2019 Algebraic Systems and Coding Theory

Font size  SML

Register update notification mail Add to favorite lecture list
Academic unit or major
Undergraduate major in Information and Communications Engineering
Instructor(s)
Kasai Kenta 
Course component(s)
Lecture
Day/Period(Room No.)
Mon3-4(W631)  Thr3-4(W631)  
Group
-
Course number
ICT.C209
Credits
2
Academic year
2019
Offered quarter
3Q
Syllabus updated
2019/9/25
Lecture notes updated
2019/11/17
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

Intercultural skills Communication skills Specialist 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 Explain construction of BCH codes, minimum distance of BCH codes, cyclic RS codes
Class 15 relation on RS codes an BCH codes, Explain relation on RS codes an BCH codes

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

Assessment criteria and methods

Grade are based on exercise, homework, mid-term and final exams.

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 (i.e., required knowledge, skills, courses, etc.)

nothing

Page Top