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Is linear sketching over F2 optimal for one-way XOR communication?

Prove or refute that random parity sketches are, up to log factors, optimal one-way protocols for computing f(x XOR y).

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1{
2 "domain": "computer-science",
3 "topics": [
4 "algorithms-complexity"
5 ],
6 "type": "proof",
7 "source": {
8 "key": "sublinear",
9 "name": "Sublinear.info",
10 "url": "https://sublinear.info/78",
11 "id": "78",
12 "curator": "Sublinear.info editors",
13 "poser": "Grigory Yaroslavtsev",
14 "status": "open",
15 "as_of": "2026-10-20",
16 "catalog": "0521c19bbd38"
17 }
18}