Biomedical Engineering Reference
In-Depth Information
Remark4.8
The. elements. in.
S
ϕ
. are. cyclic,. and. two. consecutive. elements. have. two.
bits. difference.. In. addition,. any. two. elements. in.
S
ϕ
. will. also. have. at.
least. two. bits. of. difference. (otherwise,. the. rule. of. Gray. coding. will. be.
violated).
Remark4.9
Based.on.Remark.4.8,.by.changing.any.particular.actual.bit.location.of.
S
ϕ
.
,0
(or.
S
ϕ
). into. the. wild. character. '*',. a. subgroup. of. schemata.
S
ξ
. is. formed..
(It. is. easily. proved. as. every. element. will. then. have. different. actual. bits.
according.to.Remark.4.8.).Therefore,.
,0
P
i
ϕ .must.be.the.same.for.all.
ϕ ∈
S
.
(
)
ϕ
,0
(or.
ϕ ∈
S
ϕ
).
,1
Example4.4
Let.
S
=
{*0 * *00 * 0, * 0 * *00 * 1,
, * 1* *11* 1}
;. we. can. divide. it. into. two.
ϕ
groups:
*0 * *00 * 0,
*0 * *00 * 1,
*0 * *01* 1,
*0 * *01* 0,
*0 * *11* 0,
*0 * *11* 1,
*0 * *10 * 1,
*0 * *10 * 0,
S
=
S
=
.
ϕ
,0
ϕ
,1
*1* *10 * 0,
*1* *10 * 1,
*1* *11* 1,
*1* *11* 0,
*1* *01* 0,
*1* *01* 1,
*1* *00 * 1
*1* *00 * 0
.
Remarks. 4.7-4.9. can. be. easily. veriied.. Furthermore,. all. elements. in.
S
ϕ
.
,0
belong.to.
Z
even
,.whereas.all.elements.in.
S
ϕ
.belong.to.
Z
odd
.in.this.example.
Without.loss.of.generality,.let.us.consider.the.case.of.
o
(φ) =
n
+ 1,.which.is.
even.. Hence,. if.
,1
ϕ ∈
Z
ϕ ∈
S
,. we. have.
.. Similarly,. if.
ϕ ∈
S
,. we. have.
i
even
ϕ
,0
ϕ
,1
ϕ ∈
.
Based.on.Remark.4.9,.one.has
Z
i
odd
P
(
ϕ
)
=
C
.for.
ϕ ∈
S
.
(4.66)
.
i
0
ϕ
,0
P
(
ϕ
)
=
C
ϕ ∈
S
.for.
.
(4.67)
j
1
ϕ
,1
.
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