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In BEC, we know for certain that the bits are either correct or erasure thus when we compute the log likelihood ratio (LLR) in belief propagation can only be inf, -inf and 0.

How can we implement the BP for a channel that is binary erasure?

I also tried to do the Galleger's algorithm. Here i do the bits-fliping to 1 or -1 whenever the incoming message of the check node is erased, 0.

Does it mean that the algorithm can only correct 50% of the error bit? Is this the way to implement the Galleger's algorithm for BEC?

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    $\begingroup$ In BEC decoding by BP makes progress in those situations, where only a single erased bit participates in a parity check. After all, on such an occasion the parity check allows you to solve for the value of the erased bit. It is a bit kludgy to think about this in terms of LLRs given that in a BEC the LLRs are either $\pm\infty$ or $0$. $\endgroup$ Feb 23 at 5:45

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You can still write loglikelihood for channel LLRs and can run the BP algorithm as well. Assume that you have an erasure channel with an erasure probabilty of $0<\epsilon<1$.

In case erasing event occurs for the bit $d_i$ and received as $r_i=e$ where $e$ denotes the erased symbols, therefore the channel LLR will be written as $$ L_c(d_i|r_i=e) = \log \left( \frac{\Pr\left\{d_i=0 | r_i=e\right\}}{\Pr\left\{d_i=1 | r_i=e\right\}} \right) = \log \left( \frac{\epsilon}{\epsilon} \right) = 0$$ If no erasing occurs and $r_i=0$ $$ L_c(d_i|r_i=1) = \log \left( \frac{\Pr\left\{d_i=0 | r_i=0\right\}}{\Pr\left\{d_i=1 | r_i=0\right\}} \right) = \log \left( \frac{1}{0} \right)= \infty$$ If no erasing occurs and $r_i=1$ $$ L_c(d_i|r_i=1) = \log \left( \frac{\Pr\left\{d_i=0 | r_i=1\right\}}{\Pr\left\{d_i=1 | r_i=1\right\}} \right) = \log \left( \frac{0}{1} \right)= -\infty$$

After calculating the channel LLRs, you can send them to the BP decoder. For practical purposes $\infty$ and $-\infty$ can be replaced with $+1$ and $-1$ "only for min-sum algorithm".

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