6.3 Asymmetric BCH Codes
133
where each M
(i)
(x) is the minimal polynomial of α
i such that
i /
∈ {a − q + 2 + c, . . . , a, a + 1, . . . , a + q − 1},
a =
q
m −1
2
, i runs through the coset representatives mod n and 0 ≤ c ≤ q − 2.
From the BCH bound, it follows that the minimum distances of the quantum code
are lower bounded by d z ≥ 2q + 2 and d x ≥ 2q − c. Applying the same method
shown in the proof of Theorem 6.3.1, we know that k 1 = n − 2m(q − 1) − 1 and
k 2 = m(2q − c − 3) + 1; so k 1 − k 2 = n − m(4q − c − 5) − 2.
Therefore, one can get an
[[n, n − m(4q − c − 5) − 2, d z ≥ (2q + 2)/d x ≥ (2q − c)]] q
AQECC.
Corollary 6.3.1 Let n = q
m
− 1, q is an odd prime power and m ≥ 3 is an integer.
Then we have
(i) there exists an
[[n, n − m(2c − l − 4) − 2, d z ≥ c/d x ≥ (c − l)]] q
AQECC, where 2 ≤ c ≤ q and 0 ≤ l ≤ c − 2;
(ii) there exists an
[[n, n − m(2c − l − 6) − 2, d z ≥ c/d x ≥ (c − l)]] q
AQECC, where q + 2 < c ≤ 2q and 0 ≤ l ≤ c − q − 3;
(iii) there exists an
[[n, n − m(4q − l − 5) − 1, d z ≥ (2q + 1)/d x ≥ (2q − l)]] q
AQECC, where 0 ≤ l ≤ q − 2.
Proof (i) It suffices to consider C 1 as the BCH code generated by
g 1 (x) = M
(0)
(x)M
(1)
(x) . . . M
(c−2)
(x),
and C 2 as the cyclic code generated by
g 2 (x) =
i
M
(i)
(x),