# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...