Partition Number of a Matrix
The log of the partition number is a lower bound on the deterministic communication complexity and the square of the log of the partition number is an upper bound. In other words, if $CC$ is the communication complexity, then we know that $\log_2 p \leq CC \leq (\log_2 p)^2$: this is Theorem 2.11. in the Kushilevitz-Nisan monograph. It is an open problem whether $CC = \Theta(\log p)$, and the biggest known gap is a factor $2$.

So $p &lt; r^{\log^d r}$ for a constant $d$ implies that the communication complexity is at most $\log^{2d+2} r$. In other words there exists such a constant $d$ if and only if the log rank conjecture is true. I believe the best we know is $p &lt; r^{C\sqrt{r}}$ for a constant $C$, by a recent result of Lovett.