This is the web site for our book, Self-Dual Codes and Invariant Theory, which was published by Springer, Berlin, in Feb 2006.

It is Volume 17 of the series Algorithms and Computation in Mathematics; ISBN 3-540-30729-X, xxiii+430 pp.

To go to the Springer site for this book, click here.

To order from Amazon: click here.

Brief description of book:

One of the most remarkable theorems in coding theory is Gleason's 1970 theorem about the weight enumerators of self-dual codes. In the past 35 years there have been hundreds of papers written about generalizations and applications of this theorem to different types of codes. This book develops a new theory which includes all the earlier generalizations at once. It is also in part an encyclopedia that gives a very extensive list of the different types of self-dual codes and their properties, including tables of the best codes as far as they are presently known. Besides self-dual codes, the book also discusses two closely-related subjects, lattices and modular forms, and quantum error-correcting codes.

The book will be of interest to people working in the areas of

- error-correcting codes
- lattices, quadratic forms and modular forms
- group theory, invariant theory, number theory
- quantum computers

That is,

- electrical engineers
- mathematicians
- computer scientists
- physicists

Table of Contents:

The authors:

• Gabriele Nebe
  Lehrstuhl D für Mathematik
  Rheinisch-Westfälische Technische Hochschule (RWTH) Aachen
  Templergraben 64
  52062 Aachen, Germany
  Email: nebe@math.rwth-aachen.de
  Home page: www.math.rwth-aachen.de/~Gabriele.Nebe/
• E. M. Rains
  Department of Mathematics
  California Institute of Technology
  Pasadena, CA 91125, U.S.A.
  Email: rains@caltech.edu
• N. J. A. Sloane
  Internet and Network Systems Research
  AT&T Shannon Labs
  180 Park Avenue
  Florham Park, NJ 07932-0971, U.S.A.
  Email: njasloane@gmail.com
  Home page: http://NeilSloane.com/

Corrections

• Page 145, In the statement of Lemma 5.4.5, line 2 should read \psi (v_{\iota }) = - \lambda (\phi ) (notice the minus-sign). The lower left entry of H_{\iota, u_{\iota }, v_{\iota}) d(1,\phi) is -\epsilon ^{-1} u_{\iota } (instead of u_{\iota }) and in the first display in the proof, the lower left entry of (H_{\iota, u_{\iota }, v_{\iota}) d(1,\phi))^2 is (2\iota -1)\epsilon ^{-1} u_{\iota} (instead of u_{\iota }).
• Page 148. In the proof of Theorem 5.4.13, the third sentence needs justification. This is supplied by a paper that can be downloaded here [New version Aug 30 2007].

  Open Problems. In this connection we would like to mention the following open problems.

  1. Is there an analogue of the universal Clifford-Weil group (5.4.6) over infinite form rings?
  2. What is the relation between the universal Clifford-Weil group and the universal central extension of the hyperbolic co-unitary group U(R, PHI) (see (5.2.11)?
  3. What is the relation between the Witt group and the Schur multiplier of U(R, PHI)? (Note that the Witt group is reminiscent of a K_0 group, while the Schur multiplier is reminiscent of K_2.)
• Section 9.1, p. 255 (formula 9.1.8):
  the first exponent in the infinite product should read m-1/2 and not (m-1)/2 .

Additional or updated references:

• Reference [49] on page 393 has now appeared: R. T. Bilous, Enumeration of the Binary Self-Dual Codes of Length 34, J. of Combin. Math. Combin. Computing, 59 (2006), 173-211.
• S. T. Dougherty, S. Y. Kim and Y. H. Park, Lifted codes and their weight enumerators, Discrete Math., 305 (2005), 123-135.
• M. Ozeki, Notes on the shadow process in self-dual codes, Discrete Math., 264 (2003), 187-200.