Synthesis of easily testable logic networks under one-type stuck-at faults at inputs and outputs of gates

The following assertions are proved: for each natural $k$ and each Boolean constant $p$, there exists a basis consisting of a Boolean function on $\max(k+1; 3)$ variables and negation of one variable (there exists a basis consisting of a Boolean function on not more than $2,5k+2$ variables and negation of this function), in which one can implement any Boolean function except a Boolean constant $p$ by a logic network which is irredundant and allows a fault detection test (a diagnostic test, respectively) with a length not exceeding $2$ under not more than $k$ stuck-at-$p$ faults at inputs and outputs of gates. It is shown that, when considering only stuck-at-$p$ faults at inputs of gates, one can reduce the mentioned bounds on lengths of tests to $1$.