5.4 Algebraic Geometry Codes
115
where d 1 (i) ≥ N i − t −
t
j=1
a j (i). Moreover, C 2 (i) has parameters
⎡
⎣ N i − t,
t
j=1
b j (i) − g i + 1, d 2 (i)
⎤
⎦
q
,
where d 2 (i) ≥ N i − t −
t
j=1
b j (i).
Setting
t
j=1
b j (i) = =(N i + 2g i + K i − t − 2)/2 and proceeding similarly as in
the proof of Theorem 5.4.3, the result follows.
Let q be a prime power and let C be an [n, k, d] q m code over F q m . Let β be a basis
of F q m over F q , and assume also that β
⊥ is a dual basis of β. Let C
⊥ be the Euclidean
dual of C. Then we have [β(C)]
⊥
= β
⊥
(C
⊥
) (see for instance [56, 94]).
Theorem 5.4.5 For any prime p, there exists a sequence (Q i ) i≥1 of asymptotically
good quantum codes over F p .
Proof Let q
2
= p
2r , p prime. Let us consider the tower of function fields T =
(F 1 , F 2 , . . .) over F q 2 , shown in [47], defined by F t = F q 2 (x 1 , . . . , x t ), where
x
q
i+1 + x i+1 = x
q
i /(x
q−1
i
+ 1),
for i = 1, . . . t − 1. This tower attains the Drinfeld-Vladut bound. We next expand
the codes C 1 (i) and C 2 (i), shown in the proof of Theorem 5.4.3, with respect to some
basis β of F q 2 over F p . Thus, we obtain codes β(C 1 (i)) and β(C 2 (i)), both over F p ,
with parameters
[2r (N i − 2), 2r (a 1 (i) + a 2 (i) − g i + 1), d
∗
1 (i)] p ,
where d
∗
1 (i) ≥ d 1 (i) ≥ N i − 2 − (a 1 (i) + a 2 (i)), and
[2r (N i − 2), 2r (b 1 (i) + b 2 (i) − g i + 1), d
∗
2 (i)] p ,
where d
∗
2 (i) ≥ d 2 (i) ≥ N i − 2 − (b 1 (i) + b 2 (i)), respectively. Since the inclusion
β(C 1 (i)) ⊂ β(C 2 (i)) holds, we apply the CSS construction to these codes, obtaining
therefore an
[[2r (N i − 2), 2r K i , D i ]] p
quantum code, where D i ≥ min{N i − 2 − (b 1 (i) + b 2 (i)), a 1 (i) + a 2 (i) − (2g i −
2)} (note that since [β(C 1 (i))]
⊥
= β
⊥
(C 1 (i))
⊥ , it follows that the minimum distance
115
where d 1 (i) ≥ N i − t −
t
j=1
a j (i). Moreover, C 2 (i) has parameters
⎡
⎣ N i − t,
t
j=1
b j (i) − g i + 1, d 2 (i)
⎤
⎦
q
,
where d 2 (i) ≥ N i − t −
t
j=1
b j (i).
Setting
t
j=1
b j (i) = =(N i + 2g i + K i − t − 2)/2 and proceeding similarly as in
the proof of Theorem 5.4.3, the result follows.
Let q be a prime power and let C be an [n, k, d] q m code over F q m . Let β be a basis
of F q m over F q , and assume also that β
⊥ is a dual basis of β. Let C
⊥ be the Euclidean
dual of C. Then we have [β(C)]
⊥
= β
⊥
(C
⊥
) (see for instance [56, 94]).
Theorem 5.4.5 For any prime p, there exists a sequence (Q i ) i≥1 of asymptotically
good quantum codes over F p .
Proof Let q
2
= p
2r , p prime. Let us consider the tower of function fields T =
(F 1 , F 2 , . . .) over F q 2 , shown in [47], defined by F t = F q 2 (x 1 , . . . , x t ), where
x
q
i+1 + x i+1 = x
q
i /(x
q−1
i
+ 1),
for i = 1, . . . t − 1. This tower attains the Drinfeld-Vladut bound. We next expand
the codes C 1 (i) and C 2 (i), shown in the proof of Theorem 5.4.3, with respect to some
basis β of F q 2 over F p . Thus, we obtain codes β(C 1 (i)) and β(C 2 (i)), both over F p ,
with parameters
[2r (N i − 2), 2r (a 1 (i) + a 2 (i) − g i + 1), d
∗
1 (i)] p ,
where d
∗
1 (i) ≥ d 1 (i) ≥ N i − 2 − (a 1 (i) + a 2 (i)), and
[2r (N i − 2), 2r (b 1 (i) + b 2 (i) − g i + 1), d
∗
2 (i)] p ,
where d
∗
2 (i) ≥ d 2 (i) ≥ N i − 2 − (b 1 (i) + b 2 (i)), respectively. Since the inclusion
β(C 1 (i)) ⊂ β(C 2 (i)) holds, we apply the CSS construction to these codes, obtaining
therefore an
[[2r (N i − 2), 2r K i , D i ]] p
quantum code, where D i ≥ min{N i − 2 − (b 1 (i) + b 2 (i)), a 1 (i) + a 2 (i) − (2g i −
2)} (note that since [β(C 1 (i))]
⊥
= β
⊥
(C 1 (i))
⊥ , it follows that the minimum distance