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Table 3.
Correlation Sum and Weight Distributions of
Family B
(Trace Number
of
γ
=1)
(a)
n
=2
t
+ 1 (an odd integer) ; Period = 2(2
n
−
1)
Subset
ℵ
No. of
Constituent
w
0
w
1
Sequences
class
¯
2(2
t
2
t
−
1
(2
t
−
1
+1)
η ∈P
;
ηγ ∈R
2
2
t
+2
t
2
2
t
P
−
1)
−
2
¯
−
2(2
t
+1)
2
t
−
1
(2
t
−
1
−
1)
η ∈Q
;
ηγ ∈S
2
2
t
−
2
t
2
2
t
Q
−
2
¯
−
2+
ω
2
t
+1
2
2
t
−
2
2
2
t
2
2
t
+2
t
R
η ∈P
;
ηγ ∈S
−
2
¯
−
2
− ω
2
t
+1
2
2
t
−
2
2
2
t
2
2
t
−
2
t
S
η ∈Q
;
ηγ ∈R
−
2
¯
2
n
B
−
2
1
η ∈<
2
>
−
2
0
(b)
n
=2
t
(an even integer); Period = 2(2
n
−
1)
¯
(2
t
−
2+
ω
2
t
) 2
t
−
2
(2
t
−
1
+1)
η ∈P
;
ηγ ∈R
2
n
−
1
+2
t
−
1
−
2 2
n
−
1
+2
t
−
1
P
¯
2
t
2
t
) 2
t
−
2
(2
t
−
1
−
2
n
−
1
−
2
t
−
1
−
2 2
n
−
1
−
2
t
−
1
Q
(
−
−
2
− ω
1)
η ∈Q
;
ηγ ∈S
¯
(2
t
2
t
) 2
t
−
2
(2
t
−
1
+1)
2
n
−
1
+2
t
−
1
−
2 2
n
−
1
−
2
t
−
1
R
−
2
− ω
η ∈P
;
ηγ ∈S
¯
(
−
2
t
−
2+
ω
2
t
) 2
t
−
2
(2
t
−
1
−
1)
η ∈Q
;
ηγ ∈R
2
n
−
1
−
2
t
−
1
−
2 2
n
−
1
+2
t
−
1
S
¯
2
n
B
−
2
1
η ∈<
2
>
−
2
0
Note that the non-trivial (off-peak) values of this function are those for which
either
i
=0
.
We recover the usual full period correlation if
L
=2
n
=
j
or
τ
−
1
.
Definition 3.
The first moment of the partial correlation function in Definition
1 is given by
2
n
−
2
1
P
i,j
(
τ, k, L
) =
P
i,j
(
τ, L
)
P
i,j
(
τ, k, L
)
k
=
2
n
−
1
k
=0
while its second absolute moment is given by
2
n
−
2
|
2
k
=
1
2
.
P
i,j
(
τ, k, L
)
|
|
P
i,j
(
τ, k, L
)
|
2
n
−
1
k
=0
We remark that the correlations are, in general, complex valued. It is quite
straightforward to obtain the first moment of the partial periodic correlation
(for all possible
i, j, τ
). In fact this applies to the partial periodic correlation of
any two complex valued sequences
provided they have the same length.
Theorem 4.
Thefirstmomentobeys
L
P
i,j
(
τ, k, L
)
k
=
1
C
i,j
(
τ
)
,
2
n
−
and therefore, for Family A, it simply takes on values proportional to the values
in Theorem 3 with the same multiplicities.
