This work focuses on non-adaptive group testing, with a primary goal of efficiently identifying a set of at most $d$ defective elements among a given set of elements using the fewest possible number of tests. Non-adaptive combinatorial group testing often employs disjunctive codes and union-free codes. This paper discusses union-free codes with fast decoding (UFFD codes), a recently introduced class of union-free codes that combine the best of both worlds -- the linear complexity decoding of disjunctive codes and the fewest number of tests of union-free codes. In our study, we distinguish two subclasses of these codes -- one subclass, denoted as $(=d)$-UFFD codes, can be used when the number of defectives $d$ is a priori known, whereas $(\le d)$-UFFD codes works for any subset of at most $d$ defectives. Previous studies have established a lower bound on the rate of these codes for $d=2$. Our contribution lies in deriving new lower bounds on the rate for both $(=d)$- and $(\le d)$-UFFD codes for an arbitrary number $d \ge 2$ of defectives. Our results show that for $d\to\infty$, the rate of $(=d)$-UFFD codes is twice as large as the best-known lower bound on the rate of $d$-disjunctive codes. In addition, the rate of $(\le d)$-UFFD code is shown to be better than the known lower bound on the rate of $d$-disjunctive codes for small values of $d$.

翻译：本文聚焦于非自适应群测试，主要目标是在给定元素集合中，使用尽可能少的测试次数高效识别至多$d$个缺陷元素。非自适应组合群测试常采用析取码和无并码。本文讨论具有快速解码能力的无并码（UFFD码），这是一类新近引入的无并码，兼具析取码的线性复杂度解码优势与无并码的最少测试次数特点。研究中，我们将此类码划分为两个子类：一类记为$(=d)$-UFFD码，适用于缺陷数$d$先验已知的情况；另一类为$(\le d)$-UFFD码，适用于任意至多$d$个缺陷的子集。已有研究建立了$d=2$时此类码率的下界。我们的贡献在于：针对任意缺陷数$d \ge 2$，推导了$(=d)$-UFFD码与$(\le d)$-UFFD码的新的码率下界。结果表明，当$d\to\infty$时，$(=d)$-UFFD码的码率是已知$d$-析取码码率最佳下界的两倍。此外，对于较小的$d$值，$(\le d)$-UFFD码的码率优于已知$d$-析取码码率的下界。