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I know that Mazur–Ulam theorem holds for normed linear spaces over $\mathbb{R}$. I wanted to know whether under some "weak" conditions on the map $f$, can we have Mazur-Ulam theorem for a vector space over ${\mathbb{F}_2}$?.

I apologize for being vague about the "weak" condition.

More generally, I am interested in characterizing the isometries of $\mathbb{F}_2 ^n$ with Hamming distance as the metric. Clearly permutation matrices and translations are isometries. But I wanted to know if there are isometries other than these?

Thank you,

Iso

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    @JyrkiLahtonen Perhaps you're now in more of a position to promote your comments to an answer?2013-06-07

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