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 extra measurements, where is the number of stabilizer generators of the quantum code and is the number of syndrome measurement errors corrected by the BCH code. Previously, the best known general method of constructing QDS codes out of quantum codes required 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.