Let $C$ be a code over the finite field $GF(2)$ with generator matrix $G$ and parity matrix $H$. Let $e+C=C'$ be a coset of code $C$. Let $S$ be a non-singular matrix and $H'=H\times S$. Finally, let $C''\subset C'$. Is there any efficient way of finding any other element of the coset $C'$ by only using $C''$ and $H'$ (with appropriate parameters for the code $C$)?
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