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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 =
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
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