Existence and Counting Bounds for High-Memory Spatially-Coupled Codes via the Combinatorial Nullstellensatz

Abstract

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 $(\gamma,\kappa)$ 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 $\gamma$, they are asymptotically tight up to a constant factor compared with known lower bounds. We further apply the Alon--F\"uredi 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.