On single diagnostic tests for logic circuits in the Zhegalkin basis

We consider a problem of synthesis of irredundant logic circuits in the basis $\{\&,\oplus,1,0\}$ which implement Boolean functions on $n$ variables and allow short single diagnostic tests regarding constant faults of type $0$ at outputs of gates. For each Boolean function, the minimal possible length value of such a test is found. In particular, it is proved that this value does not exceed two.