  The surface code is one of the most successful approaches to topological
quantum error-correction. It boasts the smallest known syndrome extraction
circuits and correspondingly largest thresholds. Defect-based logical encodings
of a new variety called twists have made it possible to implement the full
Clifford group without state distillation. Here we investigate a patch-based
encoding involving a modified twist. In our modified formulation, the resulting
codes, called triangle codes for the shape of their planar layout, have only
weight-four checks and relatively simple syndrome extraction circuits that
maintain a high, near surface-code-level threshold. They also use 25% fewer
physical qubits per logical qubit than the surface code. Moreover, benefiting
from the twist, we can implement all Clifford gates by lattice surgery without
the need for state distillation. By a surgical transformation to the surface
code, we also develop a scheme of doing all Clifford gates on surface code
patches in an atypical planar layout, though with less qubit efficiency than
the triangle code. Finally, we remark that logical qubits encoded in triangle
codes are naturally amenable to logical tomography, and the smallest triangle
code can demonstrate high-pseudothreshold fault-tolerance to depolarizing noise
using just 13 physical qubits.
