2024 Faculty Courses School of Engineering Undergraduate major in Information and Communications Engineering
Algebraic Systems and Coding Theory
- Academic unit or major
- Undergraduate major in Information and Communications Engineering
- Instructor(s)
- Kenta Kasai
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Mon / 3-4 Thu
- Class
- -
- Course Code
- ICT.C209
- Number of credits
- 200
- Course offered
- 2024
- Offered quarter
- 3Q
- Syllabus updated
- Mar 17, 2025
- Language
- Japanese
Syllabus
Course overview and goals
Starting from some mathematical basic concepts of algebraic structure: group, ring and fields, this course lectures linear codes which are subspace whose scalar is finite fields, and their algebraic decoding.
Course description and aims
Goal: Understand the groups, rings, fields, field and its properties forming the basis of algebra, and learn the theory system on the method of constructing code space with high error correction capability. Learn systematically about the algebra and its application necessary for the construction method and decoding method of the code, centering on the Reed-Solomon code which is the most widely used error correction code.
Keywords
Error correction code, encoding and decoding, algebra, group, ring, finite field, minimum distance, linear code, Hamming code, RS code, BCH code
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
This class will be conducted face-to-face.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | code space, channels, Hamming distance, minimum distance, bounded-distance decoding, MAP decoding, ML decoding | Explain code space, channels, Hamming distance, minimum distance, bounded-distance decoding, MAP decoding, ML decoding |
| Class 2 | generator matrix, parity-check matrix, dimension, Singleton bound | Explain generator matrix, parity-check matrix, dimension, Singleton bound |
| Class 3 | parity-check matrix and minimum distance, Hamming code | Explain parity-check matrix and minimum distance, Hamming code |
| Class 4 | coset, syndrome, syndrome decoder | Explain coset, syndrome, syndrome decoder |
| Class 5 | bounds on codes, weight distribution, MacWilliams identity | Explain |
| Class 6 | group, cyclic group, sub-group, coset, quotient ring | Explain group, cyclic group, sub-group, coset, quotient ring |
| Class 7 | ring, homomorphism, isomorphism, ideal, integer ring, Euclidean algorithm, uniqueness of prime factorizations | Explain ring, isomorphism, ideal, integer ring, Euclidean algorithm, uniqueness of prime factorizations |
| Class 8 | finite field, polynomial ring, primitive element, order | Explain finite field, polynomial ring, primitive element, order |
| Class 9 | construction of finite fields, structure of finite fields, minimum polynomial | Explain construction of finite fields, structure of finite fields, minimum polynomial |
| Class 10 | construction of RS codes | Explain construction of RS codes |
| Class 11 | Vandermonde's matrices, generator and parity-check matrix of RS codes | Explain Vandermonde's matrices, generator and parity-check matrix of RS codes |
| Class 12 | decoding of RS codes | Explain decoding of RS codes |
| Class 13 | cyclic codes, generator and parity-check matrices | Explain cyclic codes, generator and parity-check matrices |
| Class 14 | construction of BCH codes, minimum distance of BCH codes, cyclic RS codes, relation on RS codes an BCH codes, | Explain construction of BCH codes, minimum distance of BCH codes, cyclic RS codes |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
Handouts are provided at each class.
Reference books, course materials, etc.
坂庭好一、渋谷智治、代数系と符号理論入門、コロナ社、2010年
植松友彦、代数系と符号理論、オーム社、2010年
Youtube https://bit.ly/2V9Ibm7
Evaluation methods and criteria
Grade is based on exams and exercises.
Related courses
- ICT.C205 ： Communication Theory (ICT)
- ICT.C201 ： Introduction to Information and Communications Engineering
- ICT.E218 ： Experiments of Information and Communications Engineering II
- ICT.C214 ： Communication Systems
Prerequisites
Nothing special is required.