All Questions
7 questions
2
votes
1
answer
185
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How many ways are there to pick $n$ points on the finite affine plane $(\Bbb F_q)^2$ such that no three are collinear?
How many ways are there to pick $n$ points on the finite affine plane $(\Bbb F_q)^2$ such that no three are collinear?
For example, how many ways can we pick $5$ points on $\Bbb F_{32}\times\Bbb F_{32}...
- 960
4
votes
1
answer
237
views
Intersection of the general linear group $\text{GL}_n(\mathbb{F}_q)$ and an affine subspace over a finite field
Let $\mathbb{F}_q$ be a finite field of $q$ elements, let $\text{M}_n(\mathbb{F}_q)$ be the vector space of $n \times n$ matrices over $\mathbb{F}_q$, let $\text{GL}_n(\mathbb{F}_q)$ be the group of $...
- 43
10
votes
1
answer
561
views
How many rich directions does a set in $\mathbb F_p^2$ determine?
$\newcommand{\F}{\mathbb F}$
A subset $P$ of the affine plane $\F_p^2$ is said to determine a direction if there is a line in this direction containing at least two points of $P$.
A set of size $|P|&...
- 23k
9
votes
0
answers
245
views
Almost blocking sets in $\mathbb F_q^2$
$\newcommand{\F}{{\mathbb F}}$
Let $q$ be an odd prime power. A blocking set in the affine plane $\F_q^2$ is a set blocking (meeting) every line.
A union of two non-parallel lines is a blocking set ...
- 23k
4
votes
1
answer
271
views
Square-free sets in $\mathbb F_2^n\oplus\mathbb F_2^n$
A square in $\mathbb F_2^n\oplus\mathbb F_2^n$ is a quadruple of the form
$$ (u,v)+\{(0,0),(0,d),(d,0),(d,d)\},\quad u,v,d\in\mathbb F_2^n,\ d\ne 0. $$
A set $A\subset\mathbb F_2^n\oplus\mathbb F_2^...
- 23k
5
votes
2
answers
139
views
Sets blocking every $2$-flat in $AG(n,2)$
The following may be well-known $-$ but not known to me:
What is the smallest possible size of a set in ${\mathbb F}_2^n$ that blocks every $2$-flat?
Here "blocks" means "have a non-empty ...
- 23k
5
votes
1
answer
459
views
$(n-2)$-blocking sets in $AG(n,2)$
Let's define $k$-blocking set in affine space $AG(n,q)$ a set that meets every coset (translate of subspace) of dimension $k$.
I have seen a lot work related to minimal $(n-1)$-blockings set.
...
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