Threshold Decoding for Disjunctive Group Testing

TL;DR

This paper explores threshold decoding methods for disjunctive group testing to efficiently determine circuit defectiveness with minimal complexity, providing bounds on statistical significance levels.

Contribution

It introduces a threshold decision rule for disjunctive group testing and analyzes its statistical properties compared to traditional decoding algorithms.

Findings

Threshold decoding simplifies the testing process.

Upper bounds on significance levels are established.

Comparison with conventional decoding algorithms is provided.

Abstract

Let $1 \le s < t$, $N \ge 1$ be integers and a complex electronic circuit of size $t$ is said to be an $s$-active, $\; s \ll t$, and can work as a system block if not more than $s$ elements of the circuit are defective. Otherwise, the circuit is said to be an $s$-defective and should be replaced by a similar $s$-active circuit. Suppose that there exists a possibility to run $N$ non-adaptive group tests to check the $s$-activity of the circuit. As usual, we say that a (disjunctive) group test yields the positive response if the group contains at least one defective element. Along with the conventional decoding algorithm based on disjunctive $s$-codes, we consider a threshold decision rule with the minimal possible decoding complexity, which is based on the simple comparison of a fixed threshold $T$, $1 \le T \le N - 1$, with the number of positive responses $p$, $0 \le p \le N$. For the both of decoding algorithms we discuss upper bounds on the $\alpha$-level of significance of the statistical test for the null hypothesis $\left\{ H_0 \,:\, \text{the circuit is $s$-active} \right\}$ verse the alternative hypothesis $\left\{ H_1 \,:\, \text{the circuit is $s$-defective} \right\}$.