All Questions
Tagged with finite-geometry reference-request
10 questions
5
votes
1
answer
341
views
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
11
votes
1
answer
269
views
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
3
votes
0
answers
127
views
$\left< 15\right>^7/15$-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
9
votes
2
answers
505
views
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
9
votes
1
answer
399
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Are bipartite Moore graphs Hamiltonian?
This is motivated by a computer-generated conjecture that bipartite distance-regular graphs are hamiltonian. I decided to check the case of Moore graphs first.
The cycles and complete bipartite graphs ...
- 9,501
5
votes
0
answers
89
views
Lines meeting a given set in a unique point
Let $p$ be a fixed prime, and suppose that $S$ is a subset of the affine plane $\mathbb F_p^2$. If $|S|\le p+1$, then by the pigeonhole principle, through any given point $s\in S$ there is a line $L=L(...
- 23k
3
votes
0
answers
100
views
Applications of finite Bolyai-Lobachevsky planes
Google scholar gives more than 200 articles comcerning finite Bolyai-Lobachevsky (BL) planes. Usually they devoted to construction of such objects (axioms may be different).
Are their any ...
- 12.3k
5
votes
1
answer
140
views
Is there literature on finite geometries with ordered lines?
A difference between finite geometries and (e.g.) Euclidean space is that "lines" in finite geometries are unordered subsets of the universe, while "lines" in Euclidean space are ordered subsets of ...
- 1,389
12
votes
2
answers
820
views
Who constructed the projective plane of order $4$ from $K_6$?
I have been trying to hunt down the original reference for the construction of the projective plane of order $4$ from the complete graph on $6$ vertices.
The reference I have at hand are Cameron and ...
- 1,418
14
votes
0
answers
552
views
Who conjectured that a transitive projective plane is Desarguesian?
The only known finite projective plane with a transitive automorphism group is the Desarguesian plane $PG(2,q)$ and it seems likely that there are no others, although this is not (quite) proved.
...
- 12.7k