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.

翻译：设计具有低错误平层的大耦合存储器准循环空间耦合LDPC（QC-SC-LDPC）码，需要消除由边扩展和提升操作引发的特定有害子结构（例如短环）。基于我们先前在文献~\cite{r15}中提出的基于团Lovász局部引理（CLLL）的设计原则和Moser--Tardos（MT）型构造性方法，本工作量化了可行设计空间的规模与结构。利用定量CLLL，我们推导了满足给定结构避免约束族的划分矩阵数量的显式下界，并进一步获得了在行/列置换下不等价解数量的界。此外，通过MT分布的Rényi熵界，我们给出了MT算法能够输出的不同解数量的可计算下界，从而为随机化构造提供了具体的多样性保证。针对消除4环候选结构的特例，我们得到了以系统参数为函数的闭式界，这为确定存储器/提升规模以及估计剩余搜索空间提供了一种基于原理的方法。