New direction on low complexity implementation of probabilistic gradient descent bit flipping decoder

Dành cho Ba Má thân yêu con, Ba Lê Trung Nhân Má Nguyễn Thị Kim Chấn Dành cho Chị em, Dành cho vợ gái yêu dấu, Cho tình thương Ba, Má dành cho Cho tình yêu Chị em, Cho tình yêu Vợ To my parents, To my brothers and sisters, To my wife and daughter, For your love, À mes parents, À mes frères et soeurs, À ma femme et ma pettite fille, 2 d v , dc M = 93 N = 155 (3, 6) R = 0.4 Z = 31 1 16 HB = @ 10 20 18 A 25 19 14 28 d v , dc N = 1296 (3, 6) R = 0.5 M = 648 Z = 54 49 1 1 43 1 1 50 1 1 27 1 1 49 B C B C B 1 10 41 1 1 52 1 32 1 1 50 50 1 C B C B C B 1 20 1 1 20 1 51 10 1 47 1 1 33 C B C B C B 24 1 1 22 53 1 1 31 1 1 18 47 1 C B C B C B 10 1 15 1 1 1 1 50 13 1 1 53 C B C B C B 1 44 1 1 1 29 40 1 16 1 13 1 C C HB = B B C B 1 1 13 41 1 1 42 1 1 48 49 1 C B C B C B 1 36 1 24 1 50 1 12 1 1 10 1 48 C B C B C B 1 47 50 1 1 1 1 1 1 28 C B C B C B 24 C 1 1 51 38 1 1 1 1 23 16 1 B C B C B 1 1 1 1 13 1 29 1 16 1 C @ A 1 35 16 1 37 1 1 1 24 1 29 d v , dc N = 1296 11 B B B 25 B B B 1 B B B 27 B B B 42 B B B 1 HB = B B B 1 B B B 29 B B B B B B 1 B B B @ 39 1 27 1 44 1 31 29 34 22 Z = 54 1 1 33 1 20 1 44 1 11 1 10 35 1 31 1 29 53 1 1 17 1 20 1 18 19 1 1 26 36 1 1 1 1 10 1 1 44 11 1 29 40 1 1 20 16 24 (4, 8) R = 0.5 M = 648 1 1 34 50 1 1 1 34 15 1 17 1 1 27 1 52 14 1 1 42 1 1 42 1 47 19 13 1 44 36 13 1 1 1 33 1 1 45 17 1 38 1 31 1 42 1 39 1 28 30 1 50 1 12 1 1 30 1 19 1 46 1 1 25 28 1 35 1 1 1 44 36 41 38 1 1 C C C C C C C C 27 C C C C C C C C C C C C C C C 41 C C C C C C C A 15 ↵ V Ni,j ↵i,j i,j new CNa,b a,b ↵a,b new i,j new a,b ↵i,j ↵a,b new ↵i,j , 81  i  nc ,  j  Z • • CNa,b ,  a  nr ,  b  Z ↵a,b new ↵i,j CNa,b ↵a,b new • V Ni,j ,  i  nc ,  j  Z i,j new a,b • V Ni,j ↵i,j i,j new ˜i,j i,j ˜i,j • v ˜i,j y = (y1,1 , y1,2 , , ync ,Z , ync ,Z )  i  nc ,  j  Z v = (v1,1 , v1,2 , , vnc ,Z , vnc ,Z ) {0, 1}N i = 1, , nc j = 1, , Z i = 1, , nc j = 1, , Z n = 1, , dv ↵n = i,j ↵i,j = {↵n }n=1 dv i,j = q(yi,j ) a = 1, , nr b = 1, , Z = {↵m }m=1 dc ↵m ↵i,j H(a ⇤ Z + b, i ⇤ Z + j) = m = 1,Y , dc (↵m0 ) ⇤ |↵m0 | m = ↵ 2↵a,b new \↵m ↵a,b new m ↵m0 2↵a,b new \↵m a,b ={ m }m=1 dc i = 1, , nc = { n }n=1 dv j = 1, , Z H(a ⇤ Z + b, i ⇤ Z + j) = n a,b i,j new ˜i,j = i,j + X n2 n i,j new n = 1, , dv ↵n = ˜i,j n ↵i,j = {↵n }n=1 dv v = {vi,j } vi,j = v MCN (l) • n i,j new (˜i,j ) l l = 1, , L MV N (l) ˜i,j = i,j 1; , dv , | i,j (.)