Asymptotic Distribution of Absorbing Sets and Fully Absorbing Sets for Regular Sparse Code Ensembles

It is well recognized that low-density parity-check (LDPC) codes can suffer from an error floor when decoded iteratively. This performance degradation is often attributed to the class of objects known as trapping sets. Past work has focused on characterizing the distribution of trapping sets for a variety of code ensembles, including regular, irregular and structured LDPC codes. As a subset of the trapping set collection, there exists a class of graphical structures called the absorbing sets. An absorbing set is a combinatorially-defined object; in particular a fully absorbing set is stable under bit-flipping decoding. By construction, there can exist trapping sets that are not stable under such a decoder. As a result, for finite-precision, iterative decoding algorithms used over additive channels, absorbing sets can describe decoding errors more accurately than the broader class of trapping sets. In this paper, we compute the normalized logarithmic asymptotic distributions of absorbing sets and fully absorbing sets, including elementary (fully) absorbing sets. The calculations are based on the trapping set enumeration method proposed by Milenkovic, Soljanin, and Whiting in [1]. We compare distributions of absorbing and trapping sets for representative code parameters of interest, and quantify the (lack of) discrepancies between the two. Good absorbing set properties are implied for known structured LDPC codes, including repeat accumulate codes and protograph-based constructions. Establishing the distribution of fully absorbing sets (especially when the discrepancy with the trapping set distribution is significant) allows one to further refine the estimates of the error rates under bit-flipping and related decoders.