Expansion Properties of Generalized ACE Codes

This work presents a generalization of the ACE algorithm proposed by Tian et. al [1] (called Generalized ACE GACE) for irregular LDPC code construction. We show that if the minimum left degree of a (dACE , η) compliant GACEconstrained LDPC code is lmin and the number of check nodes M is large enough, it is a [ α, 1 − 1+2 lmin ] , 2 > 0 expander with probability 1− O(f(M)) where f(M) = max [ 1/M η lmin (1+2)+22 , 1/M (dACE+1)2 ] for some α > 0. Here, dACE and η are the parameters of the construction. This is in contrast with a randomly constructed code that is an expander with the same parameters with probability 1−O(1/M22). Thus, at the block lengths of engineering interest the GACE algorithm is significantly more likely to produce good codes. We also discuss an improved algorithm and some trade-offs between the (dACE , η) parameters. We conclude by observing that while it is good for an irregular LDPC code with a given degree distribution to have high girth, in practice it is a hard goal to achieve at moderate block lengths and thus, it is sufficient to ensure that only cycles consisting of low-degree variable nodes are avoided.