  Low-Density Parity-Check (LDPC) codes received much attention recently due to
their capacity-approaching performance. The iterative message-passing algorithm
is a widely adopted decoding algorithm for LDPC codes \cite{Kschischang01}. An
important design issue for LDPC codes is designing codes with fast decoding
speed while maintaining capacity-approaching performance. In another words, it
is desirable that the code can be successfully decoded in few number of
decoding iterations, at the same time, achieves a significant portion of the
channel capacity. Despite of its importance, this design issue received little
attention so far. In this paper, we address this design issue for the case of
binary erasure channel.
  We prove that density-efficient capacity-approaching LDPC codes satisfy a so
called "flatness condition". We show an asymptotic approximation to the number
of decoding iterations. Based on these facts, we propose an approximated
optimization approach to finding the codes with good decoding speed. We further
show that the optimal codes in the sense of decoding speed are
"right-concentrated". That is, the degrees of check nodes concentrate around
the average right degree.
