From 1d4784a0ad8e4f6ef8b46a57d3ac59c2a422a6ff Mon Sep 17 00:00:00 2001 From: Taksh Date: Wed, 16 Sep 2026 18:06:04 +0530 Subject: [PATCH] =?UTF-8?q?Record=20R=5F3=3D10=20and=20Wel2019=20R=5F8?= =?UTF-8?q?=E2=89=A41008=20on=2044a.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- constants/44a.md | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/constants/44a.md b/constants/44a.md index 2a506b63..5a0eef06 100644 --- a/constants/44a.md +++ b/constants/44a.md @@ -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