Combinatorial designs constructed from or with coding theory

Abstract

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.