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Proof.
We evaluate the unnormalized sum in the above definition, where for con-
venience we denote the coset representatives by
γ
0
,...,γ
2
n
,
and drop summation
limits whenever convenient. We have:
2
n
2
n
−
2
P
i,j
(
τ, k, L
)[
P
i,j
(
τ, k, L
)]
∗
k,τ
=0
j
=0
which can be rewritten as
2
n
2
n
−
2
L
−
1
ω
s
i
(
k⊕t⊕τ
)
−s
j
(
k⊕t
)
−s
i
(
l⊕t⊕τ
)+
s
j
(
l⊕t
)
=
t,τ
=0
j
=0
k,l
=0
2
n
2
n
2
n
−
2
L
−
1
−
2
ω
T
[(
β
k
−β
l
)(
β
τ
γ
i
−γ
j
)
β
t
]
.
=
j
=0
τ
=0
t
=0
k,l
=0
We now separate the case
k
=
l,
and note that we can use complex conjugate
symmetry of the (
β
k
− β
l
)termstorewritethesumas
⎛
⎝
⎞
⎠
+
2
n
2
n
−
2
L
(2
n
−
1)
1
j
=0
τ
=0
j,τ
R
e
S
((
β
k
γ
j
))
β
l
)(
β
τ
γ
i
−
−
+
0
≤
k
=
l
≤
L
−
1
j,τ
R
=
L
(2
n
1)
2
(2
n
+1)+2
S
(
β
τ
γ
i
−
−
e
{
γ
j
)
}
0
≤l<k≤L−
1
where the argument of the sum
S
(
) simplifies since it is invariant under multi-
plication by a nonzero unit. The sum on the right hand side can be evaluated by
considering the equation
γ
=
β
τ
γ
i
−
·
γ
j
,
and asking how many solutions this equa-
tion has for a fixed
γ
i
.
By following the argument in the proof of Theorem 6 in [1],
it can be seen that a modified distribution will occur where the term correspond-
ing to
γ
j
is missing from the Standard Normalized Correlation Distribution from
Definition 3. Using this and normalizing we get the claimed result.
Note that from the local second partial correlation moment, it is straightforward
to obtain the global partial correlation moment defined below, and its distribu-
tion stated in the theorem below, after a renormalization of sums.
Definition 6.
We define the global second partial correlation moment for Fam-
ily A as:
2
n
−
2
|
2
=
1
2
P
(
L
)
|
|
P
i,j
(
τ, k, L
)
|
(2
2
n
−
1)
2
k,τ
=0
γ
i
,γ
j
∈Γ
v
Theorem 12.
The global second partial correlation moment for Family A is
given by
|
2
=
L
+
L
(
L
−
1)
P
(
L
)
|
−
1)
2
.
(2
2
n
