On the binary adder channel with complete feedback, with an application to quantitative group testing

TL;DR

This paper precisely determines the optimal symmetric zero-error capacity of the binary adder channel with complete feedback, and applies this result to asymptotically minimize the number of tests in a quantitative group testing problem involving two defectives.

Contribution

It provides the exact value of the symmetric zero-error capacity for the binary adder channel with feedback and links this to an optimal testing strategy in group testing.

Findings

Optimal symmetric rate point is approximately 0.78974.

Minimum number of tests asymptotically equals (log_2 n) / r.

Established a connection between channel capacity and group testing efficiency.

Abstract

We determine the exact value of the optimal symmetric rate point $(r, r)$ in the Dueck zero-error capacity region of the binary adder channel with complete feedback. We proved that the average zero-error capacity $r = h(1/2-\delta) \approx 0.78974$, where $h(\cdot)$ is the binary entropy function and $\delta = 1/(2\log_2(2+\sqrt3))$. Our motivation is a problem in quantitative group testing. Given a set of $n$ elements two of which are defective, the quantitative group testing problem asks for the identification of these two defectives through a series of tests. Each test gives the number of defectives contained in the tested subset, and the outcomes of previous tests are assumed known at the time of designing the current test. We establish that the minimum number of tests is asymptotic to $(\log_2 n) / r$ as $n \to \infty$.