Minimum weights needed to derandomize weight assignment by isolation lemma

Under isolation lemma if you have a graph with $2n$ vertices and $m$ edges an isolating weight assignment can be obtained by assigning edges weights randomly from $\{1,2,\dots,2m-1,2m\}$. A weight assignment is isolating if there is an unique perfect matching with minimum sum weight.

Is $\Omega(m)$ weight necessary or would it be possible that there is a (deterministic) scheme that can get the job done in $O(\sqrt m)$?

Note that the best weights we know uses $O(2^m)$.