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i CD CD 10 1 CD DVD 1

ii DVB-S2 LDPC 2 2 sum-product sum-product sum-product LDPC

iii, 2010 6

iv 1 1 11 1 12 4 13 10 14 11 12 2 13 21 13 22 15 23 16 24 21 23 3 2 24 31 2 24 32 2 25 33 32 36 4 2 2 38 41 38 42 39 43 42 44 45 49 5 51 51 51 52 55 53 57 59

v 6 F q 60 61 60 62 62 63 67 64 68 70 7 I 71 71 71 72 73 73 75 74 76 83 8 II 85 81 85 82 88 83 89 84 BCH 92 95 9 97 91 97 92 100 93 103 94 108 95 110 112 10 114 101 114 102 117 103 123 128 11 sum-product 129 111 129 112 sum-product 132

vi 143 12 MAP 144 121 MAP 144 122 (FB) 147 123 FB MAP 151 124 155 157 13 LDPC I 159 131 LDPC 159 132 LDPC 165 171 14 LDPC II 172 141 172 142 sum-product 174 143 sum-product 178 144 183 184 15 185 151 185 152 188 153 193 154 2 196 201 203 205 221

vii I[condition] [a, b] condition 1 0 a b = := A A A, A A A B A B A B A B A\B A B a A a A B A B A O(g(n)) n f(n) C g(n) C f(n) =O(g(n)) d h (a, b) a, b w h (a) a ln(x) F 2 F q R tanh A t Z (n,w) Prob[event] arg max x D f(x) arg min x D f(x) x A mod B 2 q q A n w 2 event f(x) x D f(x) x D x A B A, B A b A A\{b} A\b 2 2 +, 1+1=2 1+1=0 0

13 2 21 X 2 X = {0, 1} n X code X n C X C n C codeword n C code length 1 X = {0, 1} X 3 = {000, 001, 010, 011, 100, 101, 110, 111} C = {000, 011, 101, 110} C 3 4 3 C = {000, 111} 3 2 2 X = {0, 1} 2 q q 2

14 2 1 code rate C R R = log 2 C log 2 X n (21) C =2 k k X = {0, 1} 2 log 2 2=1 21 R R = log 2 C log 2 X n = log 2 2k log 2 2 n = k n (22) 2 n R n C minimum distance C C d =min{d h (a, b) :a, b C, a b} (23) 3 C = {000, 011, 101, 110} 2 3 {000, 111} d =3 C X n 1 7 F q 2 2 3 k =1 n =3 1/3

22 15 4 22 3 C = {000, 011, 101, 110} 4 {0, 1, 2, 3} 0 000, 1 011, 2 101, 3 110 encoder M L M =[1,L] C L bijection φ : M C φ φ m 1 m 2 (m 1,m 2 M) φ(m 1 ) φ(m 2 ) c C φ(m) =c m M φ 3 φ 1 φ 1 m M m = φ 1 (φ(m)) ˆX ˆM 3 φ(m) = 000, m =0 (24) 111, m =1 3 φ M C φ M C f : X Y y Y, x X, f(x) =y a, b X(a b),f(a) f(b) f

16 2 m 0 000 m =1 111 21 M = {0, 1} k {0, 1} n 2 2 n 2 k C 21 2 21 φ(m) 2 500 4 23 Y Y n Y Y ˆX ˆX ˆM = φ 1 ( ˆX) 231 Y ˆX M C {c 1, c 2,,c M } 4

23 17 R(c i )(i [1,M]) R(c i ) Y n R(c i )(i [1,M]) R(c i ) decoding rule 1 Y R(c i ) i j [1,M] Y / R(c j ) ˆX = c i 2 Y R(c i ) ˆX = 3 Y Y R(c j1 ),Y R(c j2 ),,Y R(c js ) c j1, c j2,,c js 5 R(c i ) c i R(c i ) c i 1 3 D1 R(000) = {000, 001, 010, 100}, R(111) = {111, 110, 101, 011} ψ : Y n C { } ψ Y ˆX = ψ(y ) 22 5

