TOOLS FOR COSET WEIGHT ENUMERATORS
9
Proof: We will apply the Proposition 3 to each type of cosets. Let Ey
y
+
pm-l
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
m-1
+
2m-h;-l)
=
22h; - 22h;-1
+
2h;-l
=
22h;-l
+
2h;-l '
The null vector belongs to
Dt
while the all-one vector does not. Since
2m-l
=
2m- 2m-l,
we obtain for any coset of any type
2N2m-l
=
N2m-l. That means
that
v;
contains half of codewords of
cj_
of weight
2m-l.
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
f-----t
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 p-ary expansion
(so, ... ,
Sm-l). 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/Js-i(Y) .
t-s
Proof:
c/Js(xy)
=
LX9 LYh(g+h)
8
=
L:x9 LYht(: )gihs-i
gEG
hEG
gEG
hEG i=O
~
t ( : ) L Xggi LYhhs-i
=
L ( : )
¢i(x)c/Js-i(Y)'
i=O
gEG
hEG
i-s
since, by LUCAS's Theorem, ( : )
1:-
0 (mod
p)
is equivalent to
i-
s .
D
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