72
5 Quantum Code Constructions
Let us present some examples of how our construction works.
Example 5.1.7 Consider that q = 5 and n = 11; hence m = ord 11 (5) = 5. The 5cosets are given by C 0 = {0}, C 1 = {1, 5, 3, 4, 9} and C 2 = {2, 10, 6, 8, 7}. If C 1 is
the cyclic code with defining set C 1 , then C 1 is a dual-containing code with parameters
[11, 6, d ≥ 4] 5 . From Lemma 5.1.6, one can get an [[11, 1, d ≥ 4]] 5 code.
Let us now take q = 17 and n = 19; so m = ord 19 (17) = 9. If C 1 is the code with
defining set C 1 = {1, 17, 4, 11, 16, 6, 7, 5, 9} we obtain an [[19, 1, d ≥ 5]] 17 code.
Similarly, we can construct an [61, 56, d ≥ 3] 9 code C 2 with defining set C 8 =
{8, 11, 38, 37, 28}. We know that C 2 is a dual-containing code, so an [[61, 51, d ≥
3]] 9 quantum code exists.
We can also construct an [67, 64, d ≥ 3] 29 dual-containing code with defining set
C 12 = {12, 13, 42}. Hence, there exists an [[67, 61, d ≥ 3]] 29 quantum code.
The existence of an [35, 31, d ≥ 3] 13 dual-containing code generates an [[35, 27,
d ≥ 3]] 13 quantum code. An [35, 31, d ≥ 3] 27 dual-containing code with defining
set C 3 = {3, 11, 17, 4} guarantees the existence of an [[35, 27, d ≥ 3]] 27 quantum
code. An [73, 70, d ≥ 3] 64 dual-containing code with defining set C 21 = {22, 21, 30}
exists, so there exists an [[73, 67, d ≥ 3]] 64 quantum code.
Example 5.1.8 In this example, we construct cyclic codes whose defining set consists of two q-cosets (the idea is the same as that presented in Theorem 5.1.10).
An [35, 27, d ≥ 4] 27 dual-containing code C with defining set consisting of C 2 and
C 3 ensures the existence of an [[35, 19, d ≥ 4]] 27 quantum code. Taking the cosets
C 14 = {14, 20, 30} and C 21 one has an [[73, 61, d ≥ 4]] 64 quantum code. Similarly,
an [[63, 51, d ≥ 3]] 11 quantum code (coset C 43 ) and an [[63, 39, d ≥ 4]] 11 code
(cosets C 43 and C 20 ) can be constructed. Analogously, an [[63, 51, d ≥ 3]] 23 and an
[[63, 45, d ≥ 4]] 23 quantum code (cosets C 4 and C 27 ) can be constructed.
5.1.6 Code Comparison
In this section, we compare the parameters of our quantum BCH codes with the ones
available in the literature. The codes available in the literature derived from Steane’s
code construction are generated by the same method presented in [148, Table I] by
considering the criterion for classical Euclidean dual-containing BCH codes given
in [4, Theorems 3 and 5].
Let us fix the notation:
• [[n, k, d]] q are the parameters of the new quantum codes;
• [[n
, k
, d
]] q = [[n
, n
− 2m((δ − 1)(1 − 1/q)), d
≥ δ]] q are the parameters
of quantum codes available in [4];
• [[n
, k
, d
]] q are the parameters of quantum BCH codes derived from Steane’s
code construction shown in [62, Corollary 4].
5 Quantum Code Constructions
Let us present some examples of how our construction works.
Example 5.1.7 Consider that q = 5 and n = 11; hence m = ord 11 (5) = 5. The 5cosets are given by C 0 = {0}, C 1 = {1, 5, 3, 4, 9} and C 2 = {2, 10, 6, 8, 7}. If C 1 is
the cyclic code with defining set C 1 , then C 1 is a dual-containing code with parameters
[11, 6, d ≥ 4] 5 . From Lemma 5.1.6, one can get an [[11, 1, d ≥ 4]] 5 code.
Let us now take q = 17 and n = 19; so m = ord 19 (17) = 9. If C 1 is the code with
defining set C 1 = {1, 17, 4, 11, 16, 6, 7, 5, 9} we obtain an [[19, 1, d ≥ 5]] 17 code.
Similarly, we can construct an [61, 56, d ≥ 3] 9 code C 2 with defining set C 8 =
{8, 11, 38, 37, 28}. We know that C 2 is a dual-containing code, so an [[61, 51, d ≥
3]] 9 quantum code exists.
We can also construct an [67, 64, d ≥ 3] 29 dual-containing code with defining set
C 12 = {12, 13, 42}. Hence, there exists an [[67, 61, d ≥ 3]] 29 quantum code.
The existence of an [35, 31, d ≥ 3] 13 dual-containing code generates an [[35, 27,
d ≥ 3]] 13 quantum code. An [35, 31, d ≥ 3] 27 dual-containing code with defining
set C 3 = {3, 11, 17, 4} guarantees the existence of an [[35, 27, d ≥ 3]] 27 quantum
code. An [73, 70, d ≥ 3] 64 dual-containing code with defining set C 21 = {22, 21, 30}
exists, so there exists an [[73, 67, d ≥ 3]] 64 quantum code.
Example 5.1.8 In this example, we construct cyclic codes whose defining set consists of two q-cosets (the idea is the same as that presented in Theorem 5.1.10).
An [35, 27, d ≥ 4] 27 dual-containing code C with defining set consisting of C 2 and
C 3 ensures the existence of an [[35, 19, d ≥ 4]] 27 quantum code. Taking the cosets
C 14 = {14, 20, 30} and C 21 one has an [[73, 61, d ≥ 4]] 64 quantum code. Similarly,
an [[63, 51, d ≥ 3]] 11 quantum code (coset C 43 ) and an [[63, 39, d ≥ 4]] 11 code
(cosets C 43 and C 20 ) can be constructed. Analogously, an [[63, 51, d ≥ 3]] 23 and an
[[63, 45, d ≥ 4]] 23 quantum code (cosets C 4 and C 27 ) can be constructed.
5.1.6 Code Comparison
In this section, we compare the parameters of our quantum BCH codes with the ones
available in the literature. The codes available in the literature derived from Steane’s
code construction are generated by the same method presented in [148, Table I] by
considering the criterion for classical Euclidean dual-containing BCH codes given
in [4, Theorems 3 and 5].
Let us fix the notation:
• [[n, k, d]] q are the parameters of the new quantum codes;
• [[n
, k
, d
]] q = [[n
, n
− 2m((δ − 1)(1 − 1/q)), d
≥ δ]] q are the parameters
of quantum codes available in [4];
• [[n
, k
, d
]] q are the parameters of quantum BCH codes derived from Steane’s
code construction shown in [62, Corollary 4].