  The site percolation problem is one of the core topics in statistical
physics. Evaluation of the percolation threshold, which separates two phases
(sometimes described as conducting and insulating), is useful for a range of
problems from core condensed matter to interdisciplinary application of
statistical physics in epidemiology or other transportation or connectivity
problems. In this paper with Newman--Ziff fast Monte Carlo algorithm and
finite-size scaling theory the random site percolation thresholds $p_c$ for a
square lattice with complex neighborhoods containing sites from the sixth
coordination zone are computed. Complex neighborhoods are those that contain
sites from various coordination zones (which are not necessarily compact). We
also present the source codes of the appropriate procedures (written in C) to
be replaced in original Newman--Ziff code. Similar to results previously found
for the honeycomb lattice, the percolation thresholds for complex neighborhoods
on a square lattice follow the power law $p_c(\zeta)\propto\zeta^{-\gamma_2}$
with $\gamma_2=0.5454(60)$, where $\zeta=\sum_i z_i r_i$ is the weighted
distance of sites in complex neighborhoods ($r_i$ and $z_i$ are the distance
from the central site and the number of sites in the coordination zone $i$,
respectively).
