  The decoding of Low-Density Parity-Check codes by the Belief Propagation (BP)
algorithm is revisited. We check the iterative algorithm for its convergence to
a codeword (termination), we run Monte Carlo simulations to find the
probability distribution function of the termination time, n_it. Tested on an
example [155, 64, 20] code, this termination curve shows a maximum and an
extended algebraic tail at the highest values of n_it. Aiming to reduce the
tail of the termination curve we consider a family of iterative algorithms
modifying the standard BP by means of a simple relaxation. The relaxation
parameter controls the convergence of the modified BP algorithm to a minimum of
the Bethe free energy. The improvement is experimentally demonstrated for
Additive-White-Gaussian-Noise channel in some range of the signal-to-noise
ratios. We also discuss the trade-off between the relaxation parameter of the
improved iterative scheme and the number of iterations.
