TOOLS FOR COSET WEIGHT ENUMERATORS
5
2.2. Cosets of codes of A. Let C be an [N, k)-code of A. We first recall the
MAc-WILLIAMS transform, which determines uniquely the weight polynomial of
the dual of C from the weight polynomial of the code C itself.
THEOREM
1. [16, p.146) Let C be a linear
[N,
k]-code. Define its weight poly-
nomial:
N
Wc(X, Y)
=
LAixN-iyi, Ai =card { c
E
C
I
w(c)
=
i},
i=O
where w(c) is the weight of c. Then the weight polynomial of the dual code Cj_
lS
(7)
1
W0 .L(X, Y)
=
kWc(X
+
(p -1)Y,X- Y).
p
Denote by Dx the
[N,
k
+
1)-code generated by C and a coset Cx = x
+
C
ofC:
Dx=
U
(ax+C).
aEK
As C
c
Dx , D;t
c
Cj_ ; then every weight of Cj_ can be a weight of D;t.
Moreover the weight distributions of x
+
C can be deduced from the weight
distribution of Cj_ and of D;t. Indeed suppose that the weight distribution of Cj_
is known and that we can compute the weight polynomial of D;t. The dimension
of Dx equals k
+
1 so that the dimension of D;t equals N- (k
+
1). Applying
(7) we obtain the weight polynomial of Dx:
1
Wvx(X, Y)
=
N-(k+l) Wvt(X
+
(p -1)Y,X- Y).
p
But, by definition, Wvx(X, Y) = (p-1)Qx(X, Y)
+
Wc(X,Y), where Qx(X, Y)
is the weight polynomial of the coset x
+
C . Then the expression of Qx(X, Y)
is as follows:
Qx(X, Y)
1
( ) N ( P W D.L (X
+
(p - 1) Y, X - Y)
p-1p
-k
X
(8)
- W0 .L(X
+
(p-1)Y,X-
Y)) .
In the following we always will assume that the polynomial
W
c
.L
(X, Y)
is
known and we want to have results on the polynomial Qx(X, Y), for every x.
Then the problem consists in the determination of the polynomials W D.L (X, Y).
Let. be a non zero weight of Cj_ and
A,\(x)
be the number of code'"words of
weight .in D;t. So A.x(x) is the number of elements of Cj_ of weight .which
are orthogonal to
x:
(9) A.x(x) =card { y
E
cj_
I
w(y) =.and y,x = 0}.
Indeed y
E
D;t if and only if y, z = 0 for all z in Dx. But z =ax+ c,
a
E
K and c
E
C; as y
E
Cj_, y,c = 0. Hence y
E
D;t is equivalent to
y,x = 0.
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