90
5 Quantum Code Constructions
where M
(i) are the minimal polynomials such that i /
∈ {b, b + 1, b + 2} and i runs
through the coset representatives mod (q
3
− 1) and b =
q
3 −1
2
. Proceeding similarly
as in the proof of Theorem 5.2.5 and applying the CSS construction to these code,
the results follows.
Applying Corollary 5.2.6 one can construct quantum codes with parameters
[[124, 116, d ≥ 3]] 5 , [[124, 110, d ≥ 4]] 5 , [[342, 334, d ≥ 3]] 7 ,
[[342, 328, d ≥ 4]] 7 , and so on.
Theorem 5.2.9 can be generalized in the following way.
Theorem 5.2.10 Let q ≥ 5 be an odd prime power. Then there exists an
[[q
m
− 1, q
m
− 6m − 3, d ≥ 5]] q CSS code.
Proof The coset C (
q m −1
2 ) contains only the element
q
m −1
2
. Moreover, proceeding
similarly as in the proof of Theorem 5.2.9, it is easy to see that the cosets C (
q m −1
2 ) ,
C (
q m −1
2 +1) , C (
q m −1
2 +2) and C (
q m −1
2 +3) are disjoint from the cosets C 1 , C 2 and C 3 and
also disjoint among them. Moreover, it is easy to see that each of these cosets has m
elements. Let C 1 be the cyclic code generated by
M
(0)
(x)M
(1)
(x)M
(2)
(x)M
(3)
(x),
and C 2 be the cyclic code generated by
i
M
(i)
(x),
where M
(i) are the minimal polynomials of α
i such that i /
∈ {b, b + 1, b + 2, b + 3}
and i runs through the coset representatives mod (q
m
− 1) and b =
q
m −1
2
. Applying
the CSS construction to these codes, we are done.
Corollary 5.2.7 Let q ≥ 5 be an odd prime power. Then there exist CSS codes with
parameters [[q
m
− 1, q
m
− 2m − 3, d ≥ 3]] q and [[q
m
− 1, q
m
− 4m − 3, d ≥ 4]] q .
Applying Theorem 5.2.10 and Corollary 5.2.7 one can construct quantum codes
with parameters [[80, 74, d ≥ 3]] 9 , [[80, 70, d ≥ 4]] 9 , [[80, 66, d ≥ 5] 9 , [[624,
614, d ≥ 3]] 5 , [[624, 606, d ≥ 4]] 5 , [[624, 598, d ≥ 5]] 5 , [[728, 720, d ≥ 3]] 9 ,
[[728, 714, d ≥ 4]] 9 , [[728, 708, d ≥ 5]] 9 and so on.
Theorem 5.2.11 Let n = q
m
− 1, where q ≥ 3 is a prime power and m ≥ 3. Then
there exists an [[n, n − m(2c − 3) − 1, d ≥ c]] q quantum code, where 2 ≤ c ≤ q
and (c − 1)q + 1 < q
m/2
− 1.
Proof Let C 1 be the cyclic code generated by
M
(0)
(x)M
(1)
(x) . . . M
(c−2)
(x),
5 Quantum Code Constructions
where M
(i) are the minimal polynomials such that i /
∈ {b, b + 1, b + 2} and i runs
through the coset representatives mod (q
3
− 1) and b =
q
3 −1
2
. Proceeding similarly
as in the proof of Theorem 5.2.5 and applying the CSS construction to these code,
the results follows.
Applying Corollary 5.2.6 one can construct quantum codes with parameters
[[124, 116, d ≥ 3]] 5 , [[124, 110, d ≥ 4]] 5 , [[342, 334, d ≥ 3]] 7 ,
[[342, 328, d ≥ 4]] 7 , and so on.
Theorem 5.2.9 can be generalized in the following way.
Theorem 5.2.10 Let q ≥ 5 be an odd prime power. Then there exists an
[[q
m
− 1, q
m
− 6m − 3, d ≥ 5]] q CSS code.
Proof The coset C (
q m −1
2 ) contains only the element
q
m −1
2
. Moreover, proceeding
similarly as in the proof of Theorem 5.2.9, it is easy to see that the cosets C (
q m −1
2 ) ,
C (
q m −1
2 +1) , C (
q m −1
2 +2) and C (
q m −1
2 +3) are disjoint from the cosets C 1 , C 2 and C 3 and
also disjoint among them. Moreover, it is easy to see that each of these cosets has m
elements. Let C 1 be the cyclic code generated by
M
(0)
(x)M
(1)
(x)M
(2)
(x)M
(3)
(x),
and C 2 be the cyclic code generated by
i
M
(i)
(x),
where M
(i) are the minimal polynomials of α
i such that i /
∈ {b, b + 1, b + 2, b + 3}
and i runs through the coset representatives mod (q
m
− 1) and b =
q
m −1
2
. Applying
the CSS construction to these codes, we are done.
Corollary 5.2.7 Let q ≥ 5 be an odd prime power. Then there exist CSS codes with
parameters [[q
m
− 1, q
m
− 2m − 3, d ≥ 3]] q and [[q
m
− 1, q
m
− 4m − 3, d ≥ 4]] q .
Applying Theorem 5.2.10 and Corollary 5.2.7 one can construct quantum codes
with parameters [[80, 74, d ≥ 3]] 9 , [[80, 70, d ≥ 4]] 9 , [[80, 66, d ≥ 5] 9 , [[624,
614, d ≥ 3]] 5 , [[624, 606, d ≥ 4]] 5 , [[624, 598, d ≥ 5]] 5 , [[728, 720, d ≥ 3]] 9 ,
[[728, 714, d ≥ 4]] 9 , [[728, 708, d ≥ 5]] 9 and so on.
Theorem 5.2.11 Let n = q
m
− 1, where q ≥ 3 is a prime power and m ≥ 3. Then
there exists an [[n, n − m(2c − 3) − 1, d ≥ c]] q quantum code, where 2 ≤ c ≤ q
and (c − 1)q + 1 < q
m/2
− 1.
Proof Let C 1 be the cyclic code generated by
M
(0)
(x)M
(1)
(x) . . . M
(c−2)
(x),