Last edited by Yozshugami

Sunday, May 10, 2020 | History

10 edition of **Self-Dual Codes and Invariant Theory (Algorithms and Computation in Mathematics)** found in the catalog.

- 35 Want to read
- 30 Currently reading

Published
**March 14, 2006**
by Springer
.

Written in English

- Algebra,
- Information Theory,
- Computer Books: General,
- Computers,
- Mathematics,
- Algebra - General,
- Group Theory,
- Mathematics / Algebra / General,
- error-correcting codes,
- invariant theory,
- lattices,
- modular forms,
- quantum codes,
- Coding theory,
- Duality theory (Mathematics),
- Invariants

The Physical Object | |
---|---|

Format | Hardcover |

Number of Pages | 430 |

ID Numbers | |

Open Library | OL9056156M |

ISBN 10 | 354030729X |

ISBN 10 | 9783540307297 |

Also surveyed is the theory that intersects self-dual codes, lattices, and invariant theory, which leads to an intriguing analogy between the Duursma zeta function and the zeta function attached to an algebraic curve over a finite field. Codes and Invariant Theory. By Gabriele Nebe, E. M. Rains and N. J. A. Sloane. of Gleason's theorem on the weight enumerators of codes which applies to arbitrary-genus weight enumerators of self-dual codes defined over a large class of finite rings and modules. The proof of the theorem uses a categorical approach, and will be the subject.

Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes, open problems. In recent years quaternary codes have attracted the attention of the coding community as several notorious binary nonlinear codes containing more codewords than any known linear codes were found to be binary images under the Gray map of linear codes over Z 4 This discovery opens the way for a broader study of quaternary codes, which constitute a rapidly growing area of coding theory.

Gleason gave analogous results for other Types of self-dual codes over ﬂnite ﬂelds Self-dual codes and invariant theory. Springer-Verlag(). [10] ,OnSiegelmodularformsofweight12,ngew. Math ()49{ [11] G. Nebe and B. B. Venkov, Unimodular lattices with long shadow. J. Number. Invariant theory is a subject with a long tradition and an astounding abil ity to rejuvenate itself whenever it reappears on the mathematical stage. Throughout the history of invariant theory, two features of it have always been at the center of attention: computation and applications. This book is about the computational aspects of invariant theory.

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Besides self-dual codes, the book also discusses two closely-related subjects, lattices and modular forms, and quantum error-correcting codes. This book, written by the leading experts in the subject, has no equivalent in the literature and will be of great Self-Dual Codes and Invariant Theory book to mathematicians, communication theorists, computer scientists and by: About this book One of the most remarkable and beautiful theorems in coding theory is Gleason's theorem about the weight enumerators of self-dual codes and their connections with invariant theory.

In the past 35 years there have been hundreds of papers written about generalizations and applications of this theorem to different types of codes. Besides self-dual codes, the book also discusses two closely-related subjects, lattices and modular forms, and quantum error-correcting codes.

This book, written by the leading experts in the subject, has no equivalent in the literature and will be of great interest to mathematicians, communication theorists, computer scientists and physicists.

Self-Dual Codes and Invariant Theory Nebe, Gabriele and Rains, Eric M. and Sloane, Neil J. () Self-Dual Codes and Invariant Theory.

Algorithms and Computation in by: Self-Dual Codes and Invariant Theory With 10 Figures and 34 Tables 4y Springer. Contents Preface v List of Symbols xiv List of Tables xxv List of Figures xxvii 1 The Type of a Self-Dual Code 1 4fu: Type II self-dual codes over Z/4Z containing 1 8Z: Self-dual codes over Z/8Z Self-dual codes and invariant theory 1 Gabriele NEBE, a;2 a RWTH Aachen University, Germany Abstract.

A formal notion of a Typ T of a self-dual linear code over a nite left R- module V is introduced which allows to give explicit generators of a nite complex matrix group, the associated Clifford-Weil group C.T/ •GLjVj.C/, such that the complete weight enumerators of self-dual isotropic codes.

