Table 1 LDPC decoding based on log likelihood ratio (LLR) belief propagation (BP) algorithm

H _mn

the value at row m and col n of H-matrix

N(m)

the set of bit nodes including the check node m

M(n)

the set of check nodes including the bit node n

Algorithm

Initialization

set F _n  = LLR for bit nodes (n = 1,2,…,N)

set Z _mn  = F _n if H _mn is 1 for each (m,n)

while (i ≤ i_max)

Check node

for each check node

Processing

where each (m,n) if H _mn is 1

$T_{\mathit{mn}}={\displaystyle \prod _{n^{\prime }\in N\left(m\right)\backslash \left\{n\right\}}}\frac{1-e^{Z_{m{n}^{\prime }}}}{1+e^{Z_{m{n}^{\prime }}}}$

$L_{\mathit{mn}}=\mathit{ln}\frac{1-T_{\mathit{mn}}}{1+T_{\mathit{mn}}}$

end for

Bit node

for each bit node

Processing

where each (m,n) if H_mn is 1

$Z_{\mathit{mn}}=F_n+{\displaystyle \sum _{m^{\prime }\in M\left(n\right)\backslash \left\{m\right\}}}{L}_{m^{\prime }n}$

$Z_n=F_n+{\displaystyle \sum _{m\in M\left(n\right)}}{L}_{\mathit{mn}}$

end for

Tentative

for each n (n = 1,2,…,N)

Decision

$\widehat{C}=\left[\begin{array}{ccc}\hfill {\widehat{C}}_1\hfill & \hfill \cdots \hfill & \hfill {\widehat{C}}_N\hfill \end{array}\right],{\widehat{C}}_n=\left\{\begin{array}{cc}\hfill 0,\hfill & \hfill Z_n\ge 0\hfill \\ \hfill 1,\hfill & \hfill Z_n<0\hfill \end{array}\right.$

end for

Parity check

if $H^2{\left[\begin{array}{ccc}\hfill {\widehat{C}}_1\hfill & \hfill \cdots \hfill & \hfill {\widehat{C}}_N\hfill \end{array}\right]}^T$ is 0

return success

i ++

end while