Problems of Information Transmission, Vol. 41, No. 4, 2005, pp. 331–348. Translated from Problemy Peredachi Informatsii, No. 4, 2005, pp. 36–56.
Original Russian Text Copyright
2005 by Zinoviev, Helleseth, Charpin.
On Cosets of Weight 4 of Binary BCH Codes
with Minimum Distance 8 and Exponential Sums
V. A. Zinoviev
, T. Helleseth
, and P. Charpin
Institute for Information Transmission Problems, RAS, Moscow
Department of Informatics, University of Bergen, Norway
INRIA, Domaine de Voluceau-Rocquencourt, France
Received October 21, 2004; in ﬁnal form, August 24, 2005
Abstract—We study coset weight distributions of binary primitive (narrow-sense) BCH codes
of length n =2
(m odd) with minimum distance 8. In the previous paper , 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.
This paper is a natural continuation of the previous paper , where we studied coset weight
distributions of binary primitive (narrow-sense) BCH codes of length n =2
,wherem is odd.
In particular, we described in  the coset weight distributions of such BCH codes for cosets of
weight j =1, 2, 3, 5, 6. Cosets of weight 4, which are not contained in the Reed–Muller code of
order m − 2, are the most diﬃcult case. Finding the number of codewords in such a coset is
connected with ﬁnding the number of solutions to a nonlinear system of equations in four variables
over a ﬁnite ﬁeld GF (2
). The authors of  considered the number of solutions to these systems
of equations from an algebraic-geometry point of view, ﬁnding the distribution of the number of
q-rational points in a family of algebraic curves. They derived upper and lower bounds for the
number of solutions to these systems of equations.
Here we use a number-theoretical approach of exponential sums. Indeed, it is well known that
the number of solutions to any system of equations over a given ﬁeld can be expressed in terms
of the corresponding exponential sum over this ﬁeld. So we deal here with exponential sums over
ﬁnite ﬁelds of characteristic 2. Our main result is an explicit expression for the number of words
of weight 4 in a coset of weight 4, which is expressed in terms of exponential sums of three types:
Kloosterman sums K(a)=
cubic sums C(a, b)=
the sums G(a, b)=
Supported by INRIA-Rocquencourt, France, the Norwegian Research Council, Grant no. 146874/420, and
the Russian Foundation for Basic Research, project no. 03-01-00098.
2005 Pleiades Publishing, Inc.