Communicated by J. Bierbrauer.
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Classical Bose–Chaudhuri–Hocquenghem (BCH) codes are an important class of cyclic codes in terms of their error-correcting capability. BCH codes that contain their Euclidean (or Hermitian) dual codes have been widely used in constructing quantum stabilizer codes. In this paper, necessary and sufficient conditions on Euclidean and Hermitian dual-containing primitive BCH codes with the maximum designed distance are shown, and necessary conditions on non-narrow-sense primitive BCH codes that do not contain their Euclidean or Hermitian dual codes are given. The results of Euclidean and Hermitian dual-containing primitive BCH codes in (Aly et al. IEEE Trans Inf Theory 53(3):1183–1188, 2007) are extended. Moreover, the dimensions of some non-narrow-sense primitive BCH codes are determined and some quantum codes are constructed.