18 2 22 ψ(y ) 232 1 ˆX = X 2 ˆX X 3 ˆX = c i Y R(c i ) R(c j ) P c P e P d P c + P e + P d =1 6 233 n C 2 n 1 > 2 > 3 2 6 9

23 19 minimum distance decoding rule C = {c 1, c 2,,c M } c i C c i R(c i ) R(c i ) = {y {0, 1} n : c C\{c i } d h (y, c i ) d h (y, c)} (25) i [1,M] C\{c i } C {c i } 23(a) ˆX Y ˆX =argmin d h(y,x) (26) X C arg min X C Y 23

20 2 234 r r 2 4 bounded distance decoding rule C = {c 1, c 2,,c M } R(c i ) = {y {0, 1} n : d h (c i, y) r}, i [1,M] (27) r d 1 r 2 d C x x 23(b) r (d 1)/2 d 1 2r 2 2 d 1 = d 1 (28) 2 2 d r( (d 1)/2 ) 235

24 21 n = 1000 2 800 2 C 7 R = 800/1000 = 4/5 2 800 n R 2 O(n2 n ) 8 n n O(n) O(n 2 ) 24 21 21 HDD CD DVD LDPC 241 7 1 8 f(n) C g(n) f(n) C g(n) n f(n) =O(g(n)) [10]

22 2 HDD CD DVD 7 8 242 9 10 243 12 LDPC 13 LDPC sum-product 13 14 244 CD DVD product code

23 21 2 C = {0000, 1010, 0101, 1111} 1 C 2 C 22 2 C = {00000, 11100, 00111, 11011} 10111 23 5 C C = {00000, 11111} C r =1 1 C 2 p 2 P c P d P e 3 P c + P d + P e =1 24 2 (a, b)(a, b {0, 1}) (a, b) (ā, b) ā, b a, b C = {00, 01} (a, b) =(0, 0) (a, b) =(0, 1)

71 7 I F q Reed-Solomon code MDS CD DVD 71 711 3 3 3 F q q n = q 1 ( ) α 0 α 1 α 2 α q 2 H = (α 0 ) 2 (α 1 ) 2 (α 2 ) 2 (α q 2 ) 2 ( ) 1 α α 2 α n 1 = 1 α 2 α 4 α 2(n 1) (71) F q C = {x F n q : Hx t = 0} (72) α F q C n = q 1

72 7 I k = n 2 d =3 F 2 8 n = 255,k = 253,d=3 C 1 C 2 α i α j = αi α 2j α j α 2i α 2i α 2j = α i+2j α j+2i = α i+j (α j α i ) (73) i, j [0,q 2] (i j) α i+j 0 α j α i 0 0 (α i,α 2i ) t (α j,α 2j ) t H 2 3 4 C 3 712 3 C 3 1 C X F n q Y = X + E E n F q i i [1,n] (error magnitude) e F q Y S = HY t X HX t =0 S = HY t = H(X + E) t = HE t HE t S = s 1 = eαi 1 (74) s 2 eα 2(i 1) s 2 /s 1 s 2 = eα2(i 1) s 1 eα i 1 = α i 1 (75) i

72 73 s 2 1/s 2 s 2 1 = e2 α 2(i 1) = e (76) s 2 eα 2(i 1) i e E Ê ˆX = Y Ê ˆX C 1 72 721 determinant F q n n A = {a ij } (i [1,n],j [1,n]) A = { n i=1 ( 1)i+j a ij A ij, n > 1 a 11, n =1 (77) j [1,n] A ij A i j (n 1) (n 1) 1 F q 2 2 A = a b c d A A = ad bc 1

