LATIN SQUARES New Developments in the Theory and Applications J. DENES Industrial and Scientific Consultant Formerly Head of Mathematics Institute for Research and Co-ordination of Computing Techniques (SZKI) Budapest, Hungary and A.D. KEEDWELL Department of Mathematical and Computing Sciences University of Surrey Guildford, United Kingdom With specialist contributions by G.B. BELYAVSKAYA A.E. BROUWER T. EVANS K. HEINRICH C.C. LINDNER D.A. PREECE 1991 NORTH-HOLLAND - AMSTERDAM NEW YORK OXFORD TOKYO
Vll CONTENTS Preface Acknowledgements xi xiii CHAPTER 1. INTRODUCTION. (J. Denes and A. D. Keedwell) (1) Basic definitions. 1 (2) Orthogonal latin squares. 2 (3) Isotopy and parastrophy. 4 CHAPTER 2. TRANSVERSALS AND COMPLETE MAPPINGS. (J. De'nes and A. D. Keedwell) (1) Basic facts and definitions. 7 (2) Partial transversals. 9 (3) Number of transversals in a latin square. 14 (4) Sets of mutually orthogonal latin squares with no common transversal. 23 (5) Sets of mutually orthogonal latin squares which are not extendible. 28 (6) Generalizations of the concepts of transversal and complete mapping. 33 ADDITIONAL REMARKS. 3 9 CHAPTER 3. SEQUENCEABLE AND R-SEQUENCEABLE GROUPS: ROW COMPLETE LATIN SQUARES. (J. De'nes and A. D. Keedwell) (1) Row-complete latin squares and sequenceable groups. 43 (2) Quasi-complete latin squares, terraces and quasisequenceable groups. 58 (3) R-sequenceable and R h -sequenceable groups. 67 (4) Super P-groups. 75 (5) Tuscan squares and a graph decomposition problem. 79 (6) More results on the sequencing and 2-sequencing of groups. 84 ADDITIONAL REMARKS. 99
Vlll Contents CHAPTER 4. LATIN SQUARES WITH AND WITHOUT SUBSQUARES OF PRESCRIBED TYPE. (K. Heinrich) 101 (1) Introduction. 102 (2) Without subsquares. 113 (3) With subsquares. 119 (4) With subsquares and orthogonal. 133 (5) Acknowledgement. 147 ADDITIONAL REMARKS BY THE EDITORS. 147 CHAPTER 5. RECURSIVE CONSTRUCTIONS OF MUTUALLY ORTHOGONAL LATIN SQUARES. (A. E. Brouwer) 149 (1) Introductory definitions. 150 (2) Pairwise balanced designs - definitions. 151 (3) Simple constructions for transversal designs. 152 (3)* Examples. 156 (4) Wilson's construction. 159 (4)* Examples. 161 (5) Weighting and holes. 162 (5)* Examples. 164 (6) Asymptotic results. 165 (7) Table of values of N(v) up to v=200. 166 ADDITIONAL REMARKS BY THE EDITORS. 166 CHAPTER 6. r-orthogonal LATIN SQUARES. (G. B. Belyavskaya) (1) Some weaker modifications of the concept of orthogonality. 169 (2) r-orthogonal latin squares and quasigroups. 171 (3) Partial admissibility of quasigroups, its connection with r-orthogonality. 177 (4) Spectra of partial orthogonality of latin squares (quasigroups). 186 (5) Near-orthogonal and perpendicular latin squares. 190 (6) r-orthogonal sets of latin squares. 195 (7) Applications of r-orthogonal latin squares and problems raised thereby. 200 CHAPTER 7. LATIN SQUARES AND UNIVERSAL ALGEBRA. (T. Evans) 203 (1) Universal algebra preliminaries. 204 (2) Varieties of latin squares. 206
Contents ix (3) Varieties of orthogonal latin squares. 208 (4) Euler's conjecture. 211 (5) Free algebras and orthogonal latin squares. 212 CHAPTER 8. EMBEDDING THEOREMS FOR PARTIAL LATIN SQUARES. (С. С Lindner) 217 (1) Introduction. 218 (2) Systems of distinct representatives. 219 (3) The theorems of Ryser and Evans (on latin rectangles and squares). 222 (4) Cruse's theorems (on commutative latin rectangles and squares). 225 (5) Embedding idempotent latin squares. 229 (6) Conjugate quasigroups and identities. 236 (7) Embedding semisymmetric and totally symmetric quasigroups. 240 (8) Embedding Mendelsohn and Steiner triple systems. 243 (9) Summary of embedding theorems. 253 (10) The Evans' conjecture. (Smetaniuk's proof.) 254 APPENDIX (1). Alternative description of Smetaniuk's proof of the Evans' conjecture. 261 APPENDIX (2). Additional Bibliography. 265 CHAPTER 9. LATIN SQUARES AND CODES. (J. De'nes and A. D. Keedwell) 267 (1) Basic facts about error-detecting and correcting codes. 268 (2) Codes based on orthogonal latin squares and their generalizations. 272 (3) Row and column complete latin squares in coding theory. 283 (4) Two-dimensional coding problems. 290 (5) Secret-sharing systems. 303 (6) Miscellaneous results. 308 ADDITIONAL REMARKS. 314 CHAPTER 10. LATIN SQUARES AS EXPERIMENTAL DESIGNS. (D. A. Preece) (1) Introduction. 317 (2) The design and and analysis of experiments.. 318
X Contents (3) Some practical examples of latin squares used as row-andcolumn designs. 322 (4) Some other uses of latin squares in experimental design. 324 (5) The use of latin squares in experiments with changing treatments. 327 (6) Other "latin" experimental designs. 329 (7) Statistical analysis of latin square designs. 331 (8) Randomization of latin square designs. 338 (9) Polycross designs. 341 CHAPTER 11. LATIN SQUARES AND GEOMETRY. (J. Denes and A. D. Keedwell) (1) Complete sets of mutually orthogonal latin squares and projective planes. 343 (2) Projective planes of orders 9, 10, 12 and 15. 346 (3) Non-desarguesian projective planes of prime order. 351 (4) Digraph complete sets of latin squares and incidence matrices. 352 (5) Complete sets of column orthogonal latin squares and affine planes. 358 (6) The Paige-Wexler latin squares. 360 (7) Miscellanea. 373 ADDENDUM. 377 CHAPTER 12. FREQUENCY SQUARES. (J. De'nes and A. D. Keedwell) (1) F-squares and orthogonal F-squares. 381 (2) Enumeration and classification of F-squares. 388 (3) Completion of partial F-squares. 3 89 (4) F-rectangles and other generalizations. 392 (5) A generalized Bose construction for orthogonal F-squares. 396 ADDITIONAL REMARKS. 398 Bibliography 399 Subject Index 444