114
5 Quantum Code Constructions
where D i ≥ min{N i − 2 − (b 1 (i) + b 2 (i)), a 1 (i) + a 2 (i) − (2g i − 2)}.
We know that the K i ’s assume all the values from 1 to N i − 2g i − 2, i.e., 0 < K i ≤
N i − 2g i − 2. For any such K i we set b 1 (i) + b 2 (i) = =(N i + 2g i + K i − 4)/2;
hence, it follows that N i − 2 − (b 1 (i) + b 2 (i)) ≥ a 1 (i) + a 2 (i) − (2g i − 2), where
a 1 (i) + a 2 (i) − (2g i − 2) ≥ (N i − K i − 2g i − 1)/2. The sequence of positive integers (K i ) i≥1 satisfies
0 < lim sup i→∞
K i
N i − 2
≤
≤ lim sup i→∞ N i /(N i − 2)
− lim sup i→∞ 2g i /(N i − 2)
+ lim sup i→∞ − 2/(N i − 2)
= 1 − 2/(
√
q − 1),
where in the last equality we utilize the fact that lim sup i→∞ N i /g i =
√
q − 1. For
each 0 < c < 1 − 2/(
√
q − 1), we can choose convenient values for K i such that
lim i→∞ K i /N i = c. Thus, lim sup i→∞ K i /(N i − 2) = c > 0. Moreover, we have
lim sup i→∞ (N i − K i − 2g i − 1)/2(N i − 2) = 1/2
1 − 2/(
√
q − 1) − c
> 0.
Therefore, there exists a sequence (Q i ) i≥1 of asymptotically good quantum codes
over F q . The proof is complete.
Remark 5.4.3 Since several works available in the literature already presented constructions of asymptotically good quantum codes derived from one-point AG codes
(see [12, 26, 27]), we do not present such constructions here.
Theorem 5.4.4 (t-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 t-point AG codes.
Proof We adopt the same notation as in the proof of Theorem 5.4.3. For each F i ,
let us consider the set of rational places P 1 (i), . . . , P n i (i), P n i +1 (i), . . . , P n i +t (i) of
F i , where N i = n i + t. Set D(i) = P 1 (i) + · · · + P n i (i), G 1 (i) = a 1 (i)P n i +1 (i) +
. . . + a t (i)P n i +t (i) and G 2 (i) = b 1 (i)P n i +1 (i) + . . . + b t (i)P n i +t (i), where a j (i) ≤
b j (i) for all j = 1, . . . , t, with 2g i − 2 <
t
j=1
a j (i) <
t
j=1
b j (i) < N i − t. Let us
consider the t-point AG codes C 1 (i) := C L (i)[D(i), G 1 (i)] and C 2 (i) := C L (i)
[D(i), G 2 (i)]. It follows that C 1 (i) ⊂ C 2 (i), and C 1 (i) has parameters
⎡
⎣ N i − t,
t
j=1
a j (i) − g i + 1, d 1 (i)
⎤
⎦
q
,
5 Quantum Code Constructions
where D i ≥ min{N i − 2 − (b 1 (i) + b 2 (i)), a 1 (i) + a 2 (i) − (2g i − 2)}.
We know that the K i ’s assume all the values from 1 to N i − 2g i − 2, i.e., 0 < K i ≤
N i − 2g i − 2. For any such K i we set b 1 (i) + b 2 (i) = =(N i + 2g i + K i − 4)/2;
hence, it follows that N i − 2 − (b 1 (i) + b 2 (i)) ≥ a 1 (i) + a 2 (i) − (2g i − 2), where
a 1 (i) + a 2 (i) − (2g i − 2) ≥ (N i − K i − 2g i − 1)/2. The sequence of positive integers (K i ) i≥1 satisfies
0 < lim sup i→∞
K i
N i − 2
≤
≤ lim sup i→∞ N i /(N i − 2)
− lim sup i→∞ 2g i /(N i − 2)
+ lim sup i→∞ − 2/(N i − 2)
= 1 − 2/(
√
q − 1),
where in the last equality we utilize the fact that lim sup i→∞ N i /g i =
√
q − 1. For
each 0 < c < 1 − 2/(
√
q − 1), we can choose convenient values for K i such that
lim i→∞ K i /N i = c. Thus, lim sup i→∞ K i /(N i − 2) = c > 0. Moreover, we have
lim sup i→∞ (N i − K i − 2g i − 1)/2(N i − 2) = 1/2
1 − 2/(
√
q − 1) − c
> 0.
Therefore, there exists a sequence (Q i ) i≥1 of asymptotically good quantum codes
over F q . The proof is complete.
Remark 5.4.3 Since several works available in the literature already presented constructions of asymptotically good quantum codes derived from one-point AG codes
(see [12, 26, 27]), we do not present such constructions here.
Theorem 5.4.4 (t-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 t-point AG codes.
Proof We adopt the same notation as in the proof of Theorem 5.4.3. For each F i ,
let us consider the set of rational places P 1 (i), . . . , P n i (i), P n i +1 (i), . . . , P n i +t (i) of
F i , where N i = n i + t. Set D(i) = P 1 (i) + · · · + P n i (i), G 1 (i) = a 1 (i)P n i +1 (i) +
. . . + a t (i)P n i +t (i) and G 2 (i) = b 1 (i)P n i +1 (i) + . . . + b t (i)P n i +t (i), where a j (i) ≤
b j (i) for all j = 1, . . . , t, with 2g i − 2 <
t
j=1
a j (i) <
t
j=1
b j (i) < N i − t. Let us
consider the t-point AG codes C 1 (i) := C L (i)[D(i), G 1 (i)] and C 2 (i) := C L (i)
[D(i), G 2 (i)]. It follows that C 1 (i) ⊂ C 2 (i), and C 1 (i) has parameters
⎡
⎣ N i − t,
t
j=1
a j (i) − g i + 1, d 1 (i)
⎤
⎦
q
,