Questions tagged [finite-geometry]
Galois geometry, finite projective and affine spaces, polar spaces, partial geometries, generalized polygons, near polygons, and other finite incidence geometries.
65 questions
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Dual of blocking sets in finite geometry
Let $V$ be an $n$-dimensional vector space over the finite field of cardinality $q$ and let $W_1,\ldots,W_m$ be hyperplanes of $V$ such that
$$V=\bigcup_{i=1}^mW_i \,\,\hbox{ and }\,\,0=\bigcap_{i=...
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Which finite projective planes can have a symmetric incidence matrix?
As the title says. Which finite projective planes admit a symmetric incidence matrix?
I am not an expert in the field at all, but I consulted with one. He claimed that $PG(2, \mathbb F_q)$ can always ...
- 237
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Non-Desarguesian finite projective planes with ≤3 (non-collinear) chosen points, and coordinatisation
It is well-known that an arbitrary projective plane can have very different symmetry group to a field plane. In particular, the symmetries are not transitive on the set of fundamental quadrangles. ...
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Arrangement of subspaces over finite fields
I'm trying to find out what is already known about the following setup.
Let $V$ be an $n$-dimensional vector space over a finite field $F_q$ (I'm mostly interested in the case where $q$ is prime), and ...
- 1,072
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Does every $C_4$-free bipartite graph lies in some finite projective plane?
A projective plane $Π$ is a 3-tuple $(P,L,I)$ where $P$ and $L$ are sets, and $I$ is a relation between $P$ and $L$, such that:
For every two elements $p_1$, $p_2\in P$, there exists a unique ...
- 9,501
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Geometric interpretation of the exceptional isomorphism $PSp(4,3)=PSU(4,2^2)$
It is well-known that there is an isomorphism between $PSp(4,3)$ (the symplectic group of dimension $4$ over $\mathbb F_3$) and $PSU(4,2^2)$ (the unitary group defined by $4\times4$ unitary matrices ...
- 9,501
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$\left< 15\right>^7/15$-womcode construction
In the article Womcodes constructed with projective geometries Frans Merkx constructed several good wom-codes (write-once memory codes, see How to reuse a "write-once" memory by Rivest & Shamir ...
- 12.3k
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Different powers of a primitive root simultaneously lying in a subspace
Let $p$ be a large prime and let $\alpha$ be a root of a primitive quadratic polynomial over $\mathbb{F}_p$. Let $N$ be an integer parameter of size proportional to $p$ and $$V = \{\alpha + b : b \in \...
- 2,283
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Is there a unique Baer subplane in a finite Desarguesian projective plane?
An order-$m$ subplane of a finite projective plane of order $n$ is called a Baer subplane if $n=m^2$.
It is known that the projective plane $PG(2,q)$ is a Baer subplane of the Desarguesian ...
- 9,501
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Moore graphs and finite projective geometry
In a comment on a blog post from 2009 about the hypothetical Moore graph(s) of degree 57 and girth 5, Gordon Royle offered the following observation (reproduced here in full for the sake of ...
- 1,645
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Birkhoff – von Neumann for "$k$-stochastic matrices"
Recall that a doubly-stochastic matrix is a square matrix with non-negative elements such the sum of the elements in every row, as well as in every column, is $1$. The set of doubly-stochastic ...
- 23.1k
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Covering the finite plane with lines
This is, essentially, a geometrically rendered version of the question I asked a week ago, with the emphases slightly shifted; it seems more natural and appealing (to me, at least) in this form.
Let ...
- 23.1k
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Forcing scalar products to avoid prescribed values
Let $p$ be a prime, and $n\ge 1$ an integer number. Suppose that the (not necessarily distinct) vectors $v_1,\dotsc,v_N \in{\mathbb F}_p^n$ satisfy the following condition:
\begin{gather}
\text{For ...
- 23.1k
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Is the finite projective plane stable as an extremal set system?
Let $\Sigma$ be a set of $|\Sigma| = n$ subsets of the universe $[n]$, each of size $k$, with the property that any two of these subsets intersect on at most one element. It is easy to see that the ...
- 1,389
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What (if anything) is the connection between the Feit-Higman Theorem and the regular plane tilings?
Here are two facts that are superficially similar.
Tiling Theorem: The only regular tilings of $\mathbb{R}^2$ are achieved by triangles, squares, and hexagons.
Feit-Higman Theorem: The only finite ...
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