Peter Jephson , 1947- Graphs, codes, and designs. This note is a continuation of a first investigation of er- ror patterns. While for the Hamming metric all perfect codes over finite fields are known, in the Johnson metric it was conjectured by Delsarte in 1970's that there are no nontrivial perfect codes. Nearly perfect binary codes and uniformly packed codes. Polarities of designs -- 7. This book is concerned with the relations between graphs, error-correcting codes and designs, in particular how techniques of graph theory and coding theory can give information about designs. Comparison of a received word with the code words yields a set of error vectors.
Codes -- 10 Cyclic codes -- 11 Threshold decoding -- 12. Self-orthogonal codes, designs and projective planes; 14. Extension of graphs -- 8. The E-mail message field is required. Many exercises and useful hints are included througout, and a large number of references are given. Although graph theory, design theory, and coding theory had their origins in various areas of applied mathematics, today they are to be found under the umbrella of discrete mathematics.
Strongly regular graphs with no triangles; 6. Nearly perfect binary codes and uniformly packed codes; 17. We prove that there are no two-perfect codes in J 2w,w with length less then 2. A major revision and expansion of a previous volume in this series, this account includes many examples and new results as well as improved treatments of older material. Self-orthogonal codes, designs and projective planes -- 14. Tuvi Etzion Construction of block designs is an important part of design theory.
These are notes deriving from lecture courses given by the authors in 1973 at Westfield College, London. . Then we define some forms of per- fectness of codes and pose the problem of finding semi-perfect codes. A major revision and expansion of a previous volume in this series, this account includes many examples and new results as well as improved treatments of older material. The E-mail message field is required.
This book will be welcomed by graduate students and research mathematicians and be valuable for advanced courses in finite combinatorics. So that non-specialists will find the treatment accessible the authors have included short introductions to the three main topics. A brief introduction todesign theory -- 2. We then prove a bound on length of such codes, similarly to Roos' bound for perfect codes in Johnson graph. Further, the correspondence between Hadamard matrices and block designs is well known.
So that non-specialists will find the treatment accessible the authors have included short introductions to the three main topics. They do not seek to consider each of these three topics individually, but rather to stress the many and varied connections between them. Contents: Cover -- Title -- Copyright -- Contents -- Preface -- 1. We show which parameters can be excluded for one-perfect codes. We start with the constant weight codes. We provide an anticode-based proof of a bound on length of Steiner system, prove that doubly Steiner system is a diameter perfect code and present a bound on length of doubly Steiner system. In the simplest situation code words are 0, 1 -vectors, as are the received messages and the error vectors.
Quadratic residue codes -- 15. Fi- nally a link is laid between error vectors, graphs and balanced incomplete block designs. Description: 1 online resource 156 pages. Comment: This is an M. Strongly regular graphs with no triangles; 6. A feature of this book is the discussion of then-recent construction of t-designs from codes.
Next, we consider binomial moments for perfect codes. Finally we describe Steiner systems and doubly Steiner systems, which are strongly connected with the constant weight and doubly constant weight codes respectively. The E-mail message field is required. We examine two-perfect codes in J 2w,w and present necessary conditions for existence of such codes. In this work we examine constant weight codes as well as doubly constant weight codes, and reduce the range of parameters in which perfect codes may exist in both cases. First we consider burstiness and distribution of error vectors without assuming cyclic conditions. Strongly regular graphs -- 3.