5.4 Algebraic Geometry Codes
111
Remark 5.4.1 In [27, 81], the authors utilized one-point AG codes to construct
good/(asymptotically good) quantum codes. In [26], the author applied two-point
AG codes to derive good/(asymptotically good) quantum codes. Note that, in this
context, Theorem 5.4.1 is a natural generalization of the one-point as well as twopoint AG code construction to the t-point (t ≥ 1) AG code construction.
Corollary 5.4.1 (One-Point codes) There exists a quantum code with parameters [[q(1 + (q − 1)m), b − a, d]] q 2 , where (q − 1)(m − 1) − 2 < a < b < q(1 +
(q − 1)m), m|(q + 1) and d ≥ min{q(1 + (q − 1)m) − b, a − (q − 1)(m − 1) +
2}.
Proof Let F = F q 2 (x, y), where y
q
+ y = x
m and m|(q + 1). It is known the genus
of F is g = (q − 1)(m − 1)/2, and the number of places of degree one is equal to N =
1 + q(1 + (q − 1)m) (see the Example 6.4.2. of [152]). Let {P 1 , P 2 , . . . , P n , P n+1 ,
. . . , P N } be these pairwise distinct places. Without loss of generality, choose the
F q 2 -rational point P N . Let D = P 1 + · · · + P N −1 be a divisor and let G 1 = a P N and
G 2 = bP N be other two divisors such that supp G 1 ∩ supp D = ∅ and supp G 2 ∩
supp D = ∅, where (q − 1)(m − 1) − 2 < a < b < q(1 + (q − 1)m). From Theorem 5.4.1, there exists a quantum code with parameters [[q(1 + (q − 1)m), b − a, d]] q 2 ,
where d ≥ min{q(1 + (q − 1)m) − b, a − (q − 1)(m − 1) + 2}. The proof is complete.
Remark 5.4.2 Note that the Hermitian curve defined by y
q
+ y = x
q+1 , over F q 2 , is
a particular case of the curve y
q
+ y = x
m considered in the proof of Corollary 5.4.1.
Corollary 5.4.2 (Two-Point codes) There exists a quantum code with parameters [[q(1 + (q − 1)m) − 1, b 1 + b 2 − a 1 − a 2 , d]] q 2 , where a i ≤ b i for i = 1, 2,
(q − 1)(m − 1) − 2 < a 1 + a 2 < b 1 + b 2 < q[1 + (q − 1)m] − 1, m|(q + 1) and
d ≥ min{q[1 + (q − 1)m] − b 1 − b 2 − 1, a 1 + a 2 − (q − 1)(m − 1) + 2}.
Proof Let D = P 1 + · · · + P N −2 be a divisor and let G 1 = a 1 P N −2 + a 2 P N −1 and
G 2 = b 1 P N −2 + b 2 P N −1 be other two divisors with supp G 1 ∩ supp D = ∅ and
supp G 2 ∩ supp D = ∅, where (q − 1)(m − 1) − 2 < a 1 + a 2 < b 1 + b 2 < q(1 +
(q − 1)m) − 1. From Theorem 5.4.1, there exists an
[[q(1 + (q − 1)m) − 1, b 1 + b 2 − a 1 − a 2 , d]] q 2
code,
where
d ≥ min{q(1 + (q − 1)m) − 1 − b 1 − b 2 , a 1 + a 2 − (q − 1)
(m − 1) + 2}. This finishes the proof.
Corollary 5.4.3 (t-Point codes, t ≥ 2) There exists a quantum code with parameters [[q(1 + (q − 1)m) − t + 1, b 1 + · · · + b t − (a 1 + · · · + a t ), d]] q 2 , where a i ≤
b i for i = 1, . . . t, (q − 1)(m − 1) − 2 < a 1 + · · · + a t < b 1 + · · · + b t < q(1 +
(q − 1)m) − t + 1, m|(q + 1) and d ≥ min{q(1 + (q − 1)m) − (b 1 + · · · + b t ) −
t + 1, a 1 + · · · + a t − (q − 1)(m − 1) + 2}.
Proof Similar to that of Corollary 5.4.2.
