Constructions of Symmetric 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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