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2. Otherwise, the autocorrelation distribution obeys the Standard Normalized
Correlation Distribution from Definition 3, except that the value
S
(
γ
i
)
and
its complex conjugate occur with frequency one less than that specified in
Definition 3.
Proof.
For the binary
m
sequence corresponding to a coset leader chosen from
the maximal ideal 2
R
the autocorrelation takes on the value
−
−
1for
τ
=0
,
and 2
n
−
1 otherwise. For the rest of the proof, we restrict ourselves to the
sequences which are not all zero divisors. Let
n
=2
t
+ 1 and consider correlation
between
S
a
and
τ
th
shift of itself. Because of the linearity, this value is correlation
sum of some
S
b
,where
b ∈ G
A
.These
b
's are exactly those in Cayley table of
N
(
a,
3
a
;
b
). From Lemma 1,
N
(
a,
3
a
;
b
) takes all values in
G
A
except
a
and 3
a
.
The correlation sum of
S
a
and
S
3
a
are conjugate of each other. This proves the
result.
The following follows immediately.
Theorem 6.
For Family A the first moment of the autocorrelation function
obeys
L
1
C
i,i
(
τ
)
,
and therefore it simply takes on values proportional to the values in Lemma 3
with the same multiplicities.
P
i,i
(
τ, k, L
)
k
=
2
n
−
Proof.
The proof is similar to that of Theorem 4.
Lemma 7.
Let
n
be an odd number, then for Family C 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
,
−
4
times.
¯
2. Otherwise, if
is
a
∈
P
,then
⎧
⎨
2(2
n
−
1)
,
1
time,
−
2
,
1
time,
θ
(
τ
)=
2(2
t
2
t
+1
(2
t
+1)
−
1)
,
−
4
times,
⎩
2(2
t
+1)
,
2
t
+1
(2
t
−
−
1)
times,
¯
3. Otherwise, if
is
a
∈
Q
,then
⎧
⎨
2(2
n
−
1)
,
1
time,
−
2
,
1
time,
θ
(
τ
)=
2(2
t
2
t
+1
(2
t
+1)
times,
−
1)
,
⎩
2(2
t
+1)
,
2
t
+1
(2
t
−
−
1)
−
4
times,
