On the design of switching circuits admitting small detection test sets

It is established that for an arbitrary nonconstant Boolean function $f(x_1,\dots,x_n)$ there exists a testable switching circuit, a) which is a circuit realizing the system $(f,\bar f)$ and admitting the single fault detection test set of power $O(n)$, b) which is a circuit realizing the function $f(x_1,\dots,x_n)\oplus x_{n+1}$ and admitting the single fault detection test set of power $O(n)$. It is also proved that almost all Boolean functions $f(x_1,\dots, x_n)$ can be realized by switching circuits which admit small detection test sets (test sets of power $O(n)$) under homogeneous faults (closures or breakings).