Problems of Information Transmission, Vol. 40, No. 2, 2004, pp. 118–134. Translated from Problemy Peredachi Informatsii, No. 2, 2004, pp. 19–36.
Original Russian Text Copyright
2004 by Zinoviev, Helleseth.
On Weight Distributions of Shifts
of Goethals-like Codes
V. A. Zinoviev
and T. Helleseth
Institute for Information Transmission Problems, RAS, Moscow
Department of Informatics, University of Bergen, Norway
Received December 10, 2002; in ﬁnal form, February 26, 2004
Abstract—We study weight distributions of shifts of codes from a well-known family: the
3-error-correcting binary nonlinear Goethals-like codes of length n =2
,wherem ≥ 6iseven.
These codes have covering radius ρ = 6. We know the weight distribution of any shift of weight
i = 1, 2, 3, 5, or 6. For a shift of weight 4, the weight distribution is uniquely deﬁned by the
number of leaders in this shift, i.e., the number of vectors of weight 4. We also consider the
weight distribution of shifts of codes with minimum distance 7 obtained by deleting any one
position of a Goethals-like code of length n.
This paper is a natural continuation of [1, 2]. We examine here weight distributions of shifts
of binary nonlinear Goethals-like codes (which we denote here by G). By a Goethals-like code,
we understand any binary distance-invariant code which has the same weight distribution as the
original Goethals code [3, 4]. Here we describe weight distributions of shifts of a code G.Weknow
the weight distributions of shifts of weights 1, 2, 3, 5, and 6. The weight distribution of a shift
of weight 4 is uniquely determined by the number of words of weight 4. We know the number
of diﬀerent common shifts of weights 1, 2, and 3, and we obtain lower bounds for the number of
diﬀerent common shifts of weights 4, 5, and 6.
In this paper we also consider the weight distribution of shifts of codes with minimum distance 7
obtained by deleting any one position of a Goethals-like code G (such codes are denoted here
). We know weight distributions of shifts of weights 1, 2, and 5. Weight distributions of shifts
of weights 3 and 4 are uniquely determined by the number of words of weight 4. For codes G
also know the number of diﬀerent common shifts of weights 1, 2, and 3. For the number of diﬀerent
common shifts of weights 4 and 5, we ﬁnd lower bounds.
The paper is organized as follows. In Section 2, following , we present main equations (A. i),
which give as their solutions the number of words of weight i in shifts of G of weight i.Forthese
reasons, we need some results from [6–11] concerning uniformly packed codes. In Section 3 we
consider shift weight distributions of codes G. Shift weight distributions of codes G
in Section 4. In Section 5 some results are given on shift weight distributions of Z
codes considered in .
Supported in part by the Norwegian Research Council, Grant no. 146874/420, and the Russian Foundation
for Basic Research, project no. 03-01-00098.
2004 MAIK “Nauka/Interperiodica”