CHAPTER 1 Mathematical Background

1.1．Algebra

1.2．Krawtchouk Polynomials

1.3．Combinatorial Theory

1.4．Probability Theory

CHAPTER 2 Shannon's Theorem

2.1．Introduction

2.2．Shannon's Theorem

2.3．On Coding Gain

2.4．Comments

2.5．Problems

CHAPTER 3 Linear Codes

3.1．Block Codes

3.2．Linear Codes

3.3．Hamming Codes

3.4．Majority Logic Decoding

3.5．Weight Enumerators

3.6．The Lee Metric

3.7．Comments

3.8．Problems

CHAPTER 4 Some Good Codes

4.1．Hadamard Codes and Generalizations

4.2．The Binary Golay Code

4.3．The Ternary Golay Code

4.4．Constructing Codes from Other Codes

4.5．Reed-Muller Codes

4.6．Kerdock Codes

4.7．Comments

4.8．Problems

CHAPTER 5 Bounds on Codes

5.1．Introduction：The Gilbert Bound

5.2．Upper Bounds

5.3．The Linear Programming Bound

5.4．Comments

5.5．Problems

CHAPTER 6 Cyclic Codes

6.1．Definitions

6.2．Generator Matrix and Check Polynomial

6.3．Zeros of a Cyclic Code

6.4．The Idempotent of a Cyclic Code

6.5．Other Representations of Cyclic Codes

6.6．BCH Codes

6.7．Decoding BCH Codes

6.8．Reed-Solomon Codes

6.9．Quadratic Residue Codes

6.10．Binary Cyclic Codes of Length 2n（n odd）

6.11．Generalized Reed-Muller Codes

6.12．Comments

6.13．Problems

CHAPTER 7 Perfect Codes and Uniformly Packed Codes

7.1．Lloyd's Theorem

7.2．The Characteristic Polynomial of a Code

7.3．Uniformly Packed Codes

7.4．Examples of Uniformly Packed Codes

7.5．Nonexistence Theorems

7.6．Comments

7.7．Problems

CHAPTER 8 Codes over Z4

8.1．Quaternary Codes

8.2．Binary Codes Derived from Codes over Z4

8.3．Galois Rings over Z4

8.4．Cyclic Codes over Z4

8.5．Problems

CHAPTER 9 Goppa Codes

9.1．Motivation

9.2．Goppa Codes

9.3．The Minimum Distance of Goppa Codes

9.4．Asymptotic Behaviour of Goppa Codes

9.5．Decoding Goppa Codes

9.6．Generalized BCH Codes

9.7．Comments

9.8．Problems

CHAPTER 10 Algebraic Geometry Codes

10.1．Introduction

10.2．Algebraic Curves

10.3．Divisors

10.4．Differentials on a Curve

10.5．The Riemann-Roch Theorem

10.6．Codes from Algebraic Curves

10.7．Some Geometric Codes

10.8．Improvement of the Gilbert-Varshamov Bound

10.9．Comments

10.10．Problems

CHAPTER 11 Asymptotically Good Algebraic Codes

11.1．A Simple Nonconstructive Example

11.2．Justesen Codes

11.3．Comments

11.4．Problems

CHAPTER 12 Arithmetic Codes

12.1．AN Codes

12.2．The Arithmetic and Modular Weight

12.3．Mandelbaum-Barrows Codes

12.4．Comments

12.5．Problems

CHAPTER 13 Convolutional Codes

13.1．Introduction

13.2．Decoding of Convolutional Codes

13.3．An Analog of the Gilbert Bound for Some Convolutional Codes

13.4．Construction of Convolutional Codes from Cyclic Block Codes

13.5．Automorphisms of Convolutional Codes

13.6．Comments

13.7．Problems

Hints and Solutions to Problems

References

Index