**Contemporary Mathematics**

Volume: 642;
2015;
240 pp;
Softcover

MSC: Primary 11; 14;

**Print ISBN: 978-1-4704-1018-6
Product Code: CONM/642**

List Price: $112.00

AMS Member Price: $89.60

MAA Member Price: $100.80

**Electronic ISBN: 978-1-4704-2566-1
Product Code: CONM/642.E**

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# Algebra for Secure and Reliable Communication Modeling

Share this page *Edited by *
*Mustapha Lahyane; Edgar Martínez-Moro*

This volume contains the proceedings of the CIMPA Research School and
Conference on Algebra for Secure and Reliable Communication Modeling,
held from October 1–13, 2012, in Morelia, State of
Michoacán, Mexico.

The papers cover several aspects of the theory of coding theory and
are gathered into three categories: general theory of linear codes,
algebraic geometry and coding theory, and constacyclic codes over
rings.

The aim of this volume is to fill the gap between the theoretical
part of algebraic geometry and the applications to problem solving and
computational modeling in engineering, signal processing and
information theory.

This book is published in cooperation with Real Sociedad Matemática Española (RSME)

#### Readership

Graduate students and research mathematicians interested in applications of computer algebra and coding theory.

# Table of Contents

## Algebra for Secure and Reliable Communication Modeling

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Preface ix10 free
- Some Applications of the Fourier Transform in Algebraic Coding Theory 116 free
- Introduction 116
- I. Fourier transform over the complex numbers for finite abelian groups 217
- 1. Linear codes and weight enumerators 217
- 2. Characters and the Fourier transform over the complex numbers 520
- 3. Frobenius rings 722
- 4. The MacWilliams identities 823
- 5. Additional MacWilliams identities 1025
- 6. Convolutions 1227
- II. Fourier transform over a field for finite groups 1631
- 7. Group algebras and representation theory 1631
- 8. Maschke’s theorem 1934
- 9. Structure of group algebras 2035
- III. Examples 3045
- 10. Representations over finite fields 3146
- 11. Finite abelian groups 3348
- 12. Some non-abelian groups of small order 3651
- References 3954

- A Semigroup Approach to Complete Decoding 4156
- Gröbner Bases Methods in Coding Theory 7388
- An Introduction to Algebraic Geometry Codes 87102
- Evaluation Codes and Plane Valuations 119134
- Vector Bundles with a View Toward Coding Theory 159174
- Algebraic-Geometric Codes from Rational Surfaces 173188
- Equivalence Classes and Structures of Constacyclic Codes Over Finite Fields 181196
- 1. Introduction 181196
- 2. Cyclotomic cosets 185200
- 3. Isometries between constacyclic codes 193208
- 4. Repeated-root constacyclic codes of length ℓ^{𝑡}𝑝^{𝑠} 197212
- 5. Repeated-root constacyclic codes of length 2ℓ^{𝑡}𝑝^{𝑠} 200215
- 6. Some Generalizations 206221
- Acknowledgements 216231
- References 216231

- On Repeated-Root Constacyclic Codes of Prime Power Length Over Polynomial Residue Rings 225240
- 1. Introduction 225240
- 2. Preliminaries 226241
- 3. Constacyclic Codes over Finite Commutative Rings 228243
- 4. The Polynomial Residue Ring ℛₐ=𝔽_{𝕡^{𝕞}}+𝕦𝔽_{𝕡^{𝕞}}+…+𝕦^{𝕒-1}𝔽_{𝕡^{𝕞}} 230245
- 5. 𝜆-constacyclic codes of length 𝑝^{𝑠} over ℛₐ 233248
- 6. Λ-constacyclic codes of length 𝑝^{𝑠} over ℛₐ 236251
- Acknowledgments 237252
- References 238253

- Back Cover Back Cover1258