Algebraic Coding Theory

Code:

M4081

Acronym:

M4081

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2016/2017 - 1S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Master in Mathematical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
M:ENM	7	Official Study Plan since 2013-2014	1	-	6	56	162

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:

1. Shannon’s Theory: Models of communication, probabilistic assumptions and Shannon’s Theorem
2. Block codes on arbitrary sets and Hamming metric
3. Isometries of codes and basic constructions
4. Bounds on codes (Singelton, Gilbert-Varshamov and Haming bound) with their classical codes: MDS, perfect and Hamming codes
5. Block codes over groups (parity check codes)
6. Block codes over fields (Revision of linear algebra over finite fields)
7. Dual code and the MacWilliams’s Theorems
8. Examples of codes and their decoding: Golay code and Reed-Solomon code
9. BCH-code (Revision of field theory)
10. Non-linear codes (Lee-metric and codes over finite rings)
11. Perspectives

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

• Lectures of 4 hours/week


• 4 home work assignment during the semester (each counts 1 point).

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	80,00
Participação presencial	0,00
Trabalho escrito	20,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Elaboração de relatório/dissertação/tese	40,00
Estudo autónomo	48,00
Frequência das aulas	48,00
Total:	136,00

Calculation formula of final grade

4 home work assignments = 4 points
final exam = 16 points

sum: 4+16=20 points

Examinations or Special Assignments

Final exam ( 0 - 16 points)
Makeup exam (0 - 16 points)

Internship work/project

4 home work assignments during the semester

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 home work assignments.