90
CHAPTER 7.
CODES OVER SUBFIELDS
and
ˇ
A
∈
K
mp,q
be the matrix
ˇ
a
1
,
1
ˇ
a
1
,
2
· · ·
ˇ
a
1
,j
· · ·
ˇ
a
1
,q
ˇ
a
2
,
1
ˇ
a
2
,
2
· · ·
ˇ
a
2
,j
· · ·
ˇ
a
1
,q
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
ˇ
a
i,
1
ˇ
a
i,
2
· · ·
ˇ
a
i,j
· · ·
ˇ
a
1
,q
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
ˇ
a
p,
1
ˇ
a
p,
2
· · ·
ˇ
a
p,j
· · ·
ˇ
a
p,q
For our constructions,
A
might be a spanning or control matrix for a linear
code over
F
.
Then the matrices
b
A
and
ˇ
A
can be thought of as spanning or
control matrices for linear codes over
K
.
It must be emphasized that these maps are highly dependent upon the choice
of the initial map
φ
, even though
φ
has been suppressed in the notation. We
shall see below that a careful choice of
φ
can be of great help.
(The general
situation in which
φ
is an arbitrary injection of
F
into
K
p
, for some field
K
and
some
p
, is of further interest. Here we will be concerned with the linear case,
but see the Problem 7.2.3 below.)
7.2
Expanded codes
If
C
is a code in
F
n
, then the code
b
C
=
{
ˆ
c

c
∈
C
}
in
K
mn
is called an
expanded code
.
expanded code
( 7.2.1)
Theorem.
If
C
is an
[
n, k, d
]
code, then
b
C
is an
[
mn, mk,
≥
d
]
code.
Proof.
The map
x
7→
ˆ
x
(induced by
φ
) is onetoone and has
\
r
a
+
s
b
=
r
ˆ
a
+
s
ˆ
b
,
for all
r, s
∈
K
and
a
,
b
∈
F
n
. Thus
b
C
is a linear code over
K
with

b
C

=

C

=

F

k
= (

K

m
)
k
=

K

mk
,
hence
b
C
has
K
dimension
mk
. (This counting argument is a cheat unless
F
is
finite, the case of greatest interest to us. Instead, one should construct a
K
basis
for
b
C
out of that for
F
and an
F
basis for
C
. We do this below, constructing a
generator matrix for
b
C
from one for
C
.)
If the coordinate
c
i
of
c
is nonzero, then ˆ
c
i
is not the zero vector of
K
m
.