111
Remark 5.4.1 In [27, 81], the authors utilized one-point AG codes to construct
good/(asymptotically good) quantum codes. In [26], the author applied two-point
AG codes to derive good/(asymptotically good) quantum codes. Note that, in this
context, Theorem 5.4.1 is a natural generalization of the one-point as well as twopoint AG code construction to the t-point (t ≥ 1) AG code construction.
Corollary 5.4.1 (One-Point codes) There exists a quantum code with parameters [[q(1 + (q − 1)m), b − a, d]] q 2 , where (q − 1)(m − 1) − 2 < a < b < q(1 +
(q − 1)m), m|(q + 1) and d ≥ min{q(1 + (q − 1)m) − b, a − (q − 1)(m − 1) +
2}.
Proof Let F = F q 2 (x, y), where y
q
+ y = x
m and m|(q + 1). It is known the genus
of F is g = (q − 1)(m − 1)/2, and the number of places of degree one is equal to N =
1 + q(1 + (q − 1)m) (see the Example 6.4.2. of [152]). Let {P 1 , P 2 , . . . , P n , P n+1 ,
. . . , P N } be these pairwise distinct places. Without loss of generality, choose the
F q 2 -rational point P N . Let D = P 1 + · · · + P N −1 be a divisor and let G 1 = a P N and
G 2 = bP N be other two divisors such that supp G 1 ∩ supp D = ∅ and supp G 2 ∩
supp D = ∅, where (q − 1)(m − 1) − 2 < a < b < q(1 + (q − 1)m). From Theorem 5.4.1, there exists a quantum code with parameters [[q(1 + (q − 1)m), b − a, d]] q 2 ,
where d ≥ min{q(1 + (q − 1)m) − b, a − (q − 1)(m − 1) + 2}. The proof is complete.
Remark 5.4.2 Note that the Hermitian curve defined by y
q
+ y = x
q+1 , over F q 2 , is
a particular case of the curve y
q
+ y = x
m considered in the proof of Corollary 5.4.1.
Corollary 5.4.2 (Two-Point codes) There exists a quantum code with parameters [[q(1 + (q − 1)m) − 1, b 1 + b 2 − a 1 − a 2 , d]] q 2 , where a i ≤ b i for i = 1, 2,
(q − 1)(m − 1) − 2 < a 1 + a 2 < b 1 + b 2 < q[1 + (q − 1)m] − 1, m|(q + 1) and
d ≥ min{q[1 + (q − 1)m] − b 1 − b 2 − 1, a 1 + a 2 − (q − 1)(m − 1) + 2}.
Proof Let D = P 1 + · · · + P N −2 be a divisor and let G 1 = a 1 P N −2 + a 2 P N −1 and
G 2 = b 1 P N −2 + b 2 P N −1 be other two divisors with supp G 1 ∩ supp D = ∅ and
supp G 2 ∩ supp D = ∅, where (q − 1)(m − 1) − 2 < a 1 + a 2 < b 1 + b 2 < q(1 +
(q − 1)m) − 1. From Theorem 5.4.1, there exists an
[[q(1 + (q − 1)m) − 1, b 1 + b 2 − a 1 − a 2 , d]] q 2
code,
where
d ≥ min{q(1 + (q − 1)m) − 1 − b 1 − b 2 , a 1 + a 2 − (q − 1)
(m − 1) + 2}. This finishes the proof.
Corollary 5.4.3 (t-Point codes, t ≥ 2) There exists a quantum code with parameters [[q(1 + (q − 1)m) − t + 1, b 1 + · · · + b t − (a 1 + · · · + a t ), d]] q 2 , where a i ≤
b i for i = 1, . . . t, (q − 1)(m − 1) − 2 < a 1 + · · · + a t < b 1 + · · · + b t < q(1 +
(q − 1)m) − t + 1, m|(q + 1) and d ≥ min{q(1 + (q − 1)m) − (b 1 + · · · + b t ) −
t + 1, a 1 + · · · + a t − (q − 1)(m − 1) + 2}.
Proof Similar to that of Corollary 5.4.2.