On the fundamental system of cycles in the bipartite graphs of LDPC code ensembles

This work introduces an information-theoretic lower bound on the number of fundamental cycles for bipartite graphs of low-density parity-check (LDPC) code ensembles. This information-theoretic bound is expressed in terms of the achievable gap to capacity when the transmission of the code ensemble takes place over a memoryless binary-input output-symmetric (MBIOS) channel. The bound shows quantitatively the necessity of cycles in bipartite graphs which represent good LDPC code ensembles. More explicitly, it shows that the number of fundamental cycles should grow at least like log 1/epsiv where epsiv designates the gap in rate to capacity, hence, it is unbounded as the gap to capacity vanishes. For the derivation of this bound, a new information-theoretic lower bound on the average right degree, which also behaves like log 1/epsiv, is derived. The interested reader is referred to the full paper version.