On the Binary Adder Channel With Complete Feedback, With an Application to Quantitative Group Testing

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
$\delta = 1/(2\log _{2}(2 + \sqrt {3}))$
. 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 $
.