  We illustrate the utility of the recently developed loop calculus for
improving the Belief Propagation (BP) algorithm. If the algorithm that
minimizes the Bethe free energy fails we modify the free energy by accounting
for a critical loop in a graphical representation of the code. The
log-likelihood specific critical loop is found by means of the loop calculus.
The general method is tested using an example of the Linear Programming (LP)
decoding, that can be viewed as a special limit of the BP decoding. Considering
the (155,64,20) code that performs over Additive-White-Gaussian-Noise channel
we show that the loop calculus improves the LP decoding and corrects all
previously found dangerous configurations of log-likelihoods related to
pseudo-codewords with low effective distance, thus reducing the code's
error-floor.
