In quantum error correction, CSS codes, named after their inventors, Robert Calderbank, Peter Shor and Andrew Steane, are a special type of Stabilizer codes constructed from classical codes with some special properties.
Let
C
1
and
C
2
be two (classical)
[
n
,
k
1
]
,
[
n
,
k
2
]
codes such, that
C
2
⊂
C
1
and
C
1
,
C
2
⊥
both have minimal distance
≥
2
t
+
1
, where
C
2
⊥
is the code dual to
C
2
. Then define
CSS
(
C
1
,
C
2
)
, the CSS code of
C
1
over
C
2
as an
[
n
,
k
1
−
k
2
,
d
]
code, with
d
≥
2
t
+
1
as follows:
Define for
x
∈
C
1
:
|
x
+
C
2
⟩
:=
1
/
|
C
2
|
∑
y
∈
C
2
|
x
+
y
⟩
, where
+
is bitwise addition modulo 2. Then
CSS
(
C
1
,
C
2
)
is defined as
{
|
x
+
C
2
⟩
∣
x
∈
C
1
}
.
