RESEARCH REPORT

Smallest defining sets of some STS(19)

Authors: Peter Adams, Abdollah Khodkar and Colin Ramsay
Centre for Discrete Mathematics and Computing
The University of Queensland
Queensland 4072, Australia.
Department of Mathematics and Department of Computer Science and Electrical Engineering

In [1], we consider the 104 Steiner triple systems on 19 points which have automorphism group order at least 9, and show that they all have smallest defining set sizes in the range 18 to 23, as summarised in the following table.

Here we give a smallest defining set for each design. Note that the element-set for the designs is {0, 1, 2, 3,..., 18}.
In the following table |SDS| stands for the number of blocks in the smallest defining set and o(G) stands for the order of the automorphism group of the design.

[1] P. Adams, A. Khodkar and C. Ramsay, ``Smallest defining sets of some STS(19)'', (preprint).

Design 19.1, o(G)=432, |SDS|=22

{0, 1, 3}, {0, 2, 4}, {0, 9, 13}, {0, 16, 17}, {1, 2, 5}, {1, 4, 7}, {1, 10, 14}, {2, 3, 6}, {2, 7, 12}, {3, 5, 7}, {3, 15, 16}, {4, 11, 16}, {4, 12, 15}, {5, 14, 18}, {6, 9, 16}, {7, 9, 17}, {7, 13, 16}, {8, 9, 18}, {8, 10, 17}, {8, 11, 15}, {8, 12, 16}, {9, 10, 11}.

Design 19.2, o(G)=108, |SDS|=23

{0, 1, 3}, {0, 2, 4}, {0, 16, 18}, {1, 6, 8}, {1, 10, 14}, {1, 11, 12}, {1, 17, 18}, {2, 7, 9}, {2, 11, 18}, {2, 13, 15}, {3, 10, 12}, {4, 9, 15}, {4, 14, 18}, {5, 10, 15}, {5, 13, 14}, {6, 10, 16}, {6, 12, 14}, {6, 13, 18}, {7, 8, 11}, {7, 12, 15}, {8, 9, 18}, {8, 12, 16}, {9, 10, 11}.

Design 19.3, o(G)=171, |SDS|=21

{0, 1, 3}, {0, 5, 9}, {0, 6, 11}, {0, 17, 18}, {1, 10, 11}, {1, 14, 15}, {2, 7, 12}, {2, 8, 18}, {2, 10, 14}, {2, 13, 15}, {3, 4, 8}, {3, 7, 14}, {3, 11, 16}, {4, 5, 11}, {4, 12, 17}, {5, 6, 13}, {5, 16, 17}, {7, 8, 10}, {8, 13, 14}, {9, 10, 12}, {10, 16, 18}.

Design 19.4, o(G)=57, |SDS|=21

{0, 7, 11}, {0, 10, 12}, {1, 2, 5}, {1, 8, 14}, {1, 11, 18}, {2, 11, 14}, {2, 13, 16}, {3, 4, 8}, {3, 5, 7}, {3, 9, 12}, {3, 13, 15}, {4, 9, 13}, {4, 10, 17}, {5, 10, 15}, {5, 13, 18}, {6, 11, 17}, {6, 14, 18}, {8, 10, 13}, {8, 12, 18}, {9, 10, 16}, {14, 15, 17}.

Design 19.5, o(G)=57, |SDS|=22

{0, 5, 8}, {0, 12, 16}, {0, 13, 14}, {1, 4, 6}, {1, 7, 13}, {1, 9, 11}, {1, 14, 15}, {1, 17, 18}, {2, 3, 6}, {2, 8, 10}, {2, 12, 17}, {3, 4, 7}, {3, 8, 12}, {3, 10, 17}, {4, 9, 18}, {4, 11, 17}, {5, 14, 18}, {6, 15, 18}, {7, 10, 11}, {8, 11, 14}, {9, 16, 17}, {11, 12, 13}.

Design 19.6, o(G)=144, |SDS|=23

{0, 1, 3}, {0, 10, 11}, {0, 12, 16}, {1, 4, 7}, {1, 6, 9}, {1, 12, 18}, {2, 8, 11}, {2, 10, 16}, {2, 13, 18}, {2, 15, 17}, {3, 5, 7}, {3, 11, 14}, {4, 15, 18}, {5, 12, 13}, {5, 14, 18}, {6, 11, 17}, {6, 13, 16}, {6, 14, 15}, {7, 10, 17}, {8, 9, 18}, {8, 14, 17}, {8, 15, 16}, {9, 10, 12}.

Design 19.7, o(G)=96, |SDS|=21

{0, 6, 7}, {0, 9, 12}, {0, 11, 14}, {0, 15, 17}, {0, 16, 18}, {1, 2, 5}, {1, 4, 7}, {1, 9, 15}, {1, 11, 16}, {2, 9, 16}, {2, 10, 18}, {3, 5, 9}, {3, 12, 17}, {4, 5, 10}, {4, 13, 16}, {4, 14, 17}, {6, 8, 14}, {6, 12, 15}, {6, 13, 18}, {7, 10, 16}, {12, 13, 14}.

Design 19.8, o(G)=24, |SDS|=21

{0, 2, 4}, {0, 6, 11}, {1, 8, 15}, {1, 10, 16}, {1, 11, 14}, {2, 3, 6}, {2, 9, 15}, {3, 8, 17}, {3, 12, 16}, {3, 13, 15}, {3, 14, 18}, {4, 5, 10}, {4, 14, 16}, {5, 7, 11}, {5, 13, 14}, {5, 16, 17}, {6, 7, 8}, {6, 9, 17}, {7, 9, 10}, {8, 9, 13}, {11, 16, 18}.

Design 19.9, o(G)=24, |SDS|=21

{0, 1, 3}, {0, 6, 11}, {0, 7, 9}, {1, 4, 6}, {1, 9, 16}, {2, 3, 6}, {2, 10, 17}, {2, 15, 16}, {3, 10, 12}, {3, 13, 15}, {4, 12, 15}, {4, 16, 18}, {5, 6, 12}, {5, 11, 16}, {6, 8, 13}, {7, 8, 18}, {7, 16, 17}, {8, 14, 15}, {9, 11, 17}, {10, 11, 18}, {12, 13, 14}.

Design 19.10, o(G)=24, |SDS|=21

{0, 2, 4}, {0, 12, 15}, {0, 13, 17}, {1, 2, 5}, {1, 7, 9}, {1, 11, 13}, {1, 16, 17}, {2, 10, 12}, {2, 16, 18}, {3, 9, 13}, {3, 12, 18}, {3, 14, 15}, {4, 12, 16}, {6, 9, 17}, {6, 11, 18}, {6, 13, 15}, {7, 8, 15}, {7, 14, 17}, {8, 11, 16}, {9, 10, 15}, {10, 13, 14}.

Design 19.11, o(G)=24, |SDS|=21

{0, 9, 16}, {1, 7, 13}, {1, 9, 18}, {1, 10, 16}, {2, 3, 6}, {2, 8, 16}, {2, 12, 18}, {3, 13, 15}, {3, 14, 18}, {4, 9, 11}, {4, 12, 15}, {5, 6, 12}, {5, 7, 11}, {5, 13, 14}, {5, 15, 17}, {6, 7, 8}, {6, 14, 15}, {7, 9, 10}, {7, 16, 17}, {8, 9, 13}, {8, 10, 14}.

