### 2020　Algebraic Systems and Coding Theory

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Undergraduate major in Information and Communications Engineering
Instructor(s)
Kasai Kenta
Course component(s)
Lecture
Mode of instruction
ZOOM
Day/Period(Room No.)
()
Group
-
Course number
ICT.C209
Credits
2
2020
Offered quarter
3Q
Syllabus updated
2020/7/29
Lecture notes updated
2020/4/15
Language used
Japanese
Access Index ### Course description and aims

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.

### Student learning outcomes

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 that will be developed

 ✔ Specialist skills Intercultural skills Communication skills Critical thinking skills ✔ Practical and/or problem-solving skills

### Class flow

Each week consists of two lectures.

### Course schedule/Required learning

Course schedule Required learning
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

### Textbook(s)

Handouts are provided at each class.

### Assessment criteria and methods 