Algebraic Coding Theory
Keywords |
Classification |
Keyword |
OFICIAL |
Mathematics |
Instance: 2017/2018 - 1S ![Requerida a integração com o Moodle Ícone do Moodle](/fcup/pt/imagens/MoodleIcon)
Cycles of Study/Courses
Teaching language
Suitable for English-speaking students
Objectives
Upon successful completion of this course, the student will:
- Know most of the classical examples of error correcting codes;
- Reproduce key results of the theory and give rigorous and detailed proofs of them.
- Construct new codes from old ones and examine their basic properties.
- Apply the basic techniques, results and concepts of the course to concrete examples and exercises.
Learning outcomes and competences
Upon successful completion of this course, the student will:
- Know most of the classical examples of error correcting codes;
- Reproduce key results of the theory and give rigorous and detailed proofs of them.
- Construct new codes from old ones and examine their basic properties.
- Apply the basic techniques, results and concepts of the course to concrete examples and exercises.
Working method
Presencial
Pre-requirements (prior knowledge) and co-requirements (common knowledge)
Linear Algebra over fields.
Theory of Finite Fields.
Program
The course will provide an introduction into the theory of error correcting codes. The following syllabus is a more algebraic approach to this course:
- Shannon’s Theory: Models of communication, probabilistic assumptions and Shannon’s Theorem
- Block codes on arbitrary sets and Hamming metric
- Isometries of codes and basic constructions
- Bounds on codes (Singelton, Gilbert-Varshamov and Haming bound) with their classical codes: MDS, perfect and Hamming codes
- Block codes over groups (parity check codes)
- Block codes over fields (Revision of linear algebra over finite fields)
- Dual code and the MacWilliams’s Theorems
- Examples of codes and their decoding: Golay code and Reed-Solomon code
- BCH-code (Revision of field theory)
Mandatory literature
Christian Lomp; Introduction to Algebraic Coding Theory, 2004 (lecture notes of the classes of the academic year 2004/2005)
Complementary Bibliography
Roman Steven;
Coding and information theory. ISBN: 0-387-97812-7
Hoffman D. G. 070;
Coding theory. ISBN: 0-8247-8611-4
Lint Jacobus H. van;
Coding theory. ISBN: 3-540-06363-3
Ling San;
Coding theory. ISBN: 0-521-82191-6
Pretzel Oliver;
Error-correcting codes and finite fields. ISBN: 0-19-859678-2
MacWilliams F. J.;
The theory of error-correcting codes. ISBN: 0-444-85193-3
Comments from the literature
The content of the lectures is not covered in a single book, but based on some lecture notes by W.Heise and T.Honold (2002, Sofia) as well as and on some lecture notes by V.Aurich (1993, Düsseldorf).
The lectured material is the sole subject of the course and the books indicated in the blbiography are considered to be auxiliar resources for the student.
Teaching methods and learning activities
Software
http://www.sagemath.org
keywords
Physical sciences > Mathematics > Algebra
Physical sciences > Mathematics > Algebra > Field theory
Technological sciences > Technology > Information technology
Evaluation Type
Distributed evaluation with final exam
Assessment Components
designation |
Weight (%) |
Exame |
100,00 |
Participação presencial |
0,00 |
Total: |
100,00 |
Calculation formula of final grade
The final grade (CF) is calculated as
CF=max( (T1+T2)/2, E)
where Ti is the grade of the ith teste and E is the grade of the grade of the (make-up) exam.
Examinations or Special Assignments
Two tests and one final exam (possibly one makeup exame)
Classification improvement
It is possible to improve the grade of the final exam in the makeup exam, but it is not possible to improve grade of the tests.