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The site percolation problem is one of the core topics in statisticalphysics. Evaluation of the percolation threshold, which separates two phases(sometimes described as conducting and insulating), is useful for a range ofproblems from core condensed matter to interdisciplinary application ofstatistical physics in epidemiology or other transportation or connectivityproblems. In this paper with Newman--Ziff fast Monte Carlo algorithm andfinite-size scaling theory the random site percolation thresholds $p_c$ for asquare lattice with complex neighborhoods containing sites from the sixthcoordination zone are computed. Complex neighborhoods are those that containsites from various coordination zones (which are not necessarily compact). Wealso present the source codes of the appropriate procedures (written in C) tobe replaced in original Newman--Ziff code. Similar to results previously foundfor the honeycomb lattice, the percolation thresholds for complex neighborhoodson 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 weighteddistance of sites in complex neighborhoods ($r_i$ and $z_i$ are the distancefrom the central site and the number of sites in the coordination zone $i$,respectively).

Random site percolation thresholds on square lattice for complex neighborhoods containing sites up to the sixth coordination zone