Questions tagged [finite-geometry]

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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 ...
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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 =\...
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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
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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 \...
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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 ...
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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 ...
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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 ...
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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 $E$...
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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
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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 $\...
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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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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 ...
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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_\...
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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 ...
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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_{...
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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 $l$ ...
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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 ...
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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 $...
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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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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=[...
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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 $A\in \{0,1\}^{m\times n}$, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $...
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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 ...
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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 ...
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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 ...
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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 ...
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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
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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 \...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $n+1$ ...
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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 ...
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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 ...
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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 ...
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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 $\mathbb{F}_p^n$ ...
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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 ...
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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 ...
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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 ...
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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 $D_k=\{x+x^k\|x\in GF(q)\}$...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...