2026-07-23T20:56:10Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:27063 2026-07-23T11:36:19Z ddc:004 open_access

Wide Replacement Products Meet Gray Codes: Toward Optimal Small-Bias Sets Cohen, Gil Cohen, Itay small-bias sets bias amplification expansion-amplification gray codes Optimal small-bias sets sit at the crossroads of coding theory and pseudorandomness. Reaching optimal parameters would, in particular, meet the long-standing goal of matching the Gilbert-Varshamov bound for binary codes in the high-distance regime. In a breakthrough, Ta-Shma [Ta-Shma, 2017] constructed near-optimal small-bias sets via the Rozenman-Wigderson expander-walk framework, using the wide-replacement product to maintain s "secure" registers and to route the walk through them. Within this framework, two barriers remain en route to optimal small-bias sets: (i) the cost of maintaining registers and (ii) limitations inherited from spectral-gap bounds for expanders. We overcome the first - arguably the more critical - barrier. Our key technical insight is that registers can be reused even after they are exposed. Using a Gray-code-style reuse schedule, we recycle the same s registers exponentially many times in s, thereby reducing the register-maintenance cost exponentially. This yields the first improvement over Ta-Shma’s construction in nearly a decade - quantitatively modest but an important first step toward a truly optimal construction. The remaining barrier is fairly standard in isolation; the challenge is to overcome it in concert with our register-reuse framework. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Gil Cohen and Itay Cohen 2026 Is Part Of LIPIcs, Volume 383, 41st Computational Complexity Conference (CCC 2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CCC.2026.21 urn:nbn:de:0030-drops-270631 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2026.21 eng https://creativecommons.org/licenses/by/4.0/legalcode