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

\begin{equation} 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} \end{equation} 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.

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:

\begin{equation} 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} \end{equation} 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.