74 7 I 1 F q n n a 1 a 2 a n a i i a 1 a 2 aa i a n = a a 1 a 2 a i a n i [1,n] a F q 2 F q n n a 1 a 2 a n 3 a 1 a 2 a i + a j a n = a 1 a 2 a i a n + a 1 a 2 a j a n F q n n A t = A 4 F q n n AB = A B 5 F q n n a 1 a 2 a n a 1 a 2 a n 0 a 1 a 2 a n 6 b 1,b 2,,b n F q B 0 B = n i=1 b i 722 1 1 1 1 x 1 x 2 x 3 x n X = x 2 1 x 2 2 x 2 3 x 2 n (78) x n 1 1 x n 1 2 x n 1 3 xn n 1 Vandermonde X 1 1 1 1 x 1 x 2 x 3 x n x 2 1 x 2 2 x 2 3 x 2 n = i x j ) (79) i>j(x x n 1 1 x n 1 2 x n 1 3 x n 1 n

73 75 79 i>j i, j [1,n] i, j x i x j 79 2 5 X X 73 Reed-Solomon code F q 1 α α 2 α n 1 1 α 2 α 4 α 2(n 1) H = 1 α 3 α 6 α 3(n 1) 1 α 2t α 4t α 2t(n 1) (710) F q C = {c F n q : Hc t = 0} (711) n = q 1 t [1, (q 2)/2 ] (712) H 2t H sub α j 1 α j 2 α j 2t α 2j 1 α 2j 2 α 2j 2t H sub = (713) α 2tj 1 α 2tj 2 α 2tj 2t j i [0,n 1] (i [1, 2t]) j k j l (k, l [1, 2t],k l) H sub 2

76 7 I α j 1 α j 2 α j 2t α 2j 1 α 2j 2 α 2j 2t H sub = α 2tj 1 α 2tj 2 α 2tj 2t 1 1 1 = α j1 α j α j 1 α j 2 α j 2t 2t α (2t 1)j 1 α (2t 1)j 2 α (2t 1)j 2t (714) 1 H sub 0 H sub 5 H 2t C k k = n 2t H 2t C d d 2t +1 d n k +1=n n +2t +1=2t +1 d =2t +1 F q H F q n = q 1 k = n 2t d = n k +1=2t +1 74 3 n O(n 3 ) t 3 WWPeterson 1960 Error-Correcting Codes

74 77 4 1960 741 n = q 1 C X =(X 0,,X n 1 ) C 0 Y =(Y 0,,Y n 1 ) F n q Y = X+E E =(E 0,,E n 1 ) F n q w h (E) =ν 1 ν t Y S =(S 1,S 2,,S 2t ) S = HY t 2 S = HY t = H(X + E) t = HE t i [1, 2t] ( ) S i = 1 α i α 2i α (n 1)i E 0 E 1 (715) E n 1 S E HQ T = S (716) Q F n q 5 E t 716 4 Berlekamp-Massey-Sakata Euclid Sugiyama-Kasahara-Hirasawa-Namekawa Welch-Berlekamp Feng-Rao Sudan Euclid O(n 2 ) t 5

78 7 I E t t S E t 716 E E E E E = {i [0,n 1] : Ei 0} (717) E E = ν t ν E E E E = {j 1,j 2,,j ν } x k = α j k (k [1,ν]) e k = Ejk (k [1,ν]) j / E E E j =0 715 ν S i = e k x i k, i [1, 2t] (718) k=1 S 1 x 1 x 2 x 3 x ν 1 x ν S 2 x 2 1 x 2 2 x 2 3 x 2 ν 1 x 2 ν = S 2t x 2t 1 x 2t 2 x 2t 3 x 2t ν 1 x 2t ν e 2 e 1 e ν (719) x k (k [1,ν]) e k (k [1,ν]) 742 σ(z) E ν {x 1 1,x 1 2,,x 1 ν } ν σ 1,,σ ν ν σ(z) =1+ σ k z k (720) k=1 σ(z) x 1 j ν σ(x 1 j )=1+ k=1 (j [1,ν]) σ k x k j = 0 (721)

