470-4202/02 – Abstract Algebra in Coding Theory (AvTK)
| Gurantor department | Department of Applied Mathematics | Credits | 4 |
| Subject guarantor | doc. Mgr. Petr Kovář, Ph.D. | Subject version guarantor | doc. Mgr. Petr Kovář, Ph.D. |
| Study level | undergraduate or graduate | Requirement | Optional |
| Year | 2 | Semester | winter |
| | Study language | English |
| Year of introduction | 2016/2017 | Year of cancellation | |
| Intended for the faculties | FEI | Intended for study types | Follow-up Master |
Subject aims expressed by acquired skills and competences
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.
Teaching methods
Lectures
Tutorials
Summary
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.
Compulsory literature:
Recommended literature:
Way of continuous check of knowledge in the course of semester
There will be two tests or the students will prepare a project.
E-learning
Other requirements
There are no further requirements on the student.
Prerequisities
Subject has no prerequisities.
Co-requisities
Subject has no co-requisities.
Subject syllabus:
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.
Conditions for subject completion
Occurrence in study plans
Occurrence in special blocks