Trapping and Absorbing Set Enumerators for Protograph-Based Generalized Low-Density Parity-Check Code Ensembles

Generalized low-density parity-check (GLDPC) codes have outstanding performance outperforming the standard low-density parity-check (LDPC) codes. However, the performance of these codes under iterative decoding can be degraded because of the presence of trapping, absorbing and fully absorbing sets in the code Tanner graphs. The (elementary) trapping and (fully) absorbing sets of protograph-based GLDPC codes are studied. The finite-length and asymptotic distributions of (elementary) trapping and (fully) absorbing sets for protograph-based GLDPC code ensembles are derived and evaluated. The asymptotic distributions can be efficiently computed by solving a system of equations. The relative minimum Δ-trapping/(fully) absorbing set size can be directly computed by adding one equation to the system of equations. This is useful for the code design where we can restrict the search to ensembles for which the minimum Δ-trapping and (fully) absorbing set sizes grow linearly with the blocklength. The normalized logarithmic asymptotic distributions of (elementary) trapping and (fully) absorbing sets for an example GLDPC code ensemble are evaluated.