This is a second volume of a monograph on the theory of block designs considered within a randomization approach. In the first volume, a general theory of the analysis of experiments in block designs based on models derived from the randomization principle has been presented. This volume is devoted to methods of constructing block designs that satisfy the statistical requirements established in Volume I. General consideration of constructional aspects of the designs is given in Chapter 6. The main distinction among the designs is made depending on whether they allow some of the contrasts of treatment parameters to be estimated with full efficiency or not. Designs that possess this property are presented in Chapter 7, other in Chapter 8. Separately, in Chapter 9, resolvable block designs are considered. Designs requiring special consideration are discussed in Chapter 10. The way of presentation of the designs and their properties should allow any choice of a block design to be made in full accordance with the purpose of the experiment. The book is aimed at an advanced audience, including students at the postgraduate level, well acquainted with basic texts on experimental design, and also research workers interested in designing and analyzing experiments with full understanding of the principles. Although the monograph does not pretend to give an exhaustive exposition of the theory of experimental design within the randomization approach, it is hoped that at least it gives some perspectives for further research on the subject. Tadeusz Calin'ski taught statistics, biometry and experimental design at the Agricultural University of Poznan' from 1953-1988. He obtained the title of Professor of Natural Sciences in 1974. He was head of the Department of Mathematical and Statistical Methods from 1968-1984. He is now associated with this department as Professor Emeritus. In 1998 Professor Calin'ski was awarded the Doctoral Degree honoris causa by the Agricultural University of Poznan'. His research interests include mathematical statistics and biometry, with applications to agriculture, natural sciences, biology and genetics. He has published over 150 articles in scientific journals. He is a member of the International Statistical Institute and of Bernoulli Society, the International Biometric Society, the Institute of Mathematical Statistics, and some Polish scientific societies. He has served on the editorial boards of the Journal of Statistical Planning and Inference, Biometrics, and several Polish scientific journals. Sanpei Kageyama has been Professor of Statistics and Discrete Mathematics in the Department of Mathematics, Hiroshima University, Japan, since 1992. His areas of research include design of experiments and combinatorics. He has published over 260 articles in scientific journals. Professor Kageyama is a Foundation Fellow as well as a council member of the Institute of Combinatorics and its Applications, Canada, a member of the International Statistical Institute, the Institute of Mathematical Statistics, and the National Committee for Statistics, Science Council of Japan. He has served on the editorial boards of the Journal of Japan Statistical Society, Utilitas Mathematics, Sankhya, Discussiones Mathematicae-Probability and Statistics, and the Journal of Statistical Planning and Inference.

This book will be of interest to mathematical statisticians and biometricians interested in block designs. The emphasis of the book is on the randomization approach to block designs. After presenting the general theory of analysis based on the randomization model in Part I, the constructional and combinatorial properties of design are described in Part II. The book includes many new or recently published materials.

This book is the modern first treatment of experimental designs, providing a comprehensive introduction to the interrelationship between the theory of optimal designs and the theory of cubature formulas in numerical analysis. It also offers original new ideas for constructing optimal designs. The book opens with some basics on reproducing kernels, and builds up to more advanced topics, including bounds for the number of cubature formula points, equivalence theorems for statistical optimalities, and the Sobolev Theorem for the cubature formula. It concludes with a functional analytic generalization of the above classical results. Although it is intended for readers who are interested in recent advances in the construction theory of optimal experimental designs, the book is also useful for researchers seeking rich interactions between optimal experimental designs and various mathematical subjects such as spherical designs in combinatorics and cubature formulas in numerical analysis, both closely related to embeddings of classical finite-dimensional Banach spaces in functional analysis and Hilbert identities in elementary number theory. Moreover, it provides a novel communication platform for "design theorists" in a wide variety of research fields.