Information Technology Reference
In-Depth Information
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
}}
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)
Search WWH ::




Custom Search