Seeking a derangement $S: \{0,1\}^4 \to \{0,1\}^4$ for which changing one bit in the input always changes at least two bits in the output

Question

Kindly asking for any hints about the following question:

Define a function $ S : \{0,1\}^4 \to \{0,1\}^4$ with this conditions:

1- For any $x$, $\ S(x) \neq x $

2- for any $x \neq y$, $\ S(x) \neq S(y).$

3- Changed one bit in the input always changing at least two bits in the output.

Answer 1

After not getting systematically, I looked for solutions (there seem tobe gazillions) by backtracking Try this $$ 0000\mapsto 0001\\ 0001\mapsto0010\\ 0010\mapsto0100\\ 0011\mapsto0111\\ 0100\mapsto0110\\ 0101\mapsto1000\\ 0110\mapsto0011\\ 0111\mapsto1101\\ 1000\mapsto1010\\ 1001\mapsto0101\\ 1010\mapsto1111\\ 1011\mapsto1100\\ 1100\mapsto1001\\ 1101\mapsto1110\\ 1110\mapsto0000\\ 1111\mapsto1011$$

Answer 2

One solution would be the function that shifts each bit to the left (except from $0000$ and $1111$). Mathematically, it can be described as \begin{align*} f(x)=\begin{cases} 1111 &\text{if }x=0000\\ 0000 &\text{if }x=1111\\ 2x \mod{15} &\text{else} \end{cases} \end{align*} EDIT: I just realized that the third condition is not fulfilled.