On binary primitive BCH codes with minimum distance 8 and exponential sums

The exact expressions for the number of codewords of weight 4 in the cosets of weight 4 of binary primitive BCH codes of length n = 2^m (m even) with minimum distance 8 is given in terms of several exponential sums, including cubic sums and Kloosterman sums. This provides a bound on the number of codewords of weight 4 in the cosets of weight 4 and also some limitations for possible values of Kloosterman sums over GF(2^m), (m even).