132
6 Asymmetric Quantum Codes
BCH bound, C has minimum distance greater than or equal to 2q. Since C is equivalent to C
⊥
2 , it follows that C
⊥
2 also has minimum distance greater than or equal to
2q. Therefore, the resulting asymmetric quantum code has minimum distances satisfying d z ≥ 2q + 2 and d x ≥ 2q. Furthermore, from Lemmas 6.3.4 and 6.3.5 and
by construction, one has C 2 C 1 .
From Lemma 6.3.3, the (2q − 2) q-cosets
C 1 , C 2 , . . . , C q−1 , C q+1 , . . . , C 2q−1
are disjoint and each of them has m elements. Since C 0 has only one element, the
defining set of C 1 has 2m(q − 1) + 1 elements.
Thus C 1 has dimension k 1 = n − 2m(q − 1) − 1. From Lemma 6.3.2, the coset
C [
q m −1
2 ] contains only one element. From Lemmas 6.3.6 and 6.3.7, the (2q − 2) qcosets C [
q m −1
2 + j] and C [
q m −1
2 −i] , where 1 ≤ i, j ≤ q − 1, are mutually disjoint and
each of them has m elements. Since C [
q m −1
2 ] has only one element and each of the
q-cosets C [
q m −1
2 + j] and C [
q m −1
2 −i] , 1 ≤ i, j ≤ q − 1, has m elements, m ≥ 3 (m ≥ 4
if q = 3), it follows that the q-coset C [
q m −1
2 ] is disjoint of the q-cosets C [
q m −1
2 + j]
and C [
q m −1
2 −i] , 1 ≤ i, j ≤ q − 1. Therefore, C 2 has dimension k 2 = m(2q − 3) + 1;
hence, the dimension of the AQECC equals k 1 − k 2 = n − m(4q − 5) − 2, where
n = q
m
− 1.
Applying the CSS construction to the codes C 1 and C 2 one obtains an
[[n, n − m(4q − 5) − 2, d z ≥ (2q + 2)/d x ≥ 2q]] q ,
AQECC, as desired.
Theorem 6.3.2 is a generalization of Theorem 6.3.1. It is one of the main results
of this subsection.
Theorem 6.3.2 Let n = q
m
− 1, where q is an odd prime power and m ≥ 3 is an
integer (if q = 3, m ≥ 4). Then there exists an
[[n, n − m(4q − c − 5) − 2, d z ≥ (2q + 2)/d x ≥ (2q − c)]] q
AQECC, where 0 ≤ c ≤ q − 2.
Proof Let C 1 = [n, k 1 , d 1 ] q be the BCH code generated by
g 1 (x) = M
(0)
(x)M
(1)
(x) . . . M
(q−1)
(x)M
(q+1)
(x) . . . M
(2q−1)
(x),
and let C 2 = [n, k 2 , d 2 ] q be the cyclic code generated by
g 2 (x) =
i
M
(i)
(x),
6 Asymmetric Quantum Codes
BCH bound, C has minimum distance greater than or equal to 2q. Since C is equivalent to C
⊥
2 , it follows that C
⊥
2 also has minimum distance greater than or equal to
2q. Therefore, the resulting asymmetric quantum code has minimum distances satisfying d z ≥ 2q + 2 and d x ≥ 2q. Furthermore, from Lemmas 6.3.4 and 6.3.5 and
by construction, one has C 2 C 1 .
From Lemma 6.3.3, the (2q − 2) q-cosets
C 1 , C 2 , . . . , C q−1 , C q+1 , . . . , C 2q−1
are disjoint and each of them has m elements. Since C 0 has only one element, the
defining set of C 1 has 2m(q − 1) + 1 elements.
Thus C 1 has dimension k 1 = n − 2m(q − 1) − 1. From Lemma 6.3.2, the coset
C [
q m −1
2 ] contains only one element. From Lemmas 6.3.6 and 6.3.7, the (2q − 2) qcosets C [
q m −1
2 + j] and C [
q m −1
2 −i] , where 1 ≤ i, j ≤ q − 1, are mutually disjoint and
each of them has m elements. Since C [
q m −1
2 ] has only one element and each of the
q-cosets C [
q m −1
2 + j] and C [
q m −1
2 −i] , 1 ≤ i, j ≤ q − 1, has m elements, m ≥ 3 (m ≥ 4
if q = 3), it follows that the q-coset C [
q m −1
2 ] is disjoint of the q-cosets C [
q m −1
2 + j]
and C [
q m −1
2 −i] , 1 ≤ i, j ≤ q − 1. Therefore, C 2 has dimension k 2 = m(2q − 3) + 1;
hence, the dimension of the AQECC equals k 1 − k 2 = n − m(4q − 5) − 2, where
n = q
m
− 1.
Applying the CSS construction to the codes C 1 and C 2 one obtains an
[[n, n − m(4q − 5) − 2, d z ≥ (2q + 2)/d x ≥ 2q]] q ,
AQECC, as desired.
Theorem 6.3.2 is a generalization of Theorem 6.3.1. It is one of the main results
of this subsection.
Theorem 6.3.2 Let n = q
m
− 1, where q is an odd prime power and m ≥ 3 is an
integer (if q = 3, m ≥ 4). Then there exists an
[[n, n − m(4q − c − 5) − 2, d z ≥ (2q + 2)/d x ≥ (2q − c)]] q
AQECC, where 0 ≤ c ≤ q − 2.
Proof Let C 1 = [n, k 1 , d 1 ] q be the BCH code generated by
g 1 (x) = M
(0)
(x)M
(1)
(x) . . . M
(q−1)
(x)M
(q+1)
(x) . . . M
(2q−1)
(x),
and let C 2 = [n, k 2 , d 2 ] q be the cyclic code generated by
g 2 (x) =
i
M
(i)
(x),