148
6 Asymmetric Quantum Codes
Theorem 6.7.1 ([80, Theorem 13]) An ((n, K , d)) q stabilizer code exists if and only
if there exists an additive code C ≤ F
2n
q of size |C| = q
n
/K such that C ≤ C
⊥ s and
swt(C
⊥ s \C) = d if K > 1 (and swt(C
⊥ s ) = d if K = 1).
6.7.1 Code Expansion
Let us recall the concept of dual basis (see Definition 4.3.5). Let β = {b 1 , b 2 , . . . , b m }
be a basis of F q m over F q . A dual basis of β is defined as β
⊥
= {b 1
∗
, b 2
∗
, . . . , b m
∗
},
where tr q m /q (b i b j
∗
) = δ i j , for all i, j ∈ {1, . . . , m}. A self-dual basis β is a basis
satisfying β = β
⊥ . If C is an [n, k, d 1 ] q m code and β = {b 1 , b 2 , . . . , b m } is a
basis of F q m over F q , then the q-ary expansion β(C) of C with respect to β is
an [mn, mk, d 2 ≥ d 1 ] q code given by β(C) := {(c i j ) i, j ∈ F q
mn
| c = (
j c i j b j ) i ∈
C}.
Let us recall the following result concerning the dual of a code obtaining by means
of code expansion.
Lemma 6.7.1 ([12, 56, 94]) Let C = [n, k, d] q m be a linear code over F q m , where
q is a prime power. Let C
⊥ be the dual of the code C. Then the dual code of the
q-ary expansion β(C) of code C with respect to the basis β is the q-ary expansion
β
⊥
(C
⊥
) of the dual code C
⊥ with respect to β
⊥ .
Theorem 6.7.2 establishes a method to construct AQECCs by expanding linear
codes.
Theorem 6.7.2 Let q be a prime power. Assume there exists an [[n, k, d z /d x ]] q m
AQECC derived from linear codes C 1 = [n, k 1 , d 1 ] q m and C 2 = [n, k 2 , d 2 ] q m . Then
there exists an AQECC with parameters [[mn, mk, d
∗
z /d
∗
x ]] q , where k = k 1 − k 2 ,
d
∗
z ≥ d 1 and d
∗
x ≥ d
⊥
2 , where d
⊥
2 is the minimum distance of the dual code C
⊥
2 .
Proof The proof presented here utilizes the same idea and generalizes the proof of
[94, Theorem 1] to all linear codes. Note first that [β(C)]
⊥
= β
⊥
(C
⊥
). Let C 1 =
[n, k 1 , d 1 ] q m and C 2 = [n, k 2 , d 2 ] q m be two linear nested codes C 2 ⊂ C 1 . Let β be any
basis of F q m over F q and let β
⊥ be its dual basis. Perform the expansions β(C 1 ) of the
codes C 1 and β(C 2 ) of C 2 with respect to β. Hence, the inclusion β(C 2 ) ⊂ β(C 1 )
holds. The codes β(C 1 ), β(C 2 ) and [β(C 2 )]
⊥ are linear. Furthermore, it follows
that β(C 1 ) = [mn, mk 1 , D 1 ≥ d 1 ] q and β(C 2 ) = [mn, mk 2 , D 2 ≥ d 2 ] q . Since C
⊥
2
has minimum distance d
⊥
2 , then β
⊥
(C
⊥
2 ) has minimum distance greater than or
equal to d
⊥
2 because β
⊥ is a basis of F q m over F q . From Lemma 6.7.1, the equality
[β(C 2 )]
⊥
= β
⊥
(C
⊥
2 ) is true; hence, [β(C 2 )]
⊥ has also minimum distance greater
than or equal to d
⊥
2 . Applying the CSS construction to the codes β(C 1 ), β(C 2 )
and [β(C 2 )]
⊥ , we obtain an [[mn, m(k 1 − k 2 ), d
∗
z /d
∗
x ]] q asymmetric quantum errorcorrecting code, where d
∗
z ≥ d 1 and d
∗
x ≥ d
⊥
2 . The proof is complete.
The following result is more general than Theorem 6.7.2.
