  Quantum data-syndrome (QDS) codes are a class of quantum error-correcting
codes that protect against errors both on the data qubits and on the syndrome
itself via redundant measurement of stabilizer group elements. One way to
define a QDS code is to choose a syndrome measurement code, a classical block
code that encodes the syndrome of the underlying quantum code by defining
additional stabilizer measurements. We propose the use of primitive
narrow-sense BCH codes as syndrome measurement codes. We show that these codes
asymptotically require $O(t\log\ell)$ extra measurements, where $\ell$ is the
number of stabilizer generators of the quantum code and $t$ is the number of
errors corrected by the BCH code. Previously, the best known general method of
constructing QDS codes out of quantum codes requires $O(t^3\log\ell)$ extra
measurements. As the number of additional syndrome measurements is a reasonable
metric for the amount of additional time a general QDS code requires, we
conclude that our construction protects against the same number of syndrome
errors with significantly less time overhead.
