# Questions tagged [finite-geometry]

Galois geometry, finite projective and affine spaces, polar spaces, partial geometries, generalized polygons, near polygons, and other finite incidence geometries.

61
questions

**9**

votes

**1**answer

112 views

### Does every $C_4$-free bipartite graph lies in some finite projective plane?

A projective plane $Π$ is a 3-tuple $(P,L,I)$ where $P$ and $L$ are sets, and $I$ is a relation between $P$ and $L$, such that:
For every two elements $p_1$, $p_2\in P$, there exists a unique ...

**15**

votes

**1**answer

274 views

### Geometric interpretation of the exceptional isomorphism $PSp(4,3)=PSU(4,2^2)$

It is well-known that there is an isomorphism between $PSp(4,3)$ (the symplectic group of dimension $4$ over $\mathbb F_3$) and $PSU(4,2^2)$ (the unitary group defined by $4\times4$ unitary matrices ...

**3**

votes

**0**answers

109 views

### $\left< 15\right>^7/15$-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 ...

**2**

votes

**0**answers

29 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 \...

**3**

votes

**1**answer

99 views

### Is there a unique Baer subplane in a finite Desarguesian projective plane?

An order-$m$ subplane of a finite projective plane of order $n$ is called a Baer subplane if $n=m^2$.
It is known that the projective plane $PG(2,q)$ is a Baer subplane of the Desarguesian ...

**6**

votes

**1**answer

197 views

### Moore Graphs and Finite Projective Geometry

In a comment on a blog post from 2009 about the hypothetical Moore graph(s) of degree 57 and girth 5, Gordon Royle offered the following observation (reproduced here in full for the sake of ...

**4**

votes

**0**answers

172 views

### 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 ...

**6**

votes

**1**answer

244 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 ...

**4**

votes

**0**answers

181 views

### 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 ...

**2**

votes

**0**answers

45 views

### 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 ...

**5**

votes

**0**answers

94 views

### 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 ...

**2**

votes

**0**answers

77 views

### 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 ...

**4**

votes

**1**answer

206 views

### Kantor's Singer cycle theorem

I'm trying to understand the proof of Kantor's Singer cycle theorem, which asserts that if $G$ is a subgroup of $\operatorname{GL}(n,q)$ containing a Singer cycle then $\operatorname{GL}(n/s,q^s) \leq ...

**3**

votes

**1**answer

142 views

### A system of linear equations related to the geometry of a finite plane

Let $\mathcal L$ denote the set of all lines in $\mathbb F_p^2$ parallel to one of the lines
$$ X:=\{(x,0)\colon x\in\mathbb F_p \},
\ Y:=\{(0,y)\colon y\in\mathbb F_p \},
\ Z:=\{(z,z)\...

**9**

votes

**1**answer

345 views

### Are bipartite Moore graphs Hamiltonian?

This is motivated by a computer-generated conjecture that bipartite distance-regular graphs are hamiltonian. I decided to check the case of Moore graphs first.
The cycles and complete bipartite ...

**10**

votes

**1**answer

446 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|&...

**2**

votes

**1**answer

149 views

### For which finite projective planes can the incidence structure be written as a circulant matrix?

It is well known that the projective plane of order $2$ can be represented by the circulant matrix $M_2:=circ(x,x,1,x,1,1,1)= \begin{pmatrix}
x&x&1&x&1&1&1\\
1&x&x&...

**9**

votes

**0**answers

228 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 ...

**16**

votes

**1**answer

3k views

### How do I see the equality $57 = 3 \times 19$ geometrically?

Consider the finite field ${\bf F}_p$ and its cubic extension ${\bf F}_{p^3}$. The multiplicative group ${\bf G}_m({\bf F}_{p^3})$ contains the multiplicative group ${\bf G}_m({\bf F}_p) \cong {\bf Z}/...

**5**

votes

**0**answers

81 views

### 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(...

**6**

votes

**1**answer

193 views

### Largest number of points one can pick in finite projective space without getting three on a line

Consider the projectivization $\mathbb P\mathbb F_p^n$ of $\mathbb F_p^n$. How large a set $B \subseteq \mathbb P \mathbb F_p^n$ can I pick so that no three points of $B$ lie on the same line?

**3**

votes

**0**answers

79 views

### 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 ...

**1**

vote

**1**answer

394 views

### Elliptic Curve, characteristic equation of Frobenius endomorphism relation to isogeny

Let E be an elliptic curve over $F_p$. Suppose that its j invarient is not supersingular and that $j\neq 0 $ or 1728.
Then the modular polynomial $\Phi_l(j,T)$ has a zero $\tilde{\jmath} \in \...

**4**

votes

**1**answer

149 views

### Generalization of finite-projective-plane with more than one intersection point

In a finite projective plane, each two points appear together in exactly one line, and each two lines intersect in exactly one point. It is known that, if each line contains $n+1$ points, then the ...

