152
6 Asymmetric Quantum Codes
bijective map φ((v|w)) = βv + β
q w from F
2(n−1)
q
onto F
n−1
q 2 is an isometry (symplectic/Hamming weights, resp.) (see also [80, Lemma 14]). Considering the inverse
map φ
−1 and the corresponding additive code φ
−1
(D
⊥ a ) ≤ F
2(n−1)
q
, it follows that
φ
−1
(D
⊥ a ) has minimum X -weight d
∗
x satisfying d
∗
x ≥ d x − 1 and the minimum Z -
weight d
∗
z is, at least, d
∗
z ≥ d z − 1. The proof is complete.
Remark 6.7.3 Notice that the procedure adopted in Theorems 6.7.6 and 6.7.7 can
be generalized by puncturing codes on two or more coordinates.
6.7.4 Code Extension
The technique of code extension was derived also in the quantum case [25, 80]. Here,
we extend to AQECCs this technique.
Let C be an [n, k, d] q linear code over F q . Recall that the extended code C
e (see
Definition 4.3.2) is the linear code given by
C
e
= {(x 1 , . . . , x n , x n+1 ) ∈ F
n+1
q |(x 1 , . . . , x n ) ∈ C, x 1 + · · · + x n + x n+1 = 0}.
The code C
e is linear and has parameters [n + 1, k, d
e
] q , where d
e
= d or d
e
=
d + 1. Recall that a vector v = (v 1 , . . . , v n ) ∈ F
n
q is called even-like if it satisfies the
equality
n
i=1
v i = 0, and odd-like otherwise. For an [n, k, d] q code C, the minimum
weight of the even-like codewords of C are called minimum even-like weight and
denoted by d even (or (d) even ). Similarly, the minimum weight of the odd-like codewords of C are called minimum odd-like weight and denoted by d odd (or (d) odd ).
Theorem 6.7.8 Assume that there exists an [[n, k, d z /d x ]] q AQECC derived from
two nested linear codes C 1 = [n, k 1 , d 1 ] q and C 2 = [n, k 2 , d 2 ] q , with C 2 CC 1 . Then
the following are true:
(a) If (d 1 ) even ≤ (d 1 ) odd , then there exists an [[n + 1, k, d
e
z /d
e
x ]] q AQECC, where
d
e
z ≥ d 1 and d
e
x ≥ (d
e
2 )
⊥ , where (d
e
2 )
⊥ is the minimum distance of the dual (C
e
2 )
⊥
of the extended code C
e
2 ;
(b) If (d 1 ) odd < (d 1 ) even , then there exists an [[n + 1, k, d
e
z /d
e
x ]] q AQECC, where
d
e
z ≥ d 1 + 1 and d
e
x ≥ (d
e
2 )
⊥ .
Proof We only show Item (b), since Item (a) is similar. We note first that the inclusion
C
e
2 ⊂ C
e
1 holds. The parameters of the extended codes C
e
1 and C
e
2 are [n + 1, k 1 , d
e
1 ] q
and [n + 1, k 2 , d
e
2 ] q , respectively, where d
e
1 = d 1 or d
e
1 = d 1 + 1. Since (d 1 ) odd <
(d 1 ) even , it follows from the remark shown in [67, pg. 15] that d
e
1 = d 1 + 1. From
hypothesis, we know that k = k 1 − k 2 ; so, the corresponding CSS code has dimension
k. Applying the CSS construction to the codes C
e
1 , C
e
2 and (C
e
2 )
⊥ , we obtain an
[[n + 1, k, d
e
z /d
e
x ]] q AQECC, where d
e
z ≥ d 1 + 1 and d
e
x ≥ (d
e
2 )
⊥ . This finishes the
proof.
6 Asymmetric Quantum Codes
bijective map φ((v|w)) = βv + β
q w from F
2(n−1)
q
onto F
n−1
q 2 is an isometry (symplectic/Hamming weights, resp.) (see also [80, Lemma 14]). Considering the inverse
map φ
−1 and the corresponding additive code φ
−1
(D
⊥ a ) ≤ F
2(n−1)
q
, it follows that
φ
−1
(D
⊥ a ) has minimum X -weight d
∗
x satisfying d
∗
x ≥ d x − 1 and the minimum Z -
weight d
∗
z is, at least, d
∗
z ≥ d z − 1. The proof is complete.
Remark 6.7.3 Notice that the procedure adopted in Theorems 6.7.6 and 6.7.7 can
be generalized by puncturing codes on two or more coordinates.
6.7.4 Code Extension
The technique of code extension was derived also in the quantum case [25, 80]. Here,
we extend to AQECCs this technique.
Let C be an [n, k, d] q linear code over F q . Recall that the extended code C
e (see
Definition 4.3.2) is the linear code given by
C
e
= {(x 1 , . . . , x n , x n+1 ) ∈ F
n+1
q |(x 1 , . . . , x n ) ∈ C, x 1 + · · · + x n + x n+1 = 0}.
The code C
e is linear and has parameters [n + 1, k, d
e
] q , where d
e
= d or d
e
=
d + 1. Recall that a vector v = (v 1 , . . . , v n ) ∈ F
n
q is called even-like if it satisfies the
equality
n
i=1
v i = 0, and odd-like otherwise. For an [n, k, d] q code C, the minimum
weight of the even-like codewords of C are called minimum even-like weight and
denoted by d even (or (d) even ). Similarly, the minimum weight of the odd-like codewords of C are called minimum odd-like weight and denoted by d odd (or (d) odd ).
Theorem 6.7.8 Assume that there exists an [[n, k, d z /d x ]] q AQECC derived from
two nested linear codes C 1 = [n, k 1 , d 1 ] q and C 2 = [n, k 2 , d 2 ] q , with C 2 CC 1 . Then
the following are true:
(a) If (d 1 ) even ≤ (d 1 ) odd , then there exists an [[n + 1, k, d
e
z /d
e
x ]] q AQECC, where
d
e
z ≥ d 1 and d
e
x ≥ (d
e
2 )
⊥ , where (d
e
2 )
⊥ is the minimum distance of the dual (C
e
2 )
⊥
of the extended code C
e
2 ;
(b) If (d 1 ) odd < (d 1 ) even , then there exists an [[n + 1, k, d
e
z /d
e
x ]] q AQECC, where
d
e
z ≥ d 1 + 1 and d
e
x ≥ (d
e
2 )
⊥ .
Proof We only show Item (b), since Item (a) is similar. We note first that the inclusion
C
e
2 ⊂ C
e
1 holds. The parameters of the extended codes C
e
1 and C
e
2 are [n + 1, k 1 , d
e
1 ] q
and [n + 1, k 2 , d
e
2 ] q , respectively, where d
e
1 = d 1 or d
e
1 = d 1 + 1. Since (d 1 ) odd <
(d 1 ) even , it follows from the remark shown in [67, pg. 15] that d
e
1 = d 1 + 1. From
hypothesis, we know that k = k 1 − k 2 ; so, the corresponding CSS code has dimension
k. Applying the CSS construction to the codes C
e
1 , C
e
2 and (C
e
2 )
⊥ , we obtain an
[[n + 1, k, d
e
z /d
e
x ]] q AQECC, where d
e
z ≥ d 1 + 1 and d
e
x ≥ (d
e
2 )
⊥ . This finishes the
proof.