Superpolynomial worst-case DRE complexity

Construct a Boolean function $f\colon\{0,1\}^n\to\{0,1\}$ whose decomposable randomized encoding complexity satisfies $\DRE(f)=n^{\omega(1)}$.

Background

The paper proves that for every n there exists a Boolean function with DRE complexity Ω(n²), improving the previous general lower bound of Ω(n²/log n). However, the authors observe that this remains exponentially smaller than the best known general upper bound, which is approximately O~(2n/2)\widetilde O(2^{n/2}).

They therefore formulate a conjecture that the worst-case DRE complexity grows faster than every fixed polynomial in n.

References

We expect the worst-case complexity to be much larger and make the following modest conjecture. \begin{conjecture}\label{conj:intro-random} There exists $f\colon0,1n\to0,1$ with $\DRE(f)=n{\omega(1)}$. \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 worst-case bound”