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In-Depth Information
2
n
.
Also,
R
∗
is a multiplicative group of order 2
n
(2
n
r
≤
−
2
,t
∈T
−
1) which is
where
G
1
is a cyclic group of order 2
n
a direct product
G
1
×E
−
1 (made up of
the nonzero elements in the Teichmuller set) generated by
β
and
E
is made up
of elements of the form 1 + 2
t
where
t
T.
The
Frobenius map
from
R
to
R
is the ring automorphism that takes any
element
c
=
a
+2
b
in the 2-adic representation to the element
c
f
=
a
2
+2
b
2
and
it generates the Galois group of
R
over
Z
4
with
f
m
the identity map. The
Trace
map
from
R
to
Z
4
is defined by
∈
T
(
c
)=
c
+
c
f
+
c
f
2
+
+
c
f
m
−
1
,
···
c
∈
R.
The trace is onto and has nice equidistribution properties. It will play a role in
the moment calculations we shall use later in the paper. If we let
f
2
(
c
)=
c
2
be
the squaring map defined on the finite field
GF
(2
m
) then the finite field trace is
given by
tr
(
c
)=
c
+
c
2
+
c
2
2
+
+
c
2
m
−
1
,
GF
(2
m
)
,
···
c
∈
and the following commutativity relationships hold:
μ
◦
f
=
f
2
◦
μ,
μ
◦
T
=
tr
◦
μ.
Let
G
C
=
of
R
defined by the
equivalence relation
α
=
β
if and only if
αG
C
=
βG
C
. The partition
T\{
0
}
. Consider the following partition
X
X
consists
of the following subsets which partition
R
:
1. 2
n
subsets corresponding to each
a
G
A
of
R
:[
a
]=
a
(
G
C
)
.
2. A subset consisting of proper zero divisors: [
e
]=
∈
2
\{
0
}
.
3. The zero subset: [
∞
]=
{
0
}
.
For
a, b, c
, define
N
(
a, b
;
c
) to be the number of times a fixed element of the
class [
c
] occurs in the Cayley table of [
a
]+[
b
]. This number is independent of the
element of [
c
] that is chosen, since in ([
a
]+[
b
]), the occurrence of any element of
[
c
] implies the occurrence of all the elements of [
c
]. The commutative property
of
R
implies
N
(
a, b
;
c
)=
N
(
b, a
;
c
). Various structural constants
N
(
a, b
;
c
), a,
b
,
c
∈X
∈X
, are computed in [10], and they are reproduced in the following lemma.
,w
;
x
)=
0
if
w
=
x
1
if
w
=
x.
Lemma 1.
1.
n
(
∞
⎧
⎨
0
if
x
=
e,
∞
2
n
2.
N
(
e, e
;
x
)=
−
1
if
x
=
∞
⎩
2
n
2
if
x
=
e.
3.
N
(
e, a
;
x
)=
0
if
x
=
a,e,
or
∞
for any
a∈G
A
1
otherwise.
−
)=
2
n
1
if
b
=3
a
for any
a,b∈G
A
0
otherwise.
−
4.
N
(
a, b
;
∞
5.
N
(
a, b
;
e
)=
0
if
b
=3
a
for any
a,b∈G
A
1
otherwise.
6.
n
(0
,
0; 0) = 0
.
7. If
a, b, c, d
∈
G
A
,then
N
(
a, b
;
c
)=
N
(
ad, bd
;
cd
)
.
