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.
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.
Error correction code, encoding and decoding, algebra, group, ring, finite field, minimum distance, linear code, Hamming code, RS code, BCH code
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | ✔ Practical and/or problem-solving skills |
Each week consists of two lectures.
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 |
Handouts are provided at each class.
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
Grade are based on exercise, homework, mid-term and final exams.
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