Communication Signal Design Lab.

한국어

송홍엽 교수의 잡글

교재준비작업(1)

2004.10.03 21:27

송홍엽 조회 수:7272 추천:236

이산수학과 유한체이론 - 주별 강의주제 목록

1. Property of Z

    Z is a commutative ring with 1.
    it is an integral domain.
    it is a Euclidean domain.
    it is a Unique Factorization domain.
    Z/(n) is a ring, domain, ED, and UFD.
    G=U(Z/(n)) is a multiplicative group of order phi(n)
              it is cyclic if and only if n=1,2,4,p^k, 2p^k.
    Z/(n) is a field if and only if n=prime
             U(Z/(p)) is cyclic of order p-1

2. Computations over Z

    Linear equation over  Z/(n)
    Chinese Remainer Theorem
    Quadratic equation over Z/(n)
    Quadratic Reciprocity Theorem - Legendre/Jacobi symbols
    Payley construction of Hadamard matrix
    Big Integer arithematic
    Fast Exponentiation

3. Some Public Key Crypto Algorithms

    Primality testing algorithm
    DLP - analysis
    ElGamal Algorithm
    RSA Algorithm
    Secrete Sharing Algorithm
    Coin-flipping over Telephone
    Public key Envelope

4. Permutations and Counting

   Definition, notation, order, cycles
   unique decomposition
   even and odd permutation
   Not-Burnside Theorem on counting

5. Vectors and Matrix

   n-tuple vector space over F
   Basis, Linear Independance
   simultaneous equation and coefficient matrix
   Gauss Elimination,    rank of a matrix,    LU decomposition
   column space, row space, orthogonal complement
        rank, nullity, basic relation
   Vandemond matrix
  
6. Linear Transformation and Matrix

   Definition of Linear Transformation
   Range space and Null space, basic relation
   relation to Matrix
   Multi-linear transformation and Determinant
   Existence of Determinant

7. Some Problem Discussions

8. midterm

9. Polynomial over GF(p) = Fp[x]

    Fp[x] is a commutative ring with 1.
    it is an integral domain.
    it is a Euclidean domain.
    it is a Unique Factorization domain.
    Fp[x]/(f(x)) is a ring, domain, ED, and UFD.
    Fp[x]/(f(x))  is a field if and only if f(x) is irreducible

10. Structure of Finite Field

   Extension and Subfield
   Additive structure
   Multiplicative structure and conjugate class
   Addone table
   Homomorphism and isomorphism

11. Irreducible Polynomial over finite field

   irreducible polynomials over F2
   Property of minimal polynomials
   Conjugates
   Trace function
   m-sequence

12. Irreducible Polynomial and Cyclotomic Polynomial

   x^{q^n}-x = ㅠ V_d(x)
   Number of irreducible polynomials
   Cyclotomic Polynomials
        over C
        over Finite Field
   Some Factoring

13. Error-correcting linear codes

   binary symmetric channel, binary erasure channel
   binary linear code and minimum distance decoding
   Binary Hamming code and decoding
   minimum distance decoding is ML decoding
   BCH code over Z/(p)
   Some nonlinear simultaneous equations for decoding of BCH code over  Z/(p)

14. Cyclic code

   Hamming code and BCH code as cyclic code
   RS codes - encoding and decoding
   GFFT approach

15. Some Problem Discussions

16. final exam


  


번호 제목 글쓴이 날짜 조회 수
공지 논문에 영어작문 주의사항 몇 가지 송홍엽 2008.05.22 8438
공지 젊은 학부생 여러분에게... 송홍엽 2008.11.20 5821
공지 우리학과 대학원생 모두에게 (특히, 박사과정들에게) 하고싶은 말입니다. 송홍엽 2014.01.20 5784
88 basis란.. (and Online-HW2) 송홍엽 2003.10.06 20647
87 Outlook Express Backup [1] 송홍엽 2006.03.04 18434
86 [교양상식] 메르센 소수--Mersenne Prime 송홍엽 2004.04.13 14047
85 [펌]Turbo code에 대한 글 송홍엽 2004.04.13 13619
84 funny story 송홍엽 2004.04.01 11381
83 하다마드 행렬에 대해서 (part 1) [1] 송홍엽 2003.10.06 10015
82 Dr. Shannon 송홍엽 2003.12.07 8668
81 SHANNON : CRACKING THE CHANNEL file 송홍엽 2004.04.13 8626
80 대학원생 생활 가이드 [퍼온글] 송홍엽 2007.10.11 8219
79 수학사 바로잡기 (2) 송홍엽 2008.06.03 7681
78 [펀글] 추장의 선언 [1] 송홍엽 2008.02.22 7287
» 교재준비작업(1) 송홍엽 2004.10.03 7272
76 유클리디안 n차원 벡터공간 (and Online-HW1) 송홍엽 2003.10.06 7226
75 아날로그 신호와 디지털 신호 송홍엽 2004.08.06 7133
74 글 잘 쓰는 이공계가 성공한다 송홍엽 2008.05.22 6659
73 수학사 바로잡기 (1) - 오일러의 36명 장교문제와 조선시대 최석정 (수정본 - 일부 오류 수정) [1] file 송홍엽 2008.06.03 6290
72 [펀글]수학의 open problems 모음. 송홍엽 2008.07.30 6242
71 [펀글] 전화기를 최초로 발명한 사람은 누구인가 송홍엽 2008.05.23 6185
70 [퍼온글] 우리전공의 학부생이 공부하면 좋을 수학과목 ?? 송홍엽 2003.03.23 6021