## Gabriele Nebe, Eric M. Rains and Neil J. A. Sloane

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.

#### 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


#### 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 .