Questions tagged [finite-geometry]
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63
questions
0
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1answer
37 views
In a finite affine space $(\mathbb{Z}/p)^3$, has at most $p^2 +p +1$ lines passing through a point.
It is a simple question not much backgrond information is required. However, I do need ton understand the proof of why.
Prove the finite affine space $(\mathbb{Z}/p)^3$, has only $p^2 +p +1$ lines ...
1
vote
1answer
27 views
if $|\mathcal{H}_1 \cap \mathcal{H}_2|> n+1$ then $\mathcal{H}_1 =\mathcal{H}_2$ for hyperovals of an even order plane
Let $\Pi$ be a projective plane of even order $2n$. Let $\mathcal{H}_1,\mathcal{H}_2$ be two hyperovals in $\Pi$. Show that if $|\mathcal{H}_1 \cap \mathcal{H}_2|> n+1$ then $\mathcal{H}_1 =\...
1
vote
1answer
45 views
line at infinity intersected with a line from points of the parabola
Suppose we are looking at a parabola as a conic in $PG(2,\mathbb{K})$ with the line at infinity denoted $\ell_\infty$. I am still working on a previous problem I posted here.
Suppose we have two ...
7
votes
1answer
152 views
groups of conics
Let $\mathcal{C}$ be a conic on the projective plane $PG(2,\mathbb{F})$ where $\mathrm{char}\mathbb{F}\neq 2$. Let $\ell$ be a line and let $N$ be a point on $\mathcal{C}\setminus \ell$. For $A,B\in \...
2
votes
0answers
23 views
What is known about collineation groups in general?
Is there any reference for collineation groups of finite projective planes?
By collineation of a projective plane $\pi=(P,L,i)$ I mean a bijective function $f:P\to P$ which preserves collinearity.
Are ...
2
votes
0answers
38 views
Sharply 2-transitive subgroups of the affine group $AGL_d(F)$
I'm attempting exercise 2.8.14 of Dixon & Mortimer. It asks you to show that for $d\geq 1$, $AGL_d(F)$ contains a sharply 2-transitive subgroup $H$. For $d=1$ this is easy, since $AGL_1(F)$ itself ...
-2
votes
1answer
45 views
Affine Plane of Order $5$ Picture?
I am unable to construct an affine plane of order $5$. I currently am able to construct an affine plane for orders $3$ and $4$ but I am not able to figure out the construction of an affine plane of ...
1
vote
2answers
50 views
Show that the harmonic conjugates are collinear - Menelaus and Ceva's Theorems with homogeneous coordinates
Suppose we have a triangle $\triangle ABC$ such that $D$ is an arbitrary point on $BC$, $E$ is an arbitrary point on $AB$ and $F$ is an arbitrary point on $AC$. Let $G$ be the harmonic conjugate of $E$...
2
votes
1answer
101 views
About Fano plane symmetries
I have been looking for realizations of order $21$ metacyclic group. I asked about this yesterday and learned some very good information about it Realization of the metacyclic group of order 21
I was ...
2
votes
1answer
271 views
Projective plane - Blocking set relative to line from an oval
Let $\pi=PG(2,n)$ be a desarguesian projective plane of odd order $n$ and $\mathcal{L}$ be a subcollection of lines of $\pi$. Then a set $B$ of points of $\pi$ is called a blocking set relative to $\...
0
votes
1answer
116 views
How many points belong to a line in a finite field?
This is the continuation of this question: What is the size of the set of lines in a finite field $\mathbb{F}_q$ of order $q$, where $q$ is a prime power?
Now I need to show that exactly $q$ points ...
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votes
1answer
43 views
What is the size of the set of lines in a finite field $\mathbb{F}_q$ of order $q$, where $q$ is a prime power?
Let $\mathbb{F}_q$ be a finite field of order $q$, where $q$ is a prime power. Show that the set $\mathcal{L} = \{\{(x,y) : ax + by = c\} : a,b,c \in \mathbb{F}_q \text{, and at least one of a,b is ...
2
votes
0answers
31 views
Dual concept to linear error-correcting codes
Let $d$ and $n$ be natural numbers, and let $\mathbb F_q$ be a finite field. For any injection $\sigma: \{1, \dots, d\} \to \{1, \dots, n\}$, we can define a corresponding linear transformation $T_\...
1
vote
2answers
35 views
On some special 5-tuples in projective space $PG(3,2)$
Projective space $PG(3,2)$ has nice 5-tuples of points like $(1,0,0,0)$, $(0,1,0,0)$, $(0,0,1,0)$, $(0,0,0,1)$, $(1,1,1,1).$ This 5-tuple is "nice" because these points with their pairwise sums cover ...
1
vote
1answer
31 views
Lower bound on line blocking sets in the finite projective plane
I am looking at Bruen and Rotschild's article on lower bounds for line blocking sets in the finite projective plane of order $q$. In their proof on page 8, they use the summations
\begin{align} \sum_{...
