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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 ...
user38495's user avatar
  • 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 ...
Seva's user avatar
  • 23k
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|&...
Seva's user avatar
  • 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 ...
Seva's user avatar
  • 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^...
Seva's user avatar
  • 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 ...
Seva's user avatar
  • 23k
11 votes
2 answers
791 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,...
Anurag's user avatar
  • 1,197
5 votes
1 answer
460 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. ...
Ashot's user avatar
  • 337
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 ...
pxdnr's user avatar
  • 133
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 $q(q+1)/2 + (q-1)/2$ ...
Anurag's user avatar
  • 1,197
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 ...
Anurag's user avatar
  • 1,197
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 ...
Turbo's user avatar
  • 13.9k
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 ...
Seva's user avatar
  • 23k