Threshold Decoding for Disjunctive Group Testing

Let 1 ≤ s < t, N ≥ 1 be integers and a complex electronic circuit of size t is said to be an s-active, s ≪ t, and can work as a system block if not more than s elements of the circuit are defective. Otherwise, the circuit is said to be an s-defective and should be replaced by a similar s-active circuit. Suppose that there exists a possibility to run N non-adaptive group tests to check the s-activity of the circuit. As usual, we say that a (disjunctive) group test yields the positive response if the group contains at least one defective element. Along with the conventional decoding algorithm based on disjunctive s-codes, we consider a threshold decision rule with the minimal possible decoding complexity, which is based on the simple comparison of a fixed threshold T , 1 ≤ T ≤ N − 1, with the number of positive responses p, 0 ≤ p ≤ N. For the both of decoding algorithms we discuss upper bounds on the α-level of significance of the statistical test for the null hypothesis {H0 : the circuit is s-active} verse the alternative hypothesis {H1 : the circuit is s-defective}. Let N ≥ 2, t ≥ 2, s and T be integers, where 1 ≤ s < t and 1 ≤ T < N. The symbol denote the equality by definition, |A| – the size of the set A and a binary code of length N and size t = ⌊2 RN ⌋, where a fixed parameter R > 0 is called a rate of the code X. The number of 1's in a binary column x = N i=1 x i , is called a weight of x. A code X is called a constant weight binary code of weight w, 1 ≤ w < N , if for any j ∈ [t], the weight |x (j)| = w. The conventional symbol u v will be used to denote the disjunctive (Boolean)