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

TL;DR

This paper develops quantitative bounds on the design space of structure-avoiding spatially-coupled LDPC codes using probabilistic and entropy-based methods, aiding in efficient code construction.

Contribution

It introduces explicit lower bounds on feasible code configurations and solution diversity, extending previous work with quantitative analysis and specialized bounds for cycle elimination.

Findings

Derived lower bounds on feasible edge-spreading and lifting assignments.

Provided bounds on the number of non-equivalent solutions under permutations.

Established a diversity guarantee for randomized code construction methods.

Abstract

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 Lov\'asz 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 feasible edge-spreading and lifting assignments 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 R\'enyi 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-cycles yield closed-form bounds as functions of system parameters, offering a principled way to select the memory and lifting degree and to estimate the remaining search space.