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74
LECTURE 7. THE ALGEBRA OF MODULAR FORMS
(i) Show that
L
C
is an integral lattice if and only if for any
x
= (
±
1
,...,±
n
)
∈
C
the number wt(
x
) = #
{
i
:
±
i
6
= 0
}
(called the
weight
of
x
) is divisible by 4. In
this case we say that
C
is a
doubly even
linear code.
(ii) Show that the discriminant of the lattice
L
C
is equal to 2
n

2
k
, where
k
= dim
C
.
(iii) Let
C
⊥
=
{
y
∈
F
n
2
:
x
·
y
= 0
,
∀
x
∈
C
}
. Show that
L
C
is integral if and only if
C
⊂
C
⊥
.
(iv) Assume
C
is doubly even. Show that
L
C
⊥
=
L
*
C
.
In particular,
L
C
is a uni
modular even lattice if and only if
C
=
C
⊥
(in this case
C
is called a
selfdual
code
).
(v) Let
C
⊂
F
n
2
be a selfdual doubly even code. Show that
n
must be divisible by
8.
7.6
Let
A
(
τ
) =
ϑ
(0;
τ
)
,B
(
τ
) =
ϑ
1
2
0
(0;
τ
).
(i) Show that
(
A
(

1
/τ
)
,B
(

1
/τ
)) = (
τ
i
)
1
/
2
(
A
(
τ
)
,B
(
τ
))
·
1
√
2
1
√
2
1
√
2

1
√
2
!
.
(ii) Show that the expression
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 Fall '09
 ONTONKONG
 Algebra

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