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

The following assertions are proved: for each natural $k$, there exists a basis consisting of Boolean functions on not more than $2k+2$ variables (on not more than $4k+2$ variables), in which one can implement any Boolean function except the constant $1$ by a logic network which is irredundant and allows a fault detection test with a length not exceeding $3$ (a diagnostic test with a length not exceeding $4$, respectively) under not more than $k$ arbitrary stuck-at faults at inputs and outputs of gates. It is shown that, when considering only arbitrary stuck-at faults at inputs of gates, one can reduce the mentioned bounds on lengths of tests to $2$.