Sep 9: Introduction to the course. History and basics of finite fields.
What is coding theory.
Sep 11: 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.
Sep 16: 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.
Sep 18: Decoding linear codes. Cosets. Coset leader. Syndrome.
Decoding algorithm. Hamming bound.
Sep 23: Dual codes and properties. Hamming codes: definition and proof
of 1-error-correction. 2-error-correcting BCH codes (introduction). Definition
of finite fields and prime finite fields.
Sep 25: Polynomials over finite fields. Examples. Irreducible polynomials and
their importance in finite fields. Unique factorization. 2-error-correcting BCH codes.
[A1 handed out.]
Sep 30: Comments about Assignment 1. Decoding algorithm for
2-error-correcting BCH codes. The multiplicative group of the nonzero elements
in a finite field is cyclic. Primitive elements. Representation of elements using
primitive elements. Example of decoding procedure.
Oct 2: Review of rings, fields and characteristic of a ring. Finite fields have
prime characteristic. Properties. The ring of polynomials. Division algorithm.
Greatest common divisors and Euclidean algorithm. Examples. Ideals, principal ideals,
maximal ideals, prime ideals and principal ideal domains. Characterizations of these
structures.
To October lectures.