Mathematics 4109/6101

Fields and Coding Theory

 

Course Outline

 

 

 

Topics covered in the class

  • Sept. 5: Introduction, Fields.
  • Sept. 10: Characteristic, Binomial Theorem, isomorphism.
  • Sept. 12: Isomorphism, prime fields, polynmial rings.
  • Sept. 17: Polynomial rings, Division algorithm, gcd, Euclidean algorithm
  • Sept. 19: Irreducible polynomials, unique factorization, residue class rings
  • Sept. 24: Residue class fields, fields extensions.
  • Sept. 26: Fields extensions; linear codes.
  • Assign#1 is out.
  • Oct. 1: Linear codes.
  • Oct. 3: Syndrome and Hamming codes. Multiplicative group of a finite field
  • Oct. 8: review of cyclic groups; multiplicative group of finite fields; primitive elements
  • Oct. 10: Gauss algorithm, size of a finite field, Mobius functions,
  • Oct. 15: Mobius functions, existence of irreducible polynomials
  • Oct. 17: subfields, a distinction between finite fields with odd characteristic and even characteristic
  • Assign #2 is out.
  • Oct. 22: automorphisms, characteristic polynomials, minimal polynomials.
  • Oct. 24: midterm
  • Oct. 29: no class
  • Oct. 31: no class
  • Nov. 5: minimial polynomials, primitive polynomials
  • Nov. 7: period of polynomials, trace and norm.
  • Nov. 12 Trace and Norm.
  • Nov. 14. Bases; Berlekamp's algorithm
  • Nov. 19. Berlekamp's algorithm
  • Nov. 21. rationale , factorization of x^n -1.
  • Nov. 26. cyclic codes.
  • Nov. 28. Cyclic codes and Hamming codes.
  • Dec. 3. double error correcting BCH codes,
  • Dec. 5 BCH codes with designed distance, Reed-Solomon codes.

    Assignment # 1

    Assignment # 2