134
6 Asymmetric Quantum Codes
where each M
(i)
(x) is the minimal polynomial of α
i such that i /
∈ {a, . . . , a +
c − 2 − l}, a =
q
m −1
2
and i runs through the coset representatives mod (q
m
−
1). Proceeding similarly as in the proof of Theorem 6.3.1 the result follows.
(ii) Let C 1 be the BCH code generated by
g 1 (x) = M
(0)
(x)M
(1)
(x) . . . M
(q−1)
(x)M
(q+1)
(x) . . . M
(c−2)
(x),
and C 2 generated by polynomials
g 2 (x) =
i
M
(i)
(x),
where each M
(i)
(x) is the minimal polynomial of α
i such that i /
∈ {a − r +
l, . . . , a − 1, a, a + 1, . . . , a + q − 1}, a =
q
m −1
2
, r is an integer such that r =
c − 2 − q, 0 ≤ l ≤ c − q − 3 and i runs through the coset representatives mod
(q
m
− 1). Proceeding similarly as in the proof of Theorem 6.3.1 the result
follows.
(iii) Let C 1 be the BCH code generated by
g 1 (x) = M
(1)
(x) . . . M
(q−1)
(x)M
(q+1)
(x) . . . M
(2q−1)
(x),
and C 2 generated by
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 + l, . . . , a, a + 1, . . . , a + q − 1}, a =
q
m −1
2
and i runs through the coset
representatives mod (q
m
− 1). Applying the CSS construction to C 1 and C 2
and proceeding similarly as in the proof of Theorem 6.3.1 the result follows.
6.3.1.1 The Case m = 3 and q = 3
We now investigate the case m = 3 and q = 3. For q = 3 and n = 3
3
− 1 = 26
the cosets are given by C 0 = {0}, C 1 = {1, 3, 9}, C 2 = {2, 6, 18}, C 4 = {4, 12, 10},
C 5 = {5, 15, 19}, C 7 = {7, 21, 11}, C 8 = {8, 24, 20},
C 13 = {13}, C 14 = {14, 16, 22}, C 17 = {17, 25, 23}.
Corollary 6.3.2 There exist quantum codes with parameters
[[26, 13, d z ≥ 5/d x ≥ 4]] 3 ,
[[26, 15, d z ≥ 5/d x ≥ 3]] 3 ,