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A. Correlation Function
Let
a
=(
a
(
t
)
,
0
t<L
) be two sequences over
Z
N
.
The correlation function between the sequences
a
and
b
at shift 0
≤
t<L
)and
b
=(
b
(
t
)
,
0
≤
≤
τ<L
is
defined by
L−
1
ω
a
(
t
+
τ
)
−b
(
t
)
,
0
R
a,b
(
τ
)=
≤
τ<N
(1)
t
=0
where
ω
is a primitive complex
N
th root of unity.
Specifically, in this paper there are two types of sequences: binary sequences
and quaternary sequences. They are corresponding to different
ω
in (1), i.e.,
ω
=
1 for binary sequences, and
ω
=
√
−
−
1 for quaternary sequences.
B. Gray Map
For arbitrary element
x
=2
x
1
+
x
2
in
Z
4
,
x
1
,x
2
∈
Z
2
,theGraymap
φ
:
Z
4
→
Z
2
×
Z
2
is given by
φ
(
x
)=(
x
1
,x
1
+
x
2
(mod 2)), i.e.,
0
→
00
,
1
→
01
,
2
→
11
,
3
→
10
.
(2)
If defining
π
(
x
)=
x
1
and
ν
(
x
)=
x
1
+
x
2
(mod 2), the Gray map can be
described by the two maps
π
and
ν
, i.e.,
φ
(
x
)=(
π
(
x
)
,ν
(
x
)). In particular, it is
easy to see that
π
(
x
+1)=
x
1
+
x
2
(mod 2)
,ν
(
x
+1)=
x
1
+1(mod2)
,
(3)
π
(
x
+3)=
x
1
+
x
2
+1(mod2)
,ν
(
x
+3)=
x
1
(mod 2)
.
(4)
Applying the Gray map to all entries of the sequence (
a
(0)
,
···
,a
(
L
−
1)), we
naturally obtain the Gray map sequence
φ
a
=(
φ
a
(
t
)
,
0
≤
t<
2
L
)oflength2
L
respectively, where
φ
a
(
t
)=
π
(
a
(
t
1
))
,t
=2
t
1
ν
(
a
(
t
1
))
,t
=2
t
1
+1
.
When
L
is odd, it is more convenient to use the modified Gray map sequence
ϕ
a
=(
ϕ
a
(
t
)
,
0
t<
2
L
) proposed by Nechaev [6], which is equivalent to the
Gray map sequence via a permutation of coordinates, and is defined by
ϕ
a
(
t
)=
π
(
a
(
t
1
))
,
≤
t
=2
t
1
ν
(
a
(
t
1
+
L
+1
2
))
,t
=2
t
1
+1
.
The following lemma establishes the connection between the correlation of
quaternary sequences and the correlation of their (modified) Gray map sequence.
Lemma 1 ([4]).
Let
(
a
(0)
,
···
,a
(
L
−
1))
and
(
b
(0)
,
···
,b
(
L
−
1))
be two qua-
ternary sequences of length
L
.Then,
2
L−
1
L−
1
1)
φ
(
a
(
t
))+
φ
(
b
(
t
))
=2
ω
a
(
t
)
−b
(
t
)
)
,
(
−
·
R
(
t
=0
t
=0