74 79 ν j=1 e jx i+ν j σ(x 1 j )(i [1,ν]) ( ) ν ν ν e j x i+ν j σ(x 1 j )= e j x i+ν j 1+ σ k x k j (722) j=1 = j=1 ν j=1 = S ν+i + e j x i+ν j + ν k=1 ν σ k k=1 j=1 e j x ν+i k j (723) ν σ k S ν+i k (724) k=1 = 0 (725) 722 721 724 718 σ(x 1 j )= 0(j [1,ν]) ν S ν+i = σ k S ν+i k, i [1,ν] (726) k=1 S 1 S 2 S 3 S ν 1 S ν S 2 S 3 S 4 S ν S ν+1 S 3 S 4 S 5 S ν+1 S ν+2 S 4 S 5 S 6 S ν+2 S ν+3 S ν 1 S ν S ν+1 S 2ν 3 S 2ν 2 σ ν σ ν 1 σ ν 2 σ ν 3 σ 2 = S ν+1 S ν+2 S ν+3 S ν+4 S 2ν 1 S ν S ν+1 S ν+2 S 2ν 2 S 2ν 1 σ 1 S 2ν (727) 727 - Berlekamp-Massey Sugiyama ν ν ν Y S i (i [1, 2t]) ν t 727 S 1,S 2,,S 2ν 727 727 (σ 1,σ 2,,σ ν )

80 7 I σ 1,σ 2,,σ ν σ(z) σ(z) 0 F q Chien search 721 x 1,x 2,,x ν 743 x 1,x 2,,x ν e 1,e 2,,e ν 718 S 1 x 1 x 2 x 3 x ν 1 x ν e 1 S 2 x 2 1 x 2 2 x 2 3 x 2 ν 1 x 2 ν e 2 = (728) S ν x ν 1 x ν 2 x ν 3 x ν ν 1 x ν ν e ν x 1 x 2 x ν 1 1 1 x 2 1 x 2 2 x 2 ν x 1 x 2 x ν = x 1 x 2 x ν x ν 1 x ν 2 x ν ν x ν 1 1 x ν 1 2 x ν 1 ν 0 (729) 1 728 e 1,e 2,,e ν 744 ν ν ν 1 τ t

74 81 D τ = S 1 S 2 S τ S 2 S 3 S τ+1 S τ S τ+1 S 2τ 1 (730) τ = ν D τ 0 τ >ν D τ =0 k [ν +1,t] e k =0,x k =0 1 1 1 e 1 x 1 0 0 T = x 1 x 2 x τ, Q 0 e 2 x 2 0 = x τ 1 1 x τ 1 2 x τ 1 τ 0 0 e τ x τ D τ D τ = TQT t = T Q T t (731) 75 τ>ν Q 0 Q =0 6 731 D τ =0 τ = ν Q 6 x 1,x 2,,x τ F q T 0 D τ 0 ν τ = t τ D τ D τ 0 τ = ν 745 Peterson algorithm Y ˆX 1 715 Y S 1,S 2,,S 2t τ := t

82 7 I 2 D τ 730 D τ = 0 τ := τ 1 3 ν := τ 727 σ 1,σ 2,,σ ν 4 σ(z) x 1,x 2,,x ν 5 728 Ê =(e 1,e 2,,e ν ) 6 Ê Y ˆX = Y Ê ν t t 746 F 2 3 53 1 α α 2 α 3 α 4 α 5 α 6 H = 1 α 2 α 4 α 6 α 8 α 10 α 12 1 α 3 α 6 α 9 α 12 α 15 α 18 1 α 4 α 8 α 12 α 16 α 20 α 24 C α 7 =1 1 α α 2 α 3 α 4 α 5 α 6 1 α 2 α 4 α 6 α 1 α 3 α 5 H = 1 α 3 α 6 α 2 α 5 α α 4 1 α 4 α α 5 α 2 α 6 α 3 (732) (733) C 2 E =(0, 0, 0,α 5, 0,α 2, 0) X C Y = X + E HY t = H(X + Y ) t = HE t

