It summarizes the main research findings on quantum network coding, while also systematically introducing readers to secure quantum network coding schemes. I think roman provides a fresh introduction to information theory and shows its inherent connections with coding theory. Roth, introduction to coding theory, cambridge university press, 2016. Most of the papers are in coding theory and design theory, reflecting the major interest of professor raychaudhuri, but there are articles on association schemes, algebraic graph theory, combinatorial geometry, and network flows as well.
Fourth, we explore the relationship between index coding and other problems such as network coding, distribute storage, and guessing. Jan 25, 2010 an algebraic approach to the association schemes of coding theory by p. An algebraic approach to the association schemes of coding theory philips. A recent text that develops the theory both quite generally and quite extensively is. Association schemes are of interest to both mathematicians and statisticians and this book was written with both audiences in mind. The acm digital library is published by the association for. With regard to coding methods, coding models and coding security, the book subsequently provides a series of quantum network coding schemes based on the integration of quantum cryptography. Keywords association schemes, codes and designs, du. Tdesigns and partially balanced incomplete block designs together with association schemes, as generalizations of balanced incomplete block designs, are included. In fact, if we had tried to cover all of these topics in full detail, then this survey article would have been a book rather than a paper. The reader is only assumed to know very basic abstract algebra. Indeed, in algebraic combinatorics, association schemes provide a unified approach to many topics, for example combinatorial designs and coding theory. Sloane, the theory of errorcorrecting codes, northholland.
The association schemes of coding theory springerlink. Dec 15, 2009 an algebraic approach to the association schemes of coding theory by p. Sloane abstract association schemes originated in s t a t i s t i c s, but have recently been used in coding theory and combinatorics by delsarte, mceliece and others to obtain strong upper bounds on the size of codes and other combinatorial objects, and to characterize those objects such a s perfect codes which meet t h e s. Secure quantum network coding theory tao shang springer. Part of the nato advanced study institutes series book series asic, volume 16. The material has been divided into three main parts. The seminar tackled the theory and application of special functions and covered topics ranging from the asymptotic estimation of special functions to association schemes and coding theory. Throughout the paper, the main concepts, methods, and results are illustrated by two examples that are of major significance in classical coding theory, namely, the hamming scheme and the johnson scheme. The theory of errorcorrecting codes northholland mathematical library, volume 16, 1977 11th reprint, 2003 preface. The primary object of the lecture notes is to develop a treatment of association schemes analogous to that which has been so successful in the theory of finite groups. In coding theory, association scheme theory is mainly concerned with the distance of a. Nato advanced study institutes series series c mathematical and physical sciences, vol 16. The theory of association schemes arose in statistics, in the theory of experimental design for. I want to study coding theory, but i dont know any text book about.
Shannon theory and coding, coding theory, decoding of linear codes and macwilliams identity, coding theory constructing new codes, coding theory bounds on codes, reed. Coding theory is the study of the properties of codes and their fitness for a specific application. This paper contains a survey of association scheme theory with its algebraic and analytical aspects and of its applications to coding theory in a wide sense. Hx, where hx is entropy of source bitrate, and cx is the bitrate after compression. An algebraic approach to association schemes and coding theory.
In algebra, association schemes generalize groups, and the theory of association schemes generalizes the character theory of linear representations of groups. Thus in coding theory we look for codes that are as large as possible, since such codes are most effective in transmitting information over noisy channels. The graphs of the partition classes of an association scheme are very special. Philips gloeilampenfabrieken, 1973 coding theory 97.
Association schemes and coding theory ieee transactions on. Third, we establish performance bounds and discuss their implications in network information theory. The author is a wellknown authority in the field of coding theory and his book serves an an excellent introduction to the field. In particular, some universal bounds are derived for codes and designs in polynomial type association schemes. Im in the last year of undergraduate mathematics course, so i would of a basic book, but t. This book has been written as lecture notes for students who need a grasp of the basic principles of linear codes. I found his presentation on the noisy coding theorem very well written. Association schemes provide the appropriate setting for treating certain problems from several different areas of algebraic combinatorics, for example, coding theory, design theory, algebraic graph theory, finite group theory, and finite geometry.
This area of discrete applied mathematics includes the study and discovery of various coding schemes that are used to increase the number of errors that can be corrected during data transmission. Association schemes and coding theory ieee transactions. Codes are studied by various scientific disciplinessuch as information theory, electrical engineering, mathematics, linguistics, and computer sciencefor the purpose of designing efficient and reliable data. The cyclotomic scheme cq,r, where r divides q1 and the quotient is even if q is odd has as points the elements of the finite field gfq. In particular, special interest is devoted to the relationship between codes and combinatorial designs. An introduction to association schemes and coding theory. In particular, no source coding scheme can be better than the entropy of the source. In 52, the method of linear programming was successfully combined with the duality of the commutative bosemesner algebra, and it has been serving as one of the underlying principles in the theory of commutative association schemes. Various techniques used by source coding schemes try to achieve the limit of entropy of the source. In mathematics, association schemes are structures that appear in many different forms in the fields of combinatorics and statistics. Some interesting results, conjectures, and problems are given. Delsarte gave a detailed treatment of subsets of association schemes.
History of the study of association schemes, a personal view. This paper gives an introduction to association schemes and the way they are used in coding theory and combinatorics. Design theory by zhexian wan overdrive rakuten overdrive. It is well written, clearly presented, and as is announced in the preface to the first edition 1 is accessible to a student with just elementary calculus and simple probability theory as a background.
The theme of association schemes is represented by new classification results and new classes of schemes related to posets. This volume collects uptodate applications of the theory of association schemes to coding and presents new properties of both polynomial and general association schemes. In coding theory, association scheme theory is mainly concerned with a class of subjects that involve the central notion of the distance of a code. An introduction to association schemes and coding theory n. Coding and information theory graduate texts in mathematics. Search for library items search for lists search for contacts search for a library. Niederreiter, editor, coding theory and cryprography, world scientific, 2002, pp. Coding theory and applications this book has been written as lecture notes for students who need a grasp of the basic principles of linear codes. Coding theory at work in cryptology and vice versa h. Coding theory began in the late 1940s with the work of golay, hamming and shannon. This paper contains a survey of association scheme theory with its algebraic and analytical aspects and of its applications to coding theory in a wide s. Another enjoyable part of the book is his treatment of linear codes.
An algebraic approach to association schemes paulhermann. Theory and application of special functions 1st edition. The book is written in a lucid style and is algebraic in nature. An algebraic approach to the association schemes of coding theory philips journal of research supplement, philips gloeilampenfabrieken eindhoven. The study of errorcontrol codes is called coding theory. The main chapters are decomposition theory, representation theory, and the theory of generators. Although it has its origins in an engineering problem, the subject has developed by using more and more sophisticated mathematical techniques. Delsarte an algebraic approach to the association schemes of coding theory, philips res. Some coding theory related to steiner triple systems are clearly explained. However, fortunately there are a number of excellent articlesbooks on the theory of commutative association schemes and delsarte theory, e. Basic properties of association schemes relevant to coding p.
The study of commutative association schemes equivalently the study of gelfand pairs or the study of multiplicityfree permutation groups in the group theory terminology. An algebraic approach to the association schemes of coding theory. Codes are used for data compression, cryptography, errorcorrection, and networking. An algebraic approach to the association schemes of coding theory by p. Read, highlight, and take notes, across web, tablet, and phone. This volume collects uptodate applications of the theory of association schemes to coding and presents new properties of both polynomial. We have somewhat updated delsartes original results which are often presented with new proofs. An algebraic approach to the association schemes of coding. Theory and application of special functions sciencedirect.
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