Design 19.12, o(G)=24, |SDS|=22

{0, 1, 3}, {0, 6, 8}, {0, 10, 16}, {0, 14, 18}, {1, 7, 10}, {1, 15, 18}, {1, 16, 17}, {2, 7, 11}, {2, 8, 13}, {3, 9, 16}, {3, 10, 11}, {4, 8, 14}, {4, 10, 17}, {4, 12, 16}, {4, 13, 15}, {5, 11, 12}, {5, 13, 14}, {5, 15, 16}, {6, 13, 18}, {7, 9, 18}, {8, 12, 17}, {9, 10, 15}.

Design 19.13, o(G)=24, |SDS|=21

{0, 6, 11}, {1, 2, 5}, {1, 7, 13}, {1, 9, 16}, {1, 12, 18}, {2, 3, 6}, {2, 9, 18}, {2, 10, 15}, {3, 8, 17}, {3, 10, 11}, {3, 12, 15}, {4, 5, 10}, {4, 12, 16}, {5, 7, 11}, {5, 16, 17}, {6, 7, 8}, {6, 10, 18}, {6, 14, 15}, {7, 16, 18}, {8, 9, 13}, {8, 10, 14}.

Design 19.14, o(G)=24, |SDS|=22

{0, 1, 3}, {0, 5, 7}, {0, 6, 8}, {0, 10, 11}, {0, 17, 18}, {1, 2, 5}, {1, 4, 6}, {1, 9, 15}, {2, 3, 6}, {2, 11, 17}, {3, 5, 8}, {3, 13, 17}, {4, 8, 13}, {4, 12, 16}, {4, 14, 15}, {6, 9, 17}, {6, 13, 16}, {7, 10, 13}, {7, 12, 14}, {7, 15, 18}, {8, 10, 15}, {9, 10, 12}.

Design 19.15, o(G)=24, |SDS|=21

{0, 1, 3}, {0, 5, 7}, {0, 10, 16}, {0, 12, 17}, {1, 7, 10}, {1, 14, 18}, {1, 16, 17}, {2, 3, 6}, {2, 15, 18}, {3, 11, 18}, {4, 10, 11}, {4, 14, 17}, {5, 13, 17}, {5, 14, 15}, {6, 9, 13}, {6, 11, 15}, {7, 8, 14}, {7, 13, 16}, {7, 15, 17}, {8, 15, 16}, {9, 11, 12}.

Design 19.16, o(G)=24, |SDS|=21

{0, 2, 4}, {0, 5, 7}, {0, 14, 15}, {1, 12, 15}, {1, 16, 17}, {2, 10, 11}, {2, 13, 16}, {2, 17, 18}, {3, 9, 16}, {3, 10, 17}, {3, 11, 12}, {4, 10, 14}, {4, 11, 18}, {5, 6, 10}, {6, 7, 13}, {6, 14, 18}, {6, 15, 16}, {7, 8, 16}, {7, 9, 15}, {8, 11, 17}, {9, 12, 13}.

Design 19.17, o(G)=24, |SDS|=21

{0, 1, 3}, {0, 5, 7}, {0, 9, 15}, {0, 10, 16}, {1, 2, 5}, {1, 7, 9}, {1, 14, 18}, {2, 9, 16}, {2, 14, 15}, {3, 11, 14}, {3, 12, 16}, {4, 8, 12}, {4, 10, 11}, {4, 13, 15}, {4, 17, 18}, {5, 13, 17}, {6, 11, 15}, {6, 14, 17}, {7, 8, 14}, {7, 13, 16}, {8, 11, 17}.

Design 19.18, o(G)=24, |SDS|=21

{0, 1, 3}, {0, 2, 4}, {0, 5, 8}, {0, 10, 16}, {0, 13, 18}, {1, 2, 5}, {1, 9, 15}, {2, 10, 13}, {3, 4, 7}, {3, 15, 18}, {3, 16, 17}, {4, 5, 10}, {4, 8, 16}, {5, 11, 16}, {6, 7, 10}, {6, 9, 14}, {6, 15, 17}, {8, 14, 15}, {9, 10, 12}, {9, 11, 17}, {12, 13, 14}.

Design 19.19, o(G)=12, |SDS|=20

{0, 1, 3}, {0, 2, 4}, {0, 6, 10}, {0, 9, 13}, {0, 11, 14}, {1, 2, 5}, {1, 4, 6}, {1, 9, 11}, {1, 13, 16}, {2, 3, 6}, {2, 9, 17}, {4, 5, 9}, {4, 8, 16}, {5, 6, 11}, {6, 9, 12}, {6, 15, 16}, {7, 8, 17}, {7, 10, 11}, {7, 15, 18}, {8, 14, 18}.

Design 19.20, o(G)=12, |SDS|=21

{0, 2, 4}, {0, 6, 10}, {0, 11, 12}, {0, 14, 15}, {0, 17, 18}, {1, 7, 8}, {1, 16, 18}, {2, 11, 18}, {3, 4, 7}, {3, 5, 8}, {3, 10, 15}, {3, 16, 17}, {4, 8, 15}, {4, 12, 17}, {5, 6, 11}, {6, 8, 16}, {6, 14, 18}, {7, 15, 16}, {8, 9, 10}, {9, 11, 14}, {10, 13, 16}.

Design 19.21, o(G)=12, |SDS|=21

{0, 2, 4}, {0, 5, 7}, {0, 6, 10}, {1, 8, 15}, {1, 9, 11}, {1, 10, 18}, {1, 13, 16}, {2, 3, 6}, {2, 7, 13}, {2, 9, 17}, {2, 14, 16}, {3, 4, 7}, {3, 13, 14}, {5, 12, 15}, {6, 7, 14}, {6, 9, 12}, {6, 15, 16}, {7, 15, 18}, {8, 10, 12}, {11, 12, 16}, {11, 13, 17}.

Design 19.22, o(G)=12, |SDS|=20

{0, 1, 3}, {0, 5, 7}, {0, 6, 10}, {0, 9, 16}, {0, 11, 12}, {0, 17, 18}, {1, 11, 15}, {2, 7, 9}, {2, 8, 14}, {3, 4, 7}, {3, 9, 18}, {4, 5, 9}, {4, 8, 15}, {4, 10, 18}, {4, 13, 14}, {5, 6, 11}, {6, 15, 17}, {7, 13, 18}, {8, 11, 17}, {9, 10, 13}.

Design 19.23, o(G)=12, |SDS|=20

{0, 2, 4}, {0, 12, 13}, {0, 16, 17}, {1, 2, 5}, {1, 10, 18}, {2, 8, 16}, {2, 11, 12}, {3, 4, 7}, {3, 9, 14}, {3, 10, 15}, {3, 11, 16}, {4, 12, 17}, {5, 10, 14}, {5, 12, 16}, {6, 7, 14}, {6, 8, 12}, {6, 9, 13}, {7, 9, 18}, {7, 10, 11}, {8, 10, 13}.

Design 19.24, o(G)=12, |SDS|=21

{0, 5, 7}, {0, 11, 12}, {0, 14, 15}, {0, 17, 18}, {1, 4, 6}, {1, 9, 17}, {1, 10, 14}, {1, 16, 18}, {2, 7, 9}, {2, 10, 17}, {3, 5, 8}, {4, 8, 15}, {4, 13, 14}, {5, 6, 11}, {5, 15, 18}, {6, 8, 16}, {6, 9, 13}, {7, 14, 17}, {8, 11, 18}, {9, 12, 14}, {10, 13, 16}.

Design 19.25, o(G)=12, |SDS|=21

{0, 5, 7}, {0, 8, 9}, {0, 11, 14}, {0, 12, 13}, {1, 2, 5}, {1, 4, 6}, {1, 10, 18}, {2, 3, 6}, {2, 7, 13}, {2, 9, 17}, {2, 14, 15}, {3, 13, 14}, {4, 8, 16}, {4, 10, 17}, {4, 11, 18}, {6, 7, 14}, {6, 15, 16}, {6, 17, 18}, {7, 9, 15}, {8, 10, 12}, {9, 14, 16}.

