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Record R_3=10 and Wel2019 R_8≤1008 on 44a #191

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Record R_3=10 and Wel2019 R_8≤1008 on 44a #191

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Expand Up @@ -41,10 +41,14 @@ The finiteness of $C_{44}$ (i.e. existence of a universal constant $C$) follows

* Best currently-known interval: $\frac{3}{2}\le C_{44}\le 4.394.$
* For monotone Boolean functions, Wellens proves a smaller constant ($1.325$) multiplying $2^{\deg(f)}$.
* $R_3 = 10$. Tarannikov–Kirienko [[TK2000](#TK2000)] prove $p'(4)=p(4)=10$ in resilient-function language; via the standard translation $R_d = p(d+1)$ (cf. [[KV2024](#KV2024)]), this is $R_3 = 10$, attained by the CHS function $\Xi_3$.
* [Wel2019], Table 2, gives $R_8 \le 1008$ (from the bound $W(f) \le 3.9375$ at degree $8$, hence $R_8 \le 256 \cdot 3.9375 = 1008 < 8 \cdot 2^7$).

## References

- **[NS1994]** Nisan, N. and Szegedy, M. *On the degree of Boolean functions as real polynomials.* Computational Complexity 4 (1994), 301–313. DOI: 10.1007/BF01263419.
- **[TK2000]** Tarannikov, Y. and Kirienko, D. *Spectral analysis of high order correlation immune functions.* IACR ePrint 2000/050. https://eprint.iacr.org/2000/050
- **[CHS2020]** Chiarelli, J.; Hatami, P.; Saks, M. *An Asymptotically Tight Bound on the Number of Relevant Variables in a Bounded Degree Boolean Function.* Combinatorica 40 (2020), 237–244. Preprint: https://arxiv.org/abs/1801.08564
- **[Wel2019]** Wellens, J. *A tighter bound on the number of relevant variables in a bounded degree Boolean function.* Preprint: https://arxiv.org/abs/1903.08214
- **[Wel2022]** Wellens, J. *Relationships between the number of inputs and other complexity measures of Boolean functions.* Discrete Analysis 2022:19. Preprint: https://arxiv.org/abs/2005.00566
- **[KV2024]** Krotov, D. S. and Valyuzhenich, A. *On degree-3 and $(n-4)$-correlation-immune perfect colorings of $n$-cubes.* Discrete Mathematics 347 (2024), 114138. Preprint: https://arxiv.org/abs/2311.05566