Suppose a magazine editor wishes to obtain a comparison of 25 automobiles by assessing the opinions of a certain number of test-drivers after each of the test-drivers evaluates 3 of the vehicles. Construct a block design for this comparison. List the values of the parameters $(v,b,r,k,\lambda)$ for the design, and state what each parameter represents.
Balanced Incomplete Block Design for magazine editor
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combinatorics
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0I have removed some unconstructive comments. – 2012-01-31
1 Answers
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Indeed you have the parameters right.
- The number of points $v=25$.
- The number of points in a block $k=3$.
- Every pair of points belongs to a unique block, so $\lambda=1$. (This is somewhat implicit in the question, since it's about BIBDs.)
- The number of blocks containing a given point $r=12$, since there are $24$ other points, and, given any point, each block contains two "other" points.
- The number of blocks $b=100$, which can be deduced from the above.
These parameters can also be achieved. A cyclic Steiner triple system on 25 nodes can be constructed from the starter: $S:=\{\{0,1,6\},\{0,2,10\},\{0,3,12\},\{0,4,11\}\}.$ These triangles are illustrated below:
We can see the each node distance occurs exactly once, so by cyclically rotating these triangles, we generate every edge exactly once.