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 ReedMuller code of order 1, which are contained in
the ReedMuller 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
ft
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 Ksubspace 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
pm2
.
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 pm2 \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.