6.7 New Codes from Old
153
6.7.5 The (u|u + v) Construction
The (u|u + v) construction is an interesting method for constructing new (classical)
linear codes from old ones. Our intention in this subsection is to apply this technique
(valid for classical linear codes as well as to quantum codes) in order to generate a
similar construction method for asymmetric quantum codes.
Let us recall how to perform this construction for classical codes (see Definition 4.3.8). Let C 1 and C 2 be two linear codes of same length both over F q with parameters [n, k 1 , d 1 ] q and [n, k 2 , d 2 ] q , respectively. From the (u|u + v) construction, a
new code C = {(u, u + v)|u ∈ C 1 , v ∈ C 2 } with parameters [2n, k 1 + k 2 , min{2d 1 , d 2 }] q
can be obtained. To simplify the notation, we denote by (C 1 |C 1 + C 2 ) the code
derived from the (u|u + v) technique applied to codes C 1 and C 2 .
We have
Theorem 6.7.9 Assume that there exist two asymmetric stabilizer codes [[n, k
∗
,
d
∗
z /d
∗
x ]] q , derived from linear codes C 1 = [n, k 1 , d 1 ] q and C 2 = [n, k 2 , d 2 ] q , with
C 2 ⊂ C 1 , and [[n, k
, d
z /d
x ]] q , derived from codes C 3 = [n, k 3 , d 3 ] q and C 4 =
[n, k 4 , d 4 ] q , where C 4 ⊂ C 3 . Then there exists an [[2n, k
∗
+ k
, d z /d x ]] q AQECC,
where d z ≥ min{2d 1 , d 3 }, d x ≥ min{2d
⊥
4 , d
⊥
2 }, with d
∗
z ≥ d 1 , d
∗
x ≥ d
⊥
2 , d
z ≥ d 3 and
d
x ≥ d
⊥
4 , where d
⊥
2 and d
⊥
4 are the minimum distances of the dual codes C
⊥
2 and
C
⊥
4 , respectively.
Proof Since the inclusions C 2 ⊂ C 1 and C 4 ⊂ C 3 hold it follows that the inclusion
(C 2 |C 2 + C 4 ) ⊂ (C 1 |C 1 + C 3 ) also holds. We know that the codes (C 2 |C 2 + C 4 )
and (C 1 |C 1 + C 3 ) have parameters [2n, k 2 + k 4 , min{2d 2 , d 4 }] q and [2n, k 1 + k 3 ,
min{2d 1 , d 3 }] q , respectively. Let us compute the minimum distance of the dual code
[(C 2 |C 2 + C 4 )]
⊥ . We know that a generator matrix of [(C 2 |C 2 + C 4 )]
⊥ is the matrix
H 2 0
−H 4 H 4
,
where H 2 and H 4 are the parity check matrices of C 2 and C 4 , respectively. The
codewords of [(C 2 |C 2 + C 4 )]
⊥ are of the form {(u − v, v)|u ∈ C
⊥
2 , v ∈ C
⊥
4 }. Let
us consider the codeword w = (u − v, v). If u = 0, then w = (−v, v); hence, the
minimum weight of [(C 2 |C 2 + C 4 )]
⊥ is equal to 2d
⊥
4 . On the other hand, if u = 0,
then it follows that
wt(w) =
= wt(u − v) + wt(v)
= d(u, v) + d(v, 0) ≥ d(u, 0)
= wt(u).
Thus, the minimum weight is given by d
⊥
2 and, consequently, the minimum distance
of [(C 2 |C 2 + C 4 )]
⊥ is min{2d
⊥
4 , d
⊥
2 }. Applying the CSS construction to (C 2 |C 2 +