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Results tagged with finite-fields
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user 34180
A finite field is a field with a finite number of elements. For each prime power $q^k$, there is a unique (up to isomorphism) finite field with $q^k$ elements. Up to isomorphism, these are the only finite fields.
6
votes
1
answer
454
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 $q(q+1)/2 + (q-1)/2$ …
- 223
10
votes
How to recognise that the polynomial method might work
I would like to add some more examples and references for the so called polynomial method that can help us recognise when it can be applied.
From what I understand so far, the polynomial method fall …
CommunityBot
- 1
4
votes
Hypersurface missing just one point
An even further generalisation of this result is the Alon-Füredi theorem, which says that if a polynomial $f$ does not vanish completely on the "grid" $S = S_1 \times \dots \times S_n$ where $S_i$'s a …
- 1,197
5
votes
Accepted
$(n-2)$-blocking sets in $AG(n,2)$
Not much is known for the general case.
Let $m(k, n, q)$ denote the minimum size of an $k$-blocking set in $AG(n, q)$. Trivially we have $m(0, n, q) = q^n$ and $m(n, n, q) = 1$. By Jamison/Brouwer-S …
CommunityBot
- 1
3
votes
Blocking sets in three dimensional finite affine spaces
Here is an improvement of the upper bound which I found in ``The polynomial method in Galois geometries'' by Simeon Ball. See page number 4.
The known constructions are somewhat crude. For exampl …
- 1,197
11
votes
2
answers
781
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
2
votes
Covering all, but $k$ points with affine subspaces
Here are some partial answers to your question.
Let $A = A_1 \times \dots \times A_n \subseteq F^n$ be a finite grid.
Alon and Furedi proved that you need at least $\sum (\# A - 1)$ hyperplanes to …
- 1,197
3
votes
On MDS code property
It has been proved by Simeon Ball that for $k \leq p$, all $[n, k, n-k+1]_q$ codes are Reed-Solomon codes, where $q = p^h$. See Corollary 9.2 in the following paper:
Ball, S. On sets of vectors of a …
- 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 al …
- 1,197