TOOLS FOR COSET WEIGHT ENUMERATORS 11
2) Suppose that a code C is such that its dual is a union of L cosets
y
+
pm(p-
1)-r+1
,
y
E
pm(p-
1)-r
,
where all cosets have the same weight enumer-
ator. Then there is only one weight distrihution for the cosets x
+
C, x
E
pr
and this result holds for non extended cyclic codes
C.
It becomes from the fact
that L/2 cosets composing Cl. are orthogonal to x, for any x. We will treat later
a special application of this fact (see Corollary 3).
3.4. Application of Formula (II). In this section we will consider ex-
tended cyclic codes C of depth r, for some
r,
such that:
pr+1
c
c
c
pr (
i.e.
pm(p-1)-r+1
c
cl.
c
pm(p-1)-r) .
Let Tl. be the defining set of the dual code Cl.. In accordance with Definition
3 and with the definition of C, Tl. has the following form:
{13)
Tl.
=
Ip(m(p-1) -r+ 1,m) \
{s1, ... ,s1},
where the
Si
are some elements of S of p-weight m(p-
1) - r.
Note that the
definingsetofCis T=Ip(r,m)U{n-sl···
,n-sz}.
Set
U={s1,··· ,sz}
and let E be the cyclic code (in the algebra R) whose nonzeros are the ol,
t
E U. The code E can be viewed in the algebra A , by adding an all-zero
column to the generator matrix. Clearly depth( E) equals depth( Cl.) equals
m(p- 1) -
r.
Then, since the dimension of E equals the number of elements of U,
which is the dimension of the quotient space cl.
;pm(p-
1)-r+1
,
we have clearly
{14)
cJ.
=
u
(y
+
pm(p-1)-r+l) .
yEE
The definitions of Cl., U and E hold in all this section. Recall that Dx is the
linear code generated by C and the coset of C containing
x;
we want to determine
the weight distribution of the dual of Dx·
PROPOSITION
5. Let a coset of C: Cx
=
x
+
C , x
E
pr \ C .
Let yin
E.
Then, for every
j
E [O,n -1]:
l
¢n(x aj(y))
=
0 if and only if L¢n-s,(x)¢s,(y)ast1
=
0.
t=1
That means that the number of shifts of y orthogonal to x is related with the
weight of the codeword y' of E defined by
{15)
Let n(y) be the number of elements of the set { aJ(Y)
I
j E
[0,
n-
1] }.
Then we
have
n(y) '
L(y) =card { a1(y)
I
x, aj(y)
=
0}
=
-(n- w(y)) ;
n
the code
v;
contains L(y) cosets a1(y)
+
pm(p-l)-r+l.
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