| Course Unit Code | 470-4202/01 |
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| Number of ECTS Credits Allocated | 4 ECTS credits |
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| Type of Course Unit * | Optional |
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| Level of Course Unit * | Second Cycle |
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| Year of Study * | Second Year |
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| Semester when the Course Unit is delivered | Winter Semester |
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| Mode of Delivery | Face-to-face |
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| Language of Instruction | Czech |
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| Prerequisites and Co-Requisites | Course succeeds to compulsory courses of previous semester |
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| Name of Lecturer(s) | Personal ID | Name |
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| KOV16 | doc. Mgr. Petr Kovář, Ph.D. |
| Summary |
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| 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 |
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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 |
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- Congruences, modular arithmetics, binary a q-ary systems, symmetries and their description
- Finite algebraic structures with a single operation, properties and applications, dihedral and cyclic groups.
- Products, isomorphisms, construction of groups, classification.
- Finite algebraic structures with two operations, polynomial rings, operations, properties.
- Fields of prime order, factor rings, examples.
- Factorization of polynomials, irreducibile polynomials.
- Construction of Galois fields, properties.
- Finite vector spaces, construction, examples and applications.
- Main coding theory problem, sample codes, applications.
- Codes as vector spaces. Hamming distance. Equivalence of codes.
- Simple linear and cyclic codes, importance and examples.
- Encoding and decoding by a linear code, probability of detecting and correcting an error.
- Further simple codes, codes and Latin squares. |
| Recommended or Required Reading |
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| Required Reading: |
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| 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.
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| Recommended Reading: |
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| 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 |
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| Lectures, Tutorials |
| Assesment methods and criteria |
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| Task Title | Task Type | Maximum Number of Points
(Act. for Subtasks) | Minimum Number of Points for Task Passing |
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| Credit and Examination | Credit and Examination | 100 (100) | 51 |
| Credit | Credit | 30 | 10 |
| Examination | Examination | 70 | 30 |