The tables on this page extend the tables given on the paper. In particular,
we give more detailed statistics on the number of resolvable designs and the
number of resolutions a design has. For completeness we also include data
on the (9,3,) designs for
=1,2,3.
Some of the resolvable designs and their resolutions are available
through the links on the table entries.
The resolvable design data on this page is presented in a format suitable
for the
GAP
toolkit.
See below for a description of the file format.
(*) The 2-(9,3,5) design table contains only information on those
resolvable 2-(9,3,5) designs that admit a resolution with a nontrivial
automorphism group.
Furthermore, only such resolutions (with a nontrivial automorphism group)
were counted to produce the ``Nres''-column.
So, it is possible (and likely) that
the resolvable designs listed contain additional resolutions with
a trivial automorphism group.
Furthermore, the actual number of resolvable 2-(9,3,5) designs is
likely to be orders of magnitude larger than 57125 because most of the
resolutions have a trivial automorphism group (see below).
The resolvable design data files are presented in ASCII format
suitable for the
GAP toolkit.
Below is an example resolvable design listed together with its resolutions.
# Design 119953: 6 resolution(s), autom. group order 72
D119953:=[[1,2,3],[1,2,3],[1,2,4],[1,2,5],[1,3,6],[1,3,7],
[1,4,5],[1,4,6],[1,4,8],[1,5,7],[1,5,9],[1,6,7],
[1,6,8],[1,7,9],[1,8,9],[1,8,9],[2,3,8],[2,3,9],
[2,4,5],[2,4,7],[2,4,9],[2,5,6],[2,5,8],[2,6,7],
[2,6,7],[2,6,8],[2,7,9],[2,8,9],[3,4,5],[3,4,5],
[3,4,7],[3,4,8],[3,5,6],[3,5,9],[3,6,7],[3,6,9],
[3,7,8],[3,8,9],[4,6,8],[4,6,9],[4,6,9],[4,7,8],
[4,7,9],[5,6,8],[5,6,9],[5,7,8],[5,7,8],[5,7,9]];
G119953:=Group([(4,5)(6,7)(8,9),
(2,3)(4,6)(5,7),
(2,8)(3,9)(5,6),
(1,2)(6,9)(7,8)]);
R119953_1:=[[1,40,46],[2,41,47],[3,37,45],[4,36,42],[5,23,43],[6,21,44],
[7,28,35],[8,17,48],[9,27,33],[10,18,39],[11,24,32],[12,19,38],
[13,20,34],[14,26,29],[15,22,31],[16,25,30]];
RG119953_1:=Group([(2,3)(4,6)(5,7),
(1,2)(6,9)(7,8),
(1,4,2,9,3,6)(5,7,8)]);
R119953_2:=[[1,40,46],[2,41,47],[3,37,45],[4,36,42],[5,23,43],[6,21,44],
[7,28,35],[8,17,48],[9,27,33],[10,18,39],[11,26,31],[12,19,38],
[13,20,34],[14,22,32],[15,24,29],[16,25,30]];
RG119953_2:=Group([(4,5)(6,7)(8,9),
(2,3)(4,6)(5,7),
(2,8)(3,9)(5,6),
(1,2)(6,9)(7,8)]);
R119953_3:=[[1,40,46],[2,43,44],[3,37,45],[4,36,42],[5,21,47],[6,23,41],
[7,28,35],[8,17,48],[9,27,33],[10,18,39],[11,24,32],[12,19,38],
[13,20,34],[14,26,29],[15,22,31],[16,25,30]];
RG119953_3:=Group([(2,8)(3,9)(5,6),
(1,3)(4,9)(5,8)]);
R119953_4:=[[1,40,46],[2,41,47],[3,37,45],[4,36,42],[5,23,43],[6,21,44],
[7,24,38],[8,17,48],[9,27,33],[10,18,39],[11,25,32],[12,28,29],
[13,20,34],[14,26,30],[15,19,35],[16,22,31]];
RG119953_4:=Group([(2,3)(4,6)(5,7),
(1,2)(6,9)(7,8),
(1,7)(2,8)(3,5)(6,9)]);
R119953_5:=[[1,40,46],[2,43,44],[3,37,45],[4,36,42],[5,21,47],[6,23,41],
[7,24,38],[8,17,48],[9,27,33],[10,18,39],[11,25,32],[12,28,29],
[13,20,34],[14,26,30],[15,19,35],[16,22,31]];
RG119953_5:=Group([(1,3)(4,9)(5,8),
(1,8)(2,7)(3,5)]);
R119953_6:=[[1,42,45],[2,43,44],[3,36,46],[4,37,40],[5,21,47],[6,23,41],
[7,24,38],[8,17,48],[9,27,33],[10,18,39],[11,25,32],[12,28,29],
[13,20,34],[14,26,30],[15,19,35],[16,22,31]];
RG119953_6:=Group([(2,3)(4,6)(5,7),
(1,4,8,6)(2,3,5,7)]);
This resolvable 2-(9,3,4) design has automorphism group order 72 and
6 nonisomorphic resolutions.
