  We perform a comprehensive study of the electron-doped $t-t'-J$ model on
cylinders with Density Matrix Renormalization Group (DMRG). We adopt both
periodic and anti-periodic boundary conditions along the circumference
direction to explore the finite size effect. We study doping levels of $1/6$,
$1/8$, and $1/12$ which represent the most interesting region in the phase
diagram of electron-doped cuprates. We find that for width-4 and 6 systems, the
ground state for fixed doping switches between anti-ferromagnetic Neel state
and stripe state under different boundary conditions and with system widths,
indicating the presence of large finite size effect in the $t-t'-J$ model. We
also have a careful analysis of the $d$-wave pairing correlations which also
changes quantitatively with boundary conditions and widths of the system.
However, the pairing correlations are enhanced when the system becomes wider
for all dopings, suggesting the existence of possible long-ranged
superconducting order in the thermodynamic limit. The width-8 results are found
to be dependent on the starting state in the DMRG calculation for the kept
states we can reach. For width-8 system only Neel (stripe) state can be
stabilized in DMRG calculation for $1/12$ ($1/6$) doping, while both stripe and
Neel states are stable in the DMRG sweep for $1/8$ doping, regardless of the
boundary conditions. These results indicate that $1/8$ doping is likely to lie
in the boundary of a phase transition between the Neel phase with lower doping
and the stripe phase with higher doping, consistent with the previous study.
The sensitivity of ground state on boundary conditions and size observed in
this work is similar to that in the $t'$- Hubbard model.