0
votes
2answers
121 views
Intuition behind lines and points in the projective plane
I've just started learning about projective plane and have trouble understand how to visualize the points and lines in the plane.
Specifically, for this lemma below, when its says "all points of $l$ ...
4
votes
1answer
128 views
Conjugacy of Singer cyclic groups in $\mathrm{P\Gamma L}$
Motivation
This is kind of a follow-up to this question on conjugacy of Singer cyclic groups in GL.
The "original" definition of a Singer cycle is not in the GL, but the following slightly different ...
1
vote
1answer
137 views
Projective Plane of order n
Let $P=(S,L)$ be a projective plane of order $n$. Let $K$ be a nonempty subset of $S$ with the property that no three points belonging to $K$ are collinear. Prove that $|K|$ is less than or equal to $...
0
votes
1answer
359 views
Understanding Witt's Theorem
I just began to learn something about classical polar spaces and now, I'm trying to understand three implications of Witt's theorem.
Let $V$ be an $m$-dimensional vector space over a field $K$ ...
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votes
1answer
50 views
Rank of a matrix constructed using the codewords of a linear block code.
Suppose that $[n, k,d]_2$ represents a linear block code. Then we have $2^k$ different codewords. Suppose that $c_1, c_2,....,c_{2^k}$ represents different code vectors. Define a matrix $A$ as,
$$A=[...
1
vote
2answers
92 views
Binary matrix with fixed inner product.
Suppose that $m,\ n$ are two positive integers such that $m<<n$. Let $a,\ b,\ c$ be the three positive integers such that $a\leq b < c$. Consider a binary matrix $A\in \{0,1\}^{m\times n}$, ...
1
vote
0answers
28 views
Signed incidence structures
I've been trying to understand GraphQL queries (e.g. Wikidata) with Formal Concept Analysis but my first stumbling block is that most between-object relations at least in the Wikidata ontology are ...
1
vote
1answer
68 views
Projective/ Finite Geometric Basics!
I'm taking intro to coding theory and am having some trouble understanding the basics of Projective Geometry, since our text does not give it much discussion. Namely, if PG(r-1,q) is the set of all ...
4
votes
0answers
73 views
Translating and inflating a set of $k$-dimensional subspaces of $\mathbb F_p^n$ to form a cover by affine hyperplanes?
Fix a prime number $p$ and consider the affine space $V = \mathbb F_p^n$. Let $k < n/2$. Consider subspaces $V_1, \ldots, V_n \subseteq V$ of dimension $k$, and take $v_i \notin V_i$. Do there ...
5
votes
1answer
580 views
Projective space definitions
My questions are as follows:
Are all these different definitions of projective space equivalent? For example, Bezout's theorem holds under all 4 definitions (with an appropriate change in ...
2
votes
2answers
77 views
no three points in a line on $\mathbb{Z}_p^2$
in finite geometry, the $\mathbb{Z}_p$-plane is $\mathbb{Z}_p^2$, I have proved there are exactly $p+1$ lines pass through a given point in $\mathbb{Z}_p^2$, and by this conclusion there are at most $...
1
vote
0answers
129 views
Is a perfect game of Set always possible?
For anyone not familiar with the game of Set, I'll refer you to the description on this question.
My question is this: The game ends when there are no more cards remaining in the deck and there are ...
6
votes
2answers
529 views
Primitive roots of unity occuring as eigenvalues of a product
I am currently trying to understand the proof of Benson's Lemma (1.9.1) in Generalized Quadrangles by Payne and Thas.
Background
We have two $k × k$ matrices $Q$ and $M$. We want to determine some ...
4
votes
1answer
97 views
Understanding Generalised Quadrangles
I have a project to do on Generalised Quadrangles, specifically GQ(2,2). The project needs to have information about the construction of GQ(2,2), to prove this construction meets the conditions of a ...
2
votes
2answers
461 views
Finite Incidence Geometry Questions
Definition of finite incidence geometry given: Consists of a finite set $P$ of points and a set of nonempty subsets of $P$ called lines, that satisfy the axioms:
(F1) Two points determine exactly one ...
1
vote
1answer
113 views
Bound of the size of point set in projective plane.
For those familiar with combinatorial design, a projective plane is a $(q^2+q+1, q+1, 1)$ design.
Geometrically, a projective plane of order $q$ is a set of points and lines through these points. ...
2
votes
1answer
173 views
Symplectic form and quadratic form
Let $W = \mathbb{F}_{2^{m + 1}} \oplus \mathbb{F}_{2^{m + 1}}$ be a $2(m+1)$-dimensional vector space over $\mathbb{Z}_2$ equipped with a symplectic form $ \langle \cdot , \cdot \rangle : W \times W \...
1
vote
2answers
108 views
Subspaces and intersections
Stuck in step 3! See my sketch of proof below the theorem I am trying to prove:
Any other approach is welcome!
Theorem: Let $W$ be a k-dimensional subspace of $V$. The number of ${k'}$-dimensional ...
