136
6 Asymmetric Quantum Codes
and
g 2 (x) =
i
M
(i)
(x),
where each M
(i)
(x) is the minimal polynomial of α
i such that i /
∈ {14, 40, 41}, and
i runs through the coset representatives mod 80.
The sequence 0, 1, 2, 3, 4, 5, 6 belongs to the defining set of C 1 ; from the BCH
bound one has d 1 ≥ 8. Analogously, the sequence 40, 41, 42, 43 belongs to the defining set of code C which is generated by the polynomial h 2 (x) = (x
n
− 1)/g 2 (x).
Since C is equivalent to C
⊥
2 , from the BCH bound, C
⊥
2 has minimum distance greater
than or equal to 5. The cosets corresponding to the code C 1 are C 0 = {0}, C 1 =
{1, 3, 9, 27}, C 2 = {2, 6, 18, 54}, C 4 = {4, 12, 36, 28}, C 5 = {5, 15, 45, 55}. The
cosets corresponding to C 2 are all cosets except the cosets C 14 = {14, 42, 46, 58},
C 40 = {40}, C 41 = {41, 43, 49, 67}. Hence, C 1 has dimension k 1 = 63 and C 2 has
dimension k 2 = 9, which means that the dimension of the quantum code is equal to
k 1 − k 2 = 54. Therefore, an [[80, 54, d z ≥ 8/d x ≥ 5]] 3 AQECC can be constructed.
Similarly, an [[80, 58, d z ≥ 6/d x ≥ 5]] 3 AQECC can be also constructed.
Example 6.4.2 Let us now consider C 1 and C 2 codes of length 124 over F 5 , generated, respectively, by the polynomials
g 1 (x) = M
(0)
(x)M
(1)
(x)M
(2)
(x)M
(3)
(x),
and
g 2 (x) =
i
M
(i)
(x),
where each M
(i)
(x) is the minimal polynomial of α
i such that i /
∈ {62, 63, 64},
and i runs through the coset representatives mod n = 124. Proceeding similarly as
above, an [[124, 107, d z ≥ 5/d x ≥ 4]] 5 asymmetric quantum code can be obtained.
Analogously, an [[124, 110, d z ≥ 5/d x ≥ 3]] 5 quantum code can be constructed and
so on.
6.4.1 Code Comparison
In this section, we compare the parameters of the asymmetric quantum BCH codes
constructed here with the best asymmetric CSS codes available in [3]. In order to do
this, let us recall a result shown in [3].
Theorem 6.4.1 ([3, Theorem 8]) Let q be a prime power and gcd(q, n) = 1,
with or d n (q) = m. Let C 1 and C 2 be two narrow-sense BCH codes of length
q
m/2
< n ≤ q
m
− 1 over F q with designed distances δ 1 and δ 2 in the range
2 ≤ δ 1 , δ 2 ≤ δ max = min{{nq
m/2
/(q
m
− 1), n} and δ 1 < δ
⊥
2 ≤ δ 2 < δ
⊥
1 . Assume
6 Asymmetric Quantum Codes
and
g 2 (x) =
i
M
(i)
(x),
where each M
(i)
(x) is the minimal polynomial of α
i such that i /
∈ {14, 40, 41}, and
i runs through the coset representatives mod 80.
The sequence 0, 1, 2, 3, 4, 5, 6 belongs to the defining set of C 1 ; from the BCH
bound one has d 1 ≥ 8. Analogously, the sequence 40, 41, 42, 43 belongs to the defining set of code C which is generated by the polynomial h 2 (x) = (x
n
− 1)/g 2 (x).
Since C is equivalent to C
⊥
2 , from the BCH bound, C
⊥
2 has minimum distance greater
than or equal to 5. The cosets corresponding to the code C 1 are C 0 = {0}, C 1 =
{1, 3, 9, 27}, C 2 = {2, 6, 18, 54}, C 4 = {4, 12, 36, 28}, C 5 = {5, 15, 45, 55}. The
cosets corresponding to C 2 are all cosets except the cosets C 14 = {14, 42, 46, 58},
C 40 = {40}, C 41 = {41, 43, 49, 67}. Hence, C 1 has dimension k 1 = 63 and C 2 has
dimension k 2 = 9, which means that the dimension of the quantum code is equal to
k 1 − k 2 = 54. Therefore, an [[80, 54, d z ≥ 8/d x ≥ 5]] 3 AQECC can be constructed.
Similarly, an [[80, 58, d z ≥ 6/d x ≥ 5]] 3 AQECC can be also constructed.
Example 6.4.2 Let us now consider C 1 and C 2 codes of length 124 over F 5 , generated, respectively, by the polynomials
g 1 (x) = M
(0)
(x)M
(1)
(x)M
(2)
(x)M
(3)
(x),
and
g 2 (x) =
i
M
(i)
(x),
where each M
(i)
(x) is the minimal polynomial of α
i such that i /
∈ {62, 63, 64},
and i runs through the coset representatives mod n = 124. Proceeding similarly as
above, an [[124, 107, d z ≥ 5/d x ≥ 4]] 5 asymmetric quantum code can be obtained.
Analogously, an [[124, 110, d z ≥ 5/d x ≥ 3]] 5 quantum code can be constructed and
so on.
6.4.1 Code Comparison
In this section, we compare the parameters of the asymmetric quantum BCH codes
constructed here with the best asymmetric CSS codes available in [3]. In order to do
this, let us recall a result shown in [3].
Theorem 6.4.1 ([3, Theorem 8]) Let q be a prime power and gcd(q, n) = 1,
with or d n (q) = m. Let C 1 and C 2 be two narrow-sense BCH codes of length
q
m/2
< n ≤ q
m
− 1 over F q with designed distances δ 1 and δ 2 in the range
2 ≤ δ 1 , δ 2 ≤ δ max = min{{nq
m/2
/(q
m
− 1), n} and δ 1 < δ
⊥
2 ≤ δ 2 < δ
⊥
1 . Assume