The implementation of the belief propagation algorithm is based on the decoding algorithm presented in [2]. For transmitted LDPC-encoded codeword, c, where $$c=(c_0,c_1,\mathrm{...},c_{n-1})$$, the input to the LDPC decoder is the log-likelihood ratio (LLR) value $$L(c_i)=\log \left(\frac{\mathrm{Pr}(c_i=0|\text{channel output for }{c}_i)}{\mathrm{Pr}(c_i=1|\text{channel output for }{c}_i)}\right)$$.

In each iteration, the key components of the algorithm are updated based on these equations:

$$L(r_{ji})=2\text{atanh}\left({\displaystyle \prod _{i^{\prime }\in V_j\backslash i}\tanh \left(\frac{1}{2}L(q_{i^{\prime }j})\right)}\right)$$,

$$L(q_{ij})=L(c_i)+{\displaystyle \sum _{j^{\prime }\in C_i\backslash j}L(r_{j^{\prime }i})}$$, initialized as $$L(q_{ij})=L(c_i)$$ before the first iteration, and

$$L(Q_i)=L(c_i)+{\displaystyle \sum _{j^{\prime }\in C_i}L(r_{j^{\prime }i})}$$.

At the end of each iteration, $$L(Q_i)$$ is an updated estimate of the LLR value for the transmitted bit $$c_i$$. The value $$L(Q_i)$$ is the soft-decision output for $$c_i$$. If $$L(Q_i)<0$$, the hard-decision output for $$c_i$$ is 1. Otherwise, the output is 0.

Index sets $$C_i\backslash j$$ and $$V_j\backslash i$$ are based on the parity-check matrix (PCM). Index sets $$C_i$$ and $$V_j$$ correspond to all nonzero elements in column i and row j of the PCM, respectively.

This figure highlights the computation of these index sets in a given PCM for i = 5 and j = 3.

To avoid infinite numbers in the algorithm equations, atanh(1) and atanh(-1) are set to 19.07 and –19.07, respectively. Due to finite precision, MATLAB^® returns 1 for tanh(19.07) and –1 for tanh(–19.07).

When the name-value pair argument 'Termination' is set to 'max', the decoding terminates after maxNumIter number of iterations. When 'Termination' is set to 'early', the decoding terminates when all parity checks are satisfied ($$H{c}^{\text{T}}=0$$) or after maxNumIter number of iterations.

The implementation of the layered belief propagation algorithm is based on the decoding algorithm presented in [3], Section II.A. The decoding loop iterates over subsets of rows (layers) of the PCM. For each row, m, in a layer and each bit index, j, the implementation updates the key components of the algorithm based on these equations:

(1) $$L(q_{mj})=L(q_j)-R_{mj}$$,

(2) $$A_{mj}={\displaystyle \sum _{\begin{array}{c}n \in N\left(m\right)\\ n\ne j\end{array}}\text{ψ}(L(q_{mn}))}$$,

(3) $$s_{mj}={\displaystyle \prod _{\begin{array}{c}n \in N\left(m\right)\\ n\ne j\end{array}}\text{sign}(L(q_{mn}))}$$,

(4) $$R_{mj}=-s_{mj}\text{ψ}(A_{mj})$$, and

(5) $$L(q_j)=L(q_{mj})+R_{mj}$$.

For each layer, the decoding equation (5) works on the combined input obtained from the current LLR inputs $$L(q_{mj})$$ and the previous layer updates $$R_{mj}$$.

Because only a subset of the nodes is updated in a layer, the layered belief propagation algorithm is faster compared to the belief propagation algorithm. To achieve the same error rate as attained with belief propagation decoding, use half the number of decoding iterations when using the layered belief propagation algorithm.

The implementation of the normalized min-sum decoding algorithm follows the layered belief propagation algorithm with equation (2) replaced by

$$A_{mj}=\underset{\begin{array}{c}n \in N\left(m\right)\\ n\ne j\end{array}}{\min }(\left|L(q_{mn}) \right|\cdot \alpha )$$,

where α is in the range (0, 1] and is the scaling factor specified by ScalingFactor. This equation is an adaptation of equation (4) presented in [4].

The implementation of the offset min-sum decoding algorithm follows the layered belief propagation algorithm with equation (2) replaced by

$$A_{mj} = \max (\underset{\begin{array}{c}n \in N\left(m\right)\\ n\ne j\end{array}}{\min } (\left|L\left(q_{mn}\right)\right|- \beta ), 0)$$,

where β ≥ 0 and is the offset specified by Offset. This equation is an adaptation of equation (5) presented in [4].