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

On concentration of measures for LDPC code ensembles

“…Therefore, the exponential decay rate in the concentration inequality of Theorem 4 scales like ( log 1 ε ) −2 and this asserts a rather strong concentration of the cardinality of the fundamental system of cycles around the average value. This concentration result complements the discussion in [12, Corollary 1] and [13] that introduced a lower bound on this average as a function of the fractional gap to capacity, and it scales like log 1 ε .…”

“…An informationtheoretic lower bound on the number of fundamental cycles for bipartite graphs of LDPC code ensembles was proved in [12, Corollary 1 on p. 8] (see also [13]). This bound was expressed in terms of the achievable gap to capacity when the communication takes place over an MBIOS channel.…”