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constructed a family of 2
n−
1
binary sequences of period 2(2
n
1) satisfying
the Welch bound on maximum out of phase correlations [7]. Compared with
the Kasami sequences,
US
sequences have almost the same maximal nontrivial
correlation value, more precisely
R
max
=2
n/
2
+2, but offer much more sequences.
Later in [4], by the Gray map of the optimal Family
−
A
of maximal length
sequences over
Z
4
, Helleseth and Kumar defined Kerdock sequences
Q
(2), com-
prised of 2
n
1) having
R
max
=2
n/
2
+2. In fact,
Kerdock sequences
Q
(2) turn out to be the Kerdock code punctured in two co-
ordinates in cyclic form given by Nechaev [6], which includes
US
sequences as
a subset. Recently aiming at doubling the size of
US
sequences, Tang, Udaya,
and Fan obtained Kerdock sequences from a distinct technique [8]. However, the
correlation distribution of Kerdock sequences is still open problem.
Most recently, Johansen, Helleseth, and Tang studied the correlation distribu-
tion of sequences of period 2(2
n
sequences of period 2(2
n
−
1) over
Z
4
[5]. During this study, the authors
developed some results for determining the correlation distribution of sequences
in Family
−
. Thanks to one result (c.f. Lemma 2 in this paper), we are able
to completely determine the correlation distribution of the binary Kerdock se-
quences using connections between the correlation of binary and quaternary
sequences under the Gray map.
A
2 Preliminaries
The Galois ring
R
=
GR
(4
,n
)with4
n
elements is the Galois extension of
degree
n
over
Z
4
.
R
is a commutative ring having the maximal ideal 2
R
.Let
μ
:
R
←
R
/
(2
R
) be the mod-2 reduction map given by
∈
R
.
Clearly,
μ
(
R
)=
R
/
(2
R
)
=
F
2
n
,where
F
2
n
is the finite field with 2
n
elements.
As a multiplicative group, the units
R
∗
in
R
has a cyclic group
G
C
of order
μ
(
x
)=
x
+2
R
,x
2
n
−
1. Let
β
be a generator of the cyclic group
G
C
, i.e.,
G
C
=
,β
2
n
−
2
1
,β,β
2
,
{
···
}
.
Then
α
=
μ
(
β
) is a primitive root of
F
2
n
.Theset
T
=
G
C
∪{
0
}
is called the
Teichmuller set, which is isomorphic to the finite field
F
2
.
The trace function
Tr
1
(
·
)mapselementsof
R
to
Z
4
, defined as
n
Tr
1
(
x
)=
(
σ
(
x
))
i
,
i
=0
where
σ
(
·
) is the automorphisms of
R
given by:
σ
(
a
+2
b
)=
a
2
+2
b
2
for a,b
∈
R
.
Let
tr
(
·
) denote the analogous trace function over
F
2
n
, defined by:
n
−
1
x
2
i
.
tr
1
(
x
)=
i
=0