83 S 1 S 2 S 3 S 4 = α 5 α 3 α 6 α 2 α 5 + α 2 α 5 α 3 α α 6 = α 3 1 α 1 (734) S 1 S 2 S 2 S 3 = S 1S 3 S 2 S 2 = α 4 1 0 (735) ν =2 727 S 1 S 2 σ 2 = S 3 (736) S 2 S 3 σ 1 S 4 α3 1 σ 2 = α (737) 1 α σ 1 1 σ 1 = α 2,σ 2 = α σ(z) = 1+σ 1 z + σ 2 z 2 =1+α 2 z + αz 2 σ(z) F 2 3 σ(z) σ(α 4 )=0 σ(α 2 )=0 α 4 = α 3 α 2 = α 5 6 x 1 = α 3,x 2 = α 5 4 6 728 α3 α 5 e 1 = α3 (738) α 6 α 3 e 2 1 e 1 = α 5,e 2 = α 2 71 F 2 3 F 2 z 3 + z +1 α 53 ( ) 1 α α 2 α 3 α 4 α 5 α 6 H = 1 α 2 α 4 α 6 α 8 α 10 α 12 6 x F q x y =1 y x y x 1 α 3 α 4 =1 α 3 = α 4 5

84 7 I E =(0,α 2, 0, 0, 0, 0, 0) 1 Y = X + E 2 Ê 72 F 3 1 2 0 0 1 1 1 2 2 ta b 73 tc d, a + x b c + y d, a c b d a b c d 1 2 3 74 F 2 3 F 2 z 3 + z +1 α 1 1 1 α α 2 α 4 α 2 α 4 α 75 ν =2,τ =2 D τ = TQT t = T Q T t 76 746 E =(0,α 2,α,0, 0, 0, 0)

159 13 LDPC I low-density parity check LDPC 1 1960 2 LDPC sum-product LDPC 2 131 LDPC LDPC 2 1311 LDPC LDPC F q F 2 2 LDPC LDPC 2 2 m n H (0 <m<n) 2 C C = {x F n 2 : Hxt = 0} H 1 C LDPC 1 Robert G Gallager LDPC 60 Information Theory and Reliable Communication 2 LSI LDPC LDPC 90 DMacKay LDPC LDPC

160 13 LDPC I C n m C R R 1 m/n LDPC m = 5000,n = 10000 2 H 1 w r =6 1 w c =3 C LDPC 131 1 3 8 LDPC 131 2 {H 1,H 2,,H n,} H n (n =1, 2,) n αn α 0 α 1 n H n 05n n n 3 H n 1 3n 1 n 1 n n 2 LDPC LDPC sum-product

131 LDPC 161 LDPC sum-product sum-product 3 sum-product 1 4 LDPC O(n) 1 O(n) 1312 2 LDPC Tanner graph 2 m n H i j h ij 2 5 G(H) =(V,E) V = V 1 V 2 V 1 = {v1,v 2,,v n }, V 2 = {c1,c 2,,c m } (131) V 1 V 2 h ij =1(i [1,m],j [1,n]) (i, j) v i c j e ij E h ij =0(i [1,m],j [1,n]) v i c j 2 G(H) H i c i j v j h ij =1 c i v j H 132 3 1 LDPC 4 n 5 2 V V 1,V 2 V 1 V 2

162 13 LDPC I 132 v 1,,v 5 0 1 c 1 v 1,v 2,v 3 v 1 + v 2 + v 3 =0 LDPC 2 n O(n) LDPC sum-product 132 v 2,v 3 c 1,c 2 4 4 1313 LDPC LDPC LDPC LDPC LDPC LDPC LDPC w c w r LDPC LDPC LDPC LDPC w c,w r LDPC LDPC LDPC LDPC LDPC

131 LDPC 163 w c w r LDPC 94 LDPC k 6 L k = {i [1,n]:deg(v i)=k} n (132) deg(v) v k % LDPC sum-product sum-product 7 1314 LDPC 2 w c =2 w r =4 H 1 6 7 1322

