Questions tagged [finite-geometry]
Galois geometry, finite projective and affine spaces, polar spaces, partial geometries, generalized polygons, near polygons, and other finite incidence geometries.
20 questions with no upvoted or accepted answers
14
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Who conjectured that a transitive projective plane is Desarguesian?
The only known finite projective plane with a transitive automorphism group is the Desarguesian plane $PG(2,q)$ and it seems likely that there are no others, although this is not (quite) proved.
...
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9
votes
0
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245
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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
8
votes
0
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The Maximum Number of Lines Contained in the Point Set of a Finite Projective Plane
Consider a finite projective plane of order $q$. Define $f(m)$ to be the maximum number of lines completely contained in any point set of size $m$, where $1 \leq m \leq q^2+q+1$. I would like to ...
user48028
7
votes
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301
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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
5
votes
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What (if anything) is the connection between the Feit-Higman Theorem and the regular plane tilings?
Here are two facts that are superficially similar.
Tiling Theorem: The only regular tilings of $\mathbb{R}^2$ are achieved by triangles, squares, and hexagons.
Feit-Higman Theorem: The only finite ...
- 1,389
5
votes
0
answers
89
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Lines meeting a given set in a unique point
Let $p$ be a fixed prime, and suppose that $S$ is a subset of the affine plane $\mathbb F_p^2$. If $|S|\le p+1$, then by the pigeonhole principle, through any given point $s\in S$ there is a line $L=L(...
- 23k
5
votes
0
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A question on hyperplanes in partial linear spaces and hypergraphs
A partial linear space (or a linear hypergraph) is a point line geometry $(P,L,I)$ where for every pair of points there is at most one line incident with both of them. A hyperplane in a partial linear ...
- 1,197
4
votes
0
answers
143
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Non-Desarguesian finite projective planes with ≤3 (non-collinear) chosen points, and coordinatisation
It is well-known that an arbitrary projective plane can have very different symmetry group to a field plane. In particular, the symmetries are not transitive on the set of fundamental quadrangles. ...
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4
votes
0
answers
250
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Birkhoff – von Neumann for "$k$-stochastic matrices"
Recall that a doubly-stochastic matrix is a square matrix with non-negative elements such the sum of the elements in every row, as well as in every column, is $1$. The set of doubly-stochastic ...
- 23k
4
votes
0
answers
192
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Forcing scalar products to avoid prescribed values
Let $p$ be a prime, and $n\ge 1$ an integer number. Suppose that the (not necessarily distinct) vectors $v_1,\dotsc,v_N \in{\mathbb F}_p^n$ satisfy the following condition:
\begin{gather}
\text{For ...
- 23k
4
votes
0
answers
242
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Sets of spreads in graphs
Let $G$ be a graph. A $k$-spread is a set of cliques of order $k$ which partition the vertex set (so $k|n$, where $n$ is the number of vertices).
A partial $k$-resolution of $G$ is a set of pairwise ...
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3
votes
0
answers
127
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$\left< 15\right>^7/15$-womcode construction
In the article Womcodes constructed with projective geometries Frans Merkx constructed several good wom-codes (write-once memory codes, see How to reuse a "write-once" memory by Rivest & Shamir ...
- 12.3k
3
votes
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answers
100
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Applications of finite Bolyai-Lobachevsky planes
Google scholar gives more than 200 articles comcerning finite Bolyai-Lobachevsky (BL) planes. Usually they devoted to construction of such objects (axioms may be different).
Are their any ...
- 12.3k
3
votes
0
answers
167
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Intersection of two trace equations over finite fields
Let $F_q$ be a finite field with $q$ elements. Let $n$ be an integer and $Tr:F_{q^n} \rightarrow F_q$ the trace function. My question is: For which integer $k$,
$$\{x: Tr(x)=0\}\cap\{x: Tr(x^k)=0\}=\{...
- 123
3
votes
0
answers
195
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Vector spaces over a field of prime order with certain hyperplanes
Let $V$ be a vector space of finite dimentional $d$ over a field of prime order $p$.
For what values of $d$ and $p$, one can find $d+1$ (pairwise distinct) hyperplanes (subspaces of dimension $d-1$) $...
- 3,738
2
votes
0
answers
35
views
Different powers of a primitive root simultaneously lying in a subspace
Let $p$ be a large prime and let $\alpha$ be a root of a primitive quadratic polynomial over $\mathbb{F}_p$. Let $N$ be an integer parameter of size proportional to $p$ and $$V = \{\alpha + b : b \in \...
- 2,283
2
votes
0
answers
59
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Is the finite projective plane stable as an extremal set system?
Let $\Sigma$ be a set of $|\Sigma| = n$ subsets of the universe $[n]$, each of size $k$, with the property that any two of these subsets intersect on at most one element. It is easy to see that the ...
- 1,389
2
votes
0
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86
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A system of homogeneous linear equations
This is the "real-life" (but slightly more technical) version of a question I have asked recently.
For a prime $p>10$, let $\mathcal L_X$, $\mathcal L_Y$, and $\mathcal L_Z$ denote the pencils of ...
- 23k
2
votes
0
answers
337
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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
2
votes
0
answers
251
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A problem in Galois Geometry
Given a prime $p$, out of $N$ vectors of length $p^k$ over $\Bbb F_2$ of Hamming weight $w^{k}$ that are chosen, how many vectors can there be with pairwise Hamming distance at least $2w^{k}$ given ...
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