Appears in collection : Elisa Lorenzo García: Coding theory

Lecture 1. Error-correcting Codes: repetition code, parity check code, Hamming code. Hamming distance. Linear codes: generator matrix, parity check matrix, dual code. Lecture 2. Decoding and error probability: Basic decoding, symmetric channel. Equivalent codes. Shannon Theorem. Weight enumerator. Lecture 3. Codes constructions and bounds: Punturing, restriction, extension, short, augmentation, direct sum, juxtaposition, product and concatenation of codes. Singleton, Griesmer, Plotkin, Hamming, Gilbert Varshamov and asymptotic bounds. Lecture 4. Cyclic codes: cyclic codes as ideals. Encoding cyclic codes. Parity check polynomial. BCH bound. Examples. Lecture 5. Algebraic geometry codes and applications: Codes on curves, Goppa codes, Reid-Solomon codes. The McEliece cryptosystem. Lecture 6: Codes on graphs: Some graph theory. Cycle code and graph code of a graph