BPP problems outside distance-r compositions of k-Hamming Distance

Construct a constant-cost randomized communication problem in BPP that does not reduce to any distance‑r composition of k‑Hamming Distance problems for any fixed r and k (i.e., cannot be computed within the distance‑r composition framework introduced in the paper).

Background

The authors unify many previously known constant-cost problems under the ‘distance‑r composition’ framework, which combines at most r unequal coordinates through a permutation-invariant aggregator. They note that finding BPP problems fundamentally outside this framework would broaden the landscape of constant-cost communication and potentially address longstanding structural questions.