Encyclopaedia of DesignTheory: A polarity

The following construction gives a square 2-(16,6,2) design:

• The points are the cells of a 4×4 grid.
• For each cell (i,j), the set of 6 cells lying either in row i or in column j but not both is a block; and these are all the blocks.

Two points in the same row, such as (1,1) and (1,2), lie in the two blocks (1,3) and (1,4); similarly for points in the same column; and two points in different rows and columns, such as (1,1) and (2,2), lie in the two blocks (1,2) and (2,1). The design is thus balanced with lambda=2, as claimed.

Now it is clear that if the point (i,j) is incident with the block labelled by (k,l), then the point (k,l) is incident with the block labelled by (i,j). (This happens if and only if i=k or j=l but not both.)

So the correspondence which interchanges each point with the block having the same label is a polarity of the design.

Moreover, this polarity has no absolute points. So there is a corresponding (16,6,2) graph (that is, a strongly regular graph with v=16, k=6, and lambda=mu=2), whose vertices are the cells, joined if they lie in the same row or column. This is the square lattice graph L₂(4).

Remarkably, the design has a different polarity with no points absolute, for which the (16,6,2) graph is the Shrikhande graph; and also a polarity with every point absolute, associated with the Clebsch graph.

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Peter J. Cameron
28 October 2002