关于一类群试模型的结果

Results on the Type Model in Group Testing

讨论了群试中比B—d模型更一般的B(m,d)模型.由B(m,d)模型的本质特征,得到了两个基本定理,并以此为基础,简洁地证明了对给定的m,n_k(d)的上界为(m+1)~k,其中k≥[log_(m+1)d],并给出了达到上界的最优试验序列G(m,n).

The paper deals with the following type model in group testing:Let X be a set of n ele-ments.Set is composed of all defectives in X.Every time,we test m subsets in X simultaneously.Let Its testing founction is where We refer to the model as model B(m,d). Definition:Let x,ys X,if there is a testing set S in a test,while either element x or y is in S,the other is not in S,x are called separate from y in the test,or the test is called yielding a separated pair,denoted by sp (x,y). Let n_k (d) denote the maximum ,from X we can find the d defectives in k tests. The paper offers the following results: Theorem 1 Let g be a group testing sequence for model B (m,d ),Then for any there must exist a test in g which yields a separated pair sp (x,y). Theorem 2 Let g be a group testing sequence.For any x,y X,if there exists a test in g in which x separates from y,then g is a group testing sequence for model B (m,d). Theorem 3 For model B (m,d). The paper presents a testing sequence G (m,n),where m is the number of the simul-taneous test sets and n is the total number of elements. G (m,n)-Testing Sequence: Input X,m. Output D Step 1 Determine k (test number)such that Step 2 Divide X into subsets Step 3 Divide Step4 Testthe processure goes back to Step 3,otherwise,the processure ends. Theorem 4 G (m,n)is the optimal group testing sequence for model B (m,d).