5.3 BCH Codes—Part III
105
5.3.4 Code Comparison
As usual, we compare here the parameters of our codes with the codes displayed
in the literature. In Table 5.12, our Hermitian quantum codes have parameters
[[n, n − 4(c − 2) − 2, d ≥ c]] q , where 3 ≤ c ≤ q
2 and n = q
4
− 1; [[n
, k
, d
]] q =
[[n
, n
− 2m(δ − 1)(1 − 1/q
2
), d
≥ δ]] q are the parameters of the Hermitian
quantum codes shown in Theorem 21 in [4], where m = ord n (q
2
) = 2 and 2 ≤ δ ≤
n(q
m
− 1)/(q
2m
− 1).
In Table 5.13, our quantum codes are obtained from Sect. 5.3.2 and have parameters [[n, k, d]] q given by
• [[n, n − 2mc − 2, d ≥ c + 2]] q , where 1 ≤ c < q
2
− 1;
• [[n, n − 2m(q
2
− 1) − 2, d ≥ q
2
+ 2]] q ;
• [[n, n − 2m(c − 1) − 2, d ≥ c + 2]] q , where q
2
+ 1 ≤ c ≤ 2q
2
− 2.
• [[n, n − 4m(q
2
− 1) − 2, d ≥ 2q
2
+ 2]] q , where n = q
2m
− 1, q ≥ 4 is a prime
power, m = ord n (q
2
) ≥ 3;
[[n
, k
, d
]] q = [[n
, n
− 2m(δ − 1)(1 − 1/q
2
), d
≥ δ]] q are the parameters
of the Hermitian quantum codes shown in Theorem 21 in [4], where m = ord n (q
2
) ≥
3 and 2 ≤ δ ≤ ≤n(q
m
− 1)/(q
2m
− 1).
In Table 5.14, our codes are derived from Sect. 5.3.3 and have parameters [[n, k, d]] q
given by
• [[n, n − m(2c − 1) − 2, d ≥ c + 2]] q , where 1 ≤ c ≤ q − 2;
• [[n, n − m(2q − 3) − 2, d ≥ q + 1]] q ;
• [[n, n − m(2q − 1) − 1, d ≥ q + 3]] q ;
• [[n, n − m(2c − 4) − 2, d ≥ c + 2]] q ;
• [[n, n − m(4q − 8) − 2, d ≥ 2q]] q ;
• [[n, n − m(4q − 5) − 2, d ≥ 2q + 2]] q , where q + 1 < c < 2q − 2.
The parameters [[n
, k
, d
]] q are the parameters of quantum BCH codes derived
from q-ary Steane’s construction (see Corollary 5.1.2) applied to narrow-sense BCH
codes. These codes were obtained by the same method presented in Table I in [148]
by considering the criterion for classical Euclidean dual-containing BCH codes of
Theorems 3 and 5 in [4].
As we can see in Tables 5.12, 5.13 and 5.14, according to the procedure described
in Remark 5.1.2, our quantum codes have parameters better than the ones exhibited
in the literature.
5.4 Algebraic Geometry Codes
In this section, we construct several families of quantum codes with good and asymptotically good parameters. These quantum codes are derived from (classical) t-point
(t ≥ 1) algebraic geometry (AG) codes by applying the CSS construction. More
105
5.3.4 Code Comparison
As usual, we compare here the parameters of our codes with the codes displayed
in the literature. In Table 5.12, our Hermitian quantum codes have parameters
[[n, n − 4(c − 2) − 2, d ≥ c]] q , where 3 ≤ c ≤ q
2 and n = q
4
− 1; [[n
, k
, d
]] q =
[[n
, n
− 2m(δ − 1)(1 − 1/q
2
), d
≥ δ]] q are the parameters of the Hermitian
quantum codes shown in Theorem 21 in [4], where m = ord n (q
2
) = 2 and 2 ≤ δ ≤
n(q
m
− 1)/(q
2m
− 1).
In Table 5.13, our quantum codes are obtained from Sect. 5.3.2 and have parameters [[n, k, d]] q given by
• [[n, n − 2mc − 2, d ≥ c + 2]] q , where 1 ≤ c < q
2
− 1;
• [[n, n − 2m(q
2
− 1) − 2, d ≥ q
2
+ 2]] q ;
• [[n, n − 2m(c − 1) − 2, d ≥ c + 2]] q , where q
2
+ 1 ≤ c ≤ 2q
2
− 2.
• [[n, n − 4m(q
2
− 1) − 2, d ≥ 2q
2
+ 2]] q , where n = q
2m
− 1, q ≥ 4 is a prime
power, m = ord n (q
2
) ≥ 3;
[[n
, k
, d
]] q = [[n
, n
− 2m(δ − 1)(1 − 1/q
2
), d
≥ δ]] q are the parameters
of the Hermitian quantum codes shown in Theorem 21 in [4], where m = ord n (q
2
) ≥
3 and 2 ≤ δ ≤ ≤n(q
m
− 1)/(q
2m
− 1).
In Table 5.14, our codes are derived from Sect. 5.3.3 and have parameters [[n, k, d]] q
given by
• [[n, n − m(2c − 1) − 2, d ≥ c + 2]] q , where 1 ≤ c ≤ q − 2;
• [[n, n − m(2q − 3) − 2, d ≥ q + 1]] q ;
• [[n, n − m(2q − 1) − 1, d ≥ q + 3]] q ;
• [[n, n − m(2c − 4) − 2, d ≥ c + 2]] q ;
• [[n, n − m(4q − 8) − 2, d ≥ 2q]] q ;
• [[n, n − m(4q − 5) − 2, d ≥ 2q + 2]] q , where q + 1 < c < 2q − 2.
The parameters [[n
, k
, d
]] q are the parameters of quantum BCH codes derived
from q-ary Steane’s construction (see Corollary 5.1.2) applied to narrow-sense BCH
codes. These codes were obtained by the same method presented in Table I in [148]
by considering the criterion for classical Euclidean dual-containing BCH codes of
Theorems 3 and 5 in [4].
As we can see in Tables 5.12, 5.13 and 5.14, according to the procedure described
in Remark 5.1.2, our quantum codes have parameters better than the ones exhibited
in the literature.
5.4 Algebraic Geometry Codes
In this section, we construct several families of quantum codes with good and asymptotically good parameters. These quantum codes are derived from (classical) t-point
(t ≥ 1) algebraic geometry (AG) codes by applying the CSS construction. More