Design theory is a branch of combinatorics. Its traditional roots are in the design of experiments, but it has found recent applications in cryptography, coding theory and communication networks. Design theory has grown to be a subject of considerable interest in  mathematics, not only in itself, but for its connections to other fields such as geometry, group theory, graph theory and coding theory.

Over the past several decades, algebra has become increasingly important in combinatorial design theory. The flow of ideas has for the most part been from algebra to design theory. Moreover, despite our successes, fundamental algebraic questions in design theory remain open. It seems that new or more sophisticated ideas and techniques will be needed to make progress on these questions. In the meantime, design theory is a fertile source of problems that are ideal for spurring the development of algorithms in the active field of computational algebra. Combinatorial designs are used to determine which patients receive which treatments in such a way that if a given response is observed, then the structure of the design would indicate the treatment that caused it. Modern applications are also found in a wide gamut of areas including; Finite geometry, tournament scheduling, lotteries, mathematical chemistry, mathematical biology, algorithm design and analysis, networking, group testing and cryptography.

This Course includes an introduction to Design Theory including a selection of topics from Latin squares, Steiner triple systems, balanced incomplete block designs, graph decompositions, projective and affine designs. The  course should allow students subsequently to read further in these areas, and to apply their knowledge of graph theory and design theory to other appropriate fields.

INTENDED LEARNING OUTCOMES

1. Demonstrate understanding of basic concepts and discussing graphic design theory.
2. Investigates of Purpose and Significance of Graphic Design.
3. Derive mathematical models for various tasks, such as Traditional medical experiments: Patients are given a single candidate treatment and are observed for a given response (i.e. dose-response).
4. Combinatorial designs are used to determine which patients receive which treatments in such a way that if a given response is observed, then the structure of the design would indicate the treatment that caused it.
5. Essentially, it's faster and cheaper to conduct exploratory research using a scheme based on a combinatorial design.
6. Implement the idea in Maple or Matlab.
7. Write efficient, well-documented Maple/Matlab code and present results in an informative way.

Prerequisites: Discrete Mathematics

CONTENTS

1. Basic definitions and properties,
2. Related structure, The incidence matrix,
3. Graphs, residual structures,
4. The Bruck - Ryser-Chowla theorem,
5. Singer groups and difference sets.
6. Arithmetical relations and Hadamard 2- designs,
7. Projective and affine planes,
8. Latin squares, nets. Hadamard matrices and Hadamard 20 design,
9. Biplanes, strongly regular graphs,
10. Cameron’s theorem and Hadamard 3-desings,
11. Steiner triple systems, The Mathieu groups.

BOOKS RECOMMENDED

1. Beth, T., Jungnickel, D., & Lenz, H. (1999). Design Theory: Volume 1. Cambridge University Press.
2. Wallis, W. D. (2016). Introduction to combinatorial designs. CRC Press.

BOOKS SUGGESTED

1. Cameron, P. J., Van Lint, J. H., & Cameron, P. J. (1991). Designs, graphs, codes and their links (Vol. 3). Cambridge: Cambridge University Press.
2. Cameron, P. J. (1999). Permutation groups (Vol. 45). Cambridge University Press
3. Lindner, C. C., & Rodger, C. A. (2017). Design theory. CRC press

RESEARCH PROJECT /PRACTICALS/LABS/ASSIGNMENTS

Research project/assignment based on numerical techniques and algorithms often involve features which require their solutions can be awarded to students during their course work period. Examples of features are

1. Accelerating convergence of numerical methods.
2. Finding numerical solution of analytical complex problems
3. Improving efficiency of numerical techniques

Assessment Criteria

• Sessional: 20 (Presentation / Assignment 10, Attendance 05, Quiz 05)
• Mid-Term Exam: 30
• Final-Term Exam: 50

Key Dates and Time of Class Meeting

Wednesday/Thurseday                                                           02:00 - 03:30 pm

Commencement of Classes                                                   March 15, 2021

Mid Term Examination                                    May 17 to 21, 2021 (Monday to Friday)

Final Term Examination                             July 12 to 16, 2021 (Monday to Friday)

Declaration of Result                                   July 27, 2021 (Friday)