164 13 LDPC I H 1 = 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 (133) H 1 8 H 2 = 0 1 1 0 1 1 0 0 (134) 1 0 0 1 0 0 1 1 H 1 H 2 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 H = (135) 0 1 1 0 1 1 0 0 1 0 0 1 0 0 1 1 2, 4 w r 1 0 n h 1 H 1 h 1 s(h 1 ) H 1 = s 2 (h 1 ) s n/wr 1 (h 1 ) (136) s( ) w r 4,2 H 2,,H wc H 1 H 2 H = (137) H wc w c,w r H 9 1315 LDPC LDPC 6 H G G 8 8! 9

132 LDPC 165 10 132 LDPC LDPC 2 sum-product 1321 2 sum-product 2 2 sumproduct 2 sum-product sum-product 2 133 2 {0, 1} 3 {0, 1,e} e 0, 1 1 p p p 2 2 m n H LDPC 133 2 10 H G O(n 2 )

166 13 LDPC I C C x 2 y =(y 1,y 2,,y n ) {0, 1,e} n c A(c) v B(v) sum-product sum-product 1 v N(v) =y i c c e 2 v c B(v) M v c M v c B(v)\c e v N(v) e M v c = e M v c B(v)\c e 3 c v A(c) M c v M c v A(c)\v e M c v = e M c v F 2 134 M c v = v A(c)\v M v c 134 sum-product 3

132 LDPC 167 4 v e b {0, 1} N(v) =e N(v) =b 2 135 a LDPC 0, 0, 0, 0, 0, 0, 0, 0 Y =(e, e, e, e, 0, 0, 0, 0) 1 2 3 4 v 1,v 2,,v 8 c 1,,c 4 a N(v) b 1 2 3 4 1 1 c 4 c 4 v 6,v 8 (M v6 c 4 = M v8 c 4 =0) v 3 M c4 v 3 = M v6 c 4 + M v8 c 4 =0 0 v 3 N(v 3 )=0 3 c 1 4 2 d 3 0 LDPC sum-product

168 13 LDPC I 135 1322 LDPC 9 sum-product density evolution sum-product 1 1 2 sum-product d k 11 p i i e q i 11 d k 15

132 LDPC 169 i e sum-product 2 4 1 p 2 i +1 e e d 1 e 12 p i+1 = pq d 1 i (138) i e q i k 1 e 13 q i =1 (1 p i ) k 1 (139) p i p i+1 = p(1 (1 p i ) k 1 ) d 1 (1310) p 0 = p 2 sum-product p i+1 = g(p i ) p 0 = p g(x) =p(1 (1 x) k 1 ) d 1 d =3,k =6 p i 136 x y y = x y = g(x) p =04 p i p 1 = g(p 0 ) (p 0,p 1 ) x =04 y = g(x) y = x x = p 1 12 sum-product p166 13 sum-product p166

170 13 LDPC I 136 p i p =04 y = g(x) (p 1,p 2 ) y = g(x) (p i,p i+1 ) 136 p i p i 0 14 p =04 sum-product 0 p =0451 y = x y = g(x) 137 p i 0 0 2 137 p i p =0451 14 p i 0 0

171 p d =3,k =6 0429 LDPC [5] 131 132 2 e, e, 0, 0, 0 sum-product 132 m n H H h 1,,h n T = {t 1,t 2,,t s } [1,n] T H(T ) =(h t1 h t2 h ts ) H(T ) 1 T sum-product 1 132 H 3 2 133 G G girth LDPC LDPC sum-product LDPC 4 4 134 1322

