Distinguish permutations

Abstract

We are looking for integer numbers $g_j$ and $x_j$ ($j=1,...,n$) such that the sums $T_{\pi }:={\sum }_{j=1}^n{g}_j\cdot x_{\pi \left(j\right)}$ are different for all permutations $\pi \in S_n$ and $\max \left\{T_{\pi }:\pi \in S_n\right\}$ is as small as possible.

5 pages, unsolved problem