Development of Non-Isomorphic 3-(12,6,4) and 2-(11,5,4) Designs
Document information
| Author | A.R. Thompson |
| School | University of Canterbury |
| Major | Philosophy |
| Year of publication | 1985 |
| Place | Christchurch |
| Document type | thesis |
| Language | English |
| Number of pages | 236 |
| Format | |
| Size | 8.71 MB |
- Combinatorial Design Theory
- Block Designs
- Non-Isomorphic Structures
Summary
I. Introduction to Non Isomorphic Designs
The document explores the development of non-isomorphic 3-(12,6,4) and 2-(11,5,4) designs, which are essential in combinatorial design theory. These designs consist of a fixed number of blocks, each containing a specific number of points. The aim is to determine all possible structures that meet the given parameters. The thesis emphasizes the significance of understanding the relationships between different block types and their implications for design existence. The introduction sets the stage for a comprehensive analysis of the designs, highlighting the challenges in identifying non-isomorphic structures. The author notes, 'Given a set of design parameters, a design possessing them may or may not exist.' This statement underscores the complexity of combinatorial designs and the necessity for systematic exploration.
1.1 Overview of Block Designs
Block designs are characterized by their structure, where a set number of blocks is formed from a larger set of points. The document introduces the concept of t-(v,k,A) designs, where 't' represents the number of points in each block, 'v' is the total number of points, 'k' is the size of each block, and 'A' is the number of occurrences of each t-tuple. The author explains that the balance of these designs is crucial, as it ensures that each combination of points appears a fixed number of times. This balance is vital for applications in statistical design and experimental setups, where the integrity of data collection relies on the uniformity of design.
II. Methodology for Design Development
The methodology section outlines the systematic approach taken to develop the non-isomorphic designs. The author describes the use of computational tools to enumerate possible designs, emphasizing the importance of automation in handling complex calculations. The thesis details the process of creating a 'skeleton' for the designs, which serves as a foundational structure upon which additional elements are added. The author states, 'With the assistance of a computer and subject to the non-occurrence of repeated blocks, these structures were balanced for the remaining points.' This highlights the integration of technology in combinatorial design, allowing for the efficient exploration of vast design spaces. The methodology not only provides a framework for the current study but also sets a precedent for future research in the field.
2.1 Enumeration of Designs
The enumeration process is critical in identifying all possible non-isomorphic designs. The author discusses the significance of eliminating isomorphic designs to ensure that each unique structure is accounted for. The document notes that 'Once the isomorphs had been eliminated, all the non-isomorphic 3-(12,6,4) designs' automorphisms were developed.' This statement emphasizes the rigorous nature of the analysis, as it requires careful consideration of design properties to avoid redundancy. The results of this enumeration process contribute to a comprehensive catalogue of designs, which serves as a valuable resource for researchers and practitioners in combinatorial design.
III. Results and Applications
The results section presents a detailed catalogue of the non-isomorphic designs developed throughout the thesis. The author provides a representative copy of each design, along with its non-trivial automorphisms. This extensive documentation is crucial for future research, as it offers a reference point for scholars exploring similar combinatorial structures. The author asserts, 'A representative copy of each of the 545 non-isomorphic 3-(12,6,4) designs along with all its non-trivial automorphisms is given.' This statement underscores the thoroughness of the research and its potential impact on the field. The practical applications of these designs extend to various domains, including statistics, computer science, and operations research, where efficient design structures are essential.
3.1 Implications for Future Research
The findings of this thesis have significant implications for future research in combinatorial design. The comprehensive nature of the catalogue allows researchers to build upon existing knowledge, facilitating advancements in the field. The author encourages further exploration of the relationships between different design types, stating that 'the relationship between the block types of the 3-designs and the 2-designs is then discussed.' This exploration can lead to new insights and methodologies that enhance the understanding of combinatorial structures. The document serves as a foundational text for scholars aiming to innovate within the realm of design theory.
Document reference
- The Development of the 3-(12,6,4) and 2-(11,5,4) Designs (A.R. Thompson)
- Decomposable 3-(12,6,4) and 2-(11,5,4) Designs (A.R. Thompson)
- Combinatorial Theory and Block Designs (Unknown)
- Automorphisms of Block Designs (Unknown)
- Non-Isomorphic Designs Enumeration (Unknown)