146
6 Asymmetric Quantum Codes
Corollary 6.6.2 There exists an
[[(q − 1)
2
, (q − d 1 )
2
− (q − d 2 )
2
, d z /d x ]] q
asymmetric quantum code, where d z ≥ max{(d 1 )
2
, q − d 2 }, d x ≥ min{(d 1 )
2
, q −
d 2 }, and d 1 , d 2 satisfy 2 ≤ d 1 ≤ d 2 < q − 1.
Exercise 6.6.1 Show Corollary 6.6.2.
Example 6.6.1 In this example we construct an [[36, 11, d z /d x ]] 7 AQECC, where
d z ≥ 8 and d x ≥ 2. To do this, it suffices to consider in Theorem 6.6.3, q = 7, d 1 = 4,
d 3 = 2, d 2 = d 4 = 5. Similarly, we can construct an [[36, 11, d z /d x ]] 7 AQECC, with
d z ≥ 6 and d x ≥ 3 and an [[36, 6, d z /d x ]] 7 AQECC, where d z ≥ 10 and d x ≥ 2. From
Corollary 6.6.2, we obtain an [[64, 7, d z /d x ]] 9 AQECC, where d z ≥ 25 and d x ≥ 3.
6.6.1 Code Comparison
Here, we compare the parameters of our codes with the ones displayed in the literature. To make this comparison, we proceed as follows: fixing the code length n
and the lower bound for the minimum distances d z and d x , the AQECC with greater
dimension is better than the other. This comparison is usual in the literature.
For example, our [[100, 55, d z /d x ]] 11 code, where d z ≥ 9 and d x ≥ 3 is better that
the [[100, 0, d
∗
z /d
∗
x ]] 11 code shown in [3], where d
∗
z ≥ 9 and d
∗
x ≥ 3. On the other
hand, our [[36, 0, d z /d x ]] 7 code, where d z ≥ 16 and d x ≥ 3 is not as good as the
[[36, 0, d
z /d
x ]] 7 code, where d
z ≥ 16 and d
x ≥ 8.
We point out that this construction does not provide AQECC with good parameters
due to the fact that the code length increases quickly in taking tensor products.
However, maybe the ideas presented here can motivate the reader to improve our
construction.
As can be seen, in the cases shown in Table 6.3, the new quantum codes have
parameters better than the ones shown in Theorem 8 in [3]. On the other hand,
the parameters of the new codes are not as good as the parameters of the quantum
codes given by the referee. Concerning Refs. [90, 94], it seems that the new code
parameters are not as good as those shown in such papers, but we do not perform the
comparison properly, since the corresponding parameters are quite distinct. Hence,
the AQECCs constructed in this subsection are comparable with AQECCs discussed
in the literature. In other words, there exist cases where our codes are better than
ones available in the literature and there exist other cases where our codes are not as
good as the ones available in the literature.
6 Asymmetric Quantum Codes
Corollary 6.6.2 There exists an
[[(q − 1)
2
, (q − d 1 )
2
− (q − d 2 )
2
, d z /d x ]] q
asymmetric quantum code, where d z ≥ max{(d 1 )
2
, q − d 2 }, d x ≥ min{(d 1 )
2
, q −
d 2 }, and d 1 , d 2 satisfy 2 ≤ d 1 ≤ d 2 < q − 1.
Exercise 6.6.1 Show Corollary 6.6.2.
Example 6.6.1 In this example we construct an [[36, 11, d z /d x ]] 7 AQECC, where
d z ≥ 8 and d x ≥ 2. To do this, it suffices to consider in Theorem 6.6.3, q = 7, d 1 = 4,
d 3 = 2, d 2 = d 4 = 5. Similarly, we can construct an [[36, 11, d z /d x ]] 7 AQECC, with
d z ≥ 6 and d x ≥ 3 and an [[36, 6, d z /d x ]] 7 AQECC, where d z ≥ 10 and d x ≥ 2. From
Corollary 6.6.2, we obtain an [[64, 7, d z /d x ]] 9 AQECC, where d z ≥ 25 and d x ≥ 3.
6.6.1 Code Comparison
Here, we compare the parameters of our codes with the ones displayed in the literature. To make this comparison, we proceed as follows: fixing the code length n
and the lower bound for the minimum distances d z and d x , the AQECC with greater
dimension is better than the other. This comparison is usual in the literature.
For example, our [[100, 55, d z /d x ]] 11 code, where d z ≥ 9 and d x ≥ 3 is better that
the [[100, 0, d
∗
z /d
∗
x ]] 11 code shown in [3], where d
∗
z ≥ 9 and d
∗
x ≥ 3. On the other
hand, our [[36, 0, d z /d x ]] 7 code, where d z ≥ 16 and d x ≥ 3 is not as good as the
[[36, 0, d
z /d
x ]] 7 code, where d
z ≥ 16 and d
x ≥ 8.
We point out that this construction does not provide AQECC with good parameters
due to the fact that the code length increases quickly in taking tensor products.
However, maybe the ideas presented here can motivate the reader to improve our
construction.
As can be seen, in the cases shown in Table 6.3, the new quantum codes have
parameters better than the ones shown in Theorem 8 in [3]. On the other hand,
the parameters of the new codes are not as good as the parameters of the quantum
codes given by the referee. Concerning Refs. [90, 94], it seems that the new code
parameters are not as good as those shown in such papers, but we do not perform the
comparison properly, since the corresponding parameters are quite distinct. Hence,
the AQECCs constructed in this subsection are comparable with AQECCs discussed
in the literature. In other words, there exist cases where our codes are better than
ones available in the literature and there exist other cases where our codes are not as
good as the ones available in the literature.