  In this paper, we propose a systematic low density generator matrix (LDGM)
code ensemble, which is defined by the Bernoulli process. We prove that, under
maximum likelihood (ML) decoding, the proposed ensemble can achieve the
capacity of binary-input output symmetric (BIOS) memoryless channels in terms
of bit error rate (BER). The proof technique reveals a new mechanism, different
from lowering down frame error rate (FER), that the BER can be lowered down by
assigning light codeword vectors to light information vectors. The finite
length performance is analyzed by deriving an upper bound and a lower bound,
both of which are shown to be tight in the high signal-to-noise ratio (SNR)
region. To improve the waterfall performance, we construct the systematic
convolutional LDGM (SC-LDGM) codes by a random splitting process. The SC-LDGM
codes are easily configurable in the sense that any rational code rate can be
realized without complex optimization. As a universal construction, the main
advantage of the SC-LDGM codes is their near-capacity performance in the
waterfall region and predictable performance in the error-floor region that can
be lowered down to any target as required by increasing the density of the
uncoupled LDGM codes. Numerical results are also provided to verify our
analysis.
