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where the reduction modulo
q
istakenintherange[
1)
,
0]. Since
q
is prime,
this is again the rational number corresponding to an
-sequence. The theorem
then follows from Theorem 2.
−
(
q
−
Note that this argument does not apply to
-sequences with prime power connec-
tion integer since the numerator (
N
T−τ
−
1)
f
may not be relatively prime to
q
.
3.1 Expected Arithmetic Correlations
In this section we investigate the expected values of the arithmetic autocor-
relations and cross-correlations and the second moments and variances of the
cross-correlations for a fixed shift. We leave the problem of computing second
moments and variances of the arithmetic autocorrelations as open problems.
We need some initial analysis for general
N
-ary sequences. Fix a period
T
.
As we have seen, the
N
-ary sequences of period
T
are the coecient sequences
a
of rational numbers of the form
a
=
−
f
N
T
−
1
N
T
with 0
≤
f
≤
−
1.
Lemma 2.
If
a
and
b
are distinct
N
-adic numbers whose coecient sequences
are periodic with period
T
,and
a
−
b
∈
Z
,then
{
a, b
}
=
{
0
,
−
1
}
.
Next fix a shift
τ
. Then the
τ
shift of
a
corresponds to a rational number
N
T−τ
f
N
T
a
(
τ
)
=
c
f,τ
+
−
,
−
1
c
f,τ
<N
T−τ
is an integer.
Now let
b
be another periodic
N
-ary sequence corresponding to the rational
number
where 0
≤
−
g
1
.
Then the arithmetic cross-correlation between
a
and
b
with shift
τ
is
b
=
N
T
−
(
τ
)=
Z
c
g,τ
+
−
N
T−τ
g
N
T
−
f
A
a
,
b
C
1
−
N
T
−
−
1
=
Z
N
T−τ
g
c
g,τ
.
−
f
1
−
(2)
N
T
−
Theorem 4.
For any
τ
, the expected arithmetic autocorrelation, averaged over
all sequences
a
of period
T
,is
(
τ
)] =
T
A
a
N
T−
gcd(
τ,T
)
.
The expected cross-correlation, averaged over all pairs of sequences
a
and
b
is
E
[
A
T
N
T
.
A
a
E
[
C
(
τ
)] =
,
b
