This course provides fundamental concepts of coding theory: algebraic structures including group, ring and field and linear codes defined as subspaces over Galois fields.
The aim of this course is to acquire the basic ideas of algebraic structures as well as introduction of algebraic coding theory.
To acquire the following abilities to:
1) understand the algebraic structure of the group, ring, and body, and explain them with specific examples .
2) understand the basic knowledge of linear codes and design typical binary linear codes.
3) be able to design, encode and decode BCH codes and Reed-Solomon codes.
Basic concepts of coding theory
group , ring , field
Construction method and arithmetic of the Galois field
Linear codes
Cyclic codes
Algebraic codes
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Fifteen classes are devoted to explanation of topics by the instructor, and eight to exercises.
Course schedule | Required learning | |
---|---|---|
Class 1 | Basic Concepts 1: encoding and decoding, and decoding region the received signal space | Explain encoding, decoding, decoding region and received signal space. |
Class 2 | Basic Concepts 2: errors and distance, maximum likelihood decoding, error control | Explain errors, distance, maximum likelihood decoding, and error control. |
Class 3 | Exercise 1: basic concepts | Review the basic concepts. |
Class 4 | Algebraic structure 1 : group , ring , field | Explain the definition of group, ring and field. |
Class 5 | Algebraic structure 2: integer ring, ideal | Explain the integer ring and ideal. |
Class 6 | Exercise 2: algebraic structure 1 | Review algebraic structure. |
Class 7 | Algebraic structure 3: polynomial ring, the polynomial ring ideal | Explain the ideal of the polynomial ring and the polynomial ring ideal. |
Class 8 | Galois field 1: Representation and construction method of the Galois field | Explain the construction method and the representation of the Galois field. |
Class 9 | Exercise 3: algebraic structure 2 | Review algebraic structure. |
Class 10 | Galois field 2: conjugate roots and minimal polynomial, arithmetic of Galois field | Explain conjugate roots, and minimal polynomial, arithmetic of the Galois field. |
Class 11 | Linear code 1: basic concepts, the generator matrix and parity-check matrix | Explain linear codes, generator matrix and parity-check matrix. |
Class 12 | Exercise 4: algebraic structure 3 | Review algebraic structure. |
Class 13 | Linear code 2: linear code decoding, modification of subcodes | Explain linear code decoding, modification of subcodes. |
Class 14 | Linear code 3: Hamming code, weight distribution | Explain Hamming code, and explain the weight distribution. |
Class 15 | Exercise 5: linear codes | Review linear code. |
Class 16 | Cyclic code 1: representation of the cyclic code, encoding of cyclic code | Explain representation of cyclic codes and encoding of cyclic codes. |
Class 17 | Cyclic code 2: roots of cyclic codes of and Fourier transform, BCH bound | cyclic code of roots and Fourier transform, and explain the BCH limit |
Class 18 | Exercise 6: cyclic code | Whatever review the cyclic code |
Class 19 | BCH code 1: BCH codes, Reed-Solomon code | BCH code, and explain the definition and nature of the Reed-Solomon code |
Class 20 | BCH code 2: BCH code decoding (1) | Explain the decoding method of |
Class 21 | 21. Exercise 7: BCH code | Reivew BCH code. |
Class 22 | BCH code 3: BCH code decoding (2) | Explain the decoding method of Reed-Solomon code. |
Class 23 | Exercise 8: comprehensive problems | Review the entire classes. |
(Eクラス)・代数系と符号理論，植松友彦，オーム社，2010年
(Oクラス)・代数系と符号理論入門，坂庭好一，渋谷智治著，コロナ社，2010年
符号理論入門，岩垂 好裕 著，昭晃堂，1992 年
符号理論入門，Vera Pless 著，伊藤 昇 訳，啓学出版，1985 年
情報数学入門，榎本 彦衛 著，新曜社，1987 年
符号理論，今井 秀樹 著，コロナ社，1990 年
The achievement will be evaluated by exercises and final exam.
None