Design 19.26, o(G)=12, |SDS|=20

{0, 5, 7}, {0, 8, 13}, {0, 9, 16}, {0, 11, 12}, {0, 17, 18}, {1, 4, 6}, {1, 7, 8}, {1, 10, 14}, {2, 8, 14}, {2, 10, 17}, {2, 13, 15}, {3, 9, 18}, {4, 5, 9}, {4, 13, 14}, {5, 6, 11}, {5, 15, 18}, {7, 14, 17}, {8, 10, 16}, {8, 12, 17}, {9, 11, 15}.

Design 19.27, o(G)=54, |SDS|=22

{0, 1, 3}, {1, 9, 13}, {1, 12, 16}, {2, 7, 11}, {2, 16, 18}, {3, 4, 5}, {3, 8, 15}, {3, 13, 17}, {4, 14, 18}, {4, 15, 16}, {5, 8, 16}, {5, 11, 18}, {5, 14, 17}, {6, 7, 14}, {6, 10, 18}, {6, 11, 12}, {7, 9, 16}, {7, 15, 18}, {8, 12, 18}, {9, 12, 15}, {10, 11, 17}, {10, 13, 14}.

Design 19.28, o(G)=54, |SDS|=21

{0, 1, 3}, {0, 2, 4}, {0, 5, 7}, {0, 6, 8}, {1, 4, 6}, {1, 13, 14}, {2, 3, 6}, {2, 9, 17}, {2, 10, 16}, {3, 7, 12}, {3, 8, 15}, {4, 8, 16}, {4, 9, 18}, {4, 11, 15}, {5, 10, 17}, {5, 12, 13}, {5, 14, 18}, {6, 10, 13}, {10, 11, 12}, {11, 14, 17}, {12, 14, 15}.

Design 19.29, o(G)=18, |SDS|=20

{0, 1, 3}, {0, 10, 18}, {1, 12, 17}, {2, 7, 11}, {2, 9, 16}, {2, 13, 17}, {3, 4, 5}, {3, 10, 15}, {3, 14, 18}, {4, 7, 13}, {4, 10, 16}, {5, 6, 9}, {5, 10, 11}, {5, 12, 13}, {5, 15, 16}, {6, 7, 14}, {6, 13, 16}, {7, 8, 9}, {7, 16, 17}, {9, 11, 13}.

Design 19.30, o(G)=18, |SDS|=20

{0, 12, 16}, {1, 2, 5}, {1, 8, 11}, {1, 12, 18}, {1, 14, 16}, {2, 3, 6}, {3, 9, 17}, {3, 10, 15}, {4, 8, 12}, {4, 9, 18}, {4, 10, 16}, {4, 11, 15}, {5, 6, 9}, {5, 8, 14}, {5, 12, 13}, {5, 17, 18}, {6, 7, 14}, {6, 10, 17}, {8, 16, 18}, {11, 12, 14}.

Design 19.31, o(G)=18, |SDS|=20

{0, 1, 3}, {1, 12, 16}, {1, 13, 18}, {2, 3, 6}, {2, 16, 18}, {3, 4, 5}, {3, 11, 14}, {3, 13, 17}, {4, 8, 10}, {4, 15, 16}, {5, 8, 16}, {5, 10, 15}, {5, 14, 17}, {6, 7, 14}, {7, 9, 16}, {7, 15, 18}, {8, 12, 18}, {9, 12, 15}, {10, 11, 17}, {10, 13, 14}.

Design 19.32, o(G)=18, |SDS|=19

{0, 1, 3}, {0, 6, 8}, {0, 9, 13}, {0, 17, 18}, {1, 9, 17}, {2, 3, 6}, {2, 8, 13}, {3, 7, 12}, {3, 8, 15}, {3, 10, 16}, {4, 9, 11}, {4, 10, 15}, {4, 12, 17}, {4, 14, 18}, {5, 11, 18}, {5, 14, 17}, {6, 7, 14}, {6, 13, 15}, {10, 11, 17}.

Design 19.33, o(G)=18, |SDS|=20

{0, 5, 7}, {0, 9, 13}, {0, 11, 14}, {1, 15, 16}, {2, 3, 6}, {2, 8, 14}, {2, 10, 16}, {2, 11, 15}, {3, 10, 15}, {3, 12, 18}, {3, 14, 16}, {4, 5, 9}, {4, 8, 15}, {4, 11, 12}, {4, 13, 16}, {5, 6, 11}, {5, 13, 18}, {6, 8, 16}, {6, 9, 18}, {12, 14, 15}.

Design 19.34, o(G)=18, |SDS|=20

{0, 5, 8}, {0, 10, 16}, {1, 4, 6}, {1, 7, 13}, {1, 10, 18}, {1, 14, 15}, {2, 7, 14}, {2, 8, 16}, {2, 13, 15}, {3, 4, 7}, {3, 5, 9}, {3, 8, 17}, {3, 15, 16}, {5, 13, 14}, {5, 16, 17}, {6, 13, 18}, {7, 10, 11}, {7, 17, 18}, {8, 9, 15}, {9, 11, 16}.

Design 19.35, o(G)=32, |SDS|=21

{0, 2, 4}, {0, 6, 8}, {1, 4, 6}, {1, 7, 11}, {1, 8, 14}, {1, 9, 10}, {1, 13, 18}, {1, 16, 17}, {2, 17, 18}, {3, 4, 7}, {3, 5, 8}, {3, 9, 17}, {3, 15, 18}, {4, 14, 18}, {4, 15, 17}, {5, 6, 10}, {5, 11, 12}, {7, 9, 15}, {7, 14, 17}, {8, 10, 12}, {9, 12, 13}.

Design 19.36, o(G)=32, |SDS|=21

{0, 1, 3}, {0, 10, 12}, {0, 13, 15}, {1, 4, 6}, {1, 8, 16}, {1, 11, 12}, {2, 9, 10}, {2, 12, 15}, {2, 16, 17}, {3, 4, 7}, {3, 10, 14}, {3, 11, 13}, {3, 12, 18}, {4, 9, 13}, {4, 14, 16}, {4, 17, 18}, {5, 15, 17}, {6, 8, 14}, {6, 9, 17}, {10, 15, 18}, {12, 13, 14}.

Design 19.37, o(G)=32, |SDS|=21

{0, 9, 12}, {0, 11, 14}, {0, 15, 17}, {1, 13, 17}, {1, 14, 16}, {2, 8, 13}, {2, 9, 16}, {2, 11, 15}, {3, 5, 9}, {3, 11, 16}, {3, 12, 17}, {4, 12, 18}, {5, 13, 15}, {5, 17, 18}, {6, 8, 14}, {6, 12, 15}, {6, 13, 18}, {7, 8, 11}, {7, 10, 16}, {7, 15, 18}, {9, 10, 14}.

Design 19.38, o(G)=16, |SDS|=22

{0, 1, 3}, {0, 2, 4}, {0, 6, 8}, {0, 14, 17}, {1, 9, 12}, {1, 13, 14}, {1, 15, 18}, {2, 3, 6}, {2, 7, 11}, {2, 8, 10}, {2, 9, 16}, {3, 13, 17}, {4, 7, 13}, {4, 8, 16}, {4, 10, 14}, {5, 12, 16}, {5, 13, 15}, {5, 14, 18}, {6, 11, 16}, {7, 8, 18}, {9, 11, 13}, {10, 15, 17}.