221 2 24, 26, 27, 31, 38, 193 2 2, 4,31, 105, 146, 183, 193, 196 2 13 2 159, 161 ACS 119, 121 AWGN 106, 120, 183 BCH 94 BCH 42, 92, 93, 187 BCJR 116, 147, 156 LDPC 22, 49, 133, 159, 162 MAP 101, 145, 176 min-sum 183 SN 107 sum-product 132, 135, 138, 160, 174 sum-product 22, 166, 175 u u + v 43 41, 78, 80 78, 82 17, 18, 31, 38, 193 72, 78, 80, 82 1 5, 14, 15, 19 11, 13, 21 32 188 155 120 55 39, 46, 60 177 26, 39, 60 129, 132 135, 137, 141 129 156 89 74, 80, 89 163 149, 153 6 163 10, 197 21 36, 43, 185 185 29 91 147, 157, 158 106 29, 67 42 52, 53 42 163 97 22, 99, 131, 140, 141 99 97 97, 106 100, 147 104, 147 54 52 25 130, 131 133, 173 187 48 104 189 24, 26, 27, 61 66 62 52, 55 163 58, 93 160 72, 73 104 38 22 132 4, 7,17 130, 173 20, 21, 28, 103, 122, 151 20 48 38, 45 29, 30, 40, 67, 68, 159 64 55 55 32, 35 33 33 35, 41 7, 18, 178 53, 85 14, 30, 38, 39, 45, 49, 72, 94, 192

222 19, 48, 105, 117 20, 48, 106 127 58, 94 193 46, 71, 76 49, 102, 103, 105, 110, 117 111 2, 39 26, 38, 62 67 99, 101, 144, 151, 173 163 100, 102, 157 123 3, 10, 12 47 46 98, 99, 129, 134 130, 135, 138, 149, 155 130 145, 156, 173, 182 3, 31 85, 86, 94 29, 42, 76, 86, 95, 187 98, 99 88 88 88, 165, 183 88 151 4 56 52 3 87 46, 76 107 32, 33, 41, 72, 77, 177, 193 116 35 32, 36, 41 MAP 145, 151 109 141 28, 43, 62, 68 85, 86, 94 LDPC 162 7, 18 22, 49 133, 148 94 29 26 197 29 27, 29, 31, 61 51, 60 15 15 29 187 159 64, 87, 178 51 193 24, 51 145 22, 71, 76, 77, 92 21 197 179, 183 17, 19, 111 123, 124, 156 125 161, 174 129 157 156 22, 156 38, 122, 152, 186, 188 15 87 88 161 175, 179 80, 82 122 1, 100 13 10, 27, 201 22, 159 107 197, 201 197, 198 9 74 98, 99 114 119, 121 114, 115, 126, 152, 153 22, 123 91 6 25 106 88 89 104 4, 6 20, 46 5, 20 5, 47 39, 40, 95 42 30 127 2 171 22, 49, 156, 177 LDPC 162

223 76, 81 22, 117, 119 109 66 133 129, 132, 135, 147, 161, 172, 174 122 3, 4, 16, 97 3, 16, 101, 156 17 4, 7, 17 6, 7, 17, 19, 49, 111 3, 13 3, 15, 16, 28, 68, 86, 87, 164 3, 15, 87, 156 6, 15, 16, 63 201 15, 27, 63 10, 12, 196, 200 10, 14, 62, 160 110 3, 4, 13, 24, 30, 45 85, 93 4, 13, 38 77 25, 28, 60 92 44 MAP 101 2, 4, 7, 14, 18, 101, 108, 111, 185 107 133, 142 133, 139 190, 191 132, 215 99, 101, 144 56 25 25 56 129, 132 135, 136, 141, 175, 179 19 147, 155 148, 153 187 132 104 131, 140 163, 168 195 2, 105 1, 3, 15, 139, 141 135 86 53 109 14, 24, 51 101, 102, 105 29 189, 190 163 97 105 14, 22, 71, 75, 85, 187 42, 44, 187 136, 142, 161, 162, 175 160 91 106 131

1991 1993 1995 1995 1997 2004 2007 2010 c 2010 2010 7 6 1 1 1-4-11 102-0071 03-3265-8341 FAX 03-3264-8709 http: wwwmorikitacojp Printed in Japan ISBN978-4-627-81731-9