  PCR testing is an invaluable diagnostic tool that has most recently seen
widespread use during the COVID-19 pandemic. A recent work by Wang, Gabrys and
Vardy proposed tropical codes as a model for group PCR testing. For a known but
arbitrary number of infected persons, a sufficient condition on the underlying
block design of a zero-error tropical code, called double disjunction, is
proposed. Despite this, the parameters for which the construction of doubly
disjunct block designs is known to exist are very limited. In this paper, we
define probabilistic tropical codes and consider random block designs that are
doubly disjunct with high probability. We also provide a deterministic
construction for a doubly disjunct block design given a disjunct block design.
We show that for certain choices of parameters, our probabilistic construction
has vanishing error. Our constructions, combined with existing methods, give us
three different ways to construct tropical codes. We compare the number of
tests required by each, and bounds on the error.
