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5 Quantum Code Constructions
5.4.2.1 Construction I
In the first result we utilize two t-point (t ≥ 1) AG codes to construct quantum codes
with good parameters.
Theorem 5.4.1 (General t-point construction, t ≥ 1) Let q be a prime power and
F/F q be an algebraic function field of genus g, with n + t pairwise distinct rational
places. Assume that a i , b i , i = 1, . . . , t, are positive integers such that a i ≤ b i for
all i, and 2g − 2 <
t
i=1
a i <
t
i=1
b i < n. Then there exists an [[n, k, d]] q quantum
code, where k =
t
i=1
b i −
t
i=1
a i and d ≥ min
n −
t
i=1
b i ,
t
i=1
a i − (2g − 2)
.
Proof Let {P 1 , P 2 , . . . , P n , P n+1 , . . . , P n+t } be the set of places of F/F q of degree
one. Let D = P 1 + . . . + P n be a divisor of F/F q . Assume also that G 1 and G 2 are
two divisors of F/F q given, respectively, by G 1 = a 1 P n+1 + · · · + a t P n+t and G 2 =
b 1 P n+1 + · · · + b t P n+t , where a i ≤ b i for all i = 1, . . . , t and 2g − 2 <
t
i=1
a i <
t
i=1
b i < n. From construction, supp G 1 ∩ supp D = ∅ and supp G 2 ∩ supp D = ∅.
Since G 1 ≤ G 2 , we have L(G 1 ) ⊂ L(G 2 ); hence, C L (D, G 1 ) ⊂ C L (D, G 2 ). From
Theorem 4.5.1, the code C 1 := C L (D, G 1 ) has parameters [n, k 1 , d 1 ] q , where d 1 ≥
n −
t
i=1
a i and k 1 =
t
i=1
a i − g + 1; the code C 2 := C L (D, G 2 ) has parameters
[n, k 2 , d 2 ] q , where d 2 ≥ n −
t
i=1
b i and k 2 =
t
i=1
b i − g + 1. On the other hand,
from Theorems 4.5.2 and 4.5.3, the dual code C
⊥
1 = C (D, G 1 ) of C 1 has parameters [n, k
⊥
1 , d
⊥
1 ] q , where d
⊥
1 ≥
t
i=1
a i − (2g − 2) and k
⊥
1 = n + g − 1 −
t
i=1
a i ;
the dual code C
⊥
2 = C (D, G 2 ) of C 2 has parameters [n, k
⊥
2 , d
⊥
2 ] q , with d
⊥
2 ≥
t
i=1
b i − (2g − 2) and k
⊥
2 = n + g − 1 −
t
i=1
b i .
Applying the CSS construction to C 1 and C 2 , we obtain an [[n, k, d]] q code,
with k = k 2 − k 1 = (
t
i=1
b i − g + 1) − (
t
i=1
a i − g + 1) =
t
i=1
b i −
t
i=1
a i and d ≥
min{d 2 , d
⊥
1 }, where d 2 ≥ n −
t
i=1
b i and d
⊥
1 ≥
t
i=1
a i − (2g − 2). The proof is
complete.