If the covering radius of a code is $r$, then for every vector $v$, there is a codeword with distance at most $r$ from $v$. One can bound the covering radius by the distance of the dual code from this paper. The proof of this uses the MacWilliams identity, which is related to Delsarte's association schemes.
Can this proof be made algorithmic? Given $v$, is there a way to identify a nearby codeword at distance at most $r$ from $v$?
If so, this provably de-quantizes the Max-XORSAT variant of DQI.
A starting question is whether there is a classical algorithmic interpretation of the MacWilliams identity.