6 Asymmetric Quantum Codes
Theorem 6.7.1 ([80, Theorem 13]) An ((n, K , d)) q stabilizer code exists if and only
if there exists an additive code C ≤ F
2n
q of size |C| = q
n
/K such that C ≤ C
⊥ s and
swt(C
⊥ s \C) = d if K > 1 (and swt(C
⊥ s ) = d if K = 1).
6.7.1 Code Expansion
Let us recall the concept of dual basis (see Definition 4.3.5). Let β = {b 1 , b 2 , . . . , b m }
be a basis of F q m over F q . A dual basis of β is defined as β
⊥
= {b 1
∗
, b 2
∗
, . . . , b m
∗
},
where tr q m /q (b i b j
∗
) = δ i j , for all i, j ∈ {1, . . . , m}. A self-dual basis β is a basis
satisfying β = β
⊥ . If C is an [n, k, d 1 ] q m code and β = {b 1 , b 2 , . . . , b m } is a
basis of F q m over F q , then the q-ary expansion β(C) of C with respect to β is
an [mn, mk, d 2 ≥ d 1 ] q code given by β(C) := {(c i j ) i, j ∈ F q
mn
| c = (
j c i j b j ) i ∈
C}.
Let us recall the following result concerning the dual of a code obtaining by means
of code expansion.
Lemma 6.7.1 ([12, 56, 94]) Let C = [n, k, d] q m be a linear code over F q m , where
q is a prime power. Let C
⊥ be the dual of the code C. Then the dual code of the
q-ary expansion β(C) of code C with respect to the basis β is the q-ary expansion
β
⊥
(C
⊥
) of the dual code C
⊥ with respect to β
⊥ .
Theorem 6.7.2 establishes a method to construct AQECCs by expanding linear
codes.
Theorem 6.7.2 Let q be a prime power. Assume there exists an [[n, k, d z /d x ]] q m
AQECC derived from linear codes C 1 = [n, k 1 , d 1 ] q m and C 2 = [n, k 2 , d 2 ] q m . Then
there exists an AQECC with parameters [[mn, mk, d
∗
z /d
∗
x ]] q , where k = k 1 − k 2 ,
d
∗
z ≥ d 1 and d
∗
x ≥ d
⊥
2 , where d
⊥
2 is the minimum distance of the dual code C
⊥
2 .
Proof The proof presented here utilizes the same idea and generalizes the proof of
[94, Theorem 1] to all linear codes. Note first that [β(C)]
⊥
= β
⊥
(C
⊥
). Let C 1 =
[n, k 1 , d 1 ] q m and C 2 = [n, k 2 , d 2 ] q m be two linear nested codes C 2 ⊂ C 1 . Let β be any
basis of F q m over F q and let β
⊥ be its dual basis. Perform the expansions β(C 1 ) of the
codes C 1 and β(C 2 ) of C 2 with respect to β. Hence, the inclusion β(C 2 ) ⊂ β(C 1 )
holds. The codes β(C 1 ), β(C 2 ) and [β(C 2 )]
⊥ are linear. Furthermore, it follows
that β(C 1 ) = [mn, mk 1 , D 1 ≥ d 1 ] q and β(C 2 ) = [mn, mk 2 , D 2 ≥ d 2 ] q . Since C
⊥
2
has minimum distance d
⊥
2 , then β
⊥
(C
⊥
2 ) has minimum distance greater than or
equal to d
⊥
2 because β
⊥ is a basis of F q m over F q . From Lemma 6.7.1, the equality
[β(C 2 )]
⊥
= β
⊥
(C
⊥
2 ) is true; hence, [β(C 2 )]
⊥ has also minimum distance greater
than or equal to d
⊥
2 . Applying the CSS construction to the codes β(C 1 ), β(C 2 )
and [β(C 2 )]
⊥ , we obtain an [[mn, m(k 1 − k 2 ), d
∗
z /d
∗
x ]] q asymmetric quantum errorcorrecting code, where d
∗
z ≥ d 1 and d
∗
x ≥ d
⊥
2 . The proof is complete.
The following result is more general than Theorem 6.7.2.