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The log rank conjecture states that there is some universal constant $c > 0$ so that

$$CC(f) = O(\log^c \text{rk}\,(M_f))$$

where $f : X \times Y \to \{0, 1\}$ is a boolean function, $CC$ denotes the deterministic communication complexity, $M_f$ is the $|X| \times |Y|$ binary matrix associated to $f$, and $\text{rk}\,(M_f)$ is the rank of $M_f$ calculated over the reals. In the literature, it mentions that this $c$, if it exists, is at least $\log_3 6 \approx 1.631\ldots$, and attributes this to an unpublished result of Kushilevitz in 1994. Does anyone know what example Kushilevitz uses to achieve this lower bound? I think it would be interesting to see what the worst case might be and it might give insight into how to solve the problem in general.

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The function is described in a footnote in Nisan and Wigderson's paper On rank vs. communication complexity. It is

$$ E(z_1 \dots z_6) = \sum_i z_i - \sum_{ij} z_{i}z_{j} + \\z_1z_3z_4 + z_1z_2z_5 + z_1z_4z_5 + z_2z_3z_4 + z_2z_3z_5 + \\ z_1z_2z_6 + z_1z_3z_6 + z_2z_4z_6 + z_3z_5z_6 + z_4z_5z_6. $$

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  • $\begingroup$ I thought it was improved by Mika Goos. $\endgroup$ – 1.. Mar 22 '17 at 9:56
  • $\begingroup$ @Turbo That's perfectly possible. This answer doesn't update itself every time an improvement is discovered. You could add a new answer with the improvement. $\endgroup$ – Yuval Filmus Mar 22 '17 at 12:16
  • $\begingroup$ @Turbo It seems the answer is in the paper Deterministic Communication vs. Partition Number -- will give it a read $\endgroup$ – MCT Mar 22 '17 at 13:54

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