Short single tests for circuits with arbitrary stuck-at faults at outputs of gates

The following results are proved: 1) any nonconstant Boolean function may be implemented by an irredundant circuit of gates in the basis $\{x\&y,$ $\overline x,x\oplus y\oplus z\}$ admitting a single fault detection test of length at most 2 with respect to arbitrary stuck-at faults at outputs of gates, 2) there exists a six-place Boolean function $\psi$ such that any nonconstant Boolean function may be implemented by an irredundant circuit of gates in the basis $\{\psi\}$ admitting a single diagnostic test of length at most 3 with respect to arbitrary stuck-at faults at outputs of gates.