Fully black-box reduction is defined as in Notions of reducibility between crytpographic primitives, O. Reingold et al.
Error-correcting code is used in the black-box abstract way in the sense that any implementation (instance) is eligible as long as it satisfies the definition of a $(\delta, r)$-good code, where the relative distance $\delta$ and the code rate $r$ are constants. The instance is used in the manner that the reduction can only control the inputs and get the outputs (black-box).
We require that the universal hashing gained is strictly universal (collision probability $=1/m$).
The question is as stated in the title and, might be better considered using circuits.
The method used in Limits on the Provable Consequences of One-way Permutations by Impagliazzo and Rudich may also provide some hint but I am not sure.