Information Technology Reference
In-Depth Information
n−d
p
,
N
γ,δ,
(0) =
p
n−
1
occurs
p
n−d−
1
times
,
and
N
γ,δ,
(0) =
runs through
F
p
n
+
d
−
1
p
n
−
d
−
1
p−
1
2
p
n−
1
(
p
n−d−
1
±
η
(
Δ
d
) occurs
±
η
(
Δ
d
)) times.
2
2
By (9) and (17),
S
(
γ,δ,
)=
η
(
Δ
d
)
p
n
+
2
(
1)
p
−
2
ω
−λ
γ,δ,
. By (18), for given
−
,
S
(
γ,δ,
)=
η
(
Δ
d
)
p
n
+
d
n−d
p
(
γ,δ
)
∈
R
d
,when(
b
1
,b
2
,
···
,b
n−d
) runs through
F
2
(
1)
p
−
2
ω
ρ
occurs
p
n−d−
1
+
η
(
ρ
)
p
n
−
d
−
1
−
−
η
(
Δ
d
)timesforeach
ρ
∈
F
p
.
2
i
=1
i
=1
n
−
2
d
n
R
2
d
:inthiscase,
Π
γ,δ
(
x
)+
Tr
1
(
x
)=
a
i
y
i
+
Case 2.3.
(
γ,δ
)
∈
b
i
y
i
.
Similarly as in Case 2.2, if there exists some
b
i
=0with
n
−
2
d<i
≤
n
,
then
N
γ,δ,
(
ρ
)=
p
n−
1
for any
ρ
∈
F
p
,and
S
(
γ,δ,
) = 0. Further, for given
F
p
n
,thereare
p
n
p
n−
2
d
(
γ,δ
)
∈
R
2
d
,when
runs through
−
choices for
such
that there is at least one
b
i
=0with
n
−
2
d<i
≤
n
.
If
b
i
=0forall
n
−
2
d<i
≤
n
, a similar analysis shows that for any
given (
γ,δ
)
∈
R
2
d
,when(
b
1
,b
2
,
···
,b
n−
2
d
) runs through
F
n−
2
d
p
,
N
γ,δ,
(0) =
1)
p
n
+2
d
−
2
1)
p
n
−
2
d
−
2
p
n−
1
+(
p
−
η
(
Δ
2
d
) occurs
p
n−
2
d−
1
+(
p
−
η
(
Δ
2
d
)times,
2
2
p
n
+2
d
−
2
p
n
−
2
d
−
2
N
γ,δ,
(0) =
p
n−
1
1)(
p
n−
2
d−
1
−
η
(
Δ
2
d
) occurs (
p
−
−
η
(
Δ
2
d
)) times,
2
2
and
S
(
γ,δ,
)=
η
(
Δ
2
d
)
p
2
+
d
ω
ρ
occurs
p
n−
2
d−
1
+
v
(
ρ
)
p
n
−
2
d
−
2
η
(
Δ
2
d
)timesfor
2
each
ρ
∈
F
p
.
For
i
R
i,j
,Theorem
3 can be proven by the above analysis, Equation (14), and Proposition 9.
∈{
0
,d,
2
d
}
and
j
∈{
1
,
−
1
}
,since
η
(
Δ
i
)=
j
for (
γ,δ
)
∈
Proof (of Theorem 3).
We only give the frequency of the codewords with weight
(
p
1)
p
n−
1
. The other cases can be proven in a similar way. The weight of
c
(
γ,δ,
)
is equal to (
p
−
1)
p
n−
1
if and only if
N
γ,δ,
(0) =
p
n−
1
, which occurs only in Cases
1, 2.2 and 2.3. The frequency is equal to
p
n
−
1+
(
p
n
p
n−d
)+
p
n−d−
1
|
−
−
R
d
|
+
(
p
n
p
n−
2
d
)
=(
p
n
1)(1 +
p
n−
2
d
(
p
m
+
d
p
m
+
p
m−
1
+
p
m−d
−
|
R
2
d
|
−
−
−
1)).
k
has 9 different weights for
d
=1,and10
different weights for
d>
1. The codewords with weight (
p
Remark 12.
By Theorem 3, the code
C
p
n
+2
d
−
2
1)(
p
n−
1
−
−
)
2
1)
p
n−
1
+
p
n
+2
d
−
2
k
or (
p
−
do not exist in
C
since
|
R
2
d,
1
|
=0.
2
The following result can also be similarly proven and we omit its proof here.
(
1)
p
−
2
,when
(
γ,δ,
)
runs
Proposition 13.
For
n
=2
m
≥
4
,let
ϕ
=
−
through
×
F
p
n
, the exponential sum
S
(
γ,δ,
)
defined in (8) has the
following distribution, where
ρ
=0
,
1
,
F
p
m
×
F
p
n
···
,p
−
1
.
⎧
⎨
p
n
,
1time
,
(
p
n
1)(1 +
p
m
+
n−d
p
m
+
n−
2
d
+
p
m
+
n−
3
d
p
n−
2
d
)times
,
0
,
−
−
−
1)(
p
n−
1
+
v
(
ρ
)
p
n
−
2
)
/
(2(
p
d
+ 1)) times
,
p
2
ω
ρ
,
d
(
p
m
+1)(
p
n
−
p
2
ω
ρ
,
v
(
ρ
)
p
n
−
2
)
/
(2(
p
d
(
p
n
+
d
2
p
n
+
p
d
)(
p
m
1)(
p
n−
1
−
−
−
−
−
1)) times
,
⎩
p
n
+
2
ϕω
ρ
,p
m−d
(
p
n
ρ
)
p
n
−
d
−
1
1)(
p
n−d−
1
+
η
(
−
−
)
/
2times
,
2
p
n
+
2
ϕω
ρ
,p
m−d
(
p
n
ρ
)
p
n
−
d
−
1
−
−
1)(
p
n−d−
1
−
η
(
−
)
/
2times
,
2
p
2
+
d
ω
ρ
,
(
p
m−d
v
(
ρ
)
p
n
−
2
d
−
2
1)(
p
n
1)(
p
n−
2
d−
1
)
/
(
p
2
d
−
−
−
−
−
1) times
.
2
