  We devise a new realization of the surface code on a rectangular lattice of
qubits utilizing single-qubit and nearest-neighbor two-qubit Pauli measurements
and three auxiliary qubits per plaquette. This realization gains substantial
advantages over prior pairwise measurement-based realizations of the surface
code. It has a short operation period of 4 steps and our performance analysis
for a standard circuit noise model yields a high fault-tolerance threshold of
approximately $0.66\% $. The syndrome extraction circuits avoid bidirectional
hook errors, so we can achieve full code distance by choosing appropriate
boundary conditions. We also construct variants of the syndrome extraction
circuits that entirely prevent hook errors, at the cost of larger circuit
depth. This achieves full distance regardless of boundary conditions, with only
a modest decrease in the threshold. Furthermore, we propose an efficient
strategy for dealing with dead components (qubits and measurements) in our
surface code realization, which can be adopted more generally for other surface
code realizations. This new surface code realization is highly optimized for
Majorana-based hardware, accounting for constraints imposed by layouts and the
implementation of measurements, making it competitive with the recently
proposed Floquet codes.