The full proof has just appeared in our book Self-Dual Codes and Invariant Theory (Springer, ). This paper is based on my talk at the conference on Algebraic Combinatorics in honor of Eiichi Bannai, held in Sendai, Japan, June 26–30, 1.

Motivation Self-dual codes are important because they intersect with • communications. One of the most remarkable theorems in coding theory is Gleason’s theorem that the weight enumerator of a binary doubly-even self-dual code is an element of the polynomial ring generated by the weight enumerators of the Hamming code of length 8 and the Golay code of length One of the most remarkable and beautiful theorems in coding theory is Gleason's theorem about the weight enumerators of self-dual codes and their connections with invariant theory.

In the past 35 years there have been hundreds of papers written about generalizations and applications of this theorem to different types of codes.3/5(1).

One of the most remarkable and beautiful theorems in coding theory is Gleason's theorem about the weight enumerators of self-dual codes and their connections with invariant theory. In the past 35 years there have been hundreds of papers written about generalizations and applications of this theorem to different types of codes.3/5(1).

The monograph [11] gives a framework to define the notion of a Type of a self-dual code in much more generality and shows how to apply invariant theory to find upper bounds on the minimum weight.

Self -dual codes and invariant theory Introduction A linear code V is self-dual if V = T i(08 of Ch. We have seen that many good codes are self-dual, among them the extended Golay codes and certain quadratic residue codes.

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. Topics treated in this chapter include (a) invariant theory and the relationship with self-dual codes, (b) lattices and connections with binary codes, and (c) optimal, divisible, and extremal codes.

Some open questions which arise are: Which polynomials F(x,y) occur as the weight enumerators of linear codes. Does there exist a binary self-dual Author: David Joyner, Jon-Lark Kim.

CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): The main theorem in this paper is a far-reaching generalization of Gleason’s theorem on the weight enumerators of codes which applies to arbitrary-genus weight enumerators of self-dual codes defined over a large class of finite rings and modules.

The proof of the theorem uses a categorical approach, and will be the. Self-dual codes and invariant theory 1 - Lehrstuhl D für One main application of Gleason's theorem is to bound the minimum weight of a self- dual code of a given Type and given length.

Codes with maximal possible minimum weight are called extremal. One of the most remarkable theorems in coding theory is Gleason's theorem about the weight enumerators of self-dual codes and their connections with invariant theory.

This book develops a new theory which is powerful enough to include all the earlier generalizations. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): A formal notion of a Typ T of a self-dual linear code over a finite left R-module V is introduced which allows to give explicit generators of a finite complex matrix group, the associated Clifford-Weil group C(T) ≤ GL|V |(C), such that the complete weight enumerators of self-dual isotropic codes of Type T span the.

Binary doubly even self-dual codes are called Type II because of the classification of self-dual divisible codes in the theorem of Gleason–Pierce, based on invariant theory [12]. By analogy, Conway and Sloane called a Type II lattice a unimodular even lattice [4].

The Kneser-Hecke-operator is a linear operator defined on the complex vector space spanned by the equivalence classes of a family of self-dual codes of fixed length. Self-dual codes and invariant theory.

of Gleason’s theorem on the weight enumerators of codes which applies to arbitrary-genus weight enumerators of self-dual codes defined over a large class of finite rings and modules.

The proof of the theorem uses a categorical approach, and will be the subject of a forthcoming book. and will be Author: G. Nebe, E. M. Rains and N. J. A. Sloane.Since the Hamming and Golay codes are self-dual, and have weights divisible by 4, these two polynomials must he invariant under Eqs.

(I) and (2) and hence under the group so we have found the two basic invariants we were looking for.Self-Dual Codes Theorem If C is a self-dual code of length n over F q then n must be even. Proof. We have dim(C) = dim(C?) and dim(C) + dim(C?) = n which Invariant Theory If C is a self-dual code then the weight enumerator is held invariant by the MacWilliams relations and .