On Universal Properties of Capacity-Approaching LDPC Ensembles

This paper provides some universal information-theoretic bounds related to capacity-approaching ensembles of low-density parity-check (LDPC) codes. These bounds refer to the behavior of the degree distributions of such ensembles, and also to the graphical complexity and the fundamental system of cycles associated with the Tanner graphs of LDPC ensembles. The transmission of these ensembles is assumed to take place over an arbitrary memoryless binary-input output-symmetric (MBIOS) channel. The universality of the bounds derived in this paper stems from the fact that they do not depend on the full characterization of the LDPC ensembles but rather depend on the achievable gap between the channel capacity and the design rate of the ensemble, and also on the required bit error (or erasure) probability at the end of the decoding process. 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 sum-product decoding; the bounds are reasonably tight for general MBIOS channels, and are tightened for the binary erasure channel. Index Terms Cycles, low-density parity-check (LDPC) codes, maximum-likelihood decoding, memoryless binary-input output-symmetric (MBIOS) channels, stability condition, sum-product decoding algorithm, Tanner graphs.