5.4 Algebraic Geometry Codes
107
Table 5.13 Code comparison
Our Hermitian codes
Hermitian codes in [4]
[[n, k, d]] q
[[n
, k
, d
]] q
m = 3, q = 4
[[4095, 4087, d ≥ 3]] 4
[[4095, 4083, d
≥ 3]] 4
[[4095, 4081, d ≥ 4]] 4
[[4095, 4077, d
≥ 4]] 4
[[4095, 4075, d ≥ 5]] 4
[[4095, 4071, d
≥ 5]] 4
[[4095, 4069, d ≥ 6]] 4
[[4095, 4065, d
≥ 6]] 4
[[4095, 4063, d ≥ 7]] 4
[[4095, 4059, d
≥ 7]] 4
[[4095, 4057, d ≥ 8]] 4
[[4095, 4053, d
≥ 8]] 4
[[4095, 4051, d ≥ 9]] 4
[[4095, 4047, d
≥ 9]] 4
[[4095, 4045, d ≥ 10]] 4
[[4095, 4041, d
≥ 10]] 4
[[4095, 4039, d ≥ 11]] 4
[[4095, 4035, d
≥ 11]] 4
[[4095, 4033d ≥ 12]] 4
[[4095, 4029, d
≥ 12]] 4
[[4095, 4027, d ≥ 13]] 4
[[4095, 4023, d
≥ 13]] 4
[[4095, 4021, d ≥ 14]] 4
[[4095, 4017, d
≥ 14]] 4
[[4095, 4015, d ≥ 15]] 4
[[4095, 4011, d
≥ 15]] 4
[[4095, 4009, d ≥ 16]] 4
[[4095, 4005, d
≥ 16]] 4
[[4095, 4003, d ≥ 18]] 4
[[4095, 3999, d
≥ 18]] 4
[[4095, 3997, d ≥ 19]] 4
[[4095, 3993, d
≥ 19]] 4
[[4095, 3949, d ≥ 27]] 4
[[4095, 3945, d
≥ 27]] 4
[[4095, 3925, d ≥ 31]] 4
[[4095, 3921, d
≥ 31]] 4
[[4095, 3919, d ≥ 32]] 4
[[4095, 3915, d
≥ 32]] 4
[[4095, 3913, d ≥ 34]] 4
[[4095, 3909, d
≥ 34]] 4
specifically, we construct two classical nested AG codes C 1 ⊂ C 2 , applying after the
CSS construction. Many of these codes have large minimum distances when compared with their code lengths, as well as they also have small Singleton defects. As
an example, we construct a family [[46, 2(t 2 − t 1 ), d]] 25 of quantum codes, where
t 1 , t 2 are positive integers where 1 < t 1 < t 2 < 23 and d ≥ min{46 − 2t 2 , 2t 1 − 2},
of length n = 46, with minimum distance in the range 2 ≤ d ≤ 20, having Singleton
defect at most four. Furthermore, by applying the CSS construction to sequences of
t-point classical AG codes constructed here, we obtain sequences of asymptotically
good quantum codes. The content presented here can be found in our paper [106].
As we know, methods and techniques of construction of quantum codes with good
parameters were extensively investigated in the literature [4, 25–27, 71, 73, 80, 89,
91, 94, 97, 100, 119, 147, 148]. Many of these works were performed by applying
one (or two or all of them) of the following techniques:
(1) the CSS construction based on linear Euclidean self-orthogonal codes or even
based on two nested linear codes [4, 25, 73, 80, 100];
107
Table 5.13 Code comparison
Our Hermitian codes
Hermitian codes in [4]
[[n, k, d]] q
[[n
, k
, d
]] q
m = 3, q = 4
[[4095, 4087, d ≥ 3]] 4
[[4095, 4083, d
≥ 3]] 4
[[4095, 4081, d ≥ 4]] 4
[[4095, 4077, d
≥ 4]] 4
[[4095, 4075, d ≥ 5]] 4
[[4095, 4071, d
≥ 5]] 4
[[4095, 4069, d ≥ 6]] 4
[[4095, 4065, d
≥ 6]] 4
[[4095, 4063, d ≥ 7]] 4
[[4095, 4059, d
≥ 7]] 4
[[4095, 4057, d ≥ 8]] 4
[[4095, 4053, d
≥ 8]] 4
[[4095, 4051, d ≥ 9]] 4
[[4095, 4047, d
≥ 9]] 4
[[4095, 4045, d ≥ 10]] 4
[[4095, 4041, d
≥ 10]] 4
[[4095, 4039, d ≥ 11]] 4
[[4095, 4035, d
≥ 11]] 4
[[4095, 4033d ≥ 12]] 4
[[4095, 4029, d
≥ 12]] 4
[[4095, 4027, d ≥ 13]] 4
[[4095, 4023, d
≥ 13]] 4
[[4095, 4021, d ≥ 14]] 4
[[4095, 4017, d
≥ 14]] 4
[[4095, 4015, d ≥ 15]] 4
[[4095, 4011, d
≥ 15]] 4
[[4095, 4009, d ≥ 16]] 4
[[4095, 4005, d
≥ 16]] 4
[[4095, 4003, d ≥ 18]] 4
[[4095, 3999, d
≥ 18]] 4
[[4095, 3997, d ≥ 19]] 4
[[4095, 3993, d
≥ 19]] 4
[[4095, 3949, d ≥ 27]] 4
[[4095, 3945, d
≥ 27]] 4
[[4095, 3925, d ≥ 31]] 4
[[4095, 3921, d
≥ 31]] 4
[[4095, 3919, d ≥ 32]] 4
[[4095, 3915, d
≥ 32]] 4
[[4095, 3913, d ≥ 34]] 4
[[4095, 3909, d
≥ 34]] 4
specifically, we construct two classical nested AG codes C 1 ⊂ C 2 , applying after the
CSS construction. Many of these codes have large minimum distances when compared with their code lengths, as well as they also have small Singleton defects. As
an example, we construct a family [[46, 2(t 2 − t 1 ), d]] 25 of quantum codes, where
t 1 , t 2 are positive integers where 1 < t 1 < t 2 < 23 and d ≥ min{46 − 2t 2 , 2t 1 − 2},
of length n = 46, with minimum distance in the range 2 ≤ d ≤ 20, having Singleton
defect at most four. Furthermore, by applying the CSS construction to sequences of
t-point classical AG codes constructed here, we obtain sequences of asymptotically
good quantum codes. The content presented here can be found in our paper [106].
As we know, methods and techniques of construction of quantum codes with good
parameters were extensively investigated in the literature [4, 25–27, 71, 73, 80, 89,
91, 94, 97, 100, 119, 147, 148]. Many of these works were performed by applying
one (or two or all of them) of the following techniques:
(1) the CSS construction based on linear Euclidean self-orthogonal codes or even
based on two nested linear codes [4, 25, 73, 80, 100];