A beastiary of sets having extremal Sidon constant, or, there must be more than one theorem somewhere here

Abstract

New sets (typically found by computer search) with Sidon constant equal to the square root of their cardinalities are given. For each integer $N$ there are only a finite number of groups of prime order containing $N$-element extreme sets. Some extreme sets appear to fit a pattern; others do not. Various conjectures and questions are given.