Design 19.39, o(G)=16, |SDS|=22

{0, 2, 4}, {0, 6, 8}, {0, 13, 17}, {0, 14, 16}, {1, 7, 10}, {1, 9, 12}, {1, 13, 14}, {1, 15, 18}, {2, 3, 6}, {2, 12, 17}, {2, 13, 15}, {3, 8, 15}, {3, 9, 10}, {3, 11, 18}, {4, 7, 13}, {4, 8, 16}, {4, 11, 17}, {5, 10, 11}, {5, 12, 16}, {7, 8, 18}, {10, 16, 18}, {11, 12, 14}.

Design 19.40, o(G)=16, |SDS|=21

{0, 2, 4}, {0, 9, 14}, {1, 2, 5}, {1, 4, 6}, {1, 11, 16}, {1, 14, 17}, {2, 12, 17}, {2, 14, 18}, {3, 8, 10}, {3, 9, 17}, {3, 13, 16}, {4, 7, 13}, {5, 8, 14}, {5, 12, 18}, {6, 12, 16}, {6, 13, 15}, {7, 8, 9}, {7, 15, 16}, {8, 15, 18}, {9, 11, 13}, {11, 12, 14}.

Design 19.41, o(G)=16, |SDS|=21

{0, 5, 7}, {0, 9, 14}, {0, 15, 17}, {0, 16, 18}, {1, 4, 6}, {1, 8, 12}, {1, 11, 18}, {1, 14, 17}, {2, 12, 17}, {2, 14, 18}, {3, 4, 5}, {3, 8, 10}, {4, 7, 13}, {5, 6, 9}, {5, 13, 15}, {6, 12, 18}, {7, 16, 17}, {8, 15, 16}, {8, 17, 18}, {9, 11, 13}, {11, 12, 14}.

Design 19.42, o(G)=16, |SDS|=21

{0, 1, 3}, {0, 5, 7}, {0, 11, 13}, {1, 2, 5}, {1, 11, 17}, {1, 13, 14}, {1, 15, 18}, {2, 12, 17}, {3, 8, 15}, {3, 9, 13}, {4, 8, 16}, {4, 9, 12}, {4, 10, 14}, {4, 17, 18}, {5, 13, 15}, {5, 14, 18}, {6, 10, 16}, {7, 9, 15}, {8, 9, 10}, {8, 13, 18}, {10, 11, 12}.

Design 19.43, o(G)=16, |SDS|=22

{0, 2, 4}, {0, 5, 7}, {0, 6, 8}, {0, 12, 13}, {0, 17, 18}, {1, 2, 5}, {1, 9, 15}, {1, 11, 14}, {2, 10, 18}, {3, 10, 14}, {3, 11, 15}, {3, 13, 17}, {4, 5, 9}, {4, 8, 13}, {4, 12, 16}, {5, 16, 17}, {6, 7, 11}, {6, 13, 16}, {7, 16, 18}, {8, 10, 15}, {8, 14, 17}, {9, 10, 12}.

Design 19.44, o(G)=16, |SDS|=22

{0, 1, 3}, {0, 6, 8}, {0, 10, 11}, {0, 15, 16}, {0, 17, 18}, {1, 4, 6}, {1, 7, 8}, {1, 13, 17}, {2, 7, 9}, {2, 10, 18}, {2, 14, 16}, {3, 4, 7}, {3, 9, 16}, {4, 8, 13}, {4, 10, 17}, {5, 11, 12}, {5, 13, 14}, {5, 16, 18}, {7, 12, 14}, {8, 10, 15}, {8, 14, 17}, {9, 11, 13}.

Design 19.45, o(G)=16, |SDS|=22

{0, 9, 15}, {0, 12, 17}, {0, 13, 18}, {1, 4, 6}, {1, 7, 9}, {1, 8, 10}, {1, 14, 18}, {2, 7, 11}, {2, 10, 17}, {2, 12, 18}, {2, 14, 15}, {3, 4, 7}, {3, 13, 14}, {4, 5, 9}, {4, 13, 15}, {4, 14, 16}, {5, 6, 10}, {5, 11, 16}, {6, 9, 13}, {7, 8, 14}, {7, 13, 16}, {8, 11, 17}.

Design 19.46, o(G)=16, |SDS|=21

{0, 5, 7}, {0, 10, 16}, {1, 2, 5}, {1, 7, 9}, {1, 11, 13}, {1, 14, 18}, {2, 3, 6}, {2, 7, 11}, {2, 12, 14}, {3, 12, 16}, {3, 13, 18}, {4, 8, 12}, {4, 13, 15}, {4, 14, 16}, {4, 17, 18}, {5, 6, 10}, {6, 16, 18}, {7, 15, 17}, {8, 9, 18}, {8, 11, 17}, {8, 15, 16}.

Design 19.47, o(G)=16, |SDS|=21

{0, 1, 3}, {0, 2, 4}, {0, 5, 7}, {0, 10, 16}, {0, 13, 18}, {1, 2, 5}, {1, 4, 6}, {1, 7, 10}, {1, 14, 18}, {2, 3, 6}, {3, 4, 7}, {3, 13, 14}, {4, 8, 12}, {4, 17, 18}, {5, 11, 16}, {5, 14, 15}, {6, 11, 15}, {7, 9, 18}, {8, 11, 17}, {8, 15, 16}, {9, 11, 12}.

Design 19.48, o(G)=16, |SDS|=21

{0, 6, 8}, {0, 9, 15}, {0, 10, 16}, {0, 12, 17}, {1, 8, 9}, {1, 12, 15}, {1, 14, 18}, {1, 16, 17}, {2, 3, 6}, {2, 7, 11}, {2, 10, 17}, {2, 12, 18}, {3, 4, 7}, {3, 5, 8}, {3, 11, 18}, {3, 13, 14}, {4, 10, 11}, {6, 11, 15}, {6, 14, 16}, {7, 9, 18}, {8, 15, 16}.

Design 19.49, o(G)=16, |SDS|=21

{0, 1, 3}, {0, 2, 4}, {0, 10, 17}, {0, 12, 18}, {1, 7, 11}, {1, 13, 18}, {1, 16, 17}, {2, 7, 12}, {2, 13, 17}, {2, 16, 18}, {3, 5, 8}, {4, 5, 9}, {4, 12, 17}, {4, 15, 16}, {5, 6, 10}, {5, 14, 15}, {6, 9, 14}, {6, 11, 18}, {7, 9, 15}, {8, 11, 17}, {10, 13, 15}.

Design 19.50, o(G)=16, |SDS|=21

{0, 5, 7}, {0, 8, 9}, {0, 12, 13}, {0, 15, 16}, {0, 17, 18}, {1, 4, 6}, {1, 11, 15}, {2, 3, 6}, {2, 12, 16}, {2, 14, 17}, {3, 4, 7}, {3, 11, 16}, {4, 13, 15}, {4, 14, 18}, {5, 10, 14}, {5, 13, 16}, {6, 7, 14}, {6, 8, 17}, {7, 8, 18}, {8, 11, 12}, {9, 11, 13}.

Design 19.51, o(G)=19, |SDS|=21

{0, 1, 3}, {0, 2, 4}, {0, 12, 17}, {1, 4, 6}, {1, 7, 9}, {1, 8, 14}, {2, 3, 6}, {2, 7, 11}, {3, 4, 7}, {3, 11, 16}, {3, 14, 15}, {4, 11, 13}, {4, 12, 18}, {5, 10, 18}, {5, 12, 14}, {5, 13, 15}, {5, 16, 17}, {6, 17, 18}, {8, 9, 12}, {10, 11, 12}, {10, 15, 16}.

