  In Linear Programming (LP) decoding of a Low-Density-Parity-Check (LDPC) code
one minimizes a linear functional, with coefficients related to log-likelihood
ratios, over a relaxation of the polytope spanned by the codewords
\cite{03FWK}. In order to quantify LP decoding, and thus to describe
performance of the error-correction scheme at moderate and large
Signal-to-Noise-Ratios (SNR), it is important to study the relaxed polytope to
understand better its vertexes, so-called pseudo-codewords, especially those
which are neighbors of the zero codeword. In this manuscript we propose a
technique to heuristically create a list of these neighbors and their
distances. Our pseudo-codeword-search algorithm starts by randomly choosing the
initial configuration of the noise. The configuration is modified through a
discrete number of steps. Each step consists of two sub-steps. Firstly, one
applies an LP decoder to the noise-configuration deriving a pseudo-codeword.
Secondly, one finds configuration of the noise equidistant from the pseudo
codeword and the zero codeword. The resulting noise configuration is used as an
entry for the next step. The iterations converge rapidly to a pseudo-codeword
neighboring the zero codeword. Repeated many times, this procedure is
characterized by the distribution function (frequency spectrum) of the
pseudo-codeword effective distance. The effective distance of the coding scheme
is approximated by the shortest distance pseudo-codeword in the spectrum. The
efficiency of the procedure is demonstrated on examples of the Tanner
$[155,64,20]$ code and Margulis $p=7$ and $p=11$ codes (672 and 2640 bits long
respectively) operating over an Additive-White-Gaussian-Noise (AWGN) channel.
