Counting and Entropy Bounds for Structure-Avoiding Spatially-Coupled LDPC Constructions

Designing large coupling memory quasi-cyclic spatially-coupled LDPC (QC-SC-LDPC) codes with low error floors requires eliminating specific harmful substructures (e.g., short cycles) induced by edge spreading and lifting. Building on our work~\cite{r15} that introduced a Clique Lovász Local Lemma (CLLL)-based design principle and a Moser--Tardos (MT)-type constructive approach, this work quantifies the size and structure of the feasible design space. Using the quantitative CLLL, we derive explicit lower bounds on the number of partition matrices satisfying a given family of structure-avoidance constraints, and further obtain bounds on the number of non-equivalent solutions under row/column permutations. Moreover, via Rényi-entropy bounds for the MT distribution, we provide a computable lower bound on the number of distinct solutions that the MT algorithm can output, giving a concrete diversity guarantee for randomized constructions. Specializations for eliminating 4-cycle candidates yield closed-form bounds as functions of system parameters, offering a principled way to size memory/lifting and to estimate the remaining search space.