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Results tagged with combinatorial-designs
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user 1266
Design theory is the subfield of combinatorics concerning the existence and construction of highly symmetric arrangements. Finite projective planes, latin squares, and Steiner triple systems are examples of designs.
4
votes
Accepted
Does there exist a finite hyperbolic geometry in which every line contains at least 3 points...
Take a 2-$(v,4,1)$ design on $v$ points and delete one block, along with the four points on it. In the original system each point is on exactly $(v-1)/3$ blocks, so if we assume $v\ge25$ the geometry …
- 12.1k
5
votes
Self-complementary block designs
These are the parameters for Hadamard 2-designs. Let $H$ be an $n\times n$ Hadamard matrix, where $n=4m$, normalized so that all entries in the first row are equal to 1. Each row apart from the first …
- 12.1k
1
vote
Accepted
Vector version of balanced incomplete block designs
The Gram matrix of your vectors is equal to $(1-\lambda)I +\lambda J$
(where $J$ is the all-ones matrix). If $\lambda\ne -1/(v-1)$, this matrix is invertible, whence your set of vectors is linearly in …
- 12.1k
8
votes
What are the major open problems in design theory nowaday?
Fix $\lambda>1$. Are there infinitely many symmetric $(v,k,\lambda)$ designs?
(The Hadamard conjectures would be at the top of my list though.)
- 12.1k
4
votes
Constructing Steiner Triple Systems Algorithmically
One standard algorithm for constructing Steiner triple systems is the "hill climbing" procedure. You will find it described in "Combinatorial algorithms: generation, enumeration, and search" by Kreher …
- 12.1k