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Let be $X$ the following finite set: $X=\{0,1,2,\ldots,63\}$. I want to find two function $f$ and $g$ , where $f,g:X \times X \to Z$. We define $x'=f(x,y)$ and $y'=g(x,y)$. We impose the following condition: can't exist two pairs: $(x,y) \neq (z,t)$ such that $(x',y')=(z',t')$. The function $f$ can not be equal with $g$. If we know $(x',y')$ then we can compute $(x,y)$. Let be $S = X \times X$ ($S=\{(x_1,y_1),(x_2,y_2),\ldots,(x_N,y_N)\}, \text{ cartesian product of X}$ ). The problem is to find $f$ and $g$ in order to have:

  1. $\sum_{i=1}^N(x-f(x_i,y_i))^2+(y-g(x_i,y_i))^2$ to be minimum
  2. Let be $T$, the set of elements $(x'_i,y'_i)$, where $x'_i \in S \text{ and } y'_i \in S $. We want that the number of elements from $T$ to be maximum.
  3. The following forms of f can not be used: $f(x,y)=\pm x$, $f(x,y)=\pm y$,$f(x,y)=\pm x\pm y$ $f(x,y)=\pm y$ and the same restrictions are for $g$.

For example $f(x,y)=2x-y$ and $g(x,y)=2y-x$ can be a possible solution. But I want to know if exist other solution.

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