Poetry of repetition: constructing verse through combinatorial design theory

Combinatorial designs - a tribute to Haim Hanani

<p>As a powerful technique for holding relations in things, combinatorics has experienced rapidly development in the past century, particularly,enumeration of configurations, combinatorial design and graph theory. How-<br /> ever, the main objective for mathematics is to bring about a quantitative analysis for other sciences, which implies a natural question on combinatorics. Thus, how combinatorics can contributes to other mathematical sciences, not just in discrete mathematics, but metric mathematics and physics? After a long time speculation, I brought the CC conjecture for advancing mathematics by combinatorics, i.e., any mathematical science can be reconstructed from or made by combinatorialization in my postdoctoral report for Chinese Academy of Sciences in 2005, and then reported it at a few conferences of China. Clearly,<br /> CC conjecture is in fact a combinatorial notion and holds by a philosophical law, i.e., all things are inherently related, not isolated. The main purpose of this report is to survey the roles of CC conjecture in developing mathematical sciences with notions, such as those of its contribution to algebra, topology,Euclidean geometry and differential geometry, non-solvable differential equations or classical mathematical systems with contradictions to mathematics,quantum fields after it appeared 10 years ago. All of these show the importance of combinatorics to mathematical sciences in the past and future.</p>

Balancing Carry-Over Effects in Tournaments, Dr. Ian Anderson, University of Glasgow Resolved Designs Viewed as Sets of Partitions, Prof. Rosemary Bailey, Queen Mary and Westfield College Combinatorics and Threshold Cryptography, Dr. Simon Blackburn, Royal Holloway and Bedford New College Block-Transitive Point-Intransitive Block Designs, Dr. Alan Camina, University of East Anglia Some Recent Developments in Difference Sets, Dr. Jonathan Jedwab, Hewlett-Packard Laboratories, Bristol and James A. Davis, University of Richmond, Configurations in Steiner Triple Systems, Prof. Mike Grannell and Prof. Terry Griggs, University of Central Lancashire A Survey of Recent Results on Optimal Linear Codes, Dr. Ray Hill, University of Salford

Decomposing Steiner Triple Systems Into Four-Line Configurations

Combinatorial design theory is a very active area of mathematical research, with many applications in communications and information theory, computer science, statistics, engineering, and life sciences. As one of the fundamental discrete structures, combinatorial designs are used in fields as diverse as error-correcting codes, statistical design of experiments, cryptography and information security, mobile and wireless communications, group testing algorithms in DNA screening, software and hardware testing, and interconnection networks. This monograph provides a tutorial on combinatorial designs, which gives an overview of the theory. Furthermore, the application of combinatorial designs to authentication and secrecy codes is described in depth. This close relationship of designs with cryptography and information security was first revealed in Shannon’s seminal paper on secrecy systems. We bring together in one source foundational and current contributions concerning design-theoretic constructions and characterizations of authentication and secrecy codes.

Combinatorial design theory is a source of simply stated, concrete, yet difficult discrete problems, with the Hadamard conjecture being a prime example. It has become clear that many of these problems are essentially algebraic in nature. This book provides a unified vision of the algebraic themes which have developed so far in design theory. These include the applications in design theory of matrix algebra, the automorphism group and its regular subgroups, the composition of smaller designs to make larger designs, and the connection between designs with regular group actions and solutions to group ring equations. Everything is explained at an elementary level in terms of orthogonality sets and pairwise combinatorial designs--new and simple combinatorial notions which cover many of the commonly studied designs. Particular attention is paid to how the main themes apply in the important new context of cocyclic development. Indeed, this book contains a comprehensive account of cocyclic Hadamard matrices. The book was written to inspire researchers, ranging from the expert to the beginning student, in algebra or design theory, to investigate the fundamental algebraic problems posed by combinatorial design theory.

The code over a finite fieldF q of orderq of a design is the subspace spanned by the incidence vectors of the blocks. It is shown here that if the design is a Steiner triple system on ν points, and if the integerd is such that 2 d −1≤ν<2 d+1−1, then the binary code of the design contains a subcode that can be shortened to the binary Hamming codeH d of length 2 d −1. Similarly the binary code of any Steiner quadruple system on ν+1 points contains a subcode that can be shortened to the Reed-Muller code ℜ(d−2,d) of orderd−2 and length 2 d , whered is as above.

Our main result is an existence and uniqueness theorem for Steiner triple systems which associates to every such system a binary code — called the "carrier" — which depends only on the order of the system and its 2-rank. When the Steiner triple system is of 2-rank less than the number of points of the system, the carrier organizes all the information necessary to construct directly all systems of the given order and $2$-rank from Steiner triple systems of a specified smaller order. The carriers are an easily understood, two-parameter family of binary codes related to the Hamming codes. We also discuss Steiner quadruple systems and prove an analogous existence and uniqueness theorem; in this case the binary code (corresponding to the carrier in the triple system case) is the dual of the code obtained from a first-order Reed-Muller code by repeating it a certain specified number of times. Some particularly intriguing possible enumerations and some general open problems are discussed. We also present applications of this coding-theoretic classification to the theory of triple and quadruple systems giving, for example, a direct proof of the fact that all triple systems are derived provided those of full 2-rank are and showing that whenever there are resolvable quadruple systems on $u$ and on $v$ points there is a resolvable quadruple system on $uv$ points. The methods used in both the classification and the applications make it abundantly clear why the number of triple and quadruple systems grows in such a staggering way and why a triple system that extends to a quadruple system has, generally, many such extensions.

