## Probability of false decoding with LDPC codes

It is well known that given a binary linear code $C$ under maximum likelihood decoding the probability of false decoding $P_C$ in a Binary Symmetric Channel with cross over probability $p$, in other words, the probability of decoding a wrong codeword is lower bounded by:

$$P_C(p)\leq\sum_{w=1}^nc_w\sum_{\lceil i=w/2\rceil }^w\binom{w}{i}p^i(1-p)^{w-i}$$ where $c_w$ is the number of words in the code with hamming weight $w$.

This bound can be computed for codes with a known weight enumerator polynomial. I would be interested in bounding, not necessarily through the bound above, the false decoding probability for instances of LDPC codes, that is not for ensembles, and under belief propagation.

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