142
6 Asymmetric Quantum Codes
Then an [[96, 62, d z ≥ 10/d x ≥ 9]] 7 AQECC is constructed. Analogously, we can
constructed AQECCs with parameters [[96, 64, d z ≥ 10/d x ≥ 8]] 7 , [[96, 70, d z ≥
10/d x ≥ 5]] 7 , [[96, 82, d z ≥ 5/d x ≥ 4]] 7 , [[96, 84, d z ≥ 5/d x ≥ 3]] 7 , [[372, 111, d z
≥ 45/d x ≥ 44]] 5 .
6.5.3 Construction II
In this subsection, we utilize GRS codes to obtain more asymmetric quantum codes.
The method of construction is similar to that of Theorem 6.5.2.
Theorem 6.5.4 Let q be a prime power. Then there exists an
[[N = mn, K = m(2k − n + c), d z ≥ d/d x ≥ (d − c)]] q
AQECC, where 1 < k < n < 2k + c ≤ q
m , k = n − d + 1, d > c + 1, c ≥ 1 and
m ≥ 1.
Proof Choose a vector a = (a 1 , a 2 , . . . , a n ), where the a i ’s are distinct elements of
F q m and a vector v = (v 1 , ..., v n ), where the v i ’s are nonzero elements of F
m
q . Let
C 1 be the GRS code GRS n,k (a, v). We know that C 1 is an [n, k = n − d + 1, d] q m
code. Assume that the inequality n < 2k + c is true. Let C 2 be the GRS code
GRS n,(n−k−c) (a, v). Then C 2 has parameters [n, n − k − c, k + c + 1] q m . From construction, C 2 ⊂ C 1 . Further, the dual code C
⊥
2 is the GRS code GRS n,(k+c) (a, u),
where u = (u 1 , u 2 , . . . , u n ) and each u i is given by u
−1
i = v i
j =i (a i − a j ) for all
i = 1, 2, . . . , n. Note that C
⊥
2 has minimum distance d − c.
Let β be a basis of F q m over F q and β
⊥ its dual basis. Since C 2 ⊂ C 1 , it follows that
β(C 2 ) ⊂ β(C 1 ). We know that the minimum distance of β(C 1 ) is greater than or equal
to d. Moreover, the minimum distance of [β(C 2 )]
⊥ is greater than or equal to d − c.
The codes β(C 1 ), β(C 2 ) and [β(C 2 )]
⊥ are linear. Applying the CSS construction to
β(C 1 ) and β(C 2 ), the code follows.
Definition 6.5.3 Given an [[n, k, d z /d x ]] q AQECC, the asymmetric quantum Singleton bound (see Lemma [140, Lemma 3.2]) asserts that k ≤ n − d z − d x + 2. If
the parameters of the AQECC satisfies the equality k = n − d z − d x + 2, then the
code is called maximum-distance-separable (MDS) or optimal.
Remark 6.5.1 According with Definition 6.5.3, if we consider m = 1 in Theorem 6.5.2, we have asymmetric quantum MDS codes.
6.5.4 Code Comparison
In this subsection, we compare the parameters of our codes with the ones available
in the literature. The criterion adopted is the usual one: for fixed values of the code
6 Asymmetric Quantum Codes
Then an [[96, 62, d z ≥ 10/d x ≥ 9]] 7 AQECC is constructed. Analogously, we can
constructed AQECCs with parameters [[96, 64, d z ≥ 10/d x ≥ 8]] 7 , [[96, 70, d z ≥
10/d x ≥ 5]] 7 , [[96, 82, d z ≥ 5/d x ≥ 4]] 7 , [[96, 84, d z ≥ 5/d x ≥ 3]] 7 , [[372, 111, d z
≥ 45/d x ≥ 44]] 5 .
6.5.3 Construction II
In this subsection, we utilize GRS codes to obtain more asymmetric quantum codes.
The method of construction is similar to that of Theorem 6.5.2.
Theorem 6.5.4 Let q be a prime power. Then there exists an
[[N = mn, K = m(2k − n + c), d z ≥ d/d x ≥ (d − c)]] q
AQECC, where 1 < k < n < 2k + c ≤ q
m , k = n − d + 1, d > c + 1, c ≥ 1 and
m ≥ 1.
Proof Choose a vector a = (a 1 , a 2 , . . . , a n ), where the a i ’s are distinct elements of
F q m and a vector v = (v 1 , ..., v n ), where the v i ’s are nonzero elements of F
m
q . Let
C 1 be the GRS code GRS n,k (a, v). We know that C 1 is an [n, k = n − d + 1, d] q m
code. Assume that the inequality n < 2k + c is true. Let C 2 be the GRS code
GRS n,(n−k−c) (a, v). Then C 2 has parameters [n, n − k − c, k + c + 1] q m . From construction, C 2 ⊂ C 1 . Further, the dual code C
⊥
2 is the GRS code GRS n,(k+c) (a, u),
where u = (u 1 , u 2 , . . . , u n ) and each u i is given by u
−1
i = v i
j =i (a i − a j ) for all
i = 1, 2, . . . , n. Note that C
⊥
2 has minimum distance d − c.
Let β be a basis of F q m over F q and β
⊥ its dual basis. Since C 2 ⊂ C 1 , it follows that
β(C 2 ) ⊂ β(C 1 ). We know that the minimum distance of β(C 1 ) is greater than or equal
to d. Moreover, the minimum distance of [β(C 2 )]
⊥ is greater than or equal to d − c.
The codes β(C 1 ), β(C 2 ) and [β(C 2 )]
⊥ are linear. Applying the CSS construction to
β(C 1 ) and β(C 2 ), the code follows.
Definition 6.5.3 Given an [[n, k, d z /d x ]] q AQECC, the asymmetric quantum Singleton bound (see Lemma [140, Lemma 3.2]) asserts that k ≤ n − d z − d x + 2. If
the parameters of the AQECC satisfies the equality k = n − d z − d x + 2, then the
code is called maximum-distance-separable (MDS) or optimal.
Remark 6.5.1 According with Definition 6.5.3, if we consider m = 1 in Theorem 6.5.2, we have asymmetric quantum MDS codes.
6.5.4 Code Comparison
In this subsection, we compare the parameters of our codes with the ones available
in the literature. The criterion adopted is the usual one: for fixed values of the code