November Lectures
Week of November 1-5
Nov 1: Gauss' algorithm: correctness and examples.
Comments about Assignment 2.
Nov 3: The number of primitive elements. The Moebius function
and properties. The number of irreducible polynomials of certain
degree. Examples.
Week of November 8-12
Nov 8: Prove that x^q^n - x is the product of all monic
irreducible polynomials of degree dividing n. Fast algorithms
for testing the irreducibility of a polynomial. Examples.
Nov 10: Irreducibility test (cont). Factorization of
polynomials: applications, complete factorization.
Distinct-degree factorization.
[A2 handed in; A3 handed out.]
Week of November 15-19
Nov 15: Comments about Assignment 3. Squarefree factorization.
Equal-degree factorization (Cantor and Zassenhaus).
Nov 17: Equal-degree factorization (Cantor and Zassenhaus).
Cyclic codes: definition and examples. Characterization of cyclic
codes as ideals. Generator polynomial and matrix.
Week of November 22-26
Nov 22: Generator polynomial and matrix (cont).
Minimal polynomials: definition and properties.
Nov 24: Minimal polynomials: definition and properties (cont).
Cyclotomic cosets; examples. Cyclotomic cosets and factors of x^n - 1.
Week of November 29 - December 3
Nov 29: Check polynomial and parity check matrix. Comments
about assignment 3. Hamming codes as examples of cyclic codes.
Computing minimal polynomials. 2-error-correcting BCH codes as
example of cyclic codes.
Dec 1: t-error-correcting BCH codes. Designed distance
and t-error-correcting BCH codes. Examples. Course evaluations.
Comments about the exam. Oral presentations.
[A3 handed in.]
Week of December 6
Dec 6:
Reed-Solomon codes.
To October lectures.