  We report results of large scale ground state density matrix renormalization
group(DMRG) calculations on $t$-$t'$-$J$ cylinders with circumferences 6 and 8.
We determine a rough phase diagram which appears to approximate the 2D system.
While for many properties, positive and negative $t'$ values ($t'/t = \pm 0.2$)
appear to correspond to electron and hole doped cuprate systems, respectively,
the behavior of superconductivity itself shows an inconsistency between the
model and the materials. The $t'<0$ (hole doped) region shows
antiferromagnetism limited to very low doping, stripes more generally, and the
familiar Fermi-surface of the hole doped cuprates. However, we find $t'<0$
strongly suppresses superconductivity. The $t'>0$ (electron doped) region shows
the expected circular Fermi pocket of holes around the $(\pi,\pi)$ point and a
broad low-doped region of coexisting antiferromagnetism and $d$-wave pairing
with a strong triplet $p$ component at wave-vector $(\pi,\pi)$ induced by the
antiferromagnetism and $d$-wave pairing. The pairing for the electron low-doped
system with $t'>0$ is strong and unambiguous in the DMRG simulations. At larger
doping another broad region with stripes in addition to weaker $d$-wave pairing
and striped $p$-wave pairing appears. In a small doping region near $x=0.08$
for $t'\sim-0.2$, we find a new type of stripe involving unpaired holes located
predominantly on chains spaced three lattice spacings apart. The undoped
two-leg ladder regions in between mimic the short-ranged spin correlations seen
in two-leg Heisenberg ladders.
