#### Date of Submission

3-28-1966

#### Date of Award

3-28-1967

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Supervisor

Mitra, Sujit Kumar

#### Abstract (Summary of the Work)

The theory and methods of con struction of statistical designs have connections with Modern algebraic systems, theory of numbers, arithmetic theory of quadratic forms, information theory and construction of codes. The properties of finite linear spaces have been used for the construction of (i) complete set of mutually orthogonal latin squares (Bose and Nair, 1941) (ii) balanced incomplete block designs (Bose 1939) (iii) partially balanced incomplete block designs (Bose and Nair 1939) (iv) designs where some effects of treatments are confounded (Bose, 1947). Primrose (1951) studied quadric surfaces and used them in the construction of balanced incomplete block designs. Roychoudhuri (1962) has gene - ralised his study on quadrics and obtained several series of partia1ly balanced incomplete block designs through linear spaces contained in a quadric. Shrikhande and Singh (1962) have observed some relations between association schemes and balanced incomplete block designs and using them they have given solutions to some practical designs.In this thesis solutions to balanced, doubly balanced and partially balanced incomplete block designs some of which are not listed in the known tables have been obtained. A detailed summary of work done in each chapter ie given at the beginning of that chapter. Below is given a brief summary of resulta in various chaptera.Chapter I deals with methods of constructing balanced incomplete block designs from ansociation schemes. The incidence matrix of the design is determined as the partitioned matrix (B1. B2;....; Bt) where Bi = (bijk) is the i-th association matrix jk and bijk =1 if the objects j and k are i-th associates or zero other- jk wise. In the same chapter a new general series of balanced incom- plete block designs is obtained through difference sets which con- tains a series of balanced incomplete block de signs given by Gasaner (1965). This series is obtained by taking a special set of elements of cartesian product of Galol fielde in the initial blocks. In chapter Il geometries imbedded in finite projective geometry PG(n, a) of a dimensions based on a Galois field of order s are investigated. The concept of generating Restricted Linear Analytic Independent set of points is introduced. It is shown that such a set generates a geometry isomorphie to a PG(r, imbedded in PG(n, s). The properties of PG(1, ) - Line segments imbedded in PG(1, a) are studied to a greater extent than higher dimensional spaces. Sinqular and nonsingular imbedded planes are defined and non- singular planes imbedded in a plane are used to construct a series of Regular Group Divisible designs. This series contains new designs not listed previously as shown in the appendix B. In the general case of imbedded geometries PG(r, s1) in PG(n, s) the trun- cated configuration of lines i used to construct Pairwise balanced designs (Bose and Shrikhande, 1960) which are useful in the atudy of orthogonal latin squares. The properties of line segmenta contained in a line are used to construct a series of doubly balanced incomplete block designs such that every triplet of treatments appears exactly once in the design. In chapter III non-degenerate quadrica in PG(2t-1, s) are studied. The form A of a non-degenerate quadric is classified as hyperbolic or elliptic according as (-1)t det A is a square or a non-square where the characteristic of the field is different from 2. Non linear configurations like cones and vertex-less cones cantained in a nondegenerate quadric in finite projective geometry are studied and the explicit member of such configuratione obtained. Their pro- perties are used to construct several series of symmetrie balanced and partially balanced incomplete block designs. A non-isomorphic solution to the wellknown hyperplanes solution of the symmetric balanced incomplete block design in obtained through tangent cones of a nondegenerate quadric Q2t, in PG(2t, s). Thie series includes a non-isomorphic solution to the symmetric design with parameters v - 15, k = 7, A = 3 for which Fisher and Yates (1963) show only one solution- (a, b, c, e, f, 1, k)- the cyclie one.

#### Control Number

ISILib-C6016

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

#### Recommended Citation

Ramanujacharyulu, Chilakamarri Dr., "Solutions of Some Balanced Doubly Balanced and Partially Balanced Statistical Designs." (1967). *Doctoral Theses*. 126.

https://digitalcommons.isical.ac.in/doctoral-theses/126

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842902