D119953:=[[1,2,3],[1,2,3],[1,2,4],[1,2,5],[1,3,6],[1,3,7],
[1,4,5],[1,4,6],[1,4,8],[1,5,7],[1,5,9],[1,6,7],
[1,6,8],[1,7,9],[1,8,9],[1,8,9],[2,3,8],[2,3,9],
[2,4,5],[2,4,7],[2,4,9],[2,5,6],[2,5,8],[2,6,7],
[2,6,7],[2,6,8],[2,7,9],[2,8,9],[3,4,5],[3,4,5],
[3,4,7],[3,4,8],[3,5,6],[3,5,9],[3,6,7],[3,6,9],
[3,7,8],[3,8,9],[4,6,8],[4,6,9],[4,6,9],[4,7,8],
[4,7,9],[5,6,8],[5,6,9],[5,7,8],[5,7,8],[5,7,9]];
(Note that there are repeated blocks.)
[ [ [ 1, 2, 3 ], [ 4, 6, 9 ], [ 5, 7, 8 ] ],
[ [ 1, 2, 3 ], [ 4, 6, 9 ], [ 5, 7, 8 ] ],
[ [ 1, 2, 4 ], [ 3, 7, 8 ], [ 5, 6, 9 ] ],
[ [ 1, 2, 5 ], [ 3, 6, 9 ], [ 4, 7, 8 ] ],
[ [ 1, 3, 6 ], [ 2, 5, 8 ], [ 4, 7, 9 ] ],
[ [ 1, 3, 7 ], [ 2, 4, 9 ], [ 5, 6, 8 ] ],
[ [ 1, 4, 5 ], [ 2, 8, 9 ], [ 3, 6, 7 ] ],
[ [ 1, 4, 6 ], [ 2, 3, 8 ], [ 5, 7, 9 ] ],
[ [ 1, 4, 8 ], [ 2, 7, 9 ], [ 3, 5, 6 ] ],
[ [ 1, 5, 7 ], [ 2, 3, 9 ], [ 4, 6, 8 ] ],
[ [ 1, 5, 9 ], [ 2, 6, 7 ], [ 3, 4, 8 ] ],
[ [ 1, 6, 7 ], [ 2, 4, 5 ], [ 3, 8, 9 ] ],
[ [ 1, 6, 8 ], [ 2, 4, 7 ], [ 3, 5, 9 ] ],
[ [ 1, 7, 9 ], [ 2, 6, 8 ], [ 3, 4, 5 ] ],
[ [ 1, 8, 9 ], [ 2, 5, 6 ], [ 3, 4, 7 ] ],
[ [ 1, 8, 9 ], [ 2, 6, 7 ], [ 3, 4, 5 ] ] ]
(Again note that there are repeated parallel classes.)
Furthermore, the automorphism group of this resolution has the following
generators: