Absorbing set characterization of array-based spatially coupled LDPC codes

Absorbing sets are combinatorially defined objects existing in the Tanner graph of a low-density parity-check (LDPC) code that have been shown to cause failures in the iterative message-passing decoder when transmission occurs over the additive white Gaussian noise channel. In this paper, we study the absorbing set properties of a class of high-rate array-based spatially coupled LDPC (SC-LDPC) codes that are constructed by coupling together L array-based LDPC block codes. We prove that the smallest absorbing sets existing in the Tanner graph of the SC-LDPC code have the same size as those in the corresponding uncoupled LDPC codes, and the number of such sets grow linearly with L. We show that spatial coupling greatly reduces the average number (per symbol) of minimal sets compared to the uncoupled codes, and we explain that this reduction is due to many absorbing sets and small cycles being `broken' by the coupling process. The large reduction in the number of minimal absorbing sets suggests that array-based SC-LDPC codes will have significantly improved decoding performance in the high signal-to-noise ratio regime compared to the corresponding uncoupled LDPC codes.