On Universal Properties of the Degree Distributions and Cycles of Capacity-Approaching LDPC Ensembles∗

This paper provides some universal information-theoretic bounds related to the degree distributions and the average cardinality of the fundamental system of cycles of low-density parity-check (LDPC) ensembles. The transmission of these ensembles is assumed to take place over an arbitrary memoryless binary-input output-symmetric (MBIOS) channel, and the bounds are expressed in terms of the gap between the design rates of these ensembles and the channel capacity. Some of these bounds hold under maximum-likelihood decoding (and hence, they also hold under any sub-optimal decoding algorithm) whereas the others hold particularly under the sum-product iterative decoding algorithm. The tightness of some of these bounds is exemplified numerically for capacity-approaching LDPC ensembles under iterative decoding; the bounds are reasonably tight for general MBIOS channels, and are tightened for the binary erasure channel where the tightness in the latter case is also demonstrated analytically.