Due to wide applications of BCH codes, the determination of their minimumdistance is of great interest. However, this is a very challenging problem forwhich few theoretical results have been reported in the last four decades. Evenfor the narrow-sense primitive BCH codes, which form the most well-studiedsubclass of BCH codes, there are very few theoretical results on the minimumdistance. In this paper, we present new results on the minimum distance ofnarrow-sense primitive BCH codes with special Bose distance. We prove that fora prime power $q$, the $q$-ary narrow-sense primitive BCH code with length$q^m-1$ and Bose distance $q^m-q^{m-1}-q^i-1$, where $\frac{m-2}{2} \le i \lem-\lfloor \frac{m}{3} \rfloor-1$, has minimum distance $q^m-q^{m-1}-q^i-1$.This is achieved by employing the beautiful theory of sets of quadratic forms,symmetric bilinear forms and alternating bilinear forms over finite fields,which can be best described using the framework of association schemes.
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