**13**

votes

**1**answer

181 views

### Status of the basis exchange condition for symplectic matroids

Let $J_n := \{1,2,3,\ldots,n,1^*,2^*,\ldots,n^*\}$ with the involution $x\mapsto x^*$ exchanging $i$ and $i^*$ for $1\leq i\leq n$. The following is supposed to be standard, but to avoid any doubt as ...

**11**

votes

**1**answer

265 views

### Is there a well-known notion of orientability for finite geometries?

I'm wondering if the notion of an orientable/non-orientable manifold has any reasonable extension that allows for a similar classification of finite geometries.
For example, the real projective plane ...

**6**

votes

**1**answer

231 views

### Large gaps in Singer's difference sets

This question is related to the question I asked earlier.
For a natural number $n$, a set $D$ of integer numbers is called a $n$-cyclic difference set if each integer number $x\notin n\mathbb Z$ can ...

**14**

votes

**1**answer

341 views

### What is the smallest cardinality of a self-linked set in a finite cyclic group?

A subset $A$ of a group $G$ is defined to be self-linked if $A\cap gA\ne\emptyset$ for all $g\in G$. This happens if and only if $AA^{-1}=G$.
For a finite group $G$ denote by $sl(G)$ the smallest ...

**4**

votes

**1**answer

262 views

### Which cyclic groups admit a difference set?

Problem 1. For which $n$ does the cyclic group $C_n$ admit a difference set $D\subset C_n$, i.e., a set such that each non-unit element $x\in C_n$ can be uniquely written as the difference $x=ab^{-1}$...

**4**

votes

**1**answer

233 views

### Large gaps in Singer planar difference sets?

By a classical result of Singer (1938), for a prime number $p$ the cyclic group $C_n$ of order $n=1+p+p^2$ contains a subset $D$ of cardinality $|D|=1+p$ such that $DD^{-1}=C_n$. Such set $D$ is ...

**5**

votes

**1**answer

104 views

### Is there literature on finite geometries with ordered lines?

A difference between finite geometries and (e.g.) Euclidean space is that "lines" in finite geometries are unordered subsets of the universe, while "lines" in Euclidean space are ordered subsets of ...

**4**

votes

**3**answers

373 views

### Does there exist a finite hyperbolic geometry in which every line contains at least 3 points, but not every line contains the same number of points?

It seems to me that the answer should be yes, but my naive attempts to come up with an example have failed.
Just to clarify, by finite hyperbolic geometry I mean a finite set of points and lines such ...

**12**

votes

**2**answers

772 views

### The exceptional isomorphism between PGL(3,2) and PSL(2,7): geometric origin?

It is well-known there is an isomorphism between $GL(3,2)=PGL(3,2)$, the automorphism group of the Fano plane (i.e. the projective plane over the finite field with two elements), and $PSL(2,7)$, which ...

**6**

votes

**1**answer

194 views

### Maximum number of elements in union of subspaces

Let $V$ be a $m$-dimensional vector space over $\mathbb{F}_q$ and $1<\ell<m-1$. Let $r$ be a positive integer such that $r\ell\leq m$.
QUESTION. What is the maximum number of elements in the ...

**4**

votes

**1**answer

246 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^...

**3**

votes

**1**answer

87 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 ...

**1**

vote

**1**answer

104 views

### Finding a set of disjoint affine subspaces such that their union is equal to a given subset of $\mathbb{F}_2^n$

Suppose I'm given a set of point $S = \{x_1, \dots, x_m \} \subseteq \mathbb{F}_2^n$, and the following task. Find a set of disjoint affine subspaces of $\mathbb{F}_2^n$, $A_1, \dots, A_k$ satisfying ...

**11**

votes

**2**answers

544 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,...

**5**

votes

**1**answer

330 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.
...

**3**

votes

**1**answer

211 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 ...

**8**

votes

**0**answers

143 views

### 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 ...

**6**

votes

**1**answer

342 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 $q(q+1)/2 + (q-1)/2$ ...

**4**

votes

**0**answers

216 views

### 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 ...

**38**

votes

**2**answers

1k views

### How close can one get to the missing finite projective planes?

This question can be interpreted as an instance of the Zarankiewicz problem. Suppose we have an $n\times n$ matrix with entries in $\{0,1\}$ with no $\begin{pmatrix}1 & 1\\ 1& 1\end{pmatrix} $ ...

**2**

votes

**1**answer

168 views

### 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 $\...

**10**

votes

**2**answers

413 views

### 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 $A \cup B$?

**9**

votes

**0**answers

355 views

### Subplanes of Finite Projective Planes

If a finite projective plane $\pi_1$ of order $m$ contains, as a sub plane, a
finite projective plane $\pi_2$ of order $n$, then $m \geq n^2$ with equality holding only in the case of a Baer sub plane....

**4**

votes

**2**answers

356 views

### 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}...

**2**

votes

**0**answers

323 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 ...

**5**

votes

**0**answers

183 views

### 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 ...