1. Cyclic Linear Codes

2. p2. 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. p3. Cyclic Linear Codes [1] Polynomials and words 1. Polynomial of degree n over K 2. Eg 4.1.1

4. p4. Cyclic Linear Codes 3. [Algorithm 4.1.8]Division algorithm 4. Eg. 4.1.9

5. p5. 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 Codeword c Polynomial c(x) 0000 1010 0101 1111 0 1+x 2 x+x 3 1+x+x 2 +x 3

6. p6. 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. p7. Cyclic Linear Codes 10. Lemma 4.1.17 11. Eg. 4.1.18

8. p8. [2]Introduction to cyclic codes 1. cyclic shift π(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 v 10110 111000 0000 1011 01011 011100 00001101

9. p9. Cyclic Linear Codes 3. Cyclic shiftπis a linear transformation S={v, π(v), π 2 (v), …, π n-1 (v)}, and C=, then v is a generator of the linear cyclic code C

10. p10. Cyclic Linear Codes 4. Cyclic Code in terms of polynomial

11. p11. 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+x n ) 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), x 2 g(x), …., x k-1 g(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. p12. Cyclic Linear Codes 7. Eg 4.2.16 the smallest linear cyclic code C of length 6 containing g(x)=1+x 3 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+x n (so 1+x n =g(x)h(x)).

13. p13. 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+x n ) 10. Eg 4.2.19 n=8, v=11011000 so v(x)=1+x+x 3 +x 4 g(x)=gcd(1+x+x 3 +x 4, 1+x 8 )=1+x 2 Thus g(x)=1+x 2 is the smallest cyclic linear code containing v(x), which has dimension of 6.

14. p14. 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. p15. Cyclic Linear Codes 2. Eg 4.3.2 C: the linear cyclic codes of length n=7 with generator polynomial g(x)=1+x+x 3, and deg(g(x))=3, => k = 4

16. p16. Cyclic Linear Codes 3. Efficient encoding for cyclic codes

17. p17. 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 r i is the word of length n-k => r i (x)=x i mod g(x) wH = (c+e)H => c(x) mod g(x) + e(x) mod g(x) = s(x)

18. p18. Cyclic Linear Codes 5. Eg 4.3.7 n=7, g(x)=1+x+x 3, n-k = 3

19. p19. Cyclic Linear Codes [4]. Finding cyclic codes 1. To construct a linear cyclic code of length n Find a factor g(x) of 1+x n, 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+x n (see Appendix B) Improper cyclic codes: K n and {0}

20. p20. Cyclic Linear Codes 2. Theorem 4.4.3 3. Coro 4.4.4

21. p21. Cyclic Linear Codes 4. Idempotent polynomials I(x) I(x) = I(x) 2 mod (1+x n ) for odd n Find a “basic” set of I(x) C i = { s=2 j i (mod n) | j=0, 1, …, r} where 1 = 2 r mod n

22. p22. 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. p23. Cyclic Linear Codes 7. Eg. 4.4.14 find all cyclic codes of length 9 The generator polynomial g(x)=gcd(I(x), 1+x 9 ) Idempotent polynomial I(x) 1 1+x+x 3 +x 4 +x 6 +x 7 1+x 3 1+x+x 2 : 1 x+x 2 +x 4 +x 5 +x 7 +x 8 x 3 +x 6 1+x+x 2 +x 4 +x 5 +x 7 +x 8 :

24. p24. 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)=x n b(x -1 ) mod 1+x n then a(x)b(x) mod 1+x n = 0 iff π k (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+x n = g(x)h(x) then C ⊥ : a linear code, dimension n-k with generator x k h(x -1 )

25. p25. Cyclic Linear Codes 4. Eg. 4.5.3 g(x)=1+x+x 3, n=7, k=7-3=4 h(x)=1+x+x 2 +x 4 h(x)generator for C ⊥ is g ⊥ (x)=x 4 h(x -1 )=x 4 (1+x -1 +x -2 +x -4 )=1+x 2 +x 3 +x 4 5. Eg. 4.5.4 g(x)=1+x+x 2, n=6, k=6-2=4 h(x)=1+x+x 3 +x 4 h(x)generator for C ⊥ is g ⊥ (x)=x 4 h(x -1 )=1+x+x 3 +x 4