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.

翻译：PCR检测是一种宝贵的诊断工具，在COVID-19疫情期间得到了广泛应用。Wang、Gabrys和Vardy近期的工作提出将热带码作为群组PCR检测的模型。针对已知但任意数量的感染个体，他们提出了一种零错误热带码底层区组设计的充分条件——称为双重分离性。尽管如此，已知存在的双重分离区组设计构造参数极为有限。本文定义概率型热带码，并考虑以高概率实现双重分离性的随机区组设计。同时，针对给定的分离区组设计，我们提出一种确定性的双重分离区组设计构造方法。研究表明，在特定参数选择下，我们的概率构造方法具有渐近零错误率。这些构造方法与现有技术相结合，提供了三种构建热带码的途径。我们对比了每种方法所需的检测次数及其错误界。