Sep 11: Introduction to the course. History and basics of finite fields.
What is coding theory.
Sep 13: Linear codes: coding and decoding schemes; parity-matrix H;
linear (n,k) code; dimension and length; code words.
Examples of codes (parity-check code and repetition code),
detecting and correcting errors. Canonical generator matrix.
Sep 18: Canonical generator matrix. Hamming distance, Hamming weight
and t-error-correcting codes.
Minimum distance of a code and its relation to
t-error-correcting codes. Alternative characterization as
linearly independent columns of the parity-check matrix H.
Decoding linear codes. Cosets.
Sep 20: Decoding linear codes. Coset leader. Syndrome.
Decoding algorithm. Hamming bound.
Dual codes and properties. [A1 handed out.]
Sep 25: The relation between syndrome and errors. Hamming codes:
definition and proof of 1-error-correction.
2-error-correcting BCH codes (introduction). Definition of
finite fields, prime finite fields, and
polynomials over finite fields. Examples.
Sep 27: Comments about Assignment 1. Irreducible polynomials and
their importance in finite fields.
Unique factorization. 2-error-correcting BCH codes (continuation).