6.3 Asymmetric BCH Codes
131
(i) the q-cosets C [
q m −1
2 +k] are mutually disjoint, where k = 1, . . . , q − 1;
(ii) the q-cosets C [
q m −1
2 −k] are mutually disjoint, where k = 1, . . . , q − 1;
(iii) the q-cosets C [
q m −1
2 +i] are disjoint from the cosets C [
q m −1
2 − j] , where 1 ≤ i, j ≤
q − 1.
Proof See [105, Lemma 3.5].
Lemma 6.3.7 Let n = q
m
− 1, where q ≥ 3 is an odd prime power and m ≥ 3 is an
integer (if q = 3, m ≥ 4). Then each of the q-cosets C [
q m −1
2 +i] and C [
q m −1
2 − j] , where
1 ≤ i, j ≤ q − 1, has m elements.
Proof See [105, Lemma 3.6].
Based on these previous results, we are now able to show how to construct asymmetric quantum codes.
Theorem 6.3.1 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 − 5) − 2, d z ≥ (2q + 2)/d x ≥ 2q]] q
AQECC.
Proof Let C 1 = [n, k 1 , d 1 ] q be the BCH code generated by the product of the minimal polynomials
g 1 (x) = M
(0)
(x)M
(1)
(x) . . . M
(q−1)
(x)M
(q+1)
(x) . . . M
(2q−1)
(x),
and C 2 = [n, k 2 , d 2 ] q be the cyclic code generated by the product of the minimal
polynomials
g 2 (x) =
i
M
(i)
(x),
where each M
(i)
(x) is the minimal polynomial of α
i such that
i /
∈ {a − q + 2, . . . , a − 1, a, a + 1, . . . , a + q − 1},
a =
q
m −1
2
and i runs through the coset representatives modulo n, where n = q
m
− 1.
We next construct asymmetric quantum BCH codes derived from codes C 1 and
C 2 by applying the CSS construction. From the BCH bound one has d 1 ≥ 2q + 2,
because the defining set of C 1 contains the sequence of 2q + 1 consecutive integers given by 0, 1, . . . , 2q. Similarly, the defining set of the code C generated by
the polynomial h 2 (x) = (x
n
− 1)/g 2 (x) contains the sequence of 2q − 1 consecutive integers given by a − q + 2, . . . , a − 1, a, a + 1, . . . , a + q, (note that, from
Lemma 6.3.2, the q-coset C [
q m −1
2 +1] contains the element
q
m −1
2
+ q). Thus, from the