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

Authors:
V. A. Zinoviev;T. Helleseth;P. Charpin
Affiliations:
Institute for Information Transmission Problems, RAS, Moscow, Russia;Department of Informatics, University of Bergen, Norway;INRIA, Domaine de Voluceau-Rocquencourt, France
Venue:
Problems of Information Transmission
Year:
2005

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Abstract

We study coset weight distributions of binary primitive (narrow-sense) BCH codes of length n = 2m (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.