Questions tagged [finite-geometry]
Galois geometry, finite projective and affine spaces, polar spaces, partial geometries, generalized polygons, near polygons, and other finite incidence geometries.
65 questions
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Incidence matrices of generalized quadrangles
Is there somewhere a database of incidence matrices of generalized quadrangles that one can download?
- 1,197
6
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2
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992
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On the joints problem in finite fields
The original version of the so-called "joints problem" consists of the following:
Let $L$ be a set of lines in $\mathbb{R}^{3}$. Determine the maximum number of "joints" determined by these lines, ...
- 1,197
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463
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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 ...
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3
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1
answer
245
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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 ...
- 1,197
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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 ...
CommunityBot
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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 ...
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4
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2
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383
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Finding the set of all $0$-$1$ vectors in an affine subspace
We are given a $0$-$1$ matrix $A$ with constant row and column sum, and we need to find out if there exists a $0$-$1$ vector in the solution space of $Ax = \mathbf{1}$ over $\mathbb{Q}$ (or $\mathbb{Z}...
CommunityBot
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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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2
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1
answer
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Is inner product preserved only by the stabiliser in a finite reflection group?
Is the following statement true for finite reflection groups?
Let $G$ be a finite reflection group acting on $\mathbb{R}^n$, let $x, y\in \mathbb{R}^n$
and let $z$ be in the orbit of $y$. If $\...
- 25.3k
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Is there a hyperplane avoiding two independent sets?
Let $V$ be a vector space over a field with $5$ elements, $A,B \subseteq V$ independent subsets. Must there be a subspace of $V$ of codimension 1 disjoint from $A \cup B$?
- 1,887
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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 ...
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3
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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\}=\{...
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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 ...
- 13.9k
12
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Who constructed the projective plane of order $4$ from $K_6$?
I have been trying to hunt down the original reference for the construction of the projective plane of order $4$ from the complete graph on $6$ vertices.
The reference I have at hand are Cameron and ...
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3
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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 $d-1$) $...
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