I am studying the NP-Problem of the codes Syndrome Decoding. The formulation is show below.
Input: a binary matrix $H$ of dimension $r \times n$ and a bit string $S$ of length $r$.
Property: there exists a set of $w'$ columns of $H$ adding to $S$ (with $0 < w' 0 \leq w$).
I understand that this problem is hard if the code is unknow. But if the code was know, for example if it was a Goppa code and we know the support of the code and the polynomial that generates that code, There is some algorithm to find several words with the same weight and the same syndomre?