  The asymptotic iterative decoding performances of low-density parity-check
(LDPC) codes using min-sum (MS) and sum-product (SP) decoding algorithms on
memoryless binary-input output-symmetric (MBIOS) channels are analyzed in this
paper. For MS decoding, the analysis is done by upper bounding the bit error
probability of the root bit of a tree code by the sequence error probability of
a subcode of the tree code assuming the transmission of the all-zero codeword.
The result is a recursive upper bound on the bit error probability after each
iteration. For SP decoding, we derive a recursively determined lower bound on
the bit error probability after each iteration. This recursive lower bound
recovers the density evolution equation of LDPC codes on the binary erasure
channel (BEC) with inequalities satisfied with equalities. A significant
implication of this result is that the performance of LDPC codes under SP
decoding on the BEC is an upper bound of the performance on all MBIOS channels
with the same uncoded bit error probability. All results hold for the more
general multi-edge type LDPC codes.
