Are there any known (non-trivial) randomized communication complexity lower bounds for natural gap problems in which the 1-inputs are linearly far from the 0-inputs? That is, partial functions $f:\{0,1\}^{2n} \to \{0,1,*\}$ such that the Hamming distance between every $(x,y) \in f^{-1}(1)$ and $(x',y') \in f^{-1}(0)$ is linear -- and that $f$ requires randomized protocols to communicate (say) $\Omega(\sqrt{n})$ bits?(For instance, the Gap-Hamming-Distance problem has $2\sqrt{n}$ distance, whereas I'm looking for $\Omega(n)$ distance; where $GHD(x,y) = 1$ if $HD(x,y) \ge n/2 + \sqrt{n}$ and $GHD(x,y) = 1$ if $HD(x,y) \le n/2 - \sqrt{n}$.)Edit: As pointed out by Igor, any communication complexity predicate can be made into a problem with linear-distance by requiring the inputs to be encoded by a good code. What interests me though, is whether there exist problems in the literature, in which the linear-distance occurs in a natural way (as the distance in the Gap-Hamming-Distance problem).
Communication complexity problems with linear distance
reference request;communication complexity