Design 19.52, o(G)=18, |SDS|=18

{0, 11, 14}, {1, 4, 6}, {1, 7, 12}, {1, 11, 17}, {2, 8, 14}, {2, 10, 16}, {2, 12, 18}, {3, 5, 8}, {3, 10, 15}, {3, 14, 16}, {4, 8, 15}, {4, 11, 12}, {5, 6, 11}, {5, 10, 14}, {6, 9, 18}, {6, 12, 17}, {6, 13, 15}, {8, 9, 10}.

Design 19.53, o(G)=18, |SDS|=19

{0, 16, 18}, {1, 4, 6}, {1, 7, 12}, {1, 8, 15}, {2, 10, 17}, {2, 14, 18}, {3, 10, 11}, {3, 12, 17}, {3, 13, 18}, {3, 15, 16}, {4, 8, 14}, {5, 10, 18}, {5, 13, 16}, {6, 7, 14}, {6, 8, 12}, {6, 9, 13}, {7, 8, 17}, {7, 11, 15}, {10, 12, 13}.

Design 19.54, o(G)=18, |SDS|=21

{0, 1, 3}, {0, 2, 4}, {0, 11, 12}, {1, 2, 5}, {1, 4, 6}, {1, 14, 18}, {2, 3, 6}, {2, 12, 14}, {3, 17, 18}, {4, 5, 9}, {4, 15, 17}, {5, 12, 18}, {6, 8, 9}, {7, 8, 16}, {7, 9, 15}, {7, 10, 18}, {7, 11, 17}, {8, 10, 17}, {8, 11, 18}, {9, 10, 12}, {11, 14, 16}.

Design 19.55, o(G)=18, |SDS|=21

{0, 1, 3}, {0, 2, 4}, {0, 11, 14}, {0, 12, 13}, {1, 9, 10}, {1, 15, 16}, {2, 17, 18}, {3, 13, 14}, {4, 8, 15}, {4, 12, 14}, {5, 10, 14}, {5, 16, 17}, {6, 8, 16}, {6, 9, 17}, {6, 12, 18}, {7, 8, 17}, {7, 9, 16}, {7, 10, 11}, {7, 15, 18}, {8, 11, 12}, {10, 13, 17}.

Design 19.56, o(G)=18, |SDS|=21

{0, 5, 8}, {0, 10, 18}, {0, 12, 16}, {1, 8, 15}, {1, 9, 17}, {1, 10, 16}, {1, 11, 14}, {2, 3, 6}, {2, 7, 14}, {2, 11, 13}, {2, 12, 17}, {3, 4, 7}, {3, 14, 17}, {4, 5, 10}, {5, 7, 11}, {5, 15, 16}, {6, 8, 17}, {6, 14, 18}, {7, 16, 17}, {9, 12, 14}, {10, 13, 15}.

Design 19.57, o(G)=12, |SDS|=21

{0, 1, 3}, {0, 9, 14}, {0, 10, 15}, {0, 11, 16}, {1, 2, 5}, {1, 8, 12}, {1, 11, 18}, {1, 14, 16}, {2, 8, 13}, {2, 10, 17}, {2, 12, 18}, {3, 7, 12}, {3, 9, 17}, {3, 11, 15}, {5, 6, 9}, {5, 8, 14}, {5, 12, 16}, {6, 10, 11}, {6, 15, 17}, {7, 17, 18}, {9, 10, 13}.

Design 19.58, o(G)=12, |SDS|=21

{0, 9, 14}, {0, 10, 15}, {0, 11, 16}, {0, 13, 18}, {1, 13, 16}, {2, 3, 6}, {2, 7, 11}, {2, 9, 16}, {2, 12, 15}, {3, 4, 5}, {3, 7, 12}, {3, 9, 17}, {4, 11, 15}, {5, 8, 14}, {5, 10, 18}, {5, 13, 15}, {6, 10, 12}, {6, 17, 18}, {7, 8, 9}, {7, 16, 17}, {10, 13, 14}.

Design 19.59, o(G)=12, |SDS|=21

{0, 1, 3}, {0, 9, 13}, {0, 16, 18}, {1, 8, 11}, {1, 10, 16}, {1, 14, 17}, {2, 12, 18}, {3, 4, 7}, {3, 11, 18}, {4, 5, 9}, {4, 12, 14}, {4, 13, 17}, {5, 10, 14}, {5, 12, 15}, {6, 7, 14}, {6, 9, 12}, {7, 10, 11}, {8, 14, 18}, {9, 14, 15}, {10, 12, 13}, {11, 13, 16}.

Design 19.60, o(G)=12, |SDS|=21

{0, 1, 3}, {0, 6, 8}, {0, 9, 14}, {0, 10, 16}, {0, 12, 18}, {1, 9, 15}, {1, 14, 18}, {2, 7, 11}, {2, 13, 18}, {3, 4, 5}, {3, 9, 17}, {4, 7, 13}, {4, 10, 17}, {4, 11, 16}, {5, 11, 13}, {5, 16, 17}, {6, 7, 14}, {6, 10, 18}, {6, 11, 17}, {7, 15, 17}, {12, 13, 14}.

Design 19.61, o(G)=12, |SDS|=21

{0, 1, 3}, {0, 6, 8}, {0, 11, 16}, {0, 12, 17}, {1, 2, 5}, {1, 8, 11}, {1, 12, 16}, {2, 9, 16}, {2, 12, 18}, {2, 13, 15}, {3, 4, 5}, {4, 7, 13}, {4, 9, 18}, {4, 14, 15}, {5, 8, 14}, {5, 12, 15}, {6, 7, 14}, {6, 16, 18}, {8, 16, 17}, {9, 10, 13}, {11, 13, 14}.

Design 19.62, o(G)=12, |SDS|=21

{0, 1, 3}, {0, 5, 7}, {0, 9, 14}, {0, 15, 17}, {1, 9, 15}, {1, 14, 18}, {2, 3, 6}, {2, 13, 16}, {3, 4, 5}, {3, 11, 15}, {3, 14, 17}, {4, 7, 13}, {4, 9, 12}, {5, 10, 15}, {5, 12, 13}, {5, 16, 17}, {6, 11, 17}, {6, 12, 18}, {7, 8, 9}, {8, 10, 12}, {8, 13, 18}.

Design 19.63, o(G)=12, |SDS|=21

{0, 2, 4}, {0, 11, 16}, {0, 13, 18}, {1, 4, 6}, {1, 13, 16}, {2, 10, 18}, {2, 14, 17}, {3, 7, 12}, {3, 8, 10}, {3, 9, 17}, {4, 7, 13}, {4, 12, 16}, {4, 14, 15}, {5, 6, 9}, {5, 8, 14}, {6, 11, 13}, {6, 15, 16}, {7, 8, 9}, {8, 15, 17}, {9, 10, 11}, {9, 12, 13}.

Design 19.64, o(G)=12, |SDS|=21

{0, 1, 3}, {0, 2, 4}, {0, 9, 17}, {0, 10, 11}, {1, 7, 8}, {1, 15, 16}, {1, 17, 18}, {2, 3, 6}, {2, 14, 17}, {3, 7, 11}, {3, 13, 15}, {4, 9, 16}, {4, 10, 14}, {4, 11, 17}, {5, 10, 17}, {5, 12, 15}, {6, 7, 13}, {6, 11, 18}, {7, 15, 18}, {8, 10, 18}, {12, 13, 14}.

