Latin Squares (eBook)

Front Cover 1
Latin Squares: New Developments in the Theory and Applications 4
Copyright Page 5
CONTENTS 8
Preface 12
Acknowledgements 14
CHAPTER 1. INTRODUCTION 16
(1) Basic definitions 16
(2) Orthogonal latin squares 17
(3) Isotopy and parastrophy 19
CHAPTER 2. TRANSVERSALS AND COMPLETE MAPPINGS 22
(1) Basic facts and definitions 22
(2) Partial transversals 24
(3) Number of transversals in a latin square 29
(4) Sets of mutually orthogonal latin squares with no common transversal 38
(5) Sets of mutually orthogonal latin squares which are not extendible 43
(6) Generalizations of the concepts of transversal and complete mapping 48
ADDITIONAL REMARKS 54
CHAPTER 3. SEQUENCEABLE AND R-SEQUENCEABLE GROUPS: ROW COMPLETE LATIN SQUARES 58
(1) Row-complete latin squares and sequenceable groups 58
(2) Quasi-complete latin squares, terraces and quasi- sequenceable groups 73
(3) R-sequenceable and Rh-sequenceable groups 82
(4) Super P-groups 90
(5) Tuscan squares and a graph decomposition problem 94
(6) More results on the sequencing and 2-sequencing of groups 99
ADDITIONAL REMARKS 114
CHAPTER 4. LATIN SQUARES WITH AND WITHOUT SUBSQUARES OF PRESCRIBED TYPE 116
(1) Introduction 117
(2) Without subsquares 128
(3) With subsquares 134
(4) With subsquares and orthogonal 148
(5) Acknowledgement 162
ADDITIONAL REMARKS BY THE EDITORS 162
CHAPTER 5. RECURSIVE CONSTRUCTIONS OF MUTUALLY ORTHOGONAL LATIN SQUARES 164
(1) Introductory definitions 165
(2) Pairwise balanced designs - definitions 166
(3) Simple constructions for transversal designs 167
(3)* Examples 171
(4) Wilson's construction 174
(4)* Examples 176
(5) Weighting and holes 177
(5)* Examples 179
(6) Asymptotic results 180
(7) Table of values of N(v) up to v=200 181
ADDITIONAL REMARKS BY THE EDITORS 181
CHAPTER 6. r-ORTHOGONAL LATIN SQUARES 184
(1) Some weaker modifications of the concept of orthogonality 184
(2) r-Orthogonal latin squares and quasigroups 186
(3) Partial admissibility of quasigroups, its connection with r-orthogonality 192
(4) Spectra of partial orthogonality of latin squares (quasigroups) 201
(5) Near-orthogonal and perpendicular latin squares 205
(6) r-Orthogonal sets of latin squares 210
(7) Applications of r-orthogonal latin squares and problems raised thereby 215
CHAPTER 7. LATIN SQUARES AND UNIVERSAL ALGEBRA 218
(1) Universal algebra preliminaries 219
(2) Varieties of latin squares 221
(3) Varieties of orthogonal latin squares 223
(4) Euler's conjecture 226
(5) Free algebras and orthogonal latin squares 227
CHAPTER 8. EMBEDDING THEOREMS FOR PARTIAL LATIN SQUARES 232
(1) Introduction 233
(2) Systems of distinct representatives 234
(3) The theorems of Ryser and Evans (on latin rectangles and squares) 237
(4) Cruse's theorems (on commutative latin rectangles and squares) 240
(5) Embedding idempotent latin squares 244
(6) Conjugate quasigroups and identities 251
(7) Embedding semisymmetric and totally symmetric quasigroups 255
(8) Embedding Mendelsohn and Steiner triple systems 258
(9) Summary of embedding theorems 268
(10) The Evans' conjecture. (Smetaniuk's proof) 269
APPENDIX (1). Alternative description of Smetaniuk's proof of the Evans' conjecture 276
APPENDIX (2). Additional Bibliography 280
CHAPTER 9. LATIN SQUARES AND CODES 282
(1) Basic facts about error-detecting and correcting codes 283
(2) Codes based on orthogonal latin squares and their generalizations 287
(3) Row and column complete latin squares in coding theory 298
(4) Two-dimensional coding problems 305
(5) Secret-sharing systems 318
(6) Miscellaneous results 323
ADDITIONAL REMARKS 329
CHAPTER 10. LATIN SQUARES AS EXPERIMENTAL DESIGNS 332
(1) Introduction 332
(2) The design and and analysis of experiments 333
(3) Some practical examples of latin squares used as row-and- column designs 337
(4) Some other uses of latin squares in experimental design 339
(5) The use of latin squares in experiments with changing treatments 342
(6) Other "latin" experimental designs 344
(7) Statistical analysis of latin square designs 346
(8) Randomization of latin square designs 353
(9) Polycross designs 356
CHAPTER 11. LATIN SQUARES AND GEOMETRY 358
(1) Complete sets of mutually orthogonal latin squares and projective planes 358
(2) Projective planes of orders 9, 10, 12 and 15 361
(3) Non-desarguesian projective planes of prime order 366
(4) Digraph complete sets of latin squares and incidence matrices 367
(5) Complete sets of column orthogonal latin squares and affine planes 373
(6) The Paige-Wexler latin squares 375
(7) Miscellanea 388
ADDENDUM. 392
CHAPTER 12. FREQUENCY SQUARES 396
(1) F-squares and orthogonal F-squares 396
(2) Enumeration and classification of F-squares 403
(3) Completion of partial F-squares 404
(4) F-rectangles and other generalizations 407
(5) A generalized Bose construction for orthogonal F-squares 411
ADDITIONAL REMARKS 413
Bibliography 414
Subject Index 459