6.3 Asymmetric BCH Codes
135
[[26, 16, d z ≥ 4/d x ≥ 3]] 3 .
Proof Let C 1 = [n, k 1 , d 1 ] 3 be the BCH code generated by
C 1 = =g 1 (x) = =M
(0)
(x)M
(1)
(x)M
(2)
(x)
and let C 2 = [n, k 2 , d 2 ] 3 be the cyclic code generated by
i
M
(i)
(x),
where each M
(i)
(x) is the minimal polynomial of α
i such that i /
∈ {5, 14}, and i runs
through the coset representatives mod 26.
The sequence 0, 1, 2, 3 belongs to the defining set of C 1 ; from the BCH bound,
it follows that d 1 ≥ 5. Moreover, it follows that k 1 = 19. The sequence 14, 15, 16
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, it follows that
C
⊥
2 has minimum distance d
⊥
2 ≥ 4. Moreover, C 2 has dimension k 2 = 6. Therefore,
an [[26, 13, d z ≥ 5/d x ≥ 4]] 3 AQECC can be constructed.
Analogously, let us consider C 1 generated by
g 1 (x) = =M
(0)
(x)M
(1)
(x)M
(2)
(x)
and C 2 generated by
i
M
(i)
(x), where each M
(i)
(x) is the minimal polynomial of
α
i such that i /
∈ {13, 14}, and i runs through the coset representatives mod 26. Then
one has an [[26, 15, d z ≥ 5/d x ≥ 3]] 3 AQECC.
Furthermore, if C 1 = =g 1 (x) = =M
(1)
(x)M
(2)
(x) and if C 2 is generated by
i
M
(i)
(x), where each M
(i)
(x) is the minimal polynomial of α
i such that i /
∈
{13, 14}, and i runs through the coset representatives mod 26, then an
[[26, 16, d z ≥ 4/d x ≥ 3]] 3 asymmetric quantum BCH code can be constructed.
6.4 Examples
In this section, we present illustrative examples to show how the proposed construction works.
Example 6.4.1 Let C 1 be the BCH code and C 2 be the cyclic code both of length
80 over F 3 , generated, respectively, by the polynomials
g 1 = M
(0)
(x)M
(1)
(x)M
(2)
(x)M
(4)
(x)M
(5)
(x),
135
[[26, 16, d z ≥ 4/d x ≥ 3]] 3 .
Proof Let C 1 = [n, k 1 , d 1 ] 3 be the BCH code generated by
C 1 = =g 1 (x) = =M
(0)
(x)M
(1)
(x)M
(2)
(x)
and let C 2 = [n, k 2 , d 2 ] 3 be the cyclic code generated by
i
M
(i)
(x),
where each M
(i)
(x) is the minimal polynomial of α
i such that i /
∈ {5, 14}, and i runs
through the coset representatives mod 26.
The sequence 0, 1, 2, 3 belongs to the defining set of C 1 ; from the BCH bound,
it follows that d 1 ≥ 5. Moreover, it follows that k 1 = 19. The sequence 14, 15, 16
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, it follows that
C
⊥
2 has minimum distance d
⊥
2 ≥ 4. Moreover, C 2 has dimension k 2 = 6. Therefore,
an [[26, 13, d z ≥ 5/d x ≥ 4]] 3 AQECC can be constructed.
Analogously, let us consider C 1 generated by
g 1 (x) = =M
(0)
(x)M
(1)
(x)M
(2)
(x)
and C 2 generated by
i
M
(i)
(x), where each M
(i)
(x) is the minimal polynomial of
α
i such that i /
∈ {13, 14}, and i runs through the coset representatives mod 26. Then
one has an [[26, 15, d z ≥ 5/d x ≥ 3]] 3 AQECC.
Furthermore, if C 1 = =g 1 (x) = =M
(1)
(x)M
(2)
(x) and if C 2 is generated by
i
M
(i)
(x), where each M
(i)
(x) is the minimal polynomial of α
i such that i /
∈
{13, 14}, and i runs through the coset representatives mod 26, then an
[[26, 16, d z ≥ 4/d x ≥ 3]] 3 asymmetric quantum BCH code can be constructed.
6.4 Examples
In this section, we present illustrative examples to show how the proposed construction works.
Example 6.4.1 Let C 1 be the BCH code and C 2 be the cyclic code both of length
80 over F 3 , generated, respectively, by the polynomials
g 1 = M
(0)
(x)M
(1)
(x)M
(2)
(x)M
(4)
(x)M
(5)
(x),