The finite-length performance of spatially-coupled low-density parity-check (SC-LDPC) codes is strongly affected by short cycle configurations and the harmful structures induced by them. This paper studies SC-LDPC code design directly at the protograph level, where the design variables are the edge-spreading assignments specified by the partition matrix. In contrast to CLLL/Moser--Tardos based constructive frameworks for QC-SC-LDPC codes, we focus on sharper nonconstructive existence and counting bounds. By encoding cycle-activation conditions as polynomial vanishing constraints over finite grids, we apply the Combinatorial Nullstellensatz to derive sufficient memory conditions for eliminating prescribed cycle-induced harmful structures. For fully connected $(γ,κ)$ base graphs, the resulting bounds explicitly characterize the memory required to destroy all $4$-cycles as well as all $4$- and $6$-cycles, and for fixed $γ$, they are asymptotically tight up to a constant factor compared with known lower bounds. We further apply the Alon--Füredi theorem to obtain lower bounds on the number of feasible edge-spreading assignments, including an explicit counting bound for assignments that eliminate all $4$-cycles and hence yield girth at least six. These results provide a refined algebraic-combinatorial characterization of the feasible design space for high-memory SC-LDPC codes, although no corresponding construction algorithm is provided.

翻译：空间耦合低密度奇偶校验（SC-LDPC）码的有限长性能受短环构型及其诱导的有害结构影响显著。本文直接在原型图层面研究SC-LDPC码设计，其中设计变量为由分割矩阵指定的边扩展分配。不同于基于CLLL/Moser-Tardos的QC-SC-LDPC码构造框架，我们聚焦于更精确的非构造性存在性与计数界。通过将环激活条件编码为有限网格上的多项式消零约束，我们应用组合零化定理推导出消除指定环诱导有害结构的充分记忆条件。对于全连接$(γ,κ)$基图，所得界显式刻画了消除所有4环以及所有4环和6环所需的记忆量，且在固定$γ$下，与已知下界相比渐近紧至常数因子。我们进一步应用Alon-Füredi定理获得可行边扩展分配数的下界，包括消除所有4环（从而得到围长至少为6）的分配数的显式计数界。这些结果为高记忆SC-LDPC码的可行设计空间提供了精细的代数组合刻画，尽管未给出相应的构造算法。