Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Low-Density Parity-Check Codes and Spatial Coupling for Quantitative Group Testing

Mgeni Makambi Mashauri, Alexandre Graell i Amat, Michael Lentmaier

TL;DR

This work tackles non-adaptive quantitative group testing by introducing an LDPC-code-based GT scheme and applying spatial coupling to both LDPC and GLDPC GT designs. It develops peeling decoding for the LDPC approach and derives density-evolution recursions to analyze performance, showing that increasing coupling memory $w$ yields threshold saturation and higher test-efficiency, with the LDPC scheme substantially outperforming GLDPC designs. Finite-length simulations corroborate the asymptotic results, confirming robust improvements in misdetection rates at practical block lengths. Overall, the paper demonstrates that simple parity-check constraints combined with spatial coupling enable efficient, scalable quantitative GT in low-prevalence regimes.

Abstract

A non-adaptive quantitative group testing (GT) scheme based on sparse codes-on-graphs in combination with low-complexity peeling decoding was introduced and analyzed by Karimi et al.. In this work, we propose a variant of this scheme based on low-density parity-check codes where the BCH codes at the constraint nodes are replaced by simple single parity-check codes. Furthermore, we apply spatial coupling to both GT schemes, perform a density evolution analysis, and compare their performance with and without coupling. Our analysis shows that both schemes improve with increasing coupling memory, and for all considered cases, it is observed that the LDPC code-based scheme substantially outperforms the original scheme. Simulation results for finite block length confirm the asymptotic density evolution thresholds.

Low-Density Parity-Check Codes and Spatial Coupling for Quantitative Group Testing

TL;DR

This work tackles non-adaptive quantitative group testing by introducing an LDPC-code-based GT scheme and applying spatial coupling to both LDPC and GLDPC GT designs. It develops peeling decoding for the LDPC approach and derives density-evolution recursions to analyze performance, showing that increasing coupling memory yields threshold saturation and higher test-efficiency, with the LDPC scheme substantially outperforming GLDPC designs. Finite-length simulations corroborate the asymptotic results, confirming robust improvements in misdetection rates at practical block lengths. Overall, the paper demonstrates that simple parity-check constraints combined with spatial coupling enable efficient, scalable quantitative GT in low-prevalence regimes.

Abstract

A non-adaptive quantitative group testing (GT) scheme based on sparse codes-on-graphs in combination with low-complexity peeling decoding was introduced and analyzed by Karimi et al.. In this work, we propose a variant of this scheme based on low-density parity-check codes where the BCH codes at the constraint nodes are replaced by simple single parity-check codes. Furthermore, we apply spatial coupling to both GT schemes, perform a density evolution analysis, and compare their performance with and without coupling. Our analysis shows that both schemes improve with increasing coupling memory, and for all considered cases, it is observed that the LDPC code-based scheme substantially outperforms the original scheme. Simulation results for finite block length confirm the asymptotic density evolution thresholds.
Paper Structure (13 sections, 3 theorems, 9 equations, 5 figures, 3 tables)

This paper contains 13 sections, 3 theorems, 9 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. System Model
  3. Preliminaries: Group Testing Based on GLDPC Codes
  4. Quantitative Group testing Based on LDPC codes
  5. Proposed GT scheme
  6. Density Evolution
  7. Group Testing with Spatial coupling
  8. Group Testing based on Spatially-Coupled LDPC Codes
  9. Group Testing based on Spatially-Coupled GLDPC Codes
  10. Numerical Results
  11. Density Evolution Thresholds
  12. Simulation Results
  13. Conclusion

Key Result

Proposition 1

The quantities $p^{(\ell)}_0$, $p^{(\ell)}_1$, $q^{(\ell)}_0$, and $q^{(\ell)}_1$ are given by the following density evolution equations:

Figures (5)

  • Figure 1: Bipartite graph corresponding to the assignment matrix in \ref{['eq:Matrix']}.
  • Figure 2: $\Omega_\mathsf{th}$ as a function of $\gamma$ for LDPC code-based and GLDPC code-based schemes. Dashed lines are for the uncoupled schemes, while solid lines are for the coupled schemes.
  • Figure 3: Misdetection rate for uncoupled (dashed) and coupled (solid) LDPC code-based GT.
  • Figure 4: Misdetection rate for uncoupled (dashed) and coupled (solid) GLDPC code-based GT with $t=3$ and $d_\mathsf{v}=3$.
  • Figure 5: Misdetection rate as a function of the number of tests per defective item for GLDPC code-based GT with $t=2$ and $d_\mathsf{v}=2$, and $\gamma=0.15\%$.

Theorems & Definitions (3)

  • Proposition 1
  • Proposition 2
  • Proposition 3