In this article, we investigate collections of `well-spread-out' projective
(and linear) subspaces. Projective k-subspaces in PG(d,F)
are in `higgledy-piggledy arrangement' if they meet each projective subspace of
co-dimension k in a generator set of points. We prove that the set
H of higgledy-piggledy k-subspaces has to contain more than
min∣F∣,∑i=0k⌊i+1d−k+i⌋ elements. We
also prove that H has to contain more than (k+1)⋅(d−k)
elements if the field F is algebraically closed.
An r-uniform weak (s,A) subspace design is a set of linear subspaces
H1,..,HN≤Fm each of rank r such that each linear subspace
W≤Fm of rank s meets at most A among them. This subspace
design is an r-uniform strong (s,A) subspace design if
∑i=1Nrank(Hi∩W)≤A for ∀W≤Fm of
rank s. We prove that if m=r+s then the dual ({H1⊥,...,HN⊥})
of an r-uniform weak (strong) subspace design of parameter (s,A) is an
s-uniform weak (strong) subspace design of parameter (r,A). We show the
connection between uniform weak subspace designs and higgledy-piggledy
subspaces proving that
A≥min∣F∣,∑i=0r−1⌊i+1s+i⌋ for
r-uniform weak or strong (s,A) subspace designs in Fr+s.
We show that the r-uniform strong (s,r⋅s+(2r)) subspace
design constructed by Guruswami and Kopprty (based on multiplicity codes) has
parameter A=r⋅s 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 (k+1)⋅(d−k)+1
over algebraically closed field is tight.Comment: 27 pages. Submitted to Designs Codes and Cryptograph