An algebraic approach to the association schemes of coding theory. This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings, kissing numbers, lattices, and association schemes. Association schemes are a central notion in algebraic combinatorics. In particular, special interest is devoted to the relationship between codes and combinatorial designs. Various techniques used by source coding schemes try to achieve the limit of entropy of the source. Association schemes group theory without groups core. An introduction to association schemes and coding theory n.
Index terms association schemes, codes and designs, du ality, linear programming, orthogonal polynomials, polynomial schemes. Hx, where hx is entropy of source bitrate, and cx is the bitrate after compression. The main chapters are decomposition theory, representation theory, and the theory of generators. 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. Indeed, in algebraic combinatorics, association schemes provide a unified approach to many topics, for example combinatorial designs and coding theory. By dualising the ppolynomiality, we say that an association scheme is qpolynomial or cometric if e j 1 jxj v j jxje 1. Association schemes, and the hamming scheme associate matrices, and the bosemesner algebra distribution vectors the linear programming bound formulation numerical results for small n asymptotic lower and upper bounds open problems jason li delsartes linear programming bound. Download citation on researchgate association schemes and coding theory. Contents 1 base class for codes 3 2 linear codes 3 channels 49 4 encoders 57.
Delsarte 52 proclaimed the importance of commutative association schemes as a unifying framework for coding theory and design. For more background on the association scheme of symmetric bilinear forms we refer to 15 and 36. This paper contains a survey of association scheme theory with its algebraic and analytical aspects and of its applications to. Association schemes group theory without groups rwth. Third, we establish performance bounds and discuss their implications in network information theory. Besides being basic structures in such theories as coding theory, design. On triple intersection numbers of association schemes.
Siam journal on applied mathematics society for industrial. Association schemes group theory without groups rwth aachen, ss 2009 ju. Keywords association schemes, codes and designs, du. Designed experiments, algebra and combinatorics by rosemary a. An algebraic approach to the association schemes of coding. Applications of association schemes in coding theory are summarized in the surveyof delsarte.
They provide a unified approach to various questions from design theory, coding theory, algebraic graph theory, discrete geometry, group theory and representation theory. The two aspects of coding theory enjoy a symbiotic relationship from the days of their origin. Hamming was studying devices to store information and wanted to design simple schemes to. This volume collects uptodate applications of the theory of association schemes to coding and presents new properties of both polynomial and general association schemes. An algebraic approach to the association schemes of coding theory philips journal of research supplement, philips gloeilampenfabrieken eindhoven. Pdf coding theory and algebraic combinatorics semantic. A recent text that develops the theory both quite generally and quite extensively is. 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. Zanna, advances of experimental social psychology vol. Codes in classical association schemes mathematics at ghent. Coding theory basics coding theory is an important study which attempts to minimize data loss due to errors introduced in transmission from noise, interference or other forces. This paper gives a brief outline of the theory of coherent con. The theory of association schemes arose in statistics, in the theory of experimental design for. Nato advanced study institutes series series c mathematical and physical sciences, vol 16.
It contains several recent results which appear for the first time in. Special emphasis is put on concepts arising from the theories of error correcting codes. However, formatting rules can vary widely between applications and fields of interest or study. Second, we discuss several coding schemes based on algebraic, graphtheoretic, and informationtheoretic tools. Association schemes were originally introduced by bose and his coworkers in the design of statistical experiments. In this section, we will prove that being hierarchical. An introduction to association schemes and coding theory. The theory of association schemes provides the most effective means for determining just how large is actually possible. Association schemes on general measure spaces and zero. This bound is a result in the theory of association schemes, so we include a brief introduction to association schemes. Qpolynomial association schemes with irrational eigenvalues. The material has been divided into three main parts.
Association schemes form one of the main objects of algebraic combinatorics, classically defined on finite sets. An algebraic approach to the association schemes of coding theory p delsarte. It introduces the concepts of eigenvalues and delsartes duality to the study of orthogonal polynomials and provides those interested in p and qpolynomial association schemes with a closed form for their parameters. The association schemes of coding theory springerlink. In coding theory and related subjects, an association scheme such as the hamming scheme should mainly be viewed as a structured. Such a classi cation may lend insight into the aforementioned conjecture for irrational schemes. Delsartes linear programming bound 15859 coding theory, fall 14 jason li december 10, 2014 contents 1 introduction a fundamental objective in coding theory is to find upper bounds on the size of a code infn q with a certain distance d. Dec 15, 2009 an algebraic approach to the association schemes of coding theory by p. 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. On some polynomials related to the weight enumerator of linear codes, siam j. Qpolynomial association schemes were introduced by delsarte in an algebraic approach to the association schemes of coding theory. I have not gone through and given citations or references for all of the results given here, but the presentation relies heavily on two sources, van.
In coding theory, association scheme theory is mainly concerned with the distance of a code. The theory of association schemes is presented from several different perspectives in the books by bannai and ito 2, brouwer et al. An algebraic approach to association schemes paulhermann. This volume collects uptodate applications of the theory of association schemes to coding and presents new properties of. The origins of association schemes lie in statistics, in the work of r. In particular, no source coding scheme can be better than the entropy of the source. 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. The subject has close connections with permutation groups d. Association schemes group theory without groups rwth aachen. A recent survey of the theory of commutative association schemes was given by martin and tanaka 37. The theme of association schemes is represented by new classification results and new classes of schemes related to posets.
It was due to the work of delsarte 1973 that association schemes were proven to be a useful tool for the study of a wide range of combinatorics including design theory, coding theory and algebraic graph theory. The hamming codes and delsartes linear programming bound. Fourth, we explore the relationship between index coding and other problems such as network coding, distribute storage, and guessing. We have somewhat updated delsartes original results which are often presented with new proofs. Coding theory lecture notes nathan kaplan and members of the tutorial september 7, 2011 these are the notes for the 2011 summer tutorial on coding theory. In this paper, we work towards classifying all association schemes having d 3 that are qpolynomial with irrational eigenvalues.
The study of errorcontrol codes is called coding theory. We explore the application of the theory of association schemes to coding. An algebraic approach to the association schemes of coding theory by p. 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. Upper bounds make us aware of the best transmission rates possible for codes of a certain size and. Representation theory of association schemes rwth publications. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far a. Pdf approximation of integration over finite groups. Niederreiter, editor, coding theory and cryprography, world scientific, 2002, pp. Hajime tanaka introduction to delsarte theory summer.
Theory of association schemes is the first conceptoriented treatment of the structure theory of 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. 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. We present in these lecture notes a survey of delsartes work on the algebraic theory of association schemes, which has influenced considerably the recent developments in coding theory.
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