TOOLS FOR COSET WEIGHT ENUMERATORS 7
3.2. Application of Formula (I). In the application we present now, For-
mula
(I)
is most interesting, because
xy
has few possible weights, since the depth
of y ism- 2; moreover the Th(Y) form the set of codewords of a given weight in
the coset y
+
pm-
1
(see Proposition 2). From now on, in this Section, p
=
2,
K and G will be respectively the finite field of order 2 and 2m. We will consider
linear codes C of depth 2 such that:
P
3
c c c
P
2
(i.e. pm-
1
c c.L c
pm-
2
) .
From Proposition 1, the code
C
is an ideal of the algebra
A.
The code
C.L
is
an union of cosets of the Reed-Muller code of order 1, which are contained in
the Reed-Muller code of order 2. These cosets are precisely described in [16,
Chapter 15]; the reader can also refer to
[8].
We only recall the results we need.
Such a coset y
+
pm-
1
is uniquely defined by the symplectic form associated
toy. Since y is in pm-
2
,
then y can be identified to a quadratic boolean function
/y:
Y
=
L
fy(g)X
9
-
i.e. y9
=
fy(g).
gEG
The associated symplectic form of
/y
is
Wy:
(u,v)EG2
f---t
Wy(u,v)=/y(O)+/y(u)+/y(v)+/y(u+v)
EK.
The kernel of 111
y
is as follows defined:
£y
= {
u
E
G'
I
Vv
E
G : Wy(u, v)
=
0} .
The set
£y
is a K-subspace of G of dimension
K
=
m- 2h, where 2h is the rank
of
Wy.
Let
Ey
=
y
+
pm-
1
;
then
Wy
=
wb
for all b E
Ey.
Moreover the
weight distribution of
Ey
only depends on h; that is
(cf.
[16, p.
441]):
where hE
[0,
lm/2J ]. Note that such a coset has exactly three weights unless
m
is even and h
=
m/2.
DEFINITION
4.
Let y
E
pm-2
.
Let 2h be the rank of the symplectic form
associated toy. We will say that the coset
Ey
=
y
+
pm-
1
is of type (h).
The proofs of Proposition 2, Lemma 1 and Proposition 3 can be found in
[10] and [12].
In the following we will always consider y E pm-2 \pm-
1
,
Ey
=
y
+
pm-
1
.
If
Cx
=
x
+
C
is any coset of
C,
then we will denote by
Dx
the code
Cx
U
C
(see Section 2.2). We suppose that the weight polynomial
of
C.L
is known and we want to determine the weight polynomial of
D;i:.
In
accordance with Formula (8), the weight distribution of the coset x
+
C is:
1
(10)
Qx(X, Y)
=
22m_k
(2Wv;-(X
+
Y,X- Y)-
Wc_~_(X
+
Y,X- Y))
where k is the dimension of C.
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