TOOLS FOR COSET WEIGHT ENUMERATORS
9
Proof: We will apply the Proposition 3 to each type of cosets. Let Ey
y
+
pml
be a coset of type
(hi);
recall that the dimension of the kernel of the
symplectic form associated
toy
is
K,
=
m 2hi. Assume that
w(y)
=
.i·
Since
w(x)
=
1,
w(xy)
=
w(y).
Thus, according to (11), we have:
N.;
22h;m(2m _
.i)
=
2
2h;m(2 m _
2
m1
+
2mh;l)
=
22h;  22h;1
+
2h;l
=
22h;l
+
2h;l '
The null vector belongs to
Dt
while the allone vector does not. Since
2ml
=
2m 2ml,
we obtain for any coset of any type
2N2ml
=
N2ml. That means
that
v;
contains half of codewords of
cj_
of weight
2ml.
D
3.3. Shifts of codewords.
From now on we will treat only extended cyclic
codes as they are defined in Section 2.1. Such codes are invariant under the
shifts. Recall that we consider codes of length N
=
pm over the finite field K
of order
p.
A shift of a given codeword x is as follows:
aj :
L
x
9
X
9
ft
L
x
9
Xo:ig ,
j E [O,pm 2].
gEG gEG
PROPOSITION
4. Set S
=
[0.,
n] ,
where n
=
pm 1 . Any s
E
S is identified
with its pary expansion
(so, ... ,
Sml). Then we define a partial order on S:
for all s,
t
in S : s

t
{:::=:}
V
i
E
[0,
m 
1] :
Si
~
ti
Let x
E
A
and y
E
A.
Then
(12)
V
s
E
S : cp8 (xy)
=
L ( : )
¢i(x)c/Jsi(Y) .
ts
Proof:
c/Js(xy)
=
LX9 LYh(g+h)
8
=
L:x9 LYht(: )gihsi
gEG
hEG
gEG
hEG i=O
~
t ( : ) L Xggi LYhhsi
=
L ( : )
¢i(x)c/Jsi(Y)'
i=O
gEG
hEG
is
since, by LUCAS's Theorem, ( : )
1:
0 (mod
p)
is equivalent to
i
s .
D
9
Proof: We will apply the Proposition 3 to each type of cosets. Let Ey
y
+
pml
be a coset of type
(hi);
recall that the dimension of the kernel of the
symplectic form associated
toy
is
K,
=
m 2hi. Assume that
w(y)
=
.i·
Since
w(x)
=
1,
w(xy)
=
w(y).
Thus, according to (11), we have:
N.;
22h;m(2m _
.i)
=
2
2h;m(2 m _
2
m1
+
2mh;l)
=
22h;  22h;1
+
2h;l
=
22h;l
+
2h;l '
The null vector belongs to
Dt
while the allone vector does not. Since
2ml
=
2m 2ml,
we obtain for any coset of any type
2N2ml
=
N2ml. That means
that
v;
contains half of codewords of
cj_
of weight
2ml.
D
3.3. Shifts of codewords.
From now on we will treat only extended cyclic
codes as they are defined in Section 2.1. Such codes are invariant under the
shifts. Recall that we consider codes of length N
=
pm over the finite field K
of order
p.
A shift of a given codeword x is as follows:
aj :
L
x
9
X
9
ft
L
x
9
Xo:ig ,
j E [O,pm 2].
gEG gEG
PROPOSITION
4. Set S
=
[0.,
n] ,
where n
=
pm 1 . Any s
E
S is identified
with its pary expansion
(so, ... ,
Sml). Then we define a partial order on S:
for all s,
t
in S : s

t
{:::=:}
V
i
E
[0,
m 
1] :
Si
~
ti
Let x
E
A
and y
E
A.
Then
(12)
V
s
E
S : cp8 (xy)
=
L ( : )
¢i(x)c/Jsi(Y) .
ts
Proof:
c/Js(xy)
=
LX9 LYh(g+h)
8
=
L:x9 LYht(: )gihsi
gEG
hEG
gEG
hEG i=O
~
t ( : ) L Xggi LYhhsi
=
L ( : )
¢i(x)c/Jsi(Y)'
i=O
gEG
hEG
is
since, by LUCAS's Theorem, ( : )
1:
0 (mod
p)
is equivalent to
i
s .
D