We present two new positive results for reliable computation using formulas over physical alphabets of size$q \gt 2$. First, we show that for logical alphabets of size$\ell = q$the threshold for denoising using gates subject to q-ary symmetric noise with error probability$\varepsilon $is strictly larger than that for Boolean computation, and we show that reliable computation is possible as long as signals remain distinguishable, i.e.$\epsilon \lt (q - 1) / q$, in the limit of large fan-in$k \rightarrow \infty $. We also determine the point at which generalized majority gates with bounded fan-in fail, and show in particular that reliable computation is possible for$\epsilon \lt (q - 1) / (q (q + 1))$in the case of q prime and fan-in$k = 3$. Secondly, we provide an example where$\ell \lt q$, showing that reliable Boolean computation,$\ell = 2$, can be performed using 2-input ternary,$q = 3$, logic gates subject to symmetric ternary noise of strength$\varepsilon \lt 1/6$by using the additional alphabet element for error signaling.

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 $.