On cosets of weight 4 of binary BCH codes of length 2^m (m odd) with minimal distance 8 and exponential sums

Pascale Charpin, Tor Helleseth and Victor Zinoviev

INRIA (projet CODES), University of Bergen, and IPIT (Moscow)
France, Norway, Russia


Problems of Information Transmission, Vol. 41, n. 4, pp. 331-348, 2005.


We study the coset weight distributions of binary primitive (in narrow sence) extended BCH codes of length $n=2^m$ ($m$ odd) with minimal distance $8$. In a previous paper, we described coset weight distributions of such codes for cosets of weight $j=1,2,3,5,6$. For the case of cosets of weight $4$ we could not find the explicit expressions for the number of codewords of weight $4$ and for the number of such cosets. Here we obtain the exact expression 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 obtain some new results on Kloosterman sums which wil be published in forthcoming papers.

Keywords : Binary BCH code, coset, coset weight distribution, system of equations over a finite field, exponential sum, cubic sum, Kloosterman sum.