LDPC Codes for Quantitative Group Testing with a Non-Binary Alphabet

TL;DR

A novel scheme based on LDPC codes for quantitative group testing that significantly outperforms its binary counterpart with limited increase in complexity is proposed and analyzed.

Abstract

We propose and analyze a novel scheme based on LDPC codes for quantitative group testing. The key underlying idea is to augment the bipartite graph by introducing hidden non-binary variables to strengthen the message-passing decoder. This is achieved by grouping items into bundles of size q within the test matrix, while keeping the testing procedure unaffected. The decoder, inspired by some works on counter braids, passes lower and upper bounds on the bundle values along the edges of the graph, with the gap between the two shrinking with the decoder iterations. Through a density evolution analysis and finite length simulations, we show that the proposed scheme significantly outperforms its binary counterpart with limited increase in complexity.

Figures (3)

• Figure 1: Graphical representation of a system with $q=2$, $d_\mathsf{c}=4$, $d_\mathsf{v}=3$, $d_\mathsf{v,x}=1$ and $d_\mathsf{v,z}=2$. Tests are represented by square with a plus sign while empty squares represents bundles. In this case $\mathcal{CN}_z=\{\mathsf{c}_1,\mathsf{c}_2,\mathsf{c}_3,\mathsf{c}_4\}$ while $\mathcal{CN}_x=\{\mathsf{c}_5,\mathsf{c}_6\}$. All tests have the same degree $d_\mathsf{c}=4$ since each edge from a bundle to a test corresponds to two edges in the overall graph.
• Figure 2: Minimum $\Omega$ for different values of $\gamma$ for various bundle sizes $q$.
• Figure 3: Simulation results showing the misdetection rate for various values of $\gamma$ for a fixed rate $\Omega=5\%$ for $n=210000$. The GLDPC results are also shown for comparison. The vertical lines from the bottom mark the thresholds.