Reed-Solomon codes, a type of BCH codes, are widely employed in communicationsystems, storage devices and consumer electronics. This fact demonstrates theimportance of BCH codes -- a family of cyclic codes -- in practice. In theory,BCH codes are among the best cyclic codes in terms of their error-correctingcapability. A subclass of BCH codes are the narrow-sense primitive BCH codes.However, the dimension and minimum distance of these codes are not known ingeneral. The objective of this paper is to determine the dimension and minimumdistances of two classes of narrow-sense primitive BCH codes with designdistances $\delta=(q-1)q^{m-1}-1-q^{\lfloor (m-1)/2\rfloor}$ and$\delta=(q-1)q^{m-1}-1-q^{\lfloor (m+1)/2\rfloor}$. The weight distributions ofsome of these BCH codes are also reported. As will be seen, the two classes ofBCH codes are sometimes optimal and sometimes among the best linear codesknown.
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