Topics covered in the class
Sept. 3: Introduction, Fields.
Sept. 8: Characteristic, Binomial Theorem, isomorphism.
Sept. 10: Isomorphism, prime fields, polynmial rings.
Sept. 15: Division algorithm, gcd, Euclidean algorithm, Irreducible polynomials,
Sept. 17: residue class rings, Residue class fields, fields extensions.
Sept. 22: Fields extensions; motivation of linear codes.
Sept. 24: Linear codes.
Oct. 29: Linear codes.
Oct. 1: Syndrome and Hamming codes. Multiplicative group of a finite field,
review of cyclic groups;
Assign#1 is out.
Oct. 6: multiplicative group of finite fields; primitive elements; Gauss algorithm, size of a finite field.
Oct. 8: Gauss algorithm, size of a finite field, Mobius functions,
Oct. 13: Mobius functions, existence of irreducible polynomials
Oct. 15: subfields, a distinction between finite fields with odd characteristic and even characteristic, automorphism
Oct. 20. Midterm.
Oct. 22. automorphisms, characteristic polynomials, minimal polynomials.
Assign #2 is out.
Oct. 26-30. Reading week.
Nov. 3: minimial polynomials, primitive polynomials
Nov. 5: period of polynomials, trace and norm.
Nov. 10 Trace and Norm.
Nov. 12. Bases; Berlekamp's algorithm
Nov. 17 . Berlekamp's algorithm
Nov. 19. rationale , factorization of x^n -1.
Nov. 24. cyclic codes.
Nov. 26. Cyclic codes and Hamming codes.
Dec. 1. Double error correcting BCH codes,
Dec. 3. BCH codes with designed distance, Reed-Solomon codes.