112
5 Quantum Code Constructions
5.4.2.2 Construction II
We construct here quantum codes derived from AG codes whose divisors are multiples of a non-rational divisor G. The first result is given in the following.
Theorem 5.4.2 (General construction) Let q be a prime power and let F/F q be
an algebraic function field of genus g, with n pairwise distinct rational places P i ,
i = 1, . . . , n. Assume that there exist pairwise distinct places Q 1 , . . . , Q t of F/F q , of
degree α i ≥ 2, respectively, i = 1, . . . , t, where t ≥ 1. Let G 1 =
t
i=1
a i Q i and G 2 =
t
i=1
b i Q i , where a i ≤ b i , for all i = 1, . . . , t, and 2g − 2 < a 1 α 1 + . . . + a t α t <
b 1 α 1 + . . . + b t α t < n. Let D = P 1 + · · · + P n be a divisor of F/F q , and consider
that supp G 1 ∩ supp D = ∅ and supp G 2 ∩ supp D = ∅. Then there exists a quantum
code with parameters [[n, k, d]] q , where k = (b 1 − a 1 )α 1 + . . . + (b t − a t )α t and
d ≥ min{n − (b 1 α 1 + . . . + b t α t ), (a 1 α 1 + . . . + a t α t ) − (2g − 2)}.
Proof Similar to that of Theorem 5.4.1.
Corollary 5.4.4 Let q be a prime power and let F/F q be a hyperelliptic function
field of genus g ≥ 2, with n pairwise distinct rational places. Then there exists an
[[n, 2(t 2 − t 1 ), d]] q code, where t 1 , t 2 are positive integers satisfying 2g − 2 < t 1 <
t 2 < n and d ≥ min{n − 2t 2 , 2t 1 − 2g + 2}.
Proof Since F is a hyperelliptic function field, there exists a place G of degree
two (see Lemma 6.2.2.(a) of [152]). Let D = P 1 + · · · + P n be a divisor, where
P i are all rational points of F. Let G 2 = t 2 G and G 1 = t 1 G, where 2g − 2 <
2t 1 < 2t 2 < n. We know that supp G 1 ∩ supp D = ∅, supp G 2 ∩ supp D = ∅ and
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 − 2t 1 and k 1 = 2t 1 − g + 1. The code C 2 :=
C L (D, G 2 ) has parameters [n, k 2 , d 2 ] q , where d 2 ≥ n − 2t 2 and k 2 = 2t 2 − g + 1.
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 ≥ 2t 1 − (2g − 2) and k
⊥
1 = n + g − 1 − 2t 1 . Analogously, the dual code C
⊥
2 = C (D, G 2 ) of C 2 has parameters [n, k
⊥
2 , d
⊥
2 ] q , where
d
⊥
2 ≥ 2t 2 − (2g − 2) and k
⊥
2 = n + g − 1 − 2t 2 .
Applying the CSS construction to C 1 and C 2 , we obtain an [[n, k, d]] q code, with
k = k 2 − k 1 = (2t 2 − g + 1) − (2t 1 − g + 1) = 2(t 2 − t 1 ) and d ≥ min{d 2 , d
⊥
1 },
where d 2 ≥ n − 2t 2 and d
⊥
1 ≥ 2t 1 − (2g − 2). The proof is complete.
Corollary 5.4.5 There exists a quantum code with parameters [[46, 2(t 2 − t 1 ), d]] 25 ,
where t 1 , t 2 are positive integers such that 1 < t 1 < t 2 < 23 and d ≥ min{46 −
2t 2 , 2t 1 − 2}.
Proof Let us consider the function field F = F q 2 (x, y) with y
q
+ y = x
m and
m|(q + 1); take m = 2 and q = 5. As the genus of F is g = 2, then F is a hyperelliptic function field (see Lemma 6.2.2.(b) of [152]), and the result follows from
Corollary 5.4.4.
5 Quantum Code Constructions
5.4.2.2 Construction II
We construct here quantum codes derived from AG codes whose divisors are multiples of a non-rational divisor G. The first result is given in the following.
Theorem 5.4.2 (General construction) Let q be a prime power and let F/F q be
an algebraic function field of genus g, with n pairwise distinct rational places P i ,
i = 1, . . . , n. Assume that there exist pairwise distinct places Q 1 , . . . , Q t of F/F q , of
degree α i ≥ 2, respectively, i = 1, . . . , t, where t ≥ 1. Let G 1 =
t
i=1
a i Q i and G 2 =
t
i=1
b i Q i , where a i ≤ b i , for all i = 1, . . . , t, and 2g − 2 < a 1 α 1 + . . . + a t α t <
b 1 α 1 + . . . + b t α t < n. Let D = P 1 + · · · + P n be a divisor of F/F q , and consider
that supp G 1 ∩ supp D = ∅ and supp G 2 ∩ supp D = ∅. Then there exists a quantum
code with parameters [[n, k, d]] q , where k = (b 1 − a 1 )α 1 + . . . + (b t − a t )α t and
d ≥ min{n − (b 1 α 1 + . . . + b t α t ), (a 1 α 1 + . . . + a t α t ) − (2g − 2)}.
Proof Similar to that of Theorem 5.4.1.
Corollary 5.4.4 Let q be a prime power and let F/F q be a hyperelliptic function
field of genus g ≥ 2, with n pairwise distinct rational places. Then there exists an
[[n, 2(t 2 − t 1 ), d]] q code, where t 1 , t 2 are positive integers satisfying 2g − 2 < t 1 <
t 2 < n and d ≥ min{n − 2t 2 , 2t 1 − 2g + 2}.
Proof Since F is a hyperelliptic function field, there exists a place G of degree
two (see Lemma 6.2.2.(a) of [152]). Let D = P 1 + · · · + P n be a divisor, where
P i are all rational points of F. Let G 2 = t 2 G and G 1 = t 1 G, where 2g − 2 <
2t 1 < 2t 2 < n. We know that supp G 1 ∩ supp D = ∅, supp G 2 ∩ supp D = ∅ and
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 − 2t 1 and k 1 = 2t 1 − g + 1. The code C 2 :=
C L (D, G 2 ) has parameters [n, k 2 , d 2 ] q , where d 2 ≥ n − 2t 2 and k 2 = 2t 2 − g + 1.
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 ≥ 2t 1 − (2g − 2) and k
⊥
1 = n + g − 1 − 2t 1 . Analogously, the dual code C
⊥
2 = C (D, G 2 ) of C 2 has parameters [n, k
⊥
2 , d
⊥
2 ] q , where
d
⊥
2 ≥ 2t 2 − (2g − 2) and k
⊥
2 = n + g − 1 − 2t 2 .
Applying the CSS construction to C 1 and C 2 , we obtain an [[n, k, d]] q code, with
k = k 2 − k 1 = (2t 2 − g + 1) − (2t 1 − g + 1) = 2(t 2 − t 1 ) and d ≥ min{d 2 , d
⊥
1 },
where d 2 ≥ n − 2t 2 and d
⊥
1 ≥ 2t 1 − (2g − 2). The proof is complete.
Corollary 5.4.5 There exists a quantum code with parameters [[46, 2(t 2 − t 1 ), d]] 25 ,
where t 1 , t 2 are positive integers such that 1 < t 1 < t 2 < 23 and d ≥ min{46 −
2t 2 , 2t 1 − 2}.
Proof Let us consider the function field F = F q 2 (x, y) with y
q
+ y = x
m and
m|(q + 1); take m = 2 and q = 5. As the genus of F is g = 2, then F is a hyperelliptic function field (see Lemma 6.2.2.(b) of [152]), and the result follows from
Corollary 5.4.4.