## 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
` Pascale.Charpin@inria.fr`

` torh@ii.uib.no`

`zinov@iitp.ru `

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

### Abstract

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.