The paper proves two main results:
There is no complete problem for the class BPP0 of communication problems that have constant-cost randomized public-coin protocols. This means there is no single randomized constant-cost communication problem Q such that all other BPP0 problems can be computed by a constant-cost deterministic protocol with access to an oracle for Q.
The k-Hamming Distance problems form an infinite hierarchy within BPP0. Specifically, the authors show that for infinitely many constants k, the k-Hamming Distance problem cannot be reduced to the (k-1)-Hamming Distance problem under constant-cost reductions. This separates 1-Hamming Distance from Equality (0-Hamming Distance), and also separates 1-Hamming Distance from 2-Hamming Distance.
The authors introduce a new Ramsey-theoretic lower bound technique that can prove separations against arbitrary oracles in BPP0, overcoming limitations of previous techniques that could only handle the Equality oracle. They also show that the k-Hamming Distance problems have certain "dimension-free" properties that imply the Integer Inner Product problems IIPd cannot be reduced to k-Hamming Distance under unbounded-size BPP reductions.
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