Threshold Disjunctive Codes

Let $1 \le s < t$, $N \ge 1$ be integers and a complex electronic circuit of size $t$ is said to be an $s$-active, $\; s \ll t$, 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 substituted for the similar $s$-active circuit. Suppose that there exists a possibility to check the $s$-activity of the circuit using $N$ non-adaptive group tests identified by a conventional disjunctive $s$-code $X$ of size~$t$ and length~$N$. As usually, we say that any group test yields the positive response if the group contains at least one defective element. In this case, there is no any interest to look for the defective elements. We need to decide on the number of the defective elements in the circuit without knowing the code~$X$. In addition, the decision has the minimal possible complexity because it is based on the simple comparison of a fixed threshold $T$, $0 \le T \le N - 1$, with the number of positive responses $p$, $0 \le p \le N$, obtained after carrying out $N$ non-adaptive tests prescribed by the disjunctive $s$-code~$X$. For the introduced group testing problem, a new class of the well-known disjunctive $s$-codes called the threshold disjunctive $s$-codes is defined. The aim of our paper is to discuss both some constructions of suboptimal threshold disjunctive $s$-codes and the best random coding bounds on the rate of threshold disjunctive $s$-codes.