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Non-reduction of Cheung’s problem to distance‑r compositions

Ascertain whether the constant-cost problem constructed by Tsun‑Ming Cheung (2024) does or does not reduce to the distance‑r composition problems introduced in this paper.

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Background

The authors note a new constant-cost problem by Cheung that generalizes k-Hamming Distance and appears to use randomness in a different way. They remark that it does not seem to reduce to their distance‑r composition framework, but this has not yet been proven. Establishing the relationship would clarify the scope of the proposed general class of constant‑cost problems.

References

After we prepared the current manuscript, Tsun-Ming Cheung [Che24] independently constructed a new constant-cost problem which generalizes $k$-Hamming Distance in a way that does not seem to reduce to our “distance-$r$ composition” problems (but this is not yet proved), and which uses randomness in a new way.

— Constant-Cost Communication is not Reducible to k-Hamming Distance (2407.20204 - Fang et al., 29 Jul 2024) in Remark 1 (remark:ben-problem), Section: Constant-Cost Communication: the Story so Far and Farther