  This paper considers the achievable rates and decoding complexity of
low-density parity-check (LDPC) codes over statistically independent parallel
channels. The paper starts with the derivation of bounds on the conditional
entropy of the transmitted codeword given the received sequence at the output
of the parallel channels; the component channels are considered to be
memoryless, binary-input, and output-symmetric (MBIOS). These results serve for
the derivation of an upper bound on the achievable rates of ensembles of LDPC
codes under optimal maximum-likelihood (ML) decoding when their transmission
takes place over parallel MBIOS channels. The paper relies on the latter bound
for obtaining upper bounds on the achievable rates of ensembles of randomly and
intentionally punctured LDPC codes over MBIOS channels. The paper also provides
a lower bound on the decoding complexity (per iteration) of ensembles of LDPC
codes under message-passing iterative decoding over parallel MBIOS channels;
the bound is given in terms of the gap between the rate of these codes for
which reliable communication is achievable and the channel capacity. The paper
presents a diagram which shows interconnections between the theorems introduced
in this paper and some other previously reported results. The setting which
serves for the derivation of the bounds on the achievable rates and decoding
complexity is general, and the bounds can be applied to other scenarios which
can be treated as different forms of communication over parallel channels.
