A Boolean function f satisfies PC(l) of order k if f(x) ⊕ f(x ⊕ α) is balanced for any α such that 1 ≤ W(α) ≤ l even if any k input bits are kept constant, where W(α) denotes the Hamming weight of α. This paper shows the first design method of such functions which provides deg(f) ≥ 3. More than that, we show how to design “balanced” such functions. High nonlinearity and large degree are also obtained. Further, we present balanced SAC(k) functions which achieve the maximum degree. Finally, we extend our technique to vector output Boolean functions.