6.4 Examples
137
S 1 ∪ . . . ∪ S δ 1 −1 = S 1 ∪ . . . ∪ S δ 2 −1 , then there exists an asymmetric quantum error
control code with parameters
[[n, n − m(δ 1 − 1)(1 − 1/q) − m(δ 2 − 1)(1 − 1/q), d
∗
z /d
∗
x ]] q ,
where d
∗
z = wt (C 2 \C
⊥
1 ) ≥ δ 2 > d
∗
x = wt (C 1 \C
⊥
2 ) ≥ δ 1 .
In Table 6.1, the parameters of the asymmetric quantum BCH codes shown in [3]
are given by [[n, k
∗
, d
∗
z /d
∗
x ]] q =
= [[n, n − m(δ 1 − 1)(1 − 1/q) − m(δ 2 − 1)(1 − 1/q), d
∗
z /d
∗
x ]] q ,
where d
∗
z = wt (C 2 \C
⊥
1 ) ≥ δ 2 > d
∗
x = wt (C 1 \C
⊥
2 ) ≥ δ 1 . Here, n = q
m
− 1 is the
code length, (q is an odd prime power), q
k
∗ is the code dimension and d
∗
z /d
∗
x are the
corresponding minimum distances with respect to phase-shift and qudit-flip errors,
respectively.
Our code parameters are denoted by [[n, k, d z ≥ d/d x ≥ (d − c)]] q and they are
given in the following:
• [[n, n − m(2c − l − 4) − 2, d z ≥ c/d x ≥ (c − l)]] q , where 2 ≤ c ≤ q and 0 ≤
l ≤ c − 2;
• [[n, n − m(2c − l − 6) − 2, d z ≥ c/d x ≥ (c − l)]] q , where q + 2 < c ≤ 2q and
0 ≤ l ≤ c − q − 3;
• [[n, n − m(4q − l − 5) − 1, d z ≥ (2q + 1)/d x ≥ (2q − l)]] q , where 0 ≤ l ≤ q −
2;
• [[n, n − m(4q − l − 5) − 2, d z ≥ (2q + 2)/d x ≥ (2q − l)]] q , where 0 ≤ l ≤ q −
2,
where n = q
m
− 1 is the code length, q is an odd prime power, q
k is the code
dimension and d z /d x are the corresponding minimum distances with respect to phaseshift and qudit-flip errors, respectively.
6.5 Reed–Solomon and GRS Codes
The aim here is to construct asymmetric quantum error-correcting codes derived
from classical Reed–Solomon (RS) and generalized Reed–Solomon (GRS) codes
by applying the CSS construction (see Lemma 6.2.1). The results presented in this
subsection are obtained from Ref. [94].