Soft-Decision Decoding for LDPC Code-Based Quantitative Group Testing

TL;DR

The paper addresses non-adaptive quantitative group testing by replacing a hard-decision decoding stage with a belief-propagation based soft-decision decoder on an LDPC-style bipartite graph. It derives explicit variable-node and constraint-node updates that operate purely on soft information and analyzes the computational complexity, showing feasibility for small to moderate check-node degrees. Through simulations, the soft BP decoder significantly improves the misdetection rate over the previous peeling decoder, achieving measurable gains in the prevalence parameter $\\delta$ across short and moderate blocklengths. This work demonstrates the practical benefits of soft-information decoding for LDPC-based quantitative group testing and points toward extensions to noisy test scenarios.

Abstract

We consider the problem of identifying defective items in a population with non-adaptive quantitative group testing. For this scenario, Mashauri et al. recently proposed a low-density parity-check (LDPC) code-based quantitative group testing scheme with a hard-decision decoding approach (akin to peeling decoding). This scheme outperforms generalized LDPC code-based quantitative group testing schemes in terms of the misdetection rate. In this work, we propose a belief-propagation-based decoder for quantitative group testing with LDPC codes, where the messages being passed are purely soft. Through extensive simulations, we show that the proposed soft-information decoder outperforms the hard-decision decoder Mashauri et al.

Figures (5)

• Figure 1: Bipartite graph representation of the assignment matrix $\bm{A}$ in \ref{['eq:A']}. The circles denote the variable nodes (VN), while the squares represent the constraint nodes (CN). The connections of the items to their respective group are color-coded.
• Figure 2: Visualization of the variable and constraint node updates shown in \ref{['eq:VN_update']} and \ref{['eq:CN_update']}, respectively. The circle represents a variable node $\mathsf{v}_i$, while the square represents a constraint node $\mathsf{c}_j$. Note that the constraint node $\mathsf{c}_j$ has the test outcome of the $j$-th pool $s_j$ as a constraint the incoming messages should fulfill.
• Figure 3: We present the performance in terms of $P_{\mathsf{MD}}$ versus $\delta$ for a regular graph with $d_\mathsf{v} =3$ and $d_\mathsf{c} =6$, that yields a rate $R = 0.5$. The dashed line shows the performance of the peeling decoder in Mas23, while the solid lines show the performance of the proposed soft decoder. The results are shown for the short-length regime $n \in \{128, 256, 1024\}$. The lines are color and marker-coded, such that the same color and marker are used for the same length $n$ and only the line style (dashed or solid) determines the decoder.
• Figure 4: In this plot, we show the performance of our proposed decoder (solid lines) and compare it to the peeling decoder in Mas23 (dashed lines) for a regular graph $d_\mathsf{v} = 3,d_\mathsf{c} =6$ for moderate lengths $n \in \{2048, 8192, 16384\}$. The plot shows that for all choices of $n$, the misdetection probability $P_{\mathsf{MD}}$ achieved by the proposed decoder is much lower compared to the one by peeling.
• Figure 5: This plot shows the results as $P_{\mathsf{MD}}$ versus $\delta$ for a regular graph with $d_\mathsf{v} =3$ and $d_\mathsf{c} = 9$ for $n \in \{4095, 16380\}$. The rate of the scheme defined by this regular graph is $R=1/3$. The performance of our proposed decoder is shown in solid lines, while the performance of the peeling decoder Mas23 is shown in dashed lines. For both choices of $n$, the proposed decoder outperforms the peeling decoder.

Theorems & Definitions (2)

• Example 1: Assignment matrix
• Example 2: Constraint node update