Design 19.65, o(G)=12, |SDS|=21

{0, 1, 3}, {0, 9, 14}, {0, 10, 13}, {1, 4, 6}, {2, 10, 14}, {2, 12, 16}, {2, 13, 17}, {3, 4, 5}, {3, 9, 13}, {3, 10, 18}, {3, 14, 17}, {4, 15, 18}, {5, 8, 14}, {5, 11, 18}, {5, 16, 17}, {6, 11, 16}, {6, 12, 18}, {7, 8, 9}, {7, 16, 18}, {9, 11, 17}, {11, 13, 14}.

Design 19.66, o(G)=12, |SDS|=21

{0, 2, 4}, {0, 5, 8}, {0, 10, 13}, {1, 9, 16}, {1, 12, 17}, {2, 9, 15}, {2, 12, 18}, {3, 11, 13}, {3, 14, 15}, {3, 17, 18}, {4, 5, 10}, {4, 16, 18}, {5, 7, 14}, {5, 11, 16}, {6, 8, 10}, {6, 9, 18}, {6, 14, 17}, {7, 11, 12}, {7, 16, 17}, {8, 13, 14}, {9, 11, 17}.

Design 19.67, o(G)=12, |SDS|=21

{0, 1, 3}, {1, 2, 5}, {1, 7, 9}, {1, 11, 13}, {1, 15, 18}, {2, 3, 6}, {2, 7, 11}, {2, 16, 18}, {3, 9, 11}, {3, 14, 15}, {4, 8, 14}, {4, 10, 17}, {4, 12, 16}, {5, 11, 12}, {5, 17, 18}, {6, 11, 15}, {6, 13, 18}, {7, 8, 15}, {7, 14, 17}, {8, 12, 17}, {9, 12, 13}.

Design 19.68, o(G)=12, |STS|=20

{}, {0, 1, 3}, {0, 5, 8}, {0, 9, 15}, {0, 13, 17}, {1, 4, 6}, {1, 11, 14}, {2, 7, 14}, {2, 8, 16}, {2, 10, 15}, {3, 5, 9}, {3, 8, 17}, {3, 13, 18}, {5, 7, 11}, {5, 16, 17}, {6, 7, 8}, {6, 10, 18}, {6, 14, 15}, {7, 9, 10}, {10, 12, 14}, {11, 17, 18}.

Design 19.69, o(G)=12, |SDS|=22

{0, 5, 7}, {0, 8, 12}, {0, 9, 13}, {0, 11, 14}, {1, 4, 6}, {1, 9, 17}, {1, 11, 15}, {1, 13, 18}, {1, 14, 16}, {2, 8, 15}, {2, 12, 16}, {2, 14, 18}, {3, 5, 8}, {3, 10, 12}, {4, 11, 18}, {4, 14, 17}, {5, 12, 17}, {5, 16, 18}, {6, 7, 14}, {6, 9, 15}, {8, 9, 11}, {10, 16, 17}.

Design 19.70, o(G)=12, |SDS|=21

{0, 1, 3}, {0, 5, 7}, {0, 10, 16}, {0, 13, 15}, {1, 4, 6}, {1, 7, 11}, {1, 13, 17}, {2, 8, 15}, {2, 9, 14}, {2, 10, 11}, {2, 16, 17}, {3, 4, 7}, {3, 12, 13}, {3, 14, 17}, {5, 13, 16}, {6, 7, 13}, {6, 9, 17}, {6, 11, 12}, {7, 15, 17}, {8, 9, 11}, {8, 10, 18}.

Design 19.71, o(G)=12, |SDS|=21

{0, 5, 7}, {0, 11, 13}, {1, 2, 5}, {1, 4, 6}, {1, 15, 18}, {2, 7, 11}, {2, 12, 17}, {3, 7, 12}, {3, 9, 13}, {3, 11, 16}, {4, 8, 16}, {4, 9, 12}, {4, 10, 14}, {4, 17, 18}, {5, 8, 11}, {5, 14, 18}, {6, 10, 13}, {6, 12, 16}, {6, 15, 17}, {9, 11, 14}, {13, 16, 17}.

Design 19.72, o(G)=12, |SDS|=22

{0, 2, 4}, {0, 14, 16}, {1, 4, 6}, {1, 7, 10}, {1, 15, 18}, {2, 8, 14}, {2, 9, 17}, {2, 13, 15}, {3, 8, 15}, {3, 10, 18}, {3, 11, 16}, {3, 14, 17}, {4, 8, 16}, {4, 9, 12}, {5, 6, 9}, {5, 12, 13}, {5, 14, 18}, {6, 10, 13}, {6, 11, 18}, {7, 8, 17}, {10, 11, 12}, {13, 16, 17}.

Design 19.73, o(G)=12, |SDS|=21

{0, 1, 3}, {0, 5, 7}, {0, 9, 14}, {0, 11, 13}, {0, 16, 18}, {1, 4, 6}, {2, 3, 6}, {2, 10, 15}, {2, 12, 17}, {3, 7, 12}, {3, 13, 16}, {3, 14, 15}, {4, 8, 11}, {4, 9, 18}, {4, 10, 17}, {5, 8, 14}, {6, 7, 14}, {6, 10, 18}, {6, 11, 17}, {7, 15, 16}, {9, 10, 11}.

Design 19.74, o(G)=12, |SDS|=21

{0, 1, 3}, {0, 6, 8}, {0, 9, 14}, {0, 11, 13}, {0, 16, 18}, {1, 4, 6}, {1, 7, 10}, {2, 10, 15}, {2, 12, 17}, {3, 11, 16}, {3, 14, 15}, {4, 7, 13}, {4, 9, 18}, {4, 10, 17}, {5, 8, 14}, {5, 11, 15}, {6, 7, 14}, {6, 10, 18}, {7, 8, 9}, {7, 17, 18}, {11, 12, 14}.

Design 19.75, o(G)=12, |SDS|=21

{0, 2, 4}, {0, 9, 14}, {0, 10, 12}, {0, 11, 13}, {0, 15, 17}, {1, 4, 6}, {1, 13, 18}, {2, 10, 15}, {3, 8, 10}, {3, 9, 17}, {3, 11, 18}, {4, 7, 13}, {4, 12, 15}, {4, 14, 16}, {5, 8, 14}, {6, 7, 14}, {6, 12, 16}, {6, 13, 15}, {7, 8, 9}, {8, 17, 18}, {12, 13, 14}.

Design 19.76, o(G)=12, |SDS|=21

{0, 1, 3}, {0, 2, 4}, {0, 9, 14}, {0, 11, 13}, {0, 16, 18}, {1, 2, 5}, {1, 13, 16}, {2, 12, 17}, {3, 4, 5}, {3, 8, 10}, {3, 9, 17}, {4, 8, 11}, {4, 9, 18}, {4, 10, 17}, {6, 7, 14}, {6, 12, 16}, {6, 13, 15}, {7, 8, 9}, {8, 17, 18}, {10, 11, 14}, {12, 13, 14}.

Design 19.77, o(G)=12, |SDS|=22

{0, 1, 3}, {0, 2, 4}, {0, 5, 7}, {0, 9, 17}, {1, 7, 10}, {2, 8, 13}, {2, 16, 18}, {3, 4, 5}, {3, 7, 12}, {3, 9, 13}, {3, 11, 14}, {4, 9, 12}, {4, 11, 17}, {4, 15, 16}, {5, 10, 15}, {5, 12, 13}, {6, 7, 14}, {6, 10, 18}, {6, 16, 17}, {7, 15, 18}, {10, 13, 14}, {12, 15, 17}.

