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Encyclopaedia of DesignTheory: A BIBD with repeated blocks

This design was discovered by J. H. van Lint, but his description was a bit different.

The 13 points of the design consist of the 12 vertices of a regular icosahedron and one extra point X.

The blocks are of two types.

Type A: for each vertex of the icosahedron, the set of its five neighbours;
Type B: for each pair of opposite edges of the icosahedron, the four vertices on these edges together with the point X.

Clearly every block contains five points. There are twelve blocks of Type A and fifteen of Type B.

The following table shows how many blocks of each type contain a pair of points. V denotes any vertex; (U₁,V₁) any pair of vertices at distance 1; (U₂,V₂) any pair of vertices at distance 2; and (U₃,V₃) any pair of opposite vertices.

Type (X,V) (U₁,V₁) (U₂,V₂) (U₃,V₃)

A 0 2 2 0

B 5 1 1 5

We see that if we take the blocks of Type A with multiplicity 2, and the blocks of Type B with multiplicity 1, then each pair of points is contained in exactly five blocks. So we have a 2-(13,5,5) design, a.k.a. a BIBD with v=13, k=5, b=39, r=15, lambda=5.

Exercise Show that the automorphism group of this design has order 240.

Reference

J. H. van Lint, Block designs with repeated blocks and (b,r,lambda)=1, J. Combinatorial Theory (A) 15 (1973), 288-309.

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Peter J. Cameron
13 December 2002