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Abstract Algebra in Coding Theory

* Exchange students do not have to consider this information when selecting suitable courses for an exchange stay.

Course Unit Code470-4202/01
Number of ECTS Credits Allocated4 ECTS credits
Type of Course Unit *Optional
Level of Course Unit *Second Cycle
Year of Study *Second Year
Semester when the Course Unit is deliveredWinter Semester
Mode of DeliveryFace-to-face
Language of InstructionCzech
Prerequisites and Co-Requisites Course succeeds to compulsory courses of previous semester
Name of Lecturer(s)Personal IDName
KOV16doc. Mgr. Petr Kovář, Ph.D.
The course serves a building block for Coding Theory. The goal is to provide an overview of methods and train relevant skills, that will be used in the Coding Theory course.
Learning Outcomes of the Course Unit
After passing the course a student will be able:
- use congruences when solving discrete problems,
- describe symmetries of real world problem using groups,
- calculate polynomial operations in modular arithmetics,
- construct selected Galois fields and simple codes based on these,
- construct simple finite vector fields,
- perform comutation on code words in vector notation,
- perform operations on selected codes in matrix notation,
- encode and decode a message in a simple code,
- detect and correct basic mistakes in transmission.
Course Contents
1. Congruences, modular arithmetics, binary a q-ary systems.
2. Symmetries and their description, dihedral and cyclic groups.
3. Finite algebraic structures with a single operation, properties and applications,
4. Products, isomorphisms, construction of groups, classification.
5. Finite algebraic structures with two operations, polynomial rings, operations, properties.
6. Fields of prime order, factor rings, examples.
7. Factorization of polynomials, irreducibile polynomials.
8. Construction of Galois fields, properties.
9. Finite vector spaces, construction, examples and applications.
10. Main coding theory problem, sample codes, applications.
11. Codes as vector spaces. Hamming distance. Equivalence of codes.
12. Simple linear and cyclic codes, importance and examples.
13. Encoding and decoding by a linear code, probability of detecting and correcting an error.
14. Further simple codes, codes and Latin squares.
Recommended or Required Reading
Required Reading:
R. HILL: A First Course in Coding Theory, Oxford University Press 2006, ISBN 0-19-853803-0.
J. STANOVSKÝ: Základy algebry, Matfyzpress 2010, ISBN 9788073781057.
J. MAREŠ: Teorie kódování. Skripta ČVUT, Praha 2008.

Recommended Reading:
J. GALLIAN: Contemporary Abstract Algebra, Cegage Learning; 8th edition 2012, ISBN 978-1133599708.
J. ADÁMEK: Kódování. Matematika pro vysoké školy technické, sešit XXXI. SNTL, Praha, 1989.
Planned learning activities and teaching methods
Lectures, Tutorials
Assesment methods and criteria
Task TitleTask TypeMaximum Number of Points
(Act. for Subtasks)
Minimum Number of Points for Task Passing
Credit and ExaminationCredit and Examination100 (100)51
        CreditCredit30 10
        ExaminationExamination70 30