On Cosets of Weight 4 of Binary BCH Codes with Minimum Distance 8 and Exponential Sums

We study coset weight distributions of binary primitive (narrow-sense) BCH codes of length $n=2^m$ ($m$ odd) with minimum distance 8. In the previous paper [1], we described coset weight distributions of such codes for cosets of weight $j=1,2,3,5,6,$. Here we obtain exact expressions for the number of codewords of weight 4 in terms of exponential sums of three types, in particular, cubic sums and Kloosterman sums. This allows us to find the coset distribution of binary primitive (narrow-sense) BCH codes with minimum distance 8 and also to obtain some new results on Kloosterman sums over finite fields of characteristic 2.