Problems of Information Transmission, Vol. 40, No. 2, 2004, pp. 147–158. Translated from Problemy Peredachi Informatsii, No. 2, 2004, pp. 50–62.
Original Russian Text Copyright
2004 by Zinoviev, Sol´e.
Quaternary Codes and Biphase Sequences
D. V. Zinoviev
and P. Sol´e
Institute for Information Transmission Problems, RAS, Moscow
Laboratoire d’Informatique, Signaux et Systemes de Sophia-Antipolis, CNRS, Nice, France
Received July 15, 2003; in ﬁnal form, January 8, 2004
Abstract—Composing the Carlet map with the inverse Gray map, a new family of cyclic qua-
ternary codes is constructed from 5-cyclic Z
-codes. This new family of codes is inspired by the
quaternary Shanbag–Kumar–Helleseth family, a recent improvement on the Delsarte–Goethals
family. We conjecture that these Z
-codes are not linear. As applications, we construct families
of low-correlation quadriphase and biphase sequences.
Pioneer works [1, 2] and award-winning paper  caused much activity on quaternary codes.
A repercussion for quadriphase sequences was the construction of a large family of sequences based
on the Delsarte–Goethals codes . The corresponding family of codes was improved in  by a
systematic use of the local Weil bound of .
The aim of the present work is to obtain Z
-codes of similar performance but with less linearity.
First, an analog of the construction of  is established over Z
. Moreover, we deﬁne a map Z
. This is a composition of the Carlet map  with the inverse Gray map. It is a
scaled isometry (see ) for the homogeneous weight and the Lee weight. It maps 5-cyclic codes into
cyclic codes; this map is the map ϕ
of . The local Weil bound, combined with a Galois property
of the map Z, is then enough to bound from below the Lee minimum distance of the quaternary
codes obtained as Z-images of these Z
-codes. However, these quaternary codes being potentially
nonlinear, a spectral analysis of the map Z, similar to that of the MSB (most signiﬁcant bit) map
of , is required. From this, we bound the correlation of the attached quadriphase sequences.
The paper is structured as follows. Section 2 gathers some background material on Galois rings.
Section 3 studies the special Gray map that we need. Section 4 studies the so-called canonical
polynomials over Galois rings, which are used in Section 5 to construct some octonary constacyclic
codes. Sections 6 and 7 estimate the correlation of quadriphase and biphase sequences. Finally, in
Section 8, the length of both sequences is doubled in the special case of an even-degree extension
2.1. Galois rings of characteristic 8. Throughout the whole paper, we set n =2
R = GR(8,m) be the Galois ring of characteristic 8 with 8
elements. Let ξ be an element in R
such that ord(ξ)=n − 1. Thus, ξ
<n− 1. Set the Teichm¨uller
Supported in part by the Russian Foundation for Basic Research, project no. 03-01-00098.
2004 MAIK “Nauka/Interperiodica”