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Relation between positive roots of $E_8$ and $\mathbb{F}_2^8 /{0}$

Let $L$ denote the root lattice of $E_8$. The group $W(E_8)$ acts (linearly) on $L/2L \cong \mathbb{F}_2^8$, and hence as a permutation of $2^8=256$ elements. This permutation action has the following ...
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