  The Hubbard model and its extensions are important microscopic models for
understanding high- $T_c$ superconductivity in cuprates. In the model with
next-nearest-neighbor hopping $t'$ (the $t'$- Hubbard model), pairing is
strongly influenced by $t'$ . In particular, a recent study on a width-4
cylinder observed quasi-long-rage superconducting order, associated with a
negative $t'$ , which was taken to imply superconductivity in the
two-dimensional (2D) limit. In this work we study more carefully pairing in the
width-4 $t'$-Hubbard model. We show that in this specific system, the pairing
symmetry with $t'<0$ is not the ordinary $d$-wave one would expect in the 2D
limit. Instead we observe a so-called plaquette d-wave pairing. The plaquette
d-wave exists only on a width-4 cylinder, and so is not representative of the
2D limit. We find that a negative $t'$ suppresses the conventional d-wave,
leading to plaquette pairing. In contrast, a different $t''$ coupling acting
diagonally on the plaquettes suppresses plaquette pairing, leading to
conventional $d$-wave pairing.
