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

Lei Huang

TL;DR

This work addresses quantifying the design space for structure-avoiding QC-SC-LDPC codes under joint edge spreading and lifting. It casts the design as a constraint-satisfaction problem and applies a quantitative clique Lovász Local Lemma to obtain explicit lower bounds on the number of feasible $(P,L)$ assignments, including non-equivalent partitions under permutations. It further develops a diversity guarantee for Moser-Tardos outputs through Rényi-entropy bounds, with closed-form specializations for eliminating short cycles such as 4-cycles, and shows that circulant power optimization subsumes MT within the OO-CPO framework. Together, these results provide principled guidance for sizing memory/lifting and for practical, diverse, structure-avoiding QC-SC-LDPC code design with improved error-floor performance.

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 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.

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

TL;DR

This work addresses quantifying the design space for structure-avoiding QC-SC-LDPC codes under joint edge spreading and lifting. It casts the design as a constraint-satisfaction problem and applies a quantitative clique Lovász Local Lemma to obtain explicit lower bounds on the number of feasible assignments, including non-equivalent partitions under permutations. It further develops a diversity guarantee for Moser-Tardos outputs through Rényi-entropy bounds, with closed-form specializations for eliminating short cycles such as 4-cycles, and shows that circulant power optimization subsumes MT within the OO-CPO framework. Together, these results provide principled guidance for sizing memory/lifting and for practical, diverse, structure-avoiding QC-SC-LDPC code design with improved error-floor performance.

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 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.
Paper Structure (13 sections, 15 theorems, 64 equations)

This paper contains 13 sections, 15 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. PRELIMINARIES
  3. QC-SC-LDPC Codes
  4. Major Harmful Structures
  5. Constraint Satisfaction Problem
  6. Lower Bound on the Number of Solutions
  7. Lower Bound on the Number of Solutions
  8. Diversity of Constructive Outputs via MT Entropy
  9. Conclusion
  10. Proof of the Theorem 3
  11. Proof of the Corollary 3
  12. Proof of the Comparison Inequality
  13. Numerical Verification for $(\gamma,\kappa)=(3,5)$

Key Result

Lemma 1

Let $\mathcal{C}_{2g}$ denote the set of length-$2g$ cycle candidates in the $(\gamma,\kappa)$ base matrix, where $g\in\mathbb{N}$ and $g\ge 2$. For $c_{2g}\in\mathcal{C}_{2g}$ represented by $(j_1,i_1,j_2,i_2,\ldots,j_g,i_g)$ (with $j_{g+1}=j_1$), the nodes $(i_k,j_k)$ and $(i_k,j_{k+1})$ for $1\le Moreover, such a protograph cycle becomes an actual cycle in the Tanner graph after $Z$-lifting if

Theorems & Definitions (20)

  • Lemma 1
  • Definition 1: Absorbing Sets
  • Proposition 1: QC-SC-LDPC design as a CSP
  • Theorem 1
  • Corollary 1: uniform spreading and uniform lifting
  • Definition 2: column and row permutation
  • Definition 3: equivalent matrices
  • Remark 1
  • Lemma 2
  • Theorem 2
  • ...and 10 more