Abstract Algebra in Coding Theory

- 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.