Questions tagged [finite-geometry]
The finite-geometry tag has no usage guidance.
51
questions
0
votes
0answers
29 views
Size of arcs in finite projective planes
How can I see that the size of an oval in a finite projective plane of order $n$ is $n+1$ when the order is odd, or $n+2$ when that order is even?
I have tried to no avail to prove it algebraically, ...
0
votes
2answers
33 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
86 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 ...
0
votes
0answers
25 views
Relation between generalized quadrangles and affine planes?
A (finite) generalized quadrangle (GQ) of order $(s,t)$ is an incidence structure $\mathcal{S}= (P, B, I)$ in which
$P$ and $B$ are disjoint (nonempty) sets of objects called points and lines, ...
1
vote
1answer
63 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
0answers
35 views
Closure of $X$ definition in Combinatorics of finite geometries
In the book Combinatorics of finite geometries by Lynn Margaret Baten, there’s a part about the definition of closure ,which is as follows :
Let $X$ be any set of points of a near-linear space $S=(P,...
0
votes
1answer
95 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 $K$ ...
-3
votes
1answer
39 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=[...
2
votes
2answers
76 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
16 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
39 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
61 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
178 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
50 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 $...
2
votes
0answers
83 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 ...
5
votes
2answers
287 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
67 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
1answer
241 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
92 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
111 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
94 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
151 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
70 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
113 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
71 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
52 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 ...
10
votes
2answers
707 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
265 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
64 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 $\mathbb{F}_p^n$ ...
0
votes
1answer
72 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
63 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 ...
5
votes
1answer
335 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
70 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
101 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
418 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
191 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
293 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
24 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 ...
2
votes
1answer
61 views
Getting ovals from hyperovals
It is said that if we are given a hyperoval we can determine possible ovals by finding the stabilizer of the hyperoval.
As an example take a translation hyperoval $D(2)=\{(1,t,f(t));t\in GF(4)\}\cup\{(...
0
votes
1answer
107 views
Finite geometry - how to determine parallel classes
I try to learn a little about finite geometry and I have now encountered the following exercise:
Exercise:
Construct the affine plane $\mathrm{AP}(\mathbb{Z}_3)$. Determine it's parallel classes and ...
3
votes
2answers
193 views
How to interpret a line equation in 4-point geometry (affine plane of order 2).
I am currently reading "Basic Notions of Algebra" by Igor Shafarevich. In the first chapter example of a coordinatization of 4-point geometry is given.
Set of axioms:
Through any two distinct points ...
3
votes
1answer
66 views
What do $l+p$ and $lp$, where $p$ is a point and $l$ is a line, mean in geometery?
I am looking at a graph theory problem that describes the partite sets of a bipartite as two copies of the $(m+1)$-dimensional vector space over the finite field $\mathbb{F}_{p^n}$ ($p$ is prime and $...
3
votes
3answers
116 views
Points necessary to intersect all lines in finite projective geometry
I'm reading about finite geometries, projective and affine.
I wonder what the smallest set of points is, given a geometry $PG(d,q)$, that intersects all lines. (or hyperplanes.) For example in the ...
1
vote
1answer
95 views
Hitting a line in a $d$ dimensional cube
Let $F$ be a finite field of order $n$, and let $d$ be an integer. A line in $F^d$ is a function $\ell: F \to F^d$ given by $\ell(t) = x + t*h$, where $x,h \in F^d$, $h \neq 0$, and $t*h = (tx_1, \...
1
vote
2answers
285 views
Finite hyperbolic geometry with ideal points
I was browsing "Thinking Geometricly: A Survey in Geometries" by Thomas Q. Sibley, 2015 and on page 388 it mentions a finite hyperbolic geometry of order 3 (3 points per line) consisting of 13 (...
3
votes
2answers
199 views
Finite projective planes
How big a set of points in general position (i.e., no three collinear) can be found in a finite projective plane of order $n$?
I hope the answers won't be too technical, as I know almost nothing ...
3
votes
2answers
82 views
Question about geometry in a finite projective space
I apologize again for a dumb question!
To add some context (though I think it'll largely be unnecessary): suppose $q$ is a prime, $F:= \displaystyle \mathbb{Z}/q\mathbb{Z}$ is a field. I've defined ...
0
votes
0answers
124 views
Generalization of a projective plane?
In the area of finite geometry, a projective plane is an incidence structure of points and lines with the following properties:
Every two points are incident with a unique line
Every two lines are ...
1
vote
1answer
208 views
Intersection of blocks of the symmetric BIBD $PG(d,q)$
The definition of a Balance Incomplete block design $(v,k,\lambda)$-BIBD can be found here.
It is a well known fact (also see the link above) that every two blocks of a symmetric $(v,k,\lambda)$-BIBD ...
0
votes
2answers
154 views
Fano geometry - order
Of any three points situated on a line, there is no more than one which lies between the other two.
I suspect this is not true in Fano geometry, but I am not entirely sure (the confustion stems from ...
