Two quantum codes are considered equivalent if they are related by local unitarys and permutations, i.e., if they are related by non-entangling gates.
Given two codes, is there an easy way to tell that they are not equivalent?
I already know one can use quantum weight enumerators, i.e., if the weight enumerators of each code are different then the codes must be inequivalent. However, if the two codes have the same weight enumerators they still could be inequivalent. For example, in https://arxiv.org/pdf/0709.1780.pdf, the authors show that there are 16 inequivalent $[[7,1,3]]$ stabilizer codes but they correspond to only 10 weight enumerators.
Is there an alternative, easy to compute, code invariant?