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 $0 < w' \leq w$).
I understand that this problem is hard if the code is unknown. But if the code was know (i.e. we know its construction), for example a Goppa code where we know the support of the code and the polynomial that generates that code. Is there some class of codes where it is easy to generate several words, of certain class of codes, with the same weight and the same syndrome and the problem above is still hard?