**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 (*p*_{γ}) configurations.

Any finite projective plane of order *n* is an (*n*2 + *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 (*n*2 - 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 (*n*2)_{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 (43_{7}) configuration does not exist. However, Gropp (1990) has provided a construction which shows that for *k* ≥ 3, a (*p*_{k}) configuration exists for all *p* ≥ 2 *l*_{k} + 1, where *l*_{k} is the length of an optimal Golomb ruler of order *k*.

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