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Cyclic Linear Codes

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Cyclic Linear Codes

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    2. OUTLINE [1] Polynomials and words [2] Introduction to cyclic codes [3] Generating and parity check matrices for cyclic codes [4] Finding cyclic codes [5] Dual cyclic codes

    3. Cyclic Linear Codes [1] Polynomials and words 1. Polynomial of degree n over K 2. Eg 4.1.1

    4. Cyclic Linear Codes 3. [Algorithm 4.1.8]Division algorithm 4. Eg. 4.1.9

    5. 5. Code represented by a set of polynomials A code C of length n can be represented as a set of polynomials over K of degree at most n-1 6. E.g 4.1.12 Cyclic Linear Codes

    6. Cyclic Linear Codes 7. f(x) and p(x) are equivalent modulo h(x) 8.Eg 4.1.15 9. Eg 4.1.16

    7. Cyclic Linear Codes 10. Lemma 4.1.17 11. Eg. 4.1.18

    8. [2]Introduction to cyclic codes 1. cyclic shift p(v) V: 010110, : 001011 2.cyclic code A code C is cyclic code(or linear cyclic code) if (1)the cyclic shift of each codeword is also a codeword and (2) C is a linear code C1=(000, 110, 101, 011} is a cyclic code C2={000, 100, 011, 111} is NOT a cyclic code V=100, =010 is not in C2 Cyclic Linear Codes

    9. Cyclic Linear Codes 3. Cyclic shiftpis a linear transformation S={v, p(v), p2(v), , p n-1(v)}, and C=<S>, then v is a generator of the linear cyclic code C

    10. Cyclic Linear Codes 4. Cyclic Code in terms of polynomial

    11. Cyclic Linear Codes 5. Lemma 4.2.12 Let C be a cyclic code let v in C. Then for any polynomial a(x), c(x)=a(x)v(x)mod(1+xn) is a codeword in C 6. Theorem 4.2.13 C: a cyclic code of length n, g(x): the generator polynomial, which is the unique nonzero polynomial of minimum degree in C. degree(g(x)) : n-k, 1. C has dimension k 2. g(x), xg(x), x2g(x), ., xk-1g(x) are a basis for C 3. If c(x) in C, c(x)=a(x)g(x) for some polynomial a(x) with degree(a(x))<k

    12. Cyclic Linear Codes 7. Eg 4.2.16 the smallest linear cyclic code C of length 6 containing g(x)=1+x3 <-> 100100 is {000000, 100100, 010010, 001001, 110110, 101101, 011011, 111111} 8. Theorem 4.2.17 g(x) is the generator polynomial for a linear cyclic code of length n if only if g(x) divides 1+xn (so 1+xn =g(x)h(x)).

    13. Cyclic Linear Codes 9. Corollary 4.2.18 The generator polynomial g(x) for the smallest cyclic code of length n containing the word v(polynomial v(x)) is g(x)=gcd(v(x), 1+xn) 10. Eg 4.2.19 n=8, v=11011000 so v(x)=1+x+x3+x4 g(x)=gcd(1+x+x3+x4 , 1+x8)=1+x2 Thus g(x)=1+x2 is the smallest cyclic linear code containing v(x), which has dimension of 6.

    14. Cyclic Linear Codes [3]. Generating and parity check matrices for cyclic code 1. Effective to find a generating matrix The simplest generator matrices (Theorem 4.2.13)

    15. Cyclic Linear Codes 2. Eg 4.3.2 C: the linear cyclic codes of length n=7 with generator polynomial g(x)=1+x+x3, and deg(g(x))=3, => k = 4

    16. Cyclic Linear Codes 3. Efficient encoding for cyclic codes

    17. Cyclic Linear Codes 4. Parity check matrix H : wH=0 if only if w is a codeword Symdrome polynomial s(x) c(x): a codeword, e(x):error polynomial, and w(x)=c(x)+e(x) s(x) = w(x) mod g(x) = e(x) mod g(x), because c(x)=a(x)g(x) H: i-th row ri is the word of length n-k => ri(x)=xi mod g(x) wH = (c+e)H => c(x) mod g(x) + e(x) mod g(x) = s(x)

    18. Cyclic Linear Codes 5. Eg 4.3.7 n=7, g(x)=1+x+x3, n-k = 3

    19. Cyclic Linear Codes [4]. Finding cyclic codes 1. To construct a linear cyclic code of length n Find a factor g(x) of 1+xn, deg(g(x)) = n-k Irreducible polynomials f(x) in K[x], deg(f(x)) >= 1 There are no a(x), b(x) such that f(x)=a(x)b(x), deg(a(x))>=1, deg(b(x))>=1 For n <= 31, the factorization of 1+xn (see Appendix B) Improper cyclic codes: Kn and {0}

    20. Cyclic Linear Codes 2. Theorem 4.4.3 3. Coro 4.4.4

    21. Cyclic Linear Codes 4. Idempotent polynomials I(x) I(x) = I(x)2 mod (1+xn) for odd n Find a basic set of I(x) Ci= { s=2j i (mod n) | j=0, 1, , r} where 1 = 2r mod n

    22. Cyclic Linear Codes 5. Eg 4.4.12 6. Theorem 4.4.13 Every cyclic code contains a unique idempotent polynomial which generates the code.(?)

    23. Cyclic Linear Codes 7. Eg. 4.4.14 find all cyclic codes of length 9

    24. Cyclic Linear Codes [5].Dual cyclic codes 1. The dual code of a cyclic code is also cyclic 2. Lemma 4.5.1 a ?> a(x), b ?> b(x) and b ?> b(x)=xnb(x-1) mod 1+xn then a(x)b(x) mod 1+xn = 0 iff pk(a).b=0 for k=0,1,n-1 3. Theorem 4.5.2 C: a linear code, length n, dimension k with generator g(x) If 1+xn = g(x)h(x) then C?: a linear code , dimension n-k with generator xkh(x-1)

    25. Cyclic Linear Codes 4. Eg. 4.5.3 g(x)=1+x+x3, n=7, k=7-3=4 h(x)=1+x+x2+x4 h(x)generator for C? is g? (x)=x4h(x-1)=x4(1+x-1+x-2+x-4 )=1+x2+x3+x4 5. Eg. 4.5.4 g(x)=1+x+x2, n=6, k=6-2=4 h(x)=1+x+x3+x4 h(x)generator for C? is g? (x)=x4h(x-1)=1+x+x3+x4

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