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Wide Replacement Products Meet Gray Codes: Toward Optimal Small-Bias Sets

Authors Gil Cohen , Itay Cohen

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LIPIcs.CCC.2026.21.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Error-correcting codes
  • Theory of computation → Random walks and Markov chains
  • Theory of computation → Expander graphs and randomness extractors
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • small-bias sets
  • bias amplification
  • expansion-amplification
  • gray codes

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Abstract

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.

Cite As Get BibTex

Gil Cohen and Itay Cohen. Wide Replacement Products Meet Gray Codes: Toward Optimal Small-Bias Sets. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 21:1-21:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CCC.2026.21

Author Details

Gil Cohen
  • Tel Aviv University, Israel
Itay Cohen
  • Tel Aviv University, Israel

Funding

  • Cohen, Gil: Supported by ERC starting grant 949499 and by the Israel Science Foundation grant 2989/24.
  • Cohen, Itay: Supported by ERC starting grant 949499.

References

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