  An attractive feature of BCH codes is that one can infer valuable information
from their design parameters (length, size of the finite field, and designed
distance), such as bounds on the minimum distance and dimension of the code. In
this paper, it is shown that one can also deduce from the design parameters
whether or not a primitive, narrow-sense BCH contains its Euclidean or
Hermitian dual code. This information is invaluable in the construction of
quantum BCH codes. A new proof is provided for the dimension of BCH codes with
small designed distance, and simple bounds on the minimum distance of such
codes and their duals are derived as a consequence. These results allow us to
derive the parameters of two families of primitive quantum BCH codes as a
function of their design parameters.
