Given $a,b\in\mathbb N$ define the set $$\chi(a,b)=\{M\in\{0,1\}^{n^a\times n^b}:\mbox{ every row of }M\mbox{ is distinct}\}.$$
Also given ${\bf{x}}=(x_1,\dots,x_{n^b})\in\mathbb Z^{n^b}$ define the quantity $$Q_{a,b}(x_1,\dots,x_{n^b})=\min\Bigg\{\|{\bf{y}}\|_\infty:\substack{{\bf{y}}^T=M{\bf{x}}^T\\ M\in\chi(a,b)\\ i\neq j\implies\mbox{ }y_i\neq y_j\\ \mbox{ }y_i-y_j=y_{i'}-y_{j'}\implies i=i'\mbox{ and }j=j'}\Bigg\}$$
If $a\leq b$ and $1\leq i\neq j\leq n\implies\mbox{ }x_i\neq x_j$ holds then is it true there is a $z\in\mathbb Z$ and a $k\in\mathbb R$ such that if $z\leq x_{min}=\min_{i\in\{1,\dots,n^b\}}x_i$ holds then we have $$Q_{a,b}(x_1,\dots,x_{n^b})\leq x_{min}^k?$$
Notes: We have $n^b$ variables $x_1,\dots,x_{n^b}$ at some $b\geq1$ such that each of $x_1,\dots,x_{n^b}$ is distinct and $z\leq\min_{i\in\{1,\dots,n^b\}}x_i$. Take the vector $$y=Mx$$ where $M$ is a matrix such that every row is distinct. If every row and pairwise difference of every row of $y=Mx$ is also distinct then is the smallest $\|{\bf{y}}\|_\infty$ that is possible always bound by a polynomial in minimum value of $(x_1,\dots,x_{n^a})$ that is valid over every $A$?
