MacWilliams F.J., Sloane N.J.A. The Theory of Error-Correcting Codes

North-Holland, 1977. — 785 p.Coding theory began in the late 1940's with the work of Golay, Hamming and Shannon. Although it has its origins in an engineering problem, the subject has developed by using more and more sophisticated mathematical techniques. It is our goal to present the theory of error-correcting codes in a simple, easily understandable manner, and yet also to cover all the important aspects of the subject. Thus the reader will find both the simpler families of codes- for example, Hamming, BCH, cyclic and Reed-Muller codes- discussed in some detail, together with encoding and decoding methods, as well as more advanced topics such as quadratic residue, Golay, Goppa, alternant, Kerdock, Preparata, and self-dual codes and association schemes.
Our treatment of bounds on the size of a code is similarly thorough. We discuss both the simpler results- the sphere-packing, Plotkin, Elias and Garshamov bounds- as well as the very powerful linear programming method and the McEiiece-Rodemich-Rumsey-Welch bound, the best asymptotic result known. An appendix gives tables of bounds and of the best codes presently known of length up to 512.
Having two authors has helped to keep things simple: by the time we both understand a chapter, it is usually transparent. Therefore this book can be used both by the beginner and by the expert, as an introductory textbook and as a reference book, and both by the engineer and the mathematician.Linear codes
Nonlinear codes, Hadamard matrices, designs and the Golay code
An introduction to BCH codes and finite fields
Finite fields
Dual codes and their weight distribution
Codes, designs and perfect codes
Cyclic codes
Cyclic codes (cont.): Idempotents and Mattson-Solomon polynomials
BCH codes
Reed-Solomon and Justesen codes
MDS codes
Alternant, Goppa and other generalized BCH codes
Reed-Muller codes
First-order Reed-Muller codes
Second-order Reed-Muller, Kerdock and Preparata codes
Quadratic-residue codes
Bounds on the size of a code
Methods for combining codes
Self-dual codes and invariant theory
The Golay codes
Association schemes
A. Tables of the best codes known
B. Finite geometries