Ω(n log n) lower bound for unrestricted OR DREs

Prove that every perfect decomposable randomized encoding of the n-bit OR function has size Ω(n log n), without imposing the symmetric-support assumption used in the paper’s structured lower bound.

Background

The paper establishes an Ω(n log n) lower bound for perfect DREs of OR_n only when the support of the encoding on the all-zero input is closed under coordinate permutations. This condition captures the classic Feige–Kilian–Naor construction and related structured constructions.

For general perfect DREs of OR_n, the paper proves only the superlinear bound DRE(OR_n)=ω(n). The authors explicitly conjecture that the unrestricted lower bound should match the O(n log n) construction.

References

We nevertheless conjecture the optimal bound below and view the structured lower bound in \Cref{thm:intro-permutation-closed} as supporting evidence. \begin{conjecture}\label{conj:or-optimal} $\DRE(OR_n)=\Omega(n\log n)$. \end{conjecture}

— Improved lower bounds for decomposable randomized encoding  (2609.18020 - Holmgren et al., 16 Sep 2026) in Section 1, subsection “Discussions and AI disclosure,” paragraph “On the input length”