Design 19.78, o(G)=12, |SDS|=22

{0, 2, 4}, {0, 9, 18}, {0, 13, 17}, {0, 14, 16}, {1, 8, 11}, {1, 12, 16}, {1, 14, 15}, {2, 3, 6}, {2, 7, 11}, {3, 4, 5}, {3, 7, 12}, {3, 8, 15}, {3, 17, 18}, {4, 14, 18}, {5, 8, 16}, {6, 10, 18}, {6, 11, 12}, {7, 8, 17}, {7, 15, 18}, {10, 13, 14}, {11, 13, 16}, {12, 15, 17}.

Design 19.79, o(G)=12, |SDS|=21

{0, 1, 3}, {0, 5, 8}, {0, 6, 11}, {0, 10, 15}, {0, 17, 18}, {1, 4, 6}, {1, 8, 11}, {1, 13, 14}, {1, 15, 18}, {2, 3, 6}, {2, 7, 14}, {2, 8, 12}, {2, 9, 17}, {2, 15, 16}, {3, 11, 13}, {4, 12, 18}, {5, 16, 17}, {6, 10, 14}, {7, 12, 17}, {7, 16, 18}, {9, 10, 16}.

Design 19.80, o(G)=12, |SDS|=20

{0, 2, 4}, {0, 5, 8}, {0, 17, 18}, {1, 2, 5}, {1, 8, 17}, {1, 12, 16}, {2, 8, 18}, {2, 11, 17}, {2, 13, 15}, {3, 7, 15}, {3, 10, 12}, {3, 14, 16}, {4, 6, 12}, {5, 6, 13}, {5, 15, 17}, {6, 8, 16}, {6, 9, 14}, {8, 9, 11}, {10, 13, 18}, {12, 13, 14}.

Design 19.81, o(G)=12, |SDS|=21

{0, 1, 3}, {0, 15, 17}, {1, 2, 5}, {1, 9, 13}, {1, 10, 11}, {1, 15, 16}, {2, 10, 18}, {3, 4, 7}, {3, 9, 12}, {3, 11, 14}, {3, 17, 18}, {5, 6, 10}, {5, 11, 16}, {5, 14, 15}, {6, 7, 12}, {6, 15, 18}, {7, 9, 17}, {7, 13, 18}, {8, 9, 18}, {8, 10, 12}, {8, 11, 15}.

Design 19.82, o(G)=12, |SDS|=21

{0, 1, 3}, {0, 6, 8}, {0, 9, 10}, {0, 12, 16}, {1, 2, 5}, {1, 11, 18}, {1, 16, 17}, {2, 8, 13}, {2, 14, 15}, {3, 9, 11}, {3, 12, 14}, {3, 17, 18}, {4, 8, 14}, {4, 13, 18}, {5, 11, 17}, {5, 13, 15}, {6, 9, 17}, {7, 9, 14}, {7, 10, 15}, {7, 13, 17}, {8, 12, 15}.

Design 19.83, o(G)=12, |SDS|=21

{0, 2, 4}, {0, 9, 16}, {1, 2, 5}, {1, 4, 6}, {1, 9, 13}, {1, 14, 16}, {2, 9, 17}, {2, 10, 16}, {2, 13, 18}, {3, 7, 12}, {3, 10, 13}, {3, 11, 15}, {3, 17, 18}, {4, 7, 13}, {4, 9, 18}, {5, 8, 12}, {5, 10, 15}, {5, 14, 17}, {6, 12, 15}, {8, 9, 10}, {10, 12, 17}.

Design 19.84, o(G)=12, |SDS|=21

{0, 6, 8}, {0, 10, 11}, {0, 16, 18}, {1, 2, 5}, {1, 4, 6}, {1, 9, 17}, {1, 12, 14}, {2, 7, 11}, {2, 9, 12}, {2, 10, 13}, {3, 4, 5}, {3, 11, 16}, {4, 9, 15}, {4, 14, 18}, {5, 6, 9}, {5, 10, 16}, {5, 13, 14}, {6, 10, 18}, {6, 15, 16}, {7, 8, 18}, {11, 13, 18}.

Design 19.85, o(G)=12, |SDS|=21

{0, 1, 3}, {0, 2, 4}, {0, 7, 9}, {0, 14, 17}, {1, 8, 16}, {1, 14, 18}, {2, 7, 14}, {2, 8, 10}, {3, 4, 7}, {3, 14, 15}, {4, 5, 10}, {4, 11, 17}, {4, 12, 13}, {5, 6, 12}, {5, 7, 11}, {5, 13, 18}, {6, 8, 17}, {6, 10, 13}, {6, 16, 18}, {9, 10, 11}, {12, 16, 17}.

Design 19.86, o(G)=9, |SDS|=21

{0, 1, 3}, {0, 2, 4}, {0, 5, 7}, {0, 6, 8}, {1, 2, 5}, {1, 14, 15}, {1, 16, 18}, {2, 7, 11}, {3, 4, 5}, {3, 8, 15}, {3, 9, 10}, {4, 7, 13}, {4, 10, 17}, {4, 11, 18}, {5, 10, 11}, {6, 11, 16}, {6, 12, 15}, {7, 9, 17}, {8, 9, 14}, {8, 12, 13}, {11, 12, 17}.

Design 19.87, o(G)=9, |SDS|=21

{0, 9, 11}, {0, 10, 15}, {1, 2, 5}, {1, 12, 17}, {1, 14, 18}, {2, 3, 6}, {2, 7, 11}, {2, 10, 16}, {2, 12, 13}, {3, 9, 18}, {3, 10, 13}, {3, 11, 15}, {4, 8, 14}, {4, 9, 17}, {4, 11, 16}, {5, 12, 15}, {6, 10, 18}, {6, 13, 16}, {7, 8, 17}, {8, 12, 18}, {10, 11, 14}.

Design 19.88, o(G)=9, |SDS|=21

{0, 5, 8}, {0, 7, 9}, {0, 13, 14}, {1, 4, 6}, {1, 7, 12}, {1, 11, 14}, {2, 11, 15}, {2, 14, 17}, {3, 5, 9}, {3, 11, 17}, {3, 12, 14}, {3, 16, 18}, {4, 8, 17}, {4, 14, 18}, {5, 11, 18}, {5, 13, 17}, {6, 7, 15}, {6, 9, 10}, {8, 9, 15}, {10, 11, 13}, {12, 17, 18}.

Design 19.89, o(G)=9, |SDS|=20

{0, 1, 3}, {0, 2, 4}, {0, 9, 12}, {1, 4, 6}, {1, 7, 10}, {1, 8, 13}, {2, 12, 16}, {2, 15, 18}, {3, 4, 7}, {3, 9, 17}, {4, 13, 17}, {4, 14, 18}, {5, 6, 10}, {5, 11, 16}, {5, 12, 15}, {6, 11, 15}, {7, 8, 16}, {7, 13, 14}, {8, 12, 18}, {9, 15, 16}.

Design 19.90, o(G)=9, |SDS|=20

{0, 1, 3}, {0, 2, 4}, {0, 8, 9}, {0, 13, 14}, {1, 16, 18}, {2, 3, 6}, {2, 8, 15}, {2, 9, 13}, {3, 9, 10}, {3, 14, 18}, {4, 12, 18}, {4, 15, 17}, {5, 6, 11}, {5, 10, 18}, {5, 12, 14}, {5, 16, 17}, {6, 17, 18}, {7, 10, 14}, {9, 11, 15}, {10, 12, 16}.

Design 19.91, o(G)=9, |SDS|=21

{0, 5, 8}, {0, 7, 9}, {0, 12, 16}, {0, 13, 15}, {1, 2, 5}, {1, 4, 6}, {1, 9, 16}, {1, 10, 18}, {2, 3, 6}, {2, 9, 17}, {2, 10, 11}, {2, 15, 18}, {3, 14, 16}, {3, 17, 18}, {4, 5, 10}, {4, 8, 16}, {4, 11, 12}, {6, 10, 15}, {8, 9, 12}, {8, 13, 14}, {11, 16, 17}.

