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The Minimum Distance of Some Narrow-Sense Primitive BCH Codes
Due to wide applications of BCH codes, the determination of their minimum
distance is of great interest. However, this is a very challenging problem for
which few theoretical results have been reported in the last four decades. Even
for the narrow-sense primitive BCH codes, which form the most well-studied
subclass of BCH codes, there are very few theoretical results on the minimum
distance. In this paper, we present new results on the minimum distance of
narrow-sense primitive BCH codes with special Bose distance. We prove that for
a prime power , the -ary narrow-sense primitive BCH code with length
and Bose distance , where , has minimum distance .
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.Comment: 40 page