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You misunderstand what "decode" does. As mentioned earlier in the paper: The system has message space $\{0,1\}^{nk}$, which we map bijectively to the cosets of $\Lambda/2\Lambda$ for $\Lambda = \Lambda(G^t)$ via some function encode that is efficient to evaluate and invert. Concretely, letting $S \in\mathbb{Z}^{nk×nk}$ be any basis of $\Lambda$, we can map ...


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