Mathematics and Its Applications

Further Computational and Constructive Design Theory

By: W.D. Wallis (Editor)

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Hardcover | 30 September 2003 | Edition Number 2

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392 Pages

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Revised

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23.5 x 15.88 x 3.18

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This volume is a sequel to our 1996 compilation, Computational and Constructive Design Theory. Again we concentrate on two closely re lated aspects of the study of combinatorial designs: design construction and computer-aided study of designs. There are at least three classes of constructive problems in design theory. The first type of problem is the construction of a specific design. This might arise because that one particular case is an exception to a general rule, the last remaining case of a problem, or the smallest unknown case. A good example is the proof that there is no projective plane of parameter 10. In that case the computations involved were not different in kind from those which have been done by human brains without electronic assistance; they were merely longer. Computers have also been useful in the study of combinatorial spec trum problems: if a class of design has certain parameters, what is the set of values that the parameters can realize? In many cases, there is a recursive construction, so that the existence of a small number of "starter" designs leads to the construction of infinite classes of designs, and computers have proven very useful in finding "starter" designs.

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Preface	p. xi
The Existence of 2-SOLSSOMs	p. 1
Introduction	p. 1
Direct Constructions	p. 4
Recursive Constructions	p. 8
Conjugate Orthogonal Diagonal Latin Squares with Missing Subsquares	p. 23
Introduction	p. 24
Preliminaries	p. 27
Direct Construction	p. 30
Recursive Construction	p. 33
A General Bound	p. 37
The Case 2 [less than or equal] n [less than or equal] 6	p. 41
Combinatorial Trades: a Survey of Recent Results	p. 47
Introduction and Preliminaries	p. 47
Trades in designs	p. 48
Trades in latin squares	p. 53
Trade uses	p. 54
Latin representations and trades used in tripartite decompositions	p. 56
[mu]-way trades	p. 58
Graphical trades	p. 60
Future directions	p. 62
Two-stage Generalized Simulated Annealing for the Construction of Change-Over Designs	p. 69
Introduction	p. 69
Notation and Definitions	p. 71
A Two Stage Generalized Simulated Annealing Algorithm	p. 73
Performance of the Two-Stage GSA	p. 75
New Lower Bounds on the Maximum Number of Mutually Orthogonal Steiner Triple Systems	p. 81
Introduction	p. 81
Starters over Finite Fields	p. 83
Automorphisms and Hill-climbing	p. 92
Conclusion	p. 99
On minimal defining sets in AG(d, 3)	p. 103
Background	p. 104
Introduction	p. 105
Further Preliminaries	p. 116
Proof of the Main Theorem	p. 121
Sets of good hyperplanes in AG(d, 3)	p. 124
Related critical sets	p. 127
Appendix	p. 129
Hadamard Matrices, Orthogonal designs and Construction Algorithms	p. 133
Algorithms for constructing Hadamard matrices	p. 134
On inequivalent Hadamard matrices	p. 163
Algorithms for constructing orthogonal designs	p. 176
Short amicable sets and Kharaghani type orthogonal designs	p. 193
Constructing a Class of Designs with Proportional Balance	p. 207
Proportionally balanced designs	p. 208
The method of construction	p. 211
Feasible parameters for [alpha] = 2, 3 and 4	p. 214
Constructions motivated by Construction 5	p. 218
Constructions Using Balanced n-ary Designs	p. 227
Introduction	p. 227
Balanced n-ary Designs	p. 229
Our Construction	p. 230
Some Examples	p. 232
Some Non-existence Results	p. 235
An Improved Construction	p. 238
Quadratic Residues	p. 240
Another Improvement	p. 243
An Application	p. 246
More Difference Families	p. 247
Sets of Steiner Triple Systems of Order 9 Revisited	p. 255
Introduction	p. 255
Large sets	p. 257
Results	p. 260
Uniformly intersecting sets	p. 263
Solving Isomorphism Problems for t-Designs	p. 277
Introduction	p. 277
Basics	p. 279
Moebius Inversions	p. 282
Extensions of designs	p. 293
Finding Double Youden Rectangles	p. 301
Introduction	p. 301
Definitions and Literature	p. 302
Examples	p. 302
Searching for a 13-cyclic 13 X 40 DYR	p. 306
Isomorphism	p. 309
Further Properties of our new DYRs	p. 313
Check for Balance	p. 314
Kirkman Triple Systems and their Generalizations: A Survey	p. 317
Introduction	p. 318
Constructions for Kirkman Triple Systems and Nearly Kirkman Triple Systems for all admissible orders	p. 334
Early Generalizations	p. 342
Resolvable Packings and Coverings of v points where v [not identical with] 0 (mod 3)	p. 348
Other Generalizations	p. 359
Conclusion and Acknowledgements	p. 363
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Mathematics and Its Applications

Further Computational and Constructive Design Theory

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Hardcover

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