Union-free codes and disjunctive codes are two combinatorial structures, which are used in nonadaptive group testing to find a set of $d$ defective elements among $n$ samples by carrying out the minimal number of tests $t$. It is known that union-free codes have a larger rate, whereas disjunctive codes provide a more efficient decoding algorithm. In this paper we introduce a new family of codes for nonadaptive group testing with fast decoding. The rate of these codes is larger than the rate of disjunctive codes, while the decoding algorithm has the same complexity. In addition, we derive a lower bound on the rate of new codes for the case of $d=2$ defectives, which is significantly better than the bound for disjunctive codes and almost as good as the bound for union-free codes.
翻译:无工会守则和分离代码是两种组合结构,用于非适应性组别测试,通过进行最低数量的测试,在一美元样本中找到一组美元缺陷元素。众所周知,无工会守则的比例较高,而脱钩守则提供了更有效的解码算法。在本文中,我们引入了非适应性组别快速解码测试的新编码组合。这些编码的比率高于分离性编码的比率,而解码算法也具有同样的复杂性。此外,我们对美元=2美元的缺陷案例的新编码率的约束程度较低,这大大优于脱钩编码的约束,几乎与无工会守则的约束值一样好。