Configuration (geometry) - Constructions of Symmetric Configurations

Constructions of Symmetric Configurations

There are several techniques for constructing configurations, generally starting from known configurations. Some of the simplest of these techniques construct symmetric (pγ) configurations.

Any finite projective plane of order n is an (n2 + n + 1)n + 1 configuration. Let π be a projective plane of order n. Remove from π a point P and all the lines of π which pass through P (but not the points which lie on those lines except for P) and remove a line l not passing through P and all the points that are on line l. The result is a configuration of type (n2 - 1)n. If, in this construction, the line l is chosen to be a line which does pass through P, then the construction results in a configuration of type (n2)n. Since projective planes are known to exist for all orders n which are powers of primes, these constructions provide infinite families of symmetric configurations.

Not all configurations are realizable, for instance, a (437) configuration does not exist. However, Gropp (1990) has provided a construction which shows that for k ≥ 3, a (pk) configuration exists for all p ≥ 2 lk + 1, where lk is the length of an optimal Golomb ruler of order k.

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