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1+2
β
k
,k
=
,
2
n
where
G
A
=
, can be used to enumerate the
cyclically distinct elements in
Family A
, since each member
γ
i
,1
{
∞
,
0
,
···
−
2
}
2
n
+1
≤
i
≤
of
Γ
ν
gives a distinct sequence in the family if we take
s
i
(
t
)=
T
(
γ
i
β
t
)
.
Each sequence in
Family A
corresponds to a class in
X
.
We conclude this section by stating the complete full period correlation distri-
bution for
Family A
, which is obtained by considering the distribution of values
taken by sums of the form
2
n
S
(
γ
)=
x∈G
1
−
2
ω
T
(
γβ
t
)
,
ω
T
(
γx
)
=
t
=0
as
γ
ranges over the ring
R,
where we count the solutions of
γ
=
γ
i
β
τ
γ
j
.
Families B and C:
If sequences generated as trace of powers of (
γβ
),
γ
−
∈
G
A
= 1, the resulting sequences are of period 2(2
n
and
γ
1)[1,10]. Families of in-
terleaved m-sequences comprises of 2
n−
1
+ 1 sequences which obey a common
linear recurrence relation over
Z
4
determined by the minimal polynomial corre-
sponding to (
γβ
), where
γ
−
= 1. An interleaved sequence
is
a
can be
expressed as
is
a
(
t
)=
T
(
a
(
γβ
)
t
)where
β
is a generator of the Teichmuller set,
and
a
∈
G
A
and
γ
=3tracenumber
of
γ
is 1.
Family C
is obtained when
γ
= 3. It can be noticed that each inter-
leaved sequence can be seen as interleaved version of two
Family A
sequences
[10]. Using this fact, sequences in
Family B
are enumerated with the following
representatives:
= 0. We call interleaved family as
Family B
when
γ
is
a
,a
.
And similarly sequences in
Family C
are enumerated with the following repre-
sentatives:
{
∈
Quotient group G
A
/
{
1
,γ
}}
is
a
,a
.
In [10], we used an association scheme over
R
to study these sequence families
properties. Here we only use related Cayley table defined on
R
.
Theorem 2 ([1,10]).
The correlation sum and weight distributions of sequences
in Family A of period
2
n
{
∈
Quotient group G
A
/
{
1
,
3
}}
−
1
are given in Table 1. These sequences are grouped
under five subsets
. The trace numbers of sequences within any
subset are same. For the first subset
P
,
Q
,
R
,
S
and
B
(binary),
w
2
=2
r−
1
,and
w
3
=0
.For
B
remaining subsets,
w
3
=2
r−
1
w
1
and
w
2
=2
r−
1
w
0
.
The following theorem describes correlation sum and weight distribution of se-
quences in
Family B
and
Family C
. It also describes the internal composition of
Family A
sequences.
−
−
1
−
Theorem 3.
The correlation sum and weight distributions of the families of
BandCofperiod
2(2
r
1)
are given in Tables 2 and 3. Like before, the se-
quences are grouped based on distinct correlation values and named with a (
¯
−
)
to distinguish from sequences in Family A which have half the period. In all
these tables the subset
·
¯
corresponds to
is
2
and for all the items except the last,
B
¯
w
2
=(2
n
−
−
w
0
,
w
3
=2
n
−
w
1
; for the last item (subset
B
),
w
3
=0
,
w
2
=2
n
.
2)