Bounds and Constructions of High-Memory Spatially-Coupled Codes

Abstract

In this paper, we apply the Clique Lov\'asz Local Lemma to provide sufficient conditions on memory and lifting degree for removing certain harmful combinatorial structures in spatially-coupled (SC) codes that negatively impact decoding performance. Additionally, we present, for the first time, a constructive algorithm based on the Moser-Tardos algorithm that ensures predictable performance. Furthermore, leveraging the properties of LLL-distribution and M-T-distribution, we establish the dependencies among the harmful structures during the construction process. We provide upper bounds on the probability change of remaining harmful structures after eliminating some of them. In particular, the elimination of 4-cycles increases the probability of 6-cycles becoming active by at most a factor of $e^{8/3}$.