Design 19.92, o(G)=9, |SDS|=20

{0, 2, 4}, {0, 6, 10}, {0, 11, 16}, {1, 8, 15}, {1, 9, 12}, {1, 13, 17}, {2, 7, 10}, {2, 9, 17}, {3, 5, 7}, {4, 10, 17}, {4, 11, 14}, {4, 13, 16}, {5, 10, 16}, {5, 12, 18}, {6, 7, 15}, {7, 8, 12}, {8, 9, 13}, {8, 11, 18}, {10, 11, 13}, {12, 14, 15}.

Design 19.93, o(G)=9, |SDS|=21

{0, 5, 8}, {0, 7, 9}, {0, 15, 16}, {1, 2, 5}, {1, 4, 6}, {1, 10, 16}, {1, 15, 18}, {2, 11, 15}, {2, 13, 18}, {3, 4, 7}, {3, 5, 9}, {3, 12, 14}, {4, 8, 9}, {5, 16, 18}, {6, 7, 16}, {6, 10, 14}, {6, 17, 18}, {7, 8, 10}, {8, 11, 12}, {8, 13, 14}, {10, 15, 17}.

Design 19.94, o(G)=9, |SDS|=21

{0, 2, 4}, {0, 8, 14}, {0, 12, 16}, {1, 7, 11}, {1, 9, 14}, {1, 10, 16}, {1, 15, 18}, {3, 5, 8}, {3, 12, 18}, {3, 13, 16}, {4, 5, 9}, {4, 8, 16}, {4, 12, 14}, {5, 10, 17}, {6, 7, 13}, {6, 8, 9}, {7, 9, 18}, {7, 10, 15}, {8, 11, 12}, {9, 10, 12}, {13, 15, 17}.

Design 19.95, o(G)=9, |SDS|=20

{0, 1, 3}, {0, 2, 4}, {0, 6, 9}, {1, 2, 5}, {1, 8, 17}, {2, 3, 6}, {2, 7, 13}, {3, 4, 8}, {3, 5, 9}, {3, 13, 18}, {4, 14, 15}, {5, 6, 12}, {5, 14, 16}, {7, 10, 17}, {8, 10, 16}, {8, 11, 18}, {8, 12, 13}, {9, 10, 14}, {10, 11, 15}, {11, 14, 17}.

Design 19.96, o(G)=9, |SDS|=21

{0, 2, 4}, {0, 6, 11}, {0, 12, 14}, {1, 2, 5}, {1, 8, 15}, {1, 9, 17}, {2, 3, 6}, {2, 11, 16}, {3, 14, 15}, {4, 8, 14}, {4, 11, 13}, {4, 15, 17}, {4, 16, 18}, {5, 6, 12}, {5, 7, 14}, {6, 13, 16}, {8, 10, 13}, {9, 11, 18}, {9, 13, 15}, {10, 12, 15}, {10, 14, 17}.

Design 19.97, o(G)=9, |SDS|=21

{0, 2, 4}, {0, 9, 13}, {0, 10, 15}, {0, 12, 18}, {1, 7, 8}, {1, 9, 14}, {1, 11, 12}, {2, 3, 6}, {2, 13, 16}, {2, 17, 18}, {3, 7, 11}, {3, 8, 14}, {4, 8, 13}, {4, 9, 16}, {4, 10, 14}, {4, 15, 17}, {5, 11, 14}, {5, 13, 17}, {6, 10, 11}, {6, 16, 17}, {7, 14, 16}.

Design 19.98, o(G)=9, |SDS|=20

{0, 1, 3}, {0, 5, 8}, {0, 10, 14}, {1, 4, 7}, {1, 8, 15}, {1, 12, 18}, {2, 7, 9}, {2, 10, 13}, {2, 11, 14}, {3, 4, 8}, {3, 7, 13}, {3, 15, 18}, {4, 12, 17}, {4, 13, 18}, {6, 14, 15}, {7, 8, 17}, {7, 10, 18}, {9, 10, 11}, {9, 12, 14}, {9, 13, 16}.

Design 19.99, o(G)=9, |SDS|=21

{0, 1, 3}, {0, 2, 4}, {0, 11, 12}, {0, 16, 18}, {1, 2, 5}, {1, 9, 15}, {1, 13, 14}, {2, 7, 11}, {3, 13, 15}, {4, 5, 9}, {4, 11, 18}, {4, 12, 17}, {5, 6, 11}, {5, 10, 15}, {6, 7, 12}, {6, 8, 9}, {7, 9, 18}, {7, 10, 13}, {7, 15, 16}, {8, 11, 17}, {8, 12, 18}.

Design 19.100, o(G)=9, |SDS|=20

{0, 1, 3}, {0, 6, 9}, {0, 10, 15}, {1, 4, 7}, {1, 8, 14}, {1, 9, 17}, {2, 3, 6}, {2, 13, 18}, {3, 4, 8}, {3, 11, 17}, {4, 13, 14}, {4, 16, 17}, {5, 6, 12}, {5, 10, 13}, {5, 11, 18}, {6, 15, 18}, {7, 8, 17}, {7, 12, 18}, {8, 10, 11}, {10, 14, 17}.

Design 19.101, o(G)=9, |SDS|=21

{0, 7, 9}, {0, 10, 18}, {0, 12, 13}, {1, 7, 10}, {1, 9, 16}, {1, 11, 18}, {1, 13, 17}, {2, 3, 6}, {2, 8, 14}, {2, 11, 15}, {2, 12, 17}, {2, 16, 18}, {3, 5, 9}, {3, 8, 12}, {3, 10, 15}, {3, 14, 18}, {5, 6, 12}, {6, 14, 15}, {6, 17, 18}, {7, 11, 17}, {7, 12, 16}.

Design 19.102, o(G)=9, |SDS|=21

{0, 2, 4}, {0, 8, 13}, {0, 12, 18}, {0, 14, 16}, {1, 10, 17}, {1, 16, 18}, {2, 8, 15}, {2, 13, 18}, {3, 11, 17}, {3, 13, 14}, {3, 15, 16}, {4, 5, 9}, {4, 10, 13}, {6, 15, 18}, {6, 16, 17}, {7, 8, 16}, {7, 10, 14}, {7, 11, 18}, {7, 15, 17}, {9, 11, 14}, {9, 12, 13}.

Design 19.103, o(G)=9, |SDS|=21

{0, 2, 4}, {0, 5, 8}, {0, 7, 9}, {0, 12, 16}, {1, 2, 5}, {1, 11, 12}, {1, 14, 15}, {2, 10, 14}, {2, 11, 16}, {3, 5, 9}, {3, 8, 10}, {3, 16, 17}, {4, 11, 18}, {4, 12, 14}, {4, 13, 15}, {6, 7, 15}, {6, 8, 13}, {6, 9, 10}, {6, 17, 18}, {8, 9, 15}, {9, 14, 17}.

Design 19.104, o(G)=9, |SDS|=21

{0, 1, 3}, {0, 6, 8}, {0, 12, 15}, {1, 2, 5}, {1, 10, 14}, {1, 17, 18}, {2, 3, 6}, {2, 8, 14}, {2, 9, 17}, {3, 8, 13}, {3, 12, 17}, {4, 7, 12}, {4, 9, 10}, {4, 11, 16}, {6, 7, 13}, {6, 14, 15}, {7, 9, 18}, {7, 15, 16}, {8, 10, 18}, {9, 11, 15}, {10, 13, 16}.