1. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU1 BCH Code a larger class of powerful random error-correcting cyclic codes a remarkable generalization of the Hamming codes for multiple-error correction Def : Let q and m be given and let α be any element of GF(q m ) of order n. Then for any positive integer t and any integer j 0, the corresponding BCH code is the cyclic code of block length n with the generator polynomial g(x) = LCM[f j 0 (x), f j 0 +1 (x), …, f j 0 +2t-1 (x)] where f j (x) is the minimal polynomial at α j ※ Let 1. n = q m – 1 |n | q m - 1 2. j 0 = 1 |α: non-primitive element (α n = 1)  | α: primitive element (α n = 1) | α, α 2, α 3, …, α 2t | primitive (or narrow-sense) BCH code|

2. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU2 ※ Let q = 2, j 0 = 1 ﹡取相鄰 power of Block length : n = 2 m – 1 or n | 2 m – 1 α 保證 d* ≥ 2t + 1 Number of parity-check digits : n – k ≤ mt  Minimum distance : d* ≥ 2t + 1 (BCH bound)   t-error-correcting BCH code designed distance g(x) = LCM [f 1 (x), f 2 (x), …, f 2t (x)] ∵ i (even) = I ’ 2 l where i ’ is an odd number and l ≥ 1 ∴ α i = (α i ’ ) 2 l is a conjugate of α i ’ ∴ α i & α i ’ have the same minimal polynomial i.e. f i (x) = f i’ (x) ∴ g(x) = LCM [ f 1 (x), f 3 (x), …, f 2t-1 (x) ] ∵ deg[ f i (x) ] ≤ m  n – k ≤ mt 2 = 12 4 = 12 2 6 = 32 8 = 12 3 10 = 52 1 Eg: t = 2 g(x) = LCM [f 1 (x), f 2 (x), f 3 (x), f 4 (x)] q = 2 = LCM [x 4 +x+1, x 4 +x+1, x 4 +x 3 +x 2 +x+1, x 4 +x+1] m = 4 = (x 4 +x+1)(x 4 +x 3 +x 2 +x+1) n = q m – 1 = 15 = x 8 +x 7 +x 6 +x 4 +1 n – k = 8  BCH(15, 7, 5) 2t + 1 ≤ d* ≤ w(g(x)) (n, t)  k ! k is unknown until after g(x) is found

3. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU3 Eg: t = 2 g(x) = LCM [f 1 (x), f 2 (x), f 3 (x), f 4 (x)] q = 4 = (x 2 +x+2)(x 2 +x+3)(x 2 +3x+1) m = 2 = x 6 +3x 5 +x 4 +2x 2 +2x+1 n = q m – 1 = 15 BCH(15, 9) t = 6 g(x) = x 14 +x 13 +…+x 2 +x+1 Q: BCH(15, 1) – it can actually correct 7 errors  simple repetition code ※ The single-error-correcting BCH code of length 2 m – 1 is a Hamming Code

4. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU4 Table 7.1 Representations of GF(2 4 ). p(z) = z 4 + z + 1 Exponential Notation Polynomial Notation Binary Notation Decimal Notation Minimal Polynomial 0000000x α0α0 100011x + 1 α1α1 z00102x 4 + x + 1 α2α2 z2z2 01004x 4 + x + 1 α3α3 z3z3 10008x 4 + x 3 + x 2 + x + 1 α4α4 z + 100113x 4 + x + 1 α5α5 z 2 + z01106x 2 + x + 1 α6α6 z 3 + z 2 110012x 4 + x 3 + x 2 + x + 1 α7α7 z 3 + z + 1101111x 4 + x 3 + 1 α8α8 z 2 + 101015x 4 + x + 1 α9α9 z 3 + z101010x 4 + x 3 + x 2 + x + 1 α 10 z 2 + z + 101117x 2 + x + 1 α 11 z 3 + z 2 + z + 1111014x 4 + x 3 + 1 α 12 z 3 + z 2 + z + 1111115x 4 + x 3 + x 2 + x + 1 α 13 z 3 + z 2 + 1110113x 4 + x 3 + 1 Α 14 z 3 + 110019x 4 + x 3 + 1

5. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU5 Table 7.2 GF(4 2 )  GF(4)[x]/z 2 +z+2 p(z) = z 2 +z+2 α α 2 α 4 α 8 α 16  α α 3 α 6 α 12 α 24 α 48  α 3  α 9 Exponential Notation Polynomial Notation Binary Notation Decimal Notation Minimal Polynomial 00000 α0α0 1011x + 1 α1α1 z104x 2 + x + 2 α2α2 z + 2126x 2 + x + 3 α3α3 3z + 23214x 2 + 3x + 1 α4α4 z + 1115x 2 + x + 2 α5α5 2022x + 2 α6α6 2z208x 2 +2x + 1 α7α7 2z + 32311x 2 + 2x + 2 α8α8 z + 3137x 2 + x + 3 α9α9 2z + 22210x 2 + 2x + 1 α 10 3033x + 3 α 11 3z3012x 2 + 3x + 3 α 12 3z + 13113x 2 + 3x + 1 α 13 2z + 1219x 2 + 2x + 2 α 14 3z + 33315x 2 + 3x + 3 α = z α 15 = 1

6. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU6 TABLE 6.1 BCH CODES GENERATED BY PRIMITIVE ELEMENTS OF ORDER LESS THAN 2 10 ( 上 ) mnktmnktmnktnktnkt 3741 63 247127501325518792557129 415111 1810 431417910 6330 72 1611 361517111 5531 53 1013 292116312 4742 531261 715 222315513 4543 212 7 1271201 152714714 3745 163 1132 83113915 2947 115 10638255247113118 2155 67 994 239212319 1359 663571 925 231311521 963 512 856 2234107225115021 453 787 21559923 4932 394 719 20769125 4843 365 6410 19978726 4754 306 5711 19187927 4665 n = 2 m - 1 For t small n – k = mt

7. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU7 TABLE 6.1 BCH CODES GENERATED BY PRIMITIVE ELEMENTS OF ORDER LESS THAN 2 10 ( 下 ) nktnktnktnktnkt 51145765113222251119343511589110239339 44873132318445499392310 43983042517546409591311 430929526166473110990312 4211028627157512811189313 4121127728148531911988314 4031226829139541012187315 3941325930130551013186316 38514250311215810231003285817 3761524136112599933 3671623837103619834 358182293894629735 349192203985639636 340202114176859537 331212024267879438

8. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU8 TABLE 6.2 GALOIS FIELD GF(2 6 ) WITH ρ(α) = 1 + α +α 6 = 0 00(0 0 0 0 0 0)α 15 α 3 +α 5 (0 0 0 1 0 1) 11(1 0 0 0 0 0)α 16 1+ α +α 4 (1 1 0 0 1 0) α αα 17 α+α 2 +α 5 (0 1 1 0 0 1) α2α2 α 2 (0 1 0 0 0 0)α 18 1+α+α 2 +α 3 (1 1 1 1 0 0) α3α3 α 3 (0 0 1 0 0 0)α 19 α+α 2 +α 3 +α 4 (0 1 1 1 1 0) α4α4 α 4 (0 0 0 1 0 0)α 20 α 2 +α 3 +α 4 +α 5 (0 0 1 1 1 1) α5α5 α 5 (0 0 0 0 0 1)α 21 1+α +α 3 +α 4 +α 5 (1 1 0 1 1 1) α6α6 1+α(1 1 0 0 0 0)α 22 1 +α 2 +α 4 +α 5 (1 0 1 0 1 1) α7α7 α+α 2 (0 1 1 0 0 0)α 23 1 +α 3 +α 5 (1 0 0 1 0 1) α8α8 α 2 +α 3 (0 0 1 1 0 0)α 24 1 +α 4 (1 0 0 0 1 0) α9α9 α 3 +α 4 (0 0 0 1 1 0)α 25 α +α 5 (0 1 0 0 0 1) α 10 α 4 +α 5 (0 0 0 0 1 1)α 26 1+α+α 2 (1 1 1 0 0 0) α 11 1+α(1 1 0 0 0 1)α 27 α+α 2 +α 3 (0 1 1 1 0 0) α 12 1 +α 2 (1 0 1 0 0 0)α 28 α 2 +α 3 +α 4 (0 0 1 1 1 0) α 13 α +α 3 (0 1 0 1 0 0)α 29 α 3 +α 4 +α 5 (0 0 0 1 1 1) α 14 α 2 +α 4 (0 0 1 0 1 0)α 30 1+α +α 4 +α 5 (1 1 0 0 1 1)

9. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU9 TABLE 6.4 GENERATOR POLYNOMIALS OF ALL THE BCH CODES OF LENGTH 63 nk tg(X) 6357 1 g 1 (X) = 1 + X + X 6 51 2 g 2 (X) = (1 + X + X 6 )(1 + X + X 2 + X 4 + X 6 ) 45 3 g 3 (X) = (1 + X + X 2 + X 5 + X 6 )g 2 (X) 39 4 g 4 (X) = (1 + X 3 + X 6 )g 3 (X) 36 5 g 5 (X) = (1 + X 2 + X 3 )g 4 (X) 30 6 g 6 (X) = (1 + X 2 + X 3 + X 5 + X 6 )g 5 (X) 24 7 g 7 (X) = (1 + X + X 3 + X 4 + X 6 )g 6 (X) 1810g 10 (X) = (1 + X 2 + X 4 + X 5 + X 6 )g 7 (X) 1611g 11 (X) = (1 + X + X 2 )g 10 (X) 1013g 13 (X) = (1 + X + X 4 + X 5 + X 6 )g 11 (X) 715g 15 (X) = (1 + X + X 3 )g 13 (X)

10. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU10 TABLE 6.2 Continued α 31 1 +α 2 +α 5 (1 0 1 0 0 1)α 47 1+α+α 2 +α 5 (1 1 1 0 0 1) α 32 1 +α 3 (1 0 0 1 0 0)α 48 1 +α 2 +α 3 (1 0 1 1 0 0) α 33 α +α 4 (0 1 0 0 1 0)α 49 α +α 3 +α 4 (0 1 0 1 1 0) α 34 α 2 +α 5 (0 0 1 0 0 1)α 50 α 2 +α 4 +α 5 (0 0 1 0 1 1) α 35 1+ α +α 3 (1 1 0 1 0 0)α 51 1+α +α 3 +α 5 (1 1 0 1 0 1) α 36 α+α 2 +α 4 (0 1 1 0 1 0)α 52 1 +α 2 +α 4 (1 0 1 0 1 0) α 37 α 2 +α 3 +α 5 (0 0 1 1 0 1)α 53 α +α 3 +α 5 (0 1 0 1 0 1) α 38 1+α +α 3 +α 4 (1 1 0 1 1 0)α 54 1+α+α 2 +α 4 (1 1 1 0 1 0) α 39 α+α 2 +α 4 +α 5 (0 1 1 0 1 1)α 55 α+α 2 +α 3 +α 5 (0 1 1 1 0 1) α 40 1+α+α 2 +α 3 +α 5 (1 1 1 1 0 1)α 56 1+α+α 2 +α 3 +α 4 (1 1 1 1 1 0) α 41 1 +α 2 +α 3 +α 4 (1 0 1 1 1 0)α 57 α+α 2 +α 3 +α 4 +α 5 (0 1 1 1 1 1) α 42 α +α 3 +α 4 +α 5 (0 1 0 1 1 1)α 58 1+α+α 2 +α 3 +α 4 +α 5 (1 1 1 1 1 1) α 43 1+α+α 2 +α 4 +α 5 (1 1 1 0 1 1)α 59 1 +α 2 +α 3 +α 4 +α 5 (1 0 1 1 1 1) α 44 1 +α 2 +α 3 +α 5 (1 0 1 1 0 1)α 60 1 +α 3 +α 4 +α 5 (1 0 0 1 1 1) α 45 1 +α 3 +α 4 (1 0 0 1 1 0)α 61 1 +α 4 +α 5 (1 0 0 0 1 1) α 46 α +α 4 +α 5 (0 1 0 0 1 1)α 62 1 +α 5 (1 0 0 0 0 1) α 63 = 1

11. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU11 TABLE 6.3 MINIMAL POLYNOMIALS OF THE ELEMENTS IN GF(2 6 ) ElementsMinimal polynomials α, α 2, α 4, α 8, α 16, α 32 1 + X + X 6 α 3, α 6, α 12, α 24, α 48, α 33 1 + X + X 2 + X 4 + X 6 α 5, α 10, α 20, α 40, α 17, α 34 1 + X + X 2 + X 5 + X 6 α 7, α 14, α 28, α 56, α 49, α 35 1 + X 3 + X 6 α 9, α 18, α 36 1 + X 2 + X 3 α 11, α 22, α 44, α 25, α 50, α 37 1 + X 2 + X 3 + X 5 + X 6 α 13, α 26, α 52, α 41, α 19, α 38 1 + X + X 3 + X 4 + X 6 α 15, α 30, α 60, α 57, α 51, α 39 1 + X 2 + X 4 + X 5 + X 6 α 21, α 42 1 + X + X 2 α 23, α 46, α 29, α 58, α 53, α 43 1 + X + X 4 + X 5 + X 6 α 27, α 54, α 45 1 + X + X 6 α 31, α 62, α 61, α 59, α 55, α 47 1 + X 5 + X 6

12. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU12 Def t-error-correcting BCH code of length n = 2 m – 1: A binary n-tuple v = (v 0, v 1, …, v n-1 ) is a code word iff the polynomial v(x) = v 0 + v 1 x + … + v n-1 x n-1 has α, α 2, α 3, …, α 2t as roots. v(α i ) = v 0 + v 1 α i + v 2 α 2i + … + v n-1 α (n-1)i = 0 1 ≤ i ≤ 2t

13. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU13 By conjugate property if x – α i | v(x) α i α j 互為 conjuqate then x – α j | v(x) Ref: Block Codes – extended 4 Proof: d* ≥ 2t + 1 (BCH bound) suppose code vector v = (v 0, v 1, …, v n-1 ) with weight  ≤ 2t Let v j1, v j2, …, v j  be the nonzero components of v

14. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU14 implies (1, 1, …,1) =

15. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU15 Syndrome S = (S 1, S 2, …, S 2t ) = rH T r = (r 0, r 1, …, r n-1 ) S i = r(α i ) r(x) = r 0 + r 1 x + … + r n-1 x n-1 r(x) = α i (x)f i (x) + b i (x) f i (x): minimal polynomial of α i ∵ f i (α i ) = 0 S i = r(α i ) = b i (α i ) Q: 與 s(x) = r(x)/g(x) 之關係

16. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU16 Peterson-Gorenstein-Zierler Decoder (1) e(X) = e 0 + e 1 X + … + e n-1 X n-1 where at most t coefficients are nonzero (2) Suppose v errors actually occur, 0 ≤ v ≤ t, in unknown locations i 1, i 2, …,i v ∴ e(X) = e i 1 X i 1 + e i 2 X i 2 + … + e i v X i v (3) S 1 = v(α) = c(α) + e(α) = e(α) = e i 1 α i 1 + e i 2 α i 2 + … + e i v α i v Let Y l = e i l : error magnitude 1 ≤ l ≤ v X l = α i l : error-location number Note : all X l are different ( ∵ α n = 1) S 1 = Y 1 X 1 + Y 2 X 2 + … + Y v X v i j, e i j, v are unknown ! Q

17. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU17 (4) For α α 2 α 3 … α 2t S 1 = Y 1 X 1 + Y 2 X 2 + … + Y v X v Y l  GF(q) S 2 = Y 1 X 1 2 + Y 2 X 2 2 + … + Y v X v 2 X l  GF(q m ) S 2 = Y 1 X 1 3 + Y 2 X 2 3 + … + Y v X v 3., S 2t = Y 1 X 1 2t + Y 2 X 2 2t + … + Y v X v 2t 1.( ＊ ) 至少有一解 ∵ Syndrome S i is defined in such way 2. 證明此解 unique ( ＊ ) nonlinear equations

18. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU18 Consider the polynomial Λ(x) ≡ (1-X 1 x)(1-X 2 x)…(1-X v x) = 1 + Λ 1 x + Λ 2 x 2 + … + Λ v x v – (1) error-locator polynomial If we knew the coefficients Λ l of Λ(x) we could find the zeros of Λ(x) to obtain the error locations X l S i  Λ l 1≤i≤2t 1≤l≤v Multiply both sides of (1) by Y l X l j+v and set x = X l -1

19. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU19 ∵ v ≤ t ∴ S 1 ~ S 2v all known

20. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU20 Thm : The Vandermonde matrix has a nonzero determinant iff all of X i 1 ≤ i ≤ u are distinct Thm : The matrix of syndromes M is nonsingular if u is equal to v, the number of errors that actually occurred. The matrix is singular if u > v. If v > t i.e. u < v 時 det(M) = ?

21. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU21

22. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU22 Enter υ(x) Compute syndromes S j = υ(α j+j 0 -1 ) j = 1 … 2t v = t det(M) = 0 v  v-1 Find error location X l (l = 1…..v) by finding zeros of Λ(x) Halt Yes No special case j 0 = 1 α α 2 … α 2t Figure 7.1 The Peterson-Gorenstein-Zierler decoder.

23. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU23 Eg: BCH(15,5) triple-error-correcting code. g(X) = X 10 + X 8 + X 5 + X 4 + X 2 + X + 1 received code polynomial  V(X) = X 7 + X 2 In GF(2 4 ) S 1 = α 7 + α 2 = α 12 S 2 = α 14 + α 4 = α 9 S 3 = α 21 + α 6 = 0 S 4 = α 28 + α 8 = α 3 S 5 = α 35 + α 10 = α 0 S 6 = α 42 + α 12 = 0

24. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU24

25. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU25 Reed-Solomon Code (maximum-distance code) (1)The symbol field GF(q) and the error-locator field GF(q m ) are the same (Y l ) (X l ) (i.e. m = 1) (2)Take α as primitive n = q m – 1 = q – 1 (3)The minimal polynomial over GF(q) of an element β in the same GF(q) is f β (X) = X – β (4)Take j 0 = 1 g(X) = (X – α)(X – α 2 )… (X – α 2t ) ∴ n – k = 2t eg1. (15,11) t = 2 GF(16) j 0 = 1 g(X) = (X – α)(X – α 2 ) (X – α 3 )(X – α 4 ) = X 4 + (Z 3 + Z 2 + 1)X 3 + (Z 3 + Z 2 )X 2 + Z 3 X + (Z 2 + Z + 1) = X 4 + α 13 X 3 + α 6 X 2 + α 3 X + α 10 n – k = 4  k = 11 (11 16-ary symbols  44 bits)

26. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU26 ※ 1. n – k + 1 = 2t + 1 ≤ d* ≤ 1 + n – k ∴ d* = 1 + n – k 2. R-S codes always have relatively short block-length as compared to other cyclic codes over the same alphabet ! eg2. (7,3) t = 2 GF(8) j 0 = 4 g(X) = (X – α 4 )(X – α 5 ) (X – α 6 )(X – α 0 ) = X 4 + (Z 2 + 1)X 3 + (Z 2 + 1)X 2 + (Z + 1)X + Z n – k = 4  k = 3 (3 8-ary symbols  9 bits) If i(X) = (Z 2 + Z)X 2 + X + (Z + 1) c(X) = i(X)g(X) = (α 4 X 2 + X + α 3 )(X 4 + α 6 X 3 + α 6 X 2 + α 3 X + α) = α 4 X 6 + αX 5 + α 6 X 4 + 0X 3 + 0X 2 + α 5 X + α 5