3
votes
1answer
184 views
Hairy Points in Infinite Graphs (and Peano Continua)
I may have to throw this over to overflow, but I figured I would try here, first. This is a question in continuum theory, but it reduces to a combinatorial/graph-theoretical question. I am highly ...
0
votes
1answer
82 views
affine vector space
In an article it is written that:
"A well known result from the theory of Boolean functions is that if the algebraic degree of a Boolean functions is less than d, then the sum over the outputs of ...
3
votes
0answers
127 views
Uniqueness of $S(3,4,10)$ Steiner system
I don't understand one step in the proof of Theorem 6.3B in the book "Permutation Groups" by J.D.Dixon-M.Mortimer.
Let 00, 01, ..., 22 be the points of affine geometry $AG_2(3)$. Then there are ...
0
votes
0answers
130 views
algebraic properties of finite affine planes
An affine plane of order $n$ has $n^2$ points, $n$ points per line, $n+1$ lines intersecting each point, every two points on one line, and $n^2+n$ lines. The plane can be partitioned into $n+1$ ...
4
votes
1answer
54 views
Construct a large set such that any k+l elements span the vectorspace
Let $\mathbb{F}_q^k$ be a $k$-dimensional vectorspace over a finite field, and let $l \ge 0$ be an integer. The question is how to construct a (maximally) large set $A \subset \mathbb{F}_q^k$ such ...
11
votes
2answers
1k views
How to imagine vector spaces (and projective spaces) over a finite field
So I have been learning about projective spaces for the last few hours, and I think I understand the basics pretty well, but there is an exercise, which I do not know how to solve at all. It comes ...
1
vote
3answers
413 views
Projective Plane for F3
I have to calculate the points of a projective Plane on $\mathbb{Z}_3^2$. I thing I understood the way how to do this for the Fano-Plane but I am not sure how to do this here because I have more than ...
2
votes
1answer
84 views
Incidence structures for vector spaces over finite fields
A vector space of the form $\mathbb{F}_p^n$, where $\mathbb{F}_p$ is the finite field of prime order $p$, can be endowed with an incidence structure, i.e. a set of points (here just $\mathbb{F}_p^n$ ...
1
vote
1answer
107 views
Projective space over $\mathbb{F}^{n+1}_q$
Consider the projective space $P(\mathbb{F}^{n+1}_q)$, the projective space constructed over $\mathbb{F}^{n+1}_q$, where $q$ is prime and $n \in \mathbb{N}$.
How many points does it have? And how ...
1
vote
1answer
76 views
Strongly regular graph over subspaces
Let $\Gamma = (V,E)$ be the graph with $V = \{ U \leq \mathbb{F}_q^4 \, | \, \dim(U) = 2\}$ and $E = \{ \{U_1,U_2\} \, | \, \dim(U_1 \cap U_2) = 1 \}$.
Show that $\Gamma$ is a strongly regular graph ...
6
votes
1answer
617 views
Connection between linearly independent vectors and projective points in general position
I'm trying to understand the connection between the notions of linear independence and general position. I have no background in geometry, so first I'll start with what I know and then I'll pose ...
3
votes
1answer
75 views
Difference Sets and Hyperovals
Maschietti's theorem is as follows:
The $q+2$ set $D(x^k)$ is a hyperoval iff $D_k^*$ is a $(q-1,q/2-1,q/4-1)$ difference set in $GF(q)^*$. Where $q=2^d$, $2\leq q-2$ and
$D_k=\{x+x^k\|x\in GF(q)\}$...
2
votes
1answer
178 views
Do the last three remaining cards in a game of Set always form a set?
Question (brief introduction to the game Set is given after the question)
When a game of Set gets to a point that there are only three cards left on the table, and all other cards were already ...
5
votes
2answers
482 views
How large can a set of pairwise disjoint 2-(7,3,1) designs (Fano planes) be?
As wikipedia defines well, the Fano plane is a small symmetric block design, specifically a 2-(7,3,1)-design. The points of the design are the points of the plane, and the blocks of the design are the ...
1
vote
1answer
267 views
How many Fano Planes Can We Build with the Numbers from $1$ to $35$
The Fano plane is the finite projective plane of order 2, having the smallest possible number of points and lines, 7 each, with 3 points on every line and 3 lines through every point.
Assume that ...
1
vote
1answer
386 views
How many non-isomorphic Fano planes exist?
The Fano plane is the finite projective plane of order 2, having the smallest possible number of points and lines, 7 each, with 3 points on every line and 3 lines through every point.
So, What i want ...
1
vote
1answer
26 views
finding equivalent hyperovals
If $H=D(x^k)$ is a hyperoval, then $D(x^t)$ is a hyperoval equvalent to $H$ for $t=1/k$, $1-k$, $1/(1-k)$, $k/(1-k)$ and $(k-1)/k$.
If I consider the Segre Hyperoval $D(x^6)$ with $q = 32 = 2^5$, how ...
