Optimal DRE size for the OR function

Determine the optimal size of a decomposable randomized encoding for the n-bit OR function, closing the gap between the known lower and upper bounds; in particular, establish whether the size is necessarily on the order of n log n.

Background

The paper studies information-theoretic decomposable randomized encodings (DREs), whose size is the total bit length of the local encodings. For the n-bit OR function, the classic construction of Feige, Kilian, and Naor gives an O(n log n) upper bound, while the paper proves only a superlinear lower bound in the unrestricted setting.

The authors note that their quantitative superlinear lower bound is numerically weak because the proof relies on Higman’s lemma. They conjecture that the upper bound is optimal, while proving the matching Ω(n log n) lower bound only under a symmetric-support assumption on the encoding of the all-zero input.

References

Unfortunately, even for the $n$-bit OR function, the optimal size is unknown: a simple construction of Feige, Kilian, and Naor gives an $O(n\log n)$ upper bound, whereas the best prior lower bound is only $2n$.

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