Some $\ZZ_3^n$-equivariant triangulations of $\CP^n$

In 1983, Banchoff and Kuhnel constructed a minimal triangulation of $\CP^2$ with 9 vertices. $\CP^3$ was first triangulated by Bagchi and Datta in 2012 with 18 vertices. Known lower bound on number of vertices of a triangulation of $\CP^n$ is $1 + \frac{(n + 1)^2}{2}$ for $n \geq 3$. We give explicit construction of some triangulations of complex projective space $\CP^n$ with $\frac{4^{n+1}-1}{3}$ vertices for all $n$. No explicit triangulation of $\CP^n$ is known for $n \geq 4$.