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In-Depth Information
Clock
p
1
1
1
At GF ( 2 )
where
p=2,
3 2 0
T =
F=
3 1 2
Cell 0
Cell 1
Cell 2
1 1
= 2
a =
0 3 2
1 0
3
1
2
0
3
3
2
2
0
No. of nonreachable states = 48
Attractor = 302
Depth = 3
132
333
........3
000
312
231
123
110
....
1
01
011
303
220
222
.............
211
131
012
300
223
120
...
113
102
210
021
111
032
203
320
331
023
212
313
001
313
230
100
..............................2
221
010
.......................................................................1
133
302
.................................................................................................................................0
Fig. 3.
Structure and state transition graph of a 3 cell GF(2
2
) Dual
SACA
position ofstates. All the reachable states in a
SACA
becomes non-reachable in
SACA
[2]. The example CA ofFig. 3 is a dual counterpart ofthe
SACA
ofFig.
2. The following Theorem characterizes a
SACA
and
SACA
.
Theorem 3
: Ifthe complement vector F ofa GF(2
p
)
SACA
with characteristic
matrix T is such that
T
n
.F
= 0, and
T
n−
1
.F
= 0, then this complemented CA
is a dual
SACA
Detailed characterization ofa
SACA
and its dual are reported in [7]
2.3
Synthesis of
SACA
and Its Dual
The algorithmic steps for synthesis of an n cell GF(2
p
)
SACA
and its dual are
noted below with illustrating example. The
Steps 1
and
2
ensures that the
resulting CA is a
SACA
- the proofis omitted for shortage ofspace.
Step 1.
Generate the dependency matrix D ofsize
n × n
whose 1
st
cell has no
dependency on its neighbors (let, selfand right) and the rest ofthe cells having
000
100
010
dependency on its left neighbor only. For a 3 cell GF(2
2
)
SACA
, D is:
Step 2.
Construct characteristic matrix T ofthe
SACA
from D by performing
elementary row/column operations such that each ofthe cells has dependency
on left, self and right neighbors.
