Questions tagged [finite-geometry]

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