All Questions
13 questions
11
votes
2
answers
788
views
Blocking sets in three dimensional finite affine spaces
What is the smallest possible size of a set of points in $\mathbb{F}_q^3$ which intersects (blocks) every line?
Clearly the union of three affine hyperplanes that intersect in a singleton, say $x = 0,...
- 1,197
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
7
votes
0
answers
300
views
Arrangement of subspaces over finite fields
I'm trying to find out what is already known about the following setup.
Let $V$ be an $n$-dimensional vector space over a finite field $F_q$ (I'm mostly interested in the case where $q$ is prime), and ...
- 1,062
6
votes
1
answer
297
views
Covering the finite plane with lines
This is, essentially, a geometrically rendered version of the question I asked a week ago, with the emphases slightly shifted; it seems more natural and appealing (to me, at least) in this form.
Let ...
- 23k
6
votes
1
answer
458
views
Applications of small Kakeya sets over finite fields
It was proved by Dvir that a Kakeya set in $\mathbb{F}_q^n$ has size at least $q^n/n!$, a bound which was later improved to $q^n/2^n$.
For $n = 2$ and $q$ odd the exact bound is $q(q+1)/2 + (q-1)/2$ ...
- 1,197
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.
...
- 337
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
4
votes
1
answer
463
views
Covering all, but $k$ points with affine subspaces
For non-negative integer $d\le n$ and $k\le 2^n$, how many affine subspaces of co-dimension $d$ are needed to cover all, but exactly $k$ elements of the vector space ${\mathbb F}_2^n$, and what are ...
- 23k
3
votes
3
answers
611
views
On MDS code property
Is there a code that is Maximum Distance Separable and not isomorphic to Reed Solomon Codes? When is a MDS code isomorphic to Reed Solomon Code?
Is there an easy test? If so, could someone provide ...
- 13.9k
3
votes
1
answer
245
views
An upper bound on the number of sets of parallel lines covering points in a finite plane?
Let $\mathbb{F}$ be a finite field of characteristic $2$. Let $L_m$ denote the set of lines in $\mathbb{F}^2$ with slope $m\in\mathbb{F}$, that is, all parallel lines of the form $y=mx+b$. Consider a ...
- 133
2
votes
0
answers
337
views
Enumerating certain types of permutation polynomials
Given a prime power $q$, I would like to enumerate (preferably up to isomorphism*) all the permutation polynomials $f(x)$ on $K = GF(q^3)$ satisfying the following conditions:
$f(ax) = af(x)$ for all ...
- 1,197