172
7 Constructions of QCCs
and d = 7. Therefore V
⊥ has parameters (17, 13, 2; 1, 7) 16 . Applying the generalized Singleton bound one has 7 = 4(2/13 + 1) + 2 + 1, so V
⊥ is maximumdistance-separable code.
7.3.2 Construction of Optimal QCCs
Here we apply Theorem 7.3.2 in order to obtain family of optimal QCCs. To do this,
let us recall a result shown in [91].
Lemma 7.3.3 Assume that q = 2
t , where t is an integer t ≥ 1. Let n = q
2
+ 1 and
a =
q
2
2
. If C is the cyclic code whose defining set Z is given by Z = C a−i ∪ · · · ∪ C a ,
where 0 ≤ i ≤
q
2
− 1, then C is Hermitian dual-containing.
Proof See [91, Lemma 4.2].
In the following theorem, we present a family of quantum convolutional MDS
codes.
Theorem 7.3.3 Assume q = 2
t , where t ≥ 3 is an integer, n = q
2
+ 1 and consider
that a =
q
2
2
. Then there exists a quantum MDS convolutional code with parameters
[(n, n − 4i, 1; 2, 2i + 3)] q , where 2 ≤ i ≤
q
2
− 2.
Proof We consider the same notation utilized in Theorem 7.3.2. We know that
gcd(n, q
2
) = 1. From Theorem 7.3.2, there exists a classical convolutional MDS
code with parameters (n, n − 2i, 2; 1, 2i + 3) q 2 , for each 2 ≤ i ≤
q
2
− 2. This code
is the Euclidean dual V
⊥ of the convolutional code V whose parameters are given
by (n, 2i, 2; 1, d f ) q 2 . The codes V
⊥ and V
⊥ h have the same degree as code (see the
proof of Theorem 7 in [7]). Additionally, it is straightforward to check that wt(V
⊥
) =
wt(V
⊥ h ), so V
⊥ h is an (n, n − 2i, 2; m
∗
, 2i + 3) q 2 code. From Lemma 7.3.3 and
from Theorem 7.1.1 Item (b), one has V ⊂ V
⊥ h . Applying Lemma 7.3.1, there
exists an [(n, n − 4i, 1; 2, d f ≥ 2i + 3)] q convolutional stabilizer code, for each
2 ≤ i ≤
q
2
− 2. Replacing the parameters of the previously constructed codes in
the quantum generalized Singleton bound (Theorem 7.3.1) one has the equality
2i + 3 = 2i
4
2n−4i
+ 1
+ 2 + 1. The proof is complete.
Example 7.3.2 To illustrate an application Theorem 7.3.3, assume that q = 8, n =
65 and i = 2. Applying Theorem 7.3.3, there exists an [(65, 57, 1; 2, 7)] 8 MDS
convolutional stabilizer code.
Taking q = 16, n = 257 and i = 2, 3, 4, 5, one has an optimal QCC with parameters [(257, 249, 1; 2, 7)] 16 , [(257, 245, 1; 2, 9)] 16 , [(257, 241, 1; 2, 11)] 16 ,
[(257, 237, 1; 2, 13)] 16 , respectively, and so on.
7 Constructions of QCCs
and d = 7. Therefore V
⊥ has parameters (17, 13, 2; 1, 7) 16 . Applying the generalized Singleton bound one has 7 = 4(2/13 + 1) + 2 + 1, so V
⊥ is maximumdistance-separable code.
7.3.2 Construction of Optimal QCCs
Here we apply Theorem 7.3.2 in order to obtain family of optimal QCCs. To do this,
let us recall a result shown in [91].
Lemma 7.3.3 Assume that q = 2
t , where t is an integer t ≥ 1. Let n = q
2
+ 1 and
a =
q
2
2
. If C is the cyclic code whose defining set Z is given by Z = C a−i ∪ · · · ∪ C a ,
where 0 ≤ i ≤
q
2
− 1, then C is Hermitian dual-containing.
Proof See [91, Lemma 4.2].
In the following theorem, we present a family of quantum convolutional MDS
codes.
Theorem 7.3.3 Assume q = 2
t , where t ≥ 3 is an integer, n = q
2
+ 1 and consider
that a =
q
2
2
. Then there exists a quantum MDS convolutional code with parameters
[(n, n − 4i, 1; 2, 2i + 3)] q , where 2 ≤ i ≤
q
2
− 2.
Proof We consider the same notation utilized in Theorem 7.3.2. We know that
gcd(n, q
2
) = 1. From Theorem 7.3.2, there exists a classical convolutional MDS
code with parameters (n, n − 2i, 2; 1, 2i + 3) q 2 , for each 2 ≤ i ≤
q
2
− 2. This code
is the Euclidean dual V
⊥ of the convolutional code V whose parameters are given
by (n, 2i, 2; 1, d f ) q 2 . The codes V
⊥ and V
⊥ h have the same degree as code (see the
proof of Theorem 7 in [7]). Additionally, it is straightforward to check that wt(V
⊥
) =
wt(V
⊥ h ), so V
⊥ h is an (n, n − 2i, 2; m
∗
, 2i + 3) q 2 code. From Lemma 7.3.3 and
from Theorem 7.1.1 Item (b), one has V ⊂ V
⊥ h . Applying Lemma 7.3.1, there
exists an [(n, n − 4i, 1; 2, d f ≥ 2i + 3)] q convolutional stabilizer code, for each
2 ≤ i ≤
q
2
− 2. Replacing the parameters of the previously constructed codes in
the quantum generalized Singleton bound (Theorem 7.3.1) one has the equality
2i + 3 = 2i
4
2n−4i
+ 1
+ 2 + 1. The proof is complete.
Example 7.3.2 To illustrate an application Theorem 7.3.3, assume that q = 8, n =
65 and i = 2. Applying Theorem 7.3.3, there exists an [(65, 57, 1; 2, 7)] 8 MDS
convolutional stabilizer code.
Taking q = 16, n = 257 and i = 2, 3, 4, 5, one has an optimal QCC with parameters [(257, 249, 1; 2, 7)] 16 , [(257, 245, 1; 2, 9)] 16 , [(257, 241, 1; 2, 11)] 16 ,
[(257, 237, 1; 2, 13)] 16 , respectively, and so on.