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where
τ
runs through 0
<τ<
2(2
n
1) and
i, j
vary from 2
n−
1
to 2
n
−
−
1,
respectively.
Case 6
.0
i<
2
n−
1
j<
2
n
and
τ
=0.
It follows from Lemma 1 and Lemma 7 that
R
i,j
(
τ
)=
2
≤
≤
(
2
n
−
2
t
=0
ω
a
i
(
t
+
τ
1
)
−a
j
(
t
)
−
3
)
,
·
R
τ
=2
τ
1
(
2
n
ω
a
i
+2
n
−
1
(
t
+
τ
1
+2
n
−
1
)
−a
j
(
t
)
−
3
)
,τ
=2
τ
1
+1
.
−
2
t
=0
2
·
R
In a similar manner to Case 4, we arrive at the following distribution
⎧
⎨
2
2
n−
2
times
0
,
2
n
+
2
,
2
2
n−
2
(2
n
R
i,j
(
τ
)=
−
2)
times
(9)
⎩
2
n
+
2
,
2
2
n−
2
(2
n
−
−
2)
times
with
τ
ranging through 0
<τ <
2(2
n
−
1),
i
and
j
varying from 0 to 2
n−
1
−
1
and 2
n−
1
to 2
n
−
1, respectively.
Case 7
.0
≤ j<
2
n−
1
≤ i<
2
n
and
τ
=0.
The correlation function is
R
i,j
(
τ
)=
2
(
2
n
−
2
ω
a
i
(
t
+
τ
1
)+3
−a
j
(
t
)
)
,
·
R
τ
=2
τ
1
t
=0
(
2
n
ω
a
i
−
2
n
−
1
(
t
+
τ
1
+2
n
−
1
)+1
−a
j
(
t
)
)
,τ
=2
τ
1
+1
,
−
2
2
·
R
t
=0
which has the same correlation distribution as (9).
References
1. Boztas, S., Hammons, R., Kumar, P.V.: 4-phase sequences with near-optimum cor-
relation properties. IEEE Trans. Inform. Theory 38, 1101-1113 (1992)
2. Fan, P.Z., Darnell, M.: Sequence Design for Communications Applications. John
Wiley, Chichester (1996)
3. Hammons, R., Kumar, P.V., Calderbank, A.N., Sloane, N.J.A., Sole, P.: The
Z
4
-
Linearity of Kerdock, Preparata, Goethals and Related Codes. IEEE Trans. Inform.
Theory 40, 301-319 (1994)
4. Helleseth, T., Kumar, P.V.: Sequences with low correlation. In: Pless, V., Huffman,
C. (eds.) Handbook of Coding Theory, Elsevier, Amsterdam (1998)
5. Johansen, A., Helleseth, T., Tang, X.H.: The correlation distribution of sequences
of period 2(2
n
−
1). IEEE Trans. Inform. Theory (to appear)
6. Nechaev, A.A.: Kerdock code in a cyclic form. Discrete Mathematics Appl. 1, 365-
384 (1991)
7. Udaya, P., Siddiqi, M.U.: Optimal biphase sequences with large linear complexity
derived from sequences over Z4. IEEE Trans. Inform. Theory 42, 206-216 (1996)
8. Tang, X.H., Udaya, P., Fan, P.Z.: Generalized binary Udaya-Siddiqi sequences. IEEE
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