Do equal distance distributions imply equivalence? Let $A$ and $B$ be two binary $(n,M,d)$ codes. We define $a_i = \#\{(w_1,w_2) \in A^2:\:d(w_1,w_2) = i\}$, and same for $b_i$. If $a_i = b_i$ for all $i$, can one deduct that $A$ and $B$ are equivalent, i.e. equal up to permutation of positions and permutation of letters in a fixed position?

As an other example I proffer families of codes like Delsarte-Goethals codes. These are Grey images of codes that are linear over $\mathbf{Z}_4$, so they are not linear binary codes (even though they are binary).
