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Applying (2), we have
1)
c
t
≤
Ψ
j
(
f
(
ξ
t
))
2
l
k
+
H−
1
−
1
k
+
H−
1
j
=0
|
(
−
μ
j
|
t
=
k
t
=
k
<
2
l
π
ln(2) + 1
2
+1
D
√
2
m
.
π
ln
4(2
m
−
1)
π
The estimate of the Theorem follows.
6 Aperiodic Autocorrelation
As in the previous section, consider periodic sequences
c
0
,c
1
,...
of period
n
−
1,
where
n
=2
m
.
Theorem 6.1.
With notation as above, and for any
τ
in the range
0
<τ <n
−
1
,
n−
2
−τ
1)
c
t
(
1)
c
t
+
τ
,
Θ
(
τ
)=
(
−
−
t
=0
where
c
t
=MSB(Tr(
f
(
ξ
t
)))
. We then have the following bound
(
l
≥
4)
on PSL:
2
l
π
ln(2) + 1
2
2
+1
D
√
2
m
.
π
ln
4(2
m
−
1)
|
Θ
(
τ
)
|≤
π
=
O
(
√
n
log
n
)
,
for large
n.
In particular
|
Θ
(
τ
)
|
Proof.
As we have
c
t
=MSB(Tr(
f
(
ξ
t
))) where
t
ranges between 0 and
n
−
2and
1)
c
t
is equal to:
by (1), we obtain that (
−
2
l
2
l
−
1
−
1
μ
(Tr(
f
(
ξ
t
))) =
μ
j
ψ
j
(Tr(
f
(
ξ
t
))) =
μ
j
Ψ
j
(
f
(
ξ
t
))
.
j
=0
j
=0
Changing the order of summation, we obtain that:
2
l
2
l
−
1
−
1
n−τ−
2
Ψ
j
1
(
f
(
ξ
t
))
Ψ
j
2
(
f
(
ξ
t
+
τ
))
.
Θ
(
τ
)=
μ
j
1
μ
j
2
j
1
=0
j
2
=0
t
=0
By definition of
Ψ
,wehave:
Ψ
j
1
(
f
(
ξ
t
))
Ψ
j
2
(
f
(
ξ
t
+
τ
)) =
Ψ
(
g
(
ξ
t
))
,
where
g
(
X
)=
j
1
f
(
X
)+
j
2
f
(
Xξ
τ
). By Lemma 4.3, since 1
,ξ
τ
are linearly in-
Z
2
l
, we obtain that
g
(
X
)
dependent over
=0.Notethatif
f
(
X
)
∈
S
D
then
f
(
Xξ
τ
)
Xξ
τ
∈
S
D
since, by Lemma 4.2 the change of variable
X
→
does not
