Smallest defining sets of some \(STS(19)\)

We describe an algorithm for finding smallest defining sets of designs. Using this algorithm, we show that the 104 \(STS(19)\) which have automorphism group order at least 9 have smallest defining set sizes in the range 18-23. The numbers of designs with smallest defining sets of 18, 19, 20, 21, 22 and 23 blocks are, respectively, 1, 2, 17, 68, 14 and 2.