5.4 Algebraic Geometry Codes
113
5.4.2.3 Construction III
In this subsection, we propose constructions of sequences of asymptotically good
quantum codes derived from AG codes.
Recall that a tower of function fields (see Definition 1.3 of [47]) over F q is a
sequence T = (F 1 , F 2 , . . .) of function fields F i /F q with the following properties:
(1) F 1 ⊆ F 2 ⊆ F 3 · · · .
(2) For each n ≥ 1, the extension F n+1 /F n is separable of degree [F n+1 : F n ] > 1.
(3) g(F j ) > 1, for some j > 1.
By the Hurwitz genus formula, the condition (3) implies that g(F n ) → ∞
for n → ∞. The tower is said to be asymptotically good if λ(T ) = lim sup i→∞
N (F i )/g(F i ) > 0, where N (F i ) and g(F i ) denote the number of F q -rational points
and the genus of F i , respectively. In the case of tower of function fields one can
replace lim sup i→∞ N (F i )/g(F i ) by lim i→∞ N (F i )/g(F i ), because the sequence
(N (F i )/g(F i )) i≥1 is convergent. We say that the tower T (over F q ) attains the
Drinfeld-Vladut bound if λ(T ) = lim sup i→∞ N (F i )/g(F i ) =
√
q − 1. To simplify
the notation we put N (F i ) = N i and g(F i ) = g i .
Let (Q i ) i≥1 be a sequence of quantum codes over F q with parameters [[n i , k i ,
d i ]] q , respectively. We say that (Q i ) i≥1 is asymptotically good if lim sup i→∞ k i /n i >
0 and lim sup i→∞ d i /n i > 0. The next result shows how to construct asymptotically
good quantum codes derived from classical two-point AG codes.
Theorem 5.4.3 (Two-point asymptotically good codes) Assume that the tower T =
(F 1 , F 2 , . . .) of function fields over F q attains the Drinfeld-Vladut bound. Then there
exists a sequence (Q i ) i≥1 of asymptotically good quantum codes, over F q , derived
from classical two-point AG codes.
Proof For each F i , let us consider the set of rational places P 1 (i), . . . , P N i −2 (i),
P N i −1 (i), P N i (i) of F i . We set the divisors D(i) = P 1 (i) + · · · + P N i −2 (i), G 1 (i) =
a 1 (i)P N i −1 (i) + a 2 (i)P N i (i) and G 2 (i) = b 1 (i)P N i −1 (i) + b 2 (i)P N i (i), where
a 1 (i) ≤ b 1 (i) and a 2 (i) ≤ b 2 (i), with 2g i − 2 < a 1 (i) + a 2 (i) < b 1 (i) + b 2 (i) <
N i − 2. Let C 1 (i) := C L (i)[D(i), G 1 (i)] and C 2 (i) := C L (i)[D(i), G 2 (i)] be the
two-point AG codes, over F q , corresponding to G 1 (i) and G 2 (i), respectively; hence,
C 1 (i) ⊂ C 2 (i). The code C 1 (i) has parameters
[N i − 2, a 1 (i) + a 2 (i) − g i + 1, d 1 (i)] q ,
where d 1 (i) ≥ N i − 2 − (a 1 (i) + a 2 (i)), and C 2 (i) has parameters
[N i − 2, b 1 (i) + b 2 (i) − g i + 1, d 2 (i)] q ,
where d 2 (i) ≥ N i − 2 − (b 1 (i) + b 2 (i)). Therefore, the corresponding CSS code has
parameters
[[N i − 2, K i = b 1 (i) + b 2 (i) − (a 1 (i) + a 2 (i)), D i ]] q ,
113
5.4.2.3 Construction III
In this subsection, we propose constructions of sequences of asymptotically good
quantum codes derived from AG codes.
Recall that a tower of function fields (see Definition 1.3 of [47]) over F q is a
sequence T = (F 1 , F 2 , . . .) of function fields F i /F q with the following properties:
(1) F 1 ⊆ F 2 ⊆ F 3 · · · .
(2) For each n ≥ 1, the extension F n+1 /F n is separable of degree [F n+1 : F n ] > 1.
(3) g(F j ) > 1, for some j > 1.
By the Hurwitz genus formula, the condition (3) implies that g(F n ) → ∞
for n → ∞. The tower is said to be asymptotically good if λ(T ) = lim sup i→∞
N (F i )/g(F i ) > 0, where N (F i ) and g(F i ) denote the number of F q -rational points
and the genus of F i , respectively. In the case of tower of function fields one can
replace lim sup i→∞ N (F i )/g(F i ) by lim i→∞ N (F i )/g(F i ), because the sequence
(N (F i )/g(F i )) i≥1 is convergent. We say that the tower T (over F q ) attains the
Drinfeld-Vladut bound if λ(T ) = lim sup i→∞ N (F i )/g(F i ) =
√
q − 1. To simplify
the notation we put N (F i ) = N i and g(F i ) = g i .
Let (Q i ) i≥1 be a sequence of quantum codes over F q with parameters [[n i , k i ,
d i ]] q , respectively. We say that (Q i ) i≥1 is asymptotically good if lim sup i→∞ k i /n i >
0 and lim sup i→∞ d i /n i > 0. The next result shows how to construct asymptotically
good quantum codes derived from classical two-point AG codes.
Theorem 5.4.3 (Two-point asymptotically good codes) Assume that the tower T =
(F 1 , F 2 , . . .) of function fields over F q attains the Drinfeld-Vladut bound. Then there
exists a sequence (Q i ) i≥1 of asymptotically good quantum codes, over F q , derived
from classical two-point AG codes.
Proof For each F i , let us consider the set of rational places P 1 (i), . . . , P N i −2 (i),
P N i −1 (i), P N i (i) of F i . We set the divisors D(i) = P 1 (i) + · · · + P N i −2 (i), G 1 (i) =
a 1 (i)P N i −1 (i) + a 2 (i)P N i (i) and G 2 (i) = b 1 (i)P N i −1 (i) + b 2 (i)P N i (i), where
a 1 (i) ≤ b 1 (i) and a 2 (i) ≤ b 2 (i), with 2g i − 2 < a 1 (i) + a 2 (i) < b 1 (i) + b 2 (i) <
N i − 2. Let C 1 (i) := C L (i)[D(i), G 1 (i)] and C 2 (i) := C L (i)[D(i), G 2 (i)] be the
two-point AG codes, over F q , corresponding to G 1 (i) and G 2 (i), respectively; hence,
C 1 (i) ⊂ C 2 (i). The code C 1 (i) has parameters
[N i − 2, a 1 (i) + a 2 (i) − g i + 1, d 1 (i)] q ,
where d 1 (i) ≥ N i − 2 − (a 1 (i) + a 2 (i)), and C 2 (i) has parameters
[N i − 2, b 1 (i) + b 2 (i) − g i + 1, d 2 (i)] q ,
where d 2 (i) ≥ N i − 2 − (b 1 (i) + b 2 (i)). Therefore, the corresponding CSS code has
parameters
[[N i − 2, K i = b 1 (i) + b 2 (i) − (a 1 (i) + a 2 (i)), D i ]] q ,