  The Shor fault-tolerant error correction (FTEC) scheme uses transversal gates
and ancilla qubits prepared in the cat state in syndrome extraction circuits to
prevent propagation of errors caused by gate faults. For a stabilizer code of
distance $d$ that can correct up to $t=\lfloor(d-1)/2\rfloor$ errors, the
traditional Shor scheme handles ancilla preparation and measurement faults by
performing syndrome measurements until the syndromes are repeated $t+1$ times
in a row; in the worst-case scenario, $(t+1)^2$ rounds of measurements are
required. In this work, we improve the Shor FTEC scheme using an adaptive
syndrome measurement technique. The syndrome for error correction is determined
based on information from the differences of syndromes obtained from
consecutive rounds. Our protocols that satisfy the strong and the weak FTEC
conditions require no more than $(t+3)^2/4-1$ rounds and $(t+3)^2/4-2$ rounds,
respectively, and are applicable to any stabilizer code. Our simulations of
FTEC protocols with the adaptive schemes on hexagonal color codes of small
distances verify that our protocols preserve the code distance, can increase
the pseudothreshold, and can decrease the average number of rounds compared to
the traditional Shor scheme. We also find that for the code of distance $d$,
our FTEC protocols with the adaptive schemes require no more than $d$ rounds on
average.
