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On the Correlation Distribution of Kerdock
Sequences
Xiaohu Tang
1
, Tor Helleseth
2
, and Aina Johansen
2
1
The Provincial Key Lab of Information Coding and Transmission, Institute of
Mobile Communications
Southwest Jiaotong University, Chengdu 610031, Sichuan, China
xhutang@ieee.org
2
The Selmer Center
Department of Informatics, University of Bergen
P.O. Box 7800, N-5020 Bergen, Norway
tor.helleseth@ii.uib.no, Aina.Johansen@ii.uib.no
Abstract.
For any even integer
n
, the binary Kerdock sequences of pe-
riod 2(2
n
−
1) are optimal with respect to the well-known Welch bound.
Until now the correlation distribution of this family has not been known.
In this paper we completely determine its correlation distribution using
connections between the correlation properties of binary sequences and
quaternary sequences under the Gray map.
Keywords:
Binary sequences, quaternary sequences, Gray map, corre-
lation distribution.
1
Introduction
CDMA systems provide several users with simultaneous access to the full chan-
nel bandwidth by assigning a unique sequence from a sequence family to each
user [2]. In order to distinguish each user and minimize interference due to com-
petition and simultaneous trac across the channel, the sequence family must
have as low maximal correlation as possible. Nevertheless, the maximal correla-
tion is subjected to the limitation of some theoretical bounds, for example the
Welch and Sidelnikov bounds. The Welch bound states that for a family of se-
quences of period
N
, the maximal value of cross correlations and out-of-phase
autocorrelations (usually refer to maximal nontrivial correlation value
R
max
), is
lower bounded by
√
N
approximately.
The well-known Kasami sequence is the first optimal family of binary sequence
with respect to the Welch bound, consisting of 2
n/
2
sequences of period 2
n
−
1,
whose
R
max
is lower bounded by 2
n/
2
+ 1, for any even integer
n
. In 1996, by
employing interleaved maximal length sequences over
Z
4
, Udaya and Siddiqi
