Title of Thesis

Generalized Polygonal Designs

Author(s)

Muhammad Hussain Tahir

Abstract
Polygonal designs, a class of partially balanced incomplete block designs (PBIBDs) with regular polygons, are useful in survey sampling in terms of balanced sampling plans excluding contiguous units (BSECs) and balanced sampling plans to avoid the selection of adjacent units (BSAs), when neighboring (contiguous or adjacent) units in a population provide similar information. The reason for using such designs is that the units that are physically close might be more similar than the distant units. By the use of such designs or plans we can select the units over the entire experimental region by avoiding the selection of units that provide essentially redundant information. In other words, these neighboring units are deliberately excluded from being sampled under the idea that they provide little new information to the sampling effort.
Searches for polygonal designs may be divided into two broad categories: those which attempts to prove the existence of polygonal designs with a given set of parameters (v; k; ¸; ®), and those which attempts to construct (or enumerate) polygonal designs with a given set of parameters (v; k; ¸; ®). In this thesis, the construction of cyclic polygonal designs is generalized for the parameters: the distance ® (orm), the concurrence (or index) parameter ¸ and the treatments v.The major reasons for introducing generalized cyclic polygonal designs in this thesis are that:
(i) the existing literature considers the existence and the construction of cyclic polygonal designs only for the limited distance ®, the concurrence parameter ¸ and the treatments v;
(ii) the existence and the construction of unequal block sized cyclic polygonal designs for distance ® ¸ 1 has not been attempted in literature.
In Chapter 1, an introduction to polygonal designs is given. A brief review on the existing work on polygonal designs is presented, and some limitations in the existing work are pointed out.
In Chapter 2, the method of cyclic shifts is briefly described, and explained that how this method helps in the development of concurrence matrix (or concurrence vector) which is the main tool for the detection of the properties of cyclic polygonal designs. The distinguishing feature of this method is that the properties of a design can easily be obtained from the sets of shifts instead of constructing the actual blocks of the design. The pattern of off-diagonal zero elements (in bold form) from the main diagonal in a concurrence matrix (or in a concurrence vector) is useful in the identification of the distance in a cyclic polygonal design. In Chapter 3, minimal cyclic polygonal designs with block size k = 3 and ¸ = 1 are constructed for distance ® = 1; 2; 3; : : : ; 16 and for v < 100 treatments. In Chapter 4, the existence and construction of cyclic polygonal designs with block size k = 3, for ¸ = 1; 2; 3; 4; 6; 12 and for ® = 1; 2; 3; 4; 5; 6 is considered, and complete solutions for v · 100 treatments are presented.
In Chapter 5, the existence and construction of minimal cyclic polygonal designs with unequal-sized blocks and ¸ = 1 is first ever introduced for distance ® ¸ 1.In Chapter 6, the thesis is summarized and future directions for the extension of cyclic polygonal designs are proposed.

1

1.1 Introduction
1.2 Polygonal designs as balanced sampling plans
1.3 Polygonal designs as incomplete block designs
1.4 A brief review of one-dimensional polygonal designs
1.5 The problem

2.1 Introduction
2.2 The method of cyclic shifts
2.3 Detection of the properties of designs from the set of shifts

3.1 Introduction
3.2 Detection of the properties of CPD(v; 3; 1; ®)’s .
3.3 Algorithms for the construction of CPD(v; 3; 1; ®)’s with ® = 1; 2
3.4 Construction of CPD(v; 3; 1; ®)’s with ® = 1; 2; : : : ; 16
3.5 Concluding remarks

4.1 Cyclic polygonal designs with k = 3 and ® = 1
4.2 Cyclic polygonal designs with k = 3 and ® = 2
4.3 Cyclic polygonal designs with block size k = 3 and ® = 3
4.4 Cyclic polygonal designs with block size k = 3 and ® = 4
4.5 Cyclic polygonal designs with block size k = 3 and ® = 5
4.6 Cyclic polygonal designs with block size k = 3 and ® = 6

5.1 Introduction
5.2 CPD(v; [3; 2]; 1; ®)’s with ® = 1; 2; 3; 4; 5; 6
5.3 CPD(v; [4; 2]; 1; ®)’s with ® = 1; 2; 3; 4
5.4 CPD(v; [4; 3]; 1; ®)’s with ® = 1; 2; 3; 4
5.5 CPD(v; [4; 3; 2]; 1; ®)’s with ® = 1; 2; 3; 4
5.6 Concluding remarks

6.1 Overall summary
6.2 Future work