  The behavior of the percolation threshold and the jamming coverage for
isotropic random sequential adsorption samples has been studied by means of
numerical simulations. A parallel algorithm that is very efficient in terms of
its speed and memory usage has been developed and applied to the model
involving large linear $k$-mers on a square lattice with periodic boundary
conditions. We have obtained the percolation thresholds and jamming
concentrations for lengths of $k$-mers up to $2^{17}$. New large $k$ regime of
the percolation threshold behavior has been identified. The structure of the
percolating and jamming states has been investigated. The theorem of
G.~Kondrat, Z.~Koza, and P.~Brzeski [Phys. Rev. E 96, 022154 (2017)] has been
generalized to the case of periodic boundary conditions. We have proved that
any cluster at jamming is percolating cluster and that percolation occurs
before jamming.
