Resumen
In this study, we explore maximum distance separable (MDS) self-dual codes over Galois rings
G
R
(
p
m
,
r
)
G
R
(
p
m
,
r
)
with
p
=
-
1
(
mod
4
)
p
=
-
1
(
mod
4
)
and odd r. Using the building-up construction, we construct MDS self-dual codes of length four and eight over
G
R
(
p
m
,
3
)
G
R
(
p
m
,
3
)
with (
p
=
3
p
=
3
and
m
=
2
,
3
,
4
,
5
,
6
m
=
2
,
3
,
4
,
5
,
6
), (
p
=
7
p
=
7
and
m
=
2
,
3
m
=
2
,
3
), (
p
=
11
p
=
11
and
m
=
2
m
=
2
), (
p
=
19
p
=
19
and
m
=
2
m
=
2
), (
p
=
23
p
=
23
and
m
=
2
m
=
2
), and (
p
=
31
p
=
31
and
m
=
2
m
=
2
). In the building-up construction, it is important to determine the existence of a square matrix U such that
U
U
T
=
-
I
U
U
T
=
-
I
, which is called an antiorthogonal matrix. We prove that there is no
2
×
2
2
×
2
antiorthogonal matrix over
G
R
(
2
m
,
r
)
G
R
(
2
m
,
r
)
with
m
=
2
m
=
2
and odd r.