5 research outputs found
Improved bounds on sizes of generalized caps in
An -general set in is a set of points such that any subset of
size is in general position. A -general set is often called a capset. In
this paper, we study the maximum size of an -general set in ,
significantly improving previous results. When and we give a
precise estimate, solving a problem raised by Bennett.Comment: Revised version. To appear in SIAM Journal on Discrete Mathematic
How many cards should you lay out in a game of EvenQuads?: A detailed study of 2-caps in AG(n,2)
A 2-cap in the affine geometry is a subset of 4 points in general
position. In this paper we classify all 2-caps in , up to affine
equivalence, for . We also provide structural results for general
. Since the EvenQuads card deck is a model for , as a consequence
we determine the probability that an arbitrary -card layout contains a quad.Comment: 36 pages, 20 figures, 5 table
On 4-general sets in finite projective spaces
A -general set in is a set of points of
spanning the whole and such that no four of them are on a
plane. Such a pointset is said to be complete if it is not contained in a
larger -general set of . In this paper upper and lower
bounds for the size of the largest and the smallest complete -general set in
, respectively, are investigated. Complete -general sets in
, , whose size is close to the theoretical upper
bound are provided. Further results are also presented, including a description
of the complete -general sets in projective spaces of small dimension over
small fields and the construction of a transitive -general set of size in ,