This feels awfully related to the [Subset-sum problem][1]

If $m=2(1,1,\dots,1)$, all the dot products will be the sum of subsets of components of $v$. If two such scalar product coincide, then you have solved an instance of subset-sum problem. 
So, your problem is at least NP-complete, in the number of dimensions.


 [1]:https://en.wikipedia.org/wiki/Subset_sum_problem