  We study the pairing properties of the two-dimensional $t$-$t'$-$t''$-$J$
model, where $t'$ and $t''$ are second and third neighbor hoppings, at a doping
level $x\approx0.1$. Recent studies of the $t$-$t'$-$J$ model find strong
pairing for $t'>0$, associated with electron doping, but an absence of pairing
for $t'<0$ associated with hole doping. This is in contrast to the cuprates,
where the highest transition temperatures appear for hole doping. Model
parameterizations for the cuprates estimate a $t''$ comparable to $t'$, which,
in principle, might fix this discrepancy. However, we find that it does not; we
observe a suppression of pairing for the hole-doped system ($t'<0, t''>0$)
while for the electron-doped system ($t'>0, t''<0$) $d$-wave pairing is robust.
Extended hoppings appear to be insufficient to make the one-band $t$-$t'$-$J$
model capable of describing the pairing in the hole doped system.