| = dv l l ˜i,j i,j ti,j l V Ni,j MV N (l) (n) = 0, 8n = V Ni,j L ↵i,j ↵i,j • ↵i,j • CNa,b MCN (l) • • • ↵a,b new CNa,b MCN (l) a,b i,j new V Ni,j MV N (l) i,j new i,j v (.) V Ni,j MV N (l) ti,j ˜i,j ˜i,j y = (y1,1 , y1,2 , , ync ,Z , ync ,Z )  i  nc ,  j  Z v = (v1,1 , v1,2 , , vnc ,Z , vnc ,Z ) {0, 1}N i = 1, , nc j = 1, , Z i = 1, , nc j = 1, , Z i = 1, , nc j = 1, , Z n = 1, , dv (n) = i,j i = 1, , nc j = 1, , Z i,j ˜i,j = q(yi,j ) = i,j ti,j = l = 1, , L V Ni,j MV N (l) ↵i,j = ˜i,j (t ) i,j i,j CNa,b MCN (l) n = 1, , dc V Ni,j MV N (l) ↵n = ↵i,j h(a ⇤ Z + b, i ⇤ Z + j) = a,b ↵new = {↵m }n=1 dc CNa,b MCN (l) n = 1,Y , dc (↵n0 ) ⇤ |↵n0 | n = ↵ 2↵a,b new \↵n n ↵n0 2↵a,b new \↵n a,b ={ n }n=1 dc CNa,b MCN (l) n = 1, , dc V Ni,j MV N (l) i,j h(a ⇤ Z + b, i ⇤ Z + j) = new = a,b (n) V Ni,j MV N (l) i,j ˜i,j = ↵i,j + new i,j (t ) = new i,j i,j ti,j = ti,j + v = {vi,j } vi,j = v (˜i,j ) 20 c57 v21 c22 c77 v142 c3 v2 c36 c89 c38 v80 v0 c87 c1 v90 v47 c48 v154 c15 v100 c66 c47 c83 c67 c58 v11 v48 c19 c33 v139 v77 c74 c35 v30 c0 c86 v151 c12 c45 c84 v28 c29 v87 v111 c63 c26 c18 v55 v76 c34 c16 v124 v15 c71 c3 c49 c90 v119 v142 c77 c2 v58 c37 c29 v28 v91 v1 c84 c33 c36 v0 c1 v117 c87 c92 v126 c51 v62 c18 c66 v131 v109 c24 c56 v47 c23 c4 v104 c19 v77 c35 c69 v3 c39 v81 c88 c57 v132 c67 20 c88 v153 c14 c82 v1 v99 c37 c47 v37 c8 v147 v121 c73 c87 c5 v63 v118 c70 c52 c1 c2 c41 v42 c13 v0 v5 c92 c36 c20 v78 c85 v35 c45 v19 c34 v140 c75 c6 v98 c32 v138 c81 v74 c16 c55 v45 c30 c64 v88 c46 ... Rate (FER) −2 10 −3 10 −4 10 BF, 300 Iterations Gal−B, 300 Iterations GDBF, 300 Iterations PGDBF, 300 Iterations MS, 20 Iterations OMS, 20 Iterations −5 10 −6 10 −7 10 0.01 0.02 0.03 0.04 0.05... dc ) = (4, 34) R = 0.88 N = 9520 Z = 140 K N M K N K LDPC encoder K bits N= K + M bits Noisy Channel LDPC decoder N bits K bits • • • 65 (M ⇥ H N) N >M H cm  m  M H hm,n = 1nN H N M 0 0 0... VNSA−IM−PGDBF CSTS−PGDBF −7 10 −8 10 0.4 0.5 0.6 0.7 0.8 0.9 1.1 1.2 1.3 Decoder complexity nomalized by the complexity of GDBF dv = dc = M = 648 N = 1296 1.4