  Percolation on a five-dimensional simple hypercubic (sc(5)) lattice with
extended neighborhoods is investigated by means of extensive Monte Carlo
simulations, using an effective single-cluster growth algorithm. The critical
exponents, including $\tau$ and $\Omega$, the asymptotic behavior of the
threshold $p_c$ and its dependence on coordination number $z$ are investigated.
Using the bond and site percolation thresholds $p_c = 0.11817145(3)$ and
$0.14079633(4)$ respectively given by Mertens and Moore [Phys. Rev. E 98,
022120 (2018)], we find critical exponents of $\tau = 2.4177(3)$, $\Omega =
0.27(2)$ through a self-consistent process. The value of $\tau$ compares
favorably with a recent five-loop renormalization predictions $2.4175(2)$ by
Borinsky et al. [Phys. Rev. D 103, 116024 (2021)], the value 2.4180(6) that
follows from the work of Zhang et al. [Physica A 580, 126124 (2021)], and the
measurement of $2.419(1)$ by Mertens and Moore. We also confirmed the bond
threshold, finding $p_c = 0.11817150(5)$. sc(5) lattices with extended
neighborhoods up to 7th nearest neighbors are studied for both bond and site
percolation. Employing the values of $\tau$ and $\Omega$ mentioned above,
thresholds are found to high precision. For bond percolation, the asymptotic
value of $zp_c$ tends to Bethe-lattice behavior ($z p_c \sim 1$), and the
finite-$z$ correction is found to be consistent with both and $zp_{c} - 1 \sim
a_1 z^{-0.88}$ and $zp_{c} - 1 \sim a_0(3 + \ln z)/z$. For site percolation,
the asymptotic analysis is close to the predicted behavior $zp_c \sim 32\eta_c
= 1.742(2)$ for large $z$, where $\eta_c = 0.05443(7)$ is the continuum
percolation threshold of five-dimensional hyperspheres given by Torquato and
Jiao [J. Chem. Phys 137, 074106 (2015)]; finite-$z$ corrections are accounted
for by taking $p_c \approx c/(z + b)$ with $c=1.722(7)$ and $b=1$.
