Constant-Cost Communication is not Reducible to k-Hamming Distance

TL;DR

This paper demonstrates that some constant-cost communication problems cannot be reduced to k-Hamming Distance, challenging previous assumptions, and connects this to a coding theory question about affine transformations of Hamming distances.

Contribution

It provides the first examples of constant-cost problems not reducible to k-Hamming Distance and establishes a link to a coding-theoretic question about affine functions.

Findings

Certain constant-cost problems are not reducible to k-Hamming Distance.

Existence of f-codes implies f must be affine.

Theoretical connection between communication complexity and coding theory.

Abstract

Every known communication problem whose randomized communication cost is constant (independent of the input size) can be reduced to $k$-Hamming Distance, that is, solved with a constant number of deterministic queries to some $k$-Hamming Distance oracle. We exhibit the first examples of constant-cost problems which cannot be reduced to $k$-Hamming Distance. To prove this separation, we relate it to a natural coding-theoretic question. For $f : \{2, 4, 6\} \to \mathbb{N}$, we say an encoding function $E : \{0, 1\}^n \to \{0, 1\}^m$ is an $f$-code if it transforms Hamming distances according to $\mathrm{dist}(E(x), E(y)) = f(\mathrm{dist}(x, y))$ whenever $f$ is defined. We prove that, if there exist $f$-codes for infinitely many $n$, then $f$ must be affine: $f(4) = (f(2) + f(6))/2$.