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Proof.
From Lemma 9 of [10], the autocorrelation of
is
a
at
τ
th
shift is given by
S
(
η
)+
S
(
γη
), where
η
=(
a
a
(
γβ
)
τ
). When
τ
=2
n
−
−
1,
η
is multiple of 2
and hence
2 will occur only once in the distribution. The case of
τ
=0leads
to trivial correlation of 2(2
n
−
¯
−
1). For the rest of the proof we assume
is
a
∈
P
.
For even values of
τ,
(
γ
)
τ
= 1. This leads to correlations of
S
(
b
)+
S
(
γb
), where
b
is the Cayley table of
N
(
a,
3
a, b
). These have been computed in Lemma 1
and the distribution of
S
(
η
)+
S
(
γη
) have been computed in Table 2. Here the
correlations take values from
G
A
except
a
and 3
a
. For odd values of
τ,
(
γ
)
τ
=
γ
.
This results in correlation sums of
c
's in
N
(
a,
3
aγ, b
). In this case,
γ
=3.Then
this corresponds to values in
N
(
a, a, b
). Combining the two cases gives the result.
The case for
is
a
¯
∈
Q
can be similarly proved.
Lemma 8.
Let
n
be an odd number, then for Family B the full period autocor-
relation function obeys:
1. If we consider the zero divisor sequence (binary
m
-sequence), then
θ
(
τ
)=
2(2
n
−
1)
,
2
times,
2
n
+1
−
2
,
−
3
times.
¯
2. Otherwise, if
is
a
∈
P
,then
⎧
⎨
2(2
n
−
1)
,
1
time,
−
2
,
1
time,
2(2
t
2
t
+1
(2
t−
1
+1)
−
1)
,
−
4
times,
θ
(
τ
)=
2(2
t
+1)
,
2
t
+1
(2
t−
1
⎩
−
−
1)
times,
2+
ω
2
t
+1
,
2
2
t
−
times,
ω
2
t
+1
,
2
2
t
−
2
−
times,
¯
3. Otherwise, if
is
a
∈
Q
,then
⎧
⎨
⎩
2(2
n
−
1)
,
1
time,
−
2
,
1
time,
2(2
t
2
t
+1
(2
t−
1
+1)
times,
−
1)
,
θ
(
τ
)=
2(2
t
+1)
,
2
t
+1
(2
t−
1
−
−
1)
−
4
times,
2+
ω
2
t
+1
,
2
2
t
−
times,
ω
2
t
+1
,
2
2
t
−
2
−
times.
¯
¯
4. Otherwise, if
is
a
∈
R
or
S
,then
⎧
⎨
2(2
n
−
1)
,
1
time,
−
2
,
1
time,
2(2
t
2
t
+1
(2
t−
1
+1)
times,
−
1)
,
θ
(
τ
)=
−
2(2
t
+1)
,
2
t
+1
(2
t−
1
−
1)
times,
⎩
2+
ω
2
t
+1
,
2
2
t
−
−
2
times,
ω
2
t
+1
,
2
2
t
−
2
−
−
2
times,
Proof.
The proof follows as in Lemma 7 except here the results of
N
(
a,
3
a, b
)
and
N
(
a,
3
γa,c
) of Lemma 1 need to be used.
Remark:
Result similar to Theorem 6 holds good for
Family B
and
Family C
.