Constructing a set of watermarks of a specific structure may be one requirement for robust watermarking. This study aims to use the structure of Steiner triple systems to generate new watermarks. That is, the new watermark is a Steiner triple system built by using two smaller ones. The structure properties are examined to recognize watermarks at the receiver site. The main advantage is no information other than the mathematical structure has to be known for watermark recognition.Theoretical proof is given to verify the proposed watermark design method, and the experiment is conducted to confirm the theoretical behavior of the generated watermarks under random noise.

Most of the lecturers in this session on information theory are interested mainly in the probabilistic side of the subject. However, the participents are undoubtedly aware of the fact that much of what has been promised by probabilistic methods, e.g. by Shannon’s Theorems, has not been realized constructively because constructions (of codes, etc.) have all been extremely regular. Very likely, this regularity limits the class from which one can choose so much that the results are not as good as one knows should be possible. Nevertheless, we have to manage with what we have. Thus it is not surprising that coding theorists have either rediscovered a number of concepts from the mathematical discipline known as combinatorics or that they have studied parts of that theory so as to apply the results to their own problems. Since the theme of this session is recent trends in information theory it seems proper to give a survey of a development of the past few years, to wit the use of methods of coding theory to expand the mathematical theory: At present the area known as theory of designs and coding theory are influencing each other. It looks like this will become a very fruitful cooperation. It is the purpose of these lectures to explain some of the connections between these subjects to an audience familiar with algebraic coding theory. We shall assume that the reader is familiar with the standard topics from the theory as presented in E.R. Berlekamp’s Algebraic Coding Theory 1 (or e.g. the author’s lectures on Coding Theory 2. A summary of what one is expected to know is given in section 2. A few years ago the author gave a similar series of lectures to an audience of experts in the theory of combinatorial designs, stressing the coding theory background. Nevertheless, much of the material for these lectures will be the same. For a complete treatment of these parts and also an extensive treatment of connections between graph theory, coding and designs we refer the reader to the published notes of these lectures 3. Since we do not assume the reader to be more than slightly familiar with combinatorial theory we shall introduce a number of combinatorial designs in section 1. For the general theory of these topics we refer the reader to M. Hall’s Combinatorial Theory 4.KeywordsLinear CodeIncidence MatrixCode WordCyclic CodeCode TheoryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

On opposite orthogonal Steiner triple systems of non-prime-power order

It is well known that when or , there exists a Steiner triple system (STS) of order n decomposable into triangles (three pairwise intersecting triples whose intersection is empty). A triangle in an STS determines naturally two more triples: the triple of “vertices” , and the triple of “midpoints” . The number of these triples in both cases, that of “vertex” triples (inner) or that of “midpoint triples” (outer), equals one-third of the number of triples in the STS. In this paper, we consider a new problem of trinal decompositions of an STS into triangles. In this problem, one asks for three distinct decompositions of an STS of order n into triangles such that the union of the three collections of inner triples (outer triples, respectively) from the three decompositions form the set of triples of an STS of the same order. These decompositions are called trinal inner and trinal outer decompositions, respectively. We settle the existence question for trinal inner decompositions completely, and for trinal outer decompositions with two possible exceptions.

Traitor tracing is indeed a significant concept in cryptography, introduced by Chor, Fiat and Naor [7] in 1994. Combinatorial designs are Mathematical Structures that are highly suitable for creating traceable codes. They offer systematic methods for code construction, ensuring properties like traceability and resistance to collusion. Here in this paper we intend a novel method to make 3 –TA codes. No such Example of 3-TA code is available in Literature. We also take into consideration one more Combinatorial Design that is Projective Plane which even always proves to be 2-Frameproof Code.

The construction of antipodal triple systems by simulated annealing

An $\\cs$-colouring of a cubic graph $G$ is an edge-colouring of $G$ by points of a Steiner triple system $\\cs$ such that the colours of any three edges meeting at a vertex form a block of $\\cs$. A Steiner triple system which colours every simple cubic graph is said to be universal. It is known that every non-trivial point-transitive Steiner triple system that is neither projective nor affine is universal. In this paper we present the following results.&#13;\n&#13;\n(1) We give a sufficient condition for a Steiner triple system $\\cs$ to be universal.&#13;\n&#13;\n(2) With the help of this condition we identify an infinite family of universal point-intransitive Steiner triple systems that contain no proper universal subsystem. Only one such system was previously known.&#13;\n&#13;\n(3) We construct an infinite family of non-universal Steiner triple systems none of which is either projective or affine, disproving a conjecture made by Holroyd and the last author in 2004.