118
5 Quantum Code Constructions
Table 5.17 Quantum codes
Codes from
Corollary 5.4.5
q
m
t 1
t 2
[[46, 36, d ≥ 4]] 25
5
2
3
21
[[46, 32, d ≥ 6]] 25
5
2
4
20
[[46, 28, d ≥ 8]] 25
5
2
5
19
[[46, 4, d ≥ 20]] 25
5
2
11
13
The quantum codes exhibited in [119] were constructed over the fields F 2 ,
F 3 , F 4 , F 5 , F 8 , F 9 . In our case, we exhibit examples of quantum codes constructed over F 9 , F 16 , F 25 , F 49 . The codes over F 9 constructed in [119] have
parameters [[15, 13, 2]] 9 , [[15, 7, 4]] 9 , [[15, 5, 5]] 9 , [[15, 1, 7]] 9 , [[243, 241, 2]] 9 ,
[[243, 219, 6]] 9 , [[243, 213, 9]] 9 . Our codes over F 9 , shown in Tables 7.2 and
5.16, have parameters [[26, 16, d ≥ 3]] 9 , [[26, 14, d ≥ 4]] 9 , [[26, 12, d ≥ 5]] 9 ,
[[26, 4, d ≥ 9]] 9 , [[26, 2, d ≥ 10]] 9 , [[27, 17, d ≥ 3]] 9 , [[27, 15, d ≥ 4]] 9 ,
[[27, 13, d ≥ 5]] 9 , [[27, 11, d ≥ 6]] 9 , [[27, 9, d ≥ 7]] 9 , [[27, 7, d ≥ 8]] 9 , [[27, 5,
d ≥ 9]] 9 , [[27, 3, d ≥ 10]] 9 and [[27, 1, d ≥ 11]] 9 . Since the parameters among
these codes are different, we do not perform a fair comparison of such codes.
Summarizing the results obtained in this section: we have constructed several
families of quantum codes with good as well as asymptotically good parameters.
Great part of our quantum codes has large minimum distances when compared with
their corresponding code lengths. Additionally, they have relatively small Singleton defects. We have also shown how to obtain sequences of asymptotically good
quantum codes derived from t-point AG codes.
5.5 Quantum Synchronizable Codes
We provide here constructions of families of quantum synchronizable codes (QSCs)
by applying the method discovered by Fujiwara [42] (see also [44]). More precisely,
we show how to construct families of QSCs derived from the sum and intersection of cyclic codes, BCH codes and from the product of cyclic codes. All these
results presented here can be found in [61]. For more details with respect to quantum
synchronizable codes we refer the reader to [42].
Before presenting our constructions of QSCs, we will review this concept. The
main goal of frame synchronization in communication systems is to ensure that
information block boundaries can be correctly determined at the receiver. To achieve
this goal, numerous synchronization techniques have been developed for classical
communication systems. However, these techniques are not applicable to quantum
communication systems since a qubit measurement typically destroys the quantum
states and thus also the corresponding quantum information. To circumvent this
5 Quantum Code Constructions
Table 5.17 Quantum codes
Codes from
Corollary 5.4.5
q
m
t 1
t 2
[[46, 36, d ≥ 4]] 25
5
2
3
21
[[46, 32, d ≥ 6]] 25
5
2
4
20
[[46, 28, d ≥ 8]] 25
5
2
5
19
[[46, 4, d ≥ 20]] 25
5
2
11
13
The quantum codes exhibited in [119] were constructed over the fields F 2 ,
F 3 , F 4 , F 5 , F 8 , F 9 . In our case, we exhibit examples of quantum codes constructed over F 9 , F 16 , F 25 , F 49 . The codes over F 9 constructed in [119] have
parameters [[15, 13, 2]] 9 , [[15, 7, 4]] 9 , [[15, 5, 5]] 9 , [[15, 1, 7]] 9 , [[243, 241, 2]] 9 ,
[[243, 219, 6]] 9 , [[243, 213, 9]] 9 . Our codes over F 9 , shown in Tables 7.2 and
5.16, have parameters [[26, 16, d ≥ 3]] 9 , [[26, 14, d ≥ 4]] 9 , [[26, 12, d ≥ 5]] 9 ,
[[26, 4, d ≥ 9]] 9 , [[26, 2, d ≥ 10]] 9 , [[27, 17, d ≥ 3]] 9 , [[27, 15, d ≥ 4]] 9 ,
[[27, 13, d ≥ 5]] 9 , [[27, 11, d ≥ 6]] 9 , [[27, 9, d ≥ 7]] 9 , [[27, 7, d ≥ 8]] 9 , [[27, 5,
d ≥ 9]] 9 , [[27, 3, d ≥ 10]] 9 and [[27, 1, d ≥ 11]] 9 . Since the parameters among
these codes are different, we do not perform a fair comparison of such codes.
Summarizing the results obtained in this section: we have constructed several
families of quantum codes with good as well as asymptotically good parameters.
Great part of our quantum codes has large minimum distances when compared with
their corresponding code lengths. Additionally, they have relatively small Singleton defects. We have also shown how to obtain sequences of asymptotically good
quantum codes derived from t-point AG codes.
5.5 Quantum Synchronizable Codes
We provide here constructions of families of quantum synchronizable codes (QSCs)
by applying the method discovered by Fujiwara [42] (see also [44]). More precisely,
we show how to construct families of QSCs derived from the sum and intersection of cyclic codes, BCH codes and from the product of cyclic codes. All these
results presented here can be found in [61]. For more details with respect to quantum
synchronizable codes we refer the reader to [42].
Before presenting our constructions of QSCs, we will review this concept. The
main goal of frame synchronization in communication systems is to ensure that
information block boundaries can be correctly determined at the receiver. To achieve
this goal, numerous synchronization techniques have been developed for classical
communication systems. However, these techniques are not applicable to quantum
communication systems since a qubit measurement typically destroys the quantum
states and thus also the corresponding quantum information. To circumvent this