determined by a large affine point set
Given a point set in an -dimensional affine space of size
, we obtain information on the structure of the set of
directions that are not determined by , and we describe an application in
the theory of partial ovoids of certain partial geometries
An extension of the direction problem
AbstractLet U be a point set in the n-dimensional affine space AG(n,q) over the finite field of q elements and 0≤k≤n−2. In this paper we extend the definition of directions determined by U: a k-dimensional subspace Sk at infinity is determined by U if there is an affine (k+1)-dimensional subspace Tk+1 through Sk such that U∩Tk+1 spans Tk+1. We examine the extremal case |U|=qn−1, and classify point sets not determining every k-subspace in certain cases
A finite word poset : In honor of Aviezri Fraenkel on the occasion of his 70th birthday
Our word posets have �nite words of bounded length as their elements, with
the words composed from a �nite alphabet. Their partial ordering follows from the
inclusion of a word as a subsequence of another word. The elemental combinatorial
properties of such posets are established. Their automorphism groups are determined
(along with similar result for the word poset studied by Burosch, Frank and
R¨ohl [4]) and a BLYM inequality is veri�ed (via the normalized matching property)
The number of directions determined by less than q points
In this article we prove a theorem about the number of direc-
tions determined by less then
q
affine points, similar to the result of
Blokhuis et. al. [3] on the number of directions determined by
q
affine
points
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