(Mainly) Nests of designs
© The research materials and data herein are the copyright of the
Centre for Discrete Mathematics and Computing,
The University of Queensland,
24 November, 1999. They may only be used with the permission of
Professor Anne Penfold Street and
Dr Greg Gamble.
Contents
The sections (= design parameters) for which data have been obtained
are listed below. Unless otherwise specified links within the sections
listed are to minimal defining sets, and their automorphism groups and
(usually) nests of the given named design. Other information (not
to do with nests) follows an Other subheading.
| 2-(6,3,2)
| 2-(6,3,4)
| 2-(7,3,1)
| 2-(7,3,2)
|
| 2-(7,3,3)
| 2-(8,4,3)1
= 3-(8,4,1)
| 2-(9,3,1)
| 2-(9,3,2)
|
| 2-(10,4,2)
| 2-(10,4,4)1
= 3-(10,4,1)
| 2-(11,5,2)
| 2-(13,3,1)
|
| 2-(13,4,1)
| 2-(15,3,1)
| 2-(15,7,3)
| 2-(16,4,1)
|
| 2-(16,6,2)
| 2-(19,9,4)
| 2-(21,5,1)
| 2-(23,11,5)
|
| 2-(25,5,1)
| 2-(31,6,1)
| 2-(31,15,7)
| 3-(8,4,1)2
= 2-(8,4,3)
|
| 3-(10,4,1)2
= 2-(10,4,4)
| 4-(11,5,1)
|
- 1Treated as a 2-design.
- 2Treated as a 3-design.
2-(6,3,2)
2-(6,3,4)
There are four 2-(6,3,4) designs (up to isomorphism), only one of which,
D1, is simple. Each design is reducible and organised so that the first
10 blocks comprise the 2-(6,3,2) design D and the last 10 blocks are
given as F^g for some permutation g. Non-defining sets and check statistics
are also included.
2-(7,3,1)
2-(7,3,2)
There are four 2-(7,3,2) designs (up to isomorphism), only one of which,
D1, is simple. Each design is reducible and organised so that the first
7 blocks comprise the 2-(7,3,1) design F and the last 7 blocks are given
as F^g for some permutation g. Non-defining sets and check statistics
are also included.
2-(7,3,3)
There are ten 2-(7,3,3) designs (up to isomorphism), only one of which,
D1, is simple and irreducible. The remaining designs, D2 through D10
are both non-simple and reducible, and all are written in the form
F U F^g U F^h for some g, h in S_7. Currently nests are missing for
2 of the 219 minimal defining sets of D9. Non-defining sets and check
statistics are given for each reducible design. There is a fairly simple
mapping to Liz Billington's classification of these designs.
2-(8,4,3)
2-(9,3,1)
Other:
2-(9,3,2)
The smallest defining sets of the reducible 2-(9,3,2) designs and
their automorphism groups only, are given. There are 9 reducible
2-(9,3,2) designs and 27 other irreducible designs (up to isomorphism).
Each (reducible) design is written in the form D1 U D1^g
for some g in S_9, where D1 is the simple 2-(9,3,1) design D1.
There is a fairly simple mapping to Liz Billington's classification of
these designs. Check statistics for all minimal defining sets are included
for designs K1, K2, K3 and K14. Check statistics for all smallest defining
sets are included for the other reducible designs: K6, K7, K15, K23 and K29.
2-(10,4,2)
2-(10,4,4)
Nests of just one smallest defining set (of length 16) and a
minimal defining set of length 17 are given.
2-(11,5,2)
2-(13,3,1)
Nests for smallest defining sets only.
2-(13,4,1)
2-(15,3,1)
We give all the nests for design P3 (the Moran-labelled version
of W1; see below).
We give the nest of just one minimal defining set of each design.
The point set is 0..9,a..e (as per Colin Ramsay). Tony Moran labelled
the points differently.
Other:
Each link below provides examples of nondefining sets S of a design D
with the property that Aut S <= Aut D. Each S is formed by deleting one
block of a minimal defining set.
2-(15,7,3)
2-(16,4,1)
2-(16,6,2)
2-(19,9,4)
2-(21,5,1)
2-(23,11,5)
Nests for all 396 smallest defining sets (of length 8) and 221
other minimal defining sets (all of length 9), up to isomorphism,
are given.
2-(25,5,1)
Nests for 27 smallest defining sets (of length 10) are given. (There
are probably about 4000 smallest defining sets, and currently there is
no estimate of how many other minimal defining sets there may be.)
2-(31,6,1)
There are 6 smallest defining sets (of length 11) and 61 other
minimal defining sets of length 12.
2-(31,15,7)
All minimal (= smallest) defining sets of PG42 only.
The number of minimal defining sets for QuadRes is very large. Currently,
248 smallest defining sets (of length 9) of an estimated 120 000, and one
other minimal defining set of length 10, are known. Only the nest for one
smallest defining set has so far been determined, and N'' has been omitted.
3-(8,4,1)
3-(10,4,1)
4-(11,5,1)
Known to have 216 smallest defining sets of length 5 and 1046 other
minimal defining sets (of length 6), up to isomorphism. The nests for all
216 smallest defining sets and the complete list of smallest and length 6
minimal defining sets, are given.
Last update: 30 May, 2002.
Greg Gamble
http://www.csee.uq.edu.au/~gregg/