  Unequal transition probabilities between input and output symbols, input
power constraints, or input symbols of unequal durations can lead to
non-uniform capacity achieving input distributions for communication channels.
Using uniform input distributions reduces the achievable rate, which is called
the shaping gap. Gallager's idea for reliable communication with zero shaping
gap is to do encoding, matching, and jointly decoding and dematching. In this
work, a scheme is proposed that consists in matching, encoding, decoding, and
dematching. Only matching is channel specific whereas coding is not. Thus
off-the-shelf LDPC codes can be applied. Analytical formulas for shaping and
coding gap of the proposed scheme are derived and it is shown that the shaping
gap can be made zero. Numerical results show that the proposed scheme allows to
operate off-the-shelf LDPC codes with zero shaping gap and a coding gap that is
unchanged compared to uniform transmission.
