  We consider Majorana fermion stabilizer codes with small number of modes and
distance. We give an upper bound on the number of logical qubits for distance
$4$ codes, and we construct Majorana fermion codes similar to the classical
Hamming code that saturate this bound. We perform numerical studies and find
other distance $4$ and $6$ codes that we conjecture have the largest possible
number of logical qubits for the given number of physical Majorana modes. Some
of these codes have more logical qubits than any Majorana fermion code derived
from a qubit stabilizer code.
