116
5 Quantum Code Constructions
of [β(C 1 (i))]
⊥ is at least a 1 (i) + a 2 (i) − (2g i − 2)). Proceeding similarly as in the
proof of Theorem 5.4.3, we get N i − 2 − (b 1 (i) + b 2 (i)) ≥ a 1 (i) + a 2 (i) − (2g i −
2) ≥ (N i − K i − 2g i + 1)/2. Consequently, we have lim sup i→∞ 2r K i /2r (N i − 2)
> 0
and
lim sup i→∞ (N i − K i − 2g i + 1)/4r (N i − 2) = 1/4r
[1 − 2/( p
r
− 1) − c] > 0, as desired. The proof is complete.
Remark 5.4.4 Although the proofs of Theorems 5.4.3 and 5.4.5 are similar to the
proofs of the corresponding results shown in [27, 81], we utilize t-point (t ≥ 2) AG
codes, whereas in such references, the authors utilized only one-point AG codes to
perform their constructions. Another difference is that in such papers the authors
utilized the technique of code concatenation to obtain quantum codes over prime
fields; here, we utilize the technique of code expansion.
5.4.3 Examples
In Tables 5.15, 5.16 and 5.17, we exhibit some quantum codes derived from Corollaries 5.4.1, 5.4.2 and 5.4.5, respectively. In these tables, q is a prime power
and a, b, a 1 , a 2 , b 1 , b 2 , t 1 , t 2 , m are positive integers satisfying some conditions.
More precisely: in Table 5.15 we consider that (q − 1)(m − 1) − 2 < a < b, b <
q(1 + (q − 1)m) and m|(q + 1); in Table 5.16, we assume that a i ≤ b i i = 1, 2, (q −
1)(m − 1) − 2 < a 1 + a 2 < b 1 + b 2 , b 1 + b 2 < q[1 + (q − 1)m] − 1 and m|(q +
1); in Table 5.17, we suppose that 1 < t 1 < t 2 < 23.
Recall that the parameters of an Q := [[n, k, d]] q quantum code satisfy the
inequality k + 2d ≤ n + 2 (quantum Singleton bound). The Singleton defect (S D Q )
of a code is defined as S D Q = n + 2 − k − 2d. In this context, we measure the performance of a code by means of the Singleton defect. We adopt this method because,
for large alphabets, it is difficult to find codes over them: “... for large q, it is difficult
to find explicit known codes to compare with ours since there are no suitable tables
for reference” (see p. 3 of [71]).
Our [[26, 16, d ≥ 3]] 9 code is better than the [[26, 14, 3]] 9 code shown in
Ref. [34], because the Singleton defect of Q 1 := [[26, 16, d ≥ 3]] 9 is S D Q 1 ≤ 6,
whereas the Singleton defect of Q 2 := [[26, 14, 3]] 9 is S D Q 2 = 8. Our
[[26, 14, d ≥ 4]] 9 code with Singleton defect at most 8 is better than the [[26, 4, 4]] 9
code shown in Ref. [34], which has Singleton defect 16. Our codes of length 46
have Singleton defect at most 4. Moreover, the quantum codes of lengths n = 26 and
n = 27, exhibited in Table 5.15, have Singleton defect at most 6.
Note that the [[27, 3, d ≥ 10]] 9 , [[27, 5, d ≥ 9]] 9 , [[65, 9, d ≥ 25]] 25 , [[175, 31,
d ≥ 64]] 49 codes have large minimum distances when compared to their code lengths.
The quantum codes shown in [71] were constructed over the field F q 2 , where q is
a power of 2, whereas we construct here quantum codes over F q for all prime power
q. Great part of the codes displayed in [71] were constructed over F 8 ; this fact does
not allow us to compare our codes with the ones shown in [71].
5 Quantum Code Constructions
of [β(C 1 (i))]
⊥ is at least a 1 (i) + a 2 (i) − (2g i − 2)). Proceeding similarly as in the
proof of Theorem 5.4.3, we get N i − 2 − (b 1 (i) + b 2 (i)) ≥ a 1 (i) + a 2 (i) − (2g i −
2) ≥ (N i − K i − 2g i + 1)/2. Consequently, we have lim sup i→∞ 2r K i /2r (N i − 2)
> 0
and
lim sup i→∞ (N i − K i − 2g i + 1)/4r (N i − 2) = 1/4r
[1 − 2/( p
r
− 1) − c] > 0, as desired. The proof is complete.
Remark 5.4.4 Although the proofs of Theorems 5.4.3 and 5.4.5 are similar to the
proofs of the corresponding results shown in [27, 81], we utilize t-point (t ≥ 2) AG
codes, whereas in such references, the authors utilized only one-point AG codes to
perform their constructions. Another difference is that in such papers the authors
utilized the technique of code concatenation to obtain quantum codes over prime
fields; here, we utilize the technique of code expansion.
5.4.3 Examples
In Tables 5.15, 5.16 and 5.17, we exhibit some quantum codes derived from Corollaries 5.4.1, 5.4.2 and 5.4.5, respectively. In these tables, q is a prime power
and a, b, a 1 , a 2 , b 1 , b 2 , t 1 , t 2 , m are positive integers satisfying some conditions.
More precisely: in Table 5.15 we consider that (q − 1)(m − 1) − 2 < a < b, b <
q(1 + (q − 1)m) and m|(q + 1); in Table 5.16, we assume that a i ≤ b i i = 1, 2, (q −
1)(m − 1) − 2 < a 1 + a 2 < b 1 + b 2 , b 1 + b 2 < q[1 + (q − 1)m] − 1 and m|(q +
1); in Table 5.17, we suppose that 1 < t 1 < t 2 < 23.
Recall that the parameters of an Q := [[n, k, d]] q quantum code satisfy the
inequality k + 2d ≤ n + 2 (quantum Singleton bound). The Singleton defect (S D Q )
of a code is defined as S D Q = n + 2 − k − 2d. In this context, we measure the performance of a code by means of the Singleton defect. We adopt this method because,
for large alphabets, it is difficult to find codes over them: “... for large q, it is difficult
to find explicit known codes to compare with ours since there are no suitable tables
for reference” (see p. 3 of [71]).
Our [[26, 16, d ≥ 3]] 9 code is better than the [[26, 14, 3]] 9 code shown in
Ref. [34], because the Singleton defect of Q 1 := [[26, 16, d ≥ 3]] 9 is S D Q 1 ≤ 6,
whereas the Singleton defect of Q 2 := [[26, 14, 3]] 9 is S D Q 2 = 8. Our
[[26, 14, d ≥ 4]] 9 code with Singleton defect at most 8 is better than the [[26, 4, 4]] 9
code shown in Ref. [34], which has Singleton defect 16. Our codes of length 46
have Singleton defect at most 4. Moreover, the quantum codes of lengths n = 26 and
n = 27, exhibited in Table 5.15, have Singleton defect at most 6.
Note that the [[27, 3, d ≥ 10]] 9 , [[27, 5, d ≥ 9]] 9 , [[65, 9, d ≥ 25]] 25 , [[175, 31,
d ≥ 64]] 49 codes have large minimum distances when compared to their code lengths.
The quantum codes shown in [71] were constructed over the field F q 2 , where q is
a power of 2, whereas we construct here quantum codes over F q for all prime power
q. Great part of the codes displayed in [71] were constructed over F 8 ; this fact does
not allow us to compare our codes with the ones shown in [71].