156
6 Asymmetric Quantum Codes
6.7.6.3 Construction III—BCH Codes
In this subsection, we construct more families of asymmetric stabilizer codes derived
from BCH codes. The first families of AQECCs derived from BCH codes were
constructed by Aly [3, Theorem 8].
Recall that a cyclic code of length n over F q is a BCH code with designed distance
δ if, for some integer b ≥ 0, one has
g(x) = lcm{M
(b)
(x), M
(b+1)
(x), . . . , M
(b+δ−2)
(x)},
i.e., g(x) is the monic polynomial of smallest degree over F q having α
b
, α
b+1
,
. . . , α
b+δ−2 as zeros.
The next result shows how to construct more AQECCs by expanding BCH codes.
Theorem 6.7.12 Suppose that n = q
m
− 1, where q = p
t is a power of an odd
prime p, t ≥ 1 and m ≥ 3 are integers (if q = 3, m ≥ 4). Then there exist AQECCs
with parameters
• [[tn, t (n − m(4q − 5) − 2), d z ≥ (2q + 2)/d x ≥ 2q]] p ;
• [[tn, t (n − m(4q − c − 5) − 2), d z ≥ (2q + 2)/d x ≥ (2q − c)]] p , where 0 ≤ c ≤
q − 2;
• [[tn, t (n − m(2c − l − 4) − 2), d z ≥ c/d x ≥ (c − l)]] p ,
where 2 ≤ c ≤ q and 0 ≤ l ≤ c − 2;
• [[tn, t (n − m(2c − l − 6) − 2), d z ≥ c/d x ≥ (c − l)]] p ,
where q + 2 < c ≤ 2q and 0 ≤ l ≤ c − q − 3;
• [[tn, t (n − m(4q − l − 5) − 1), d z ≥ (2q + 1)/d x ≥ (2q − l)]] p , where 0 ≤ l ≤
q − 2.
Proof Consider the codes constructed in [92, Theorems 4 and 5 and Corollary 1].
These codes are derived from two distinct nested cyclic codes C 2 ⊂ C 1 . Thus, applying Theorem 6.7.2 the result holds.
Theorem 6.7.13 Let q = p
t be a power of a prime p, t ≥ 1, gcd(q, n) = 1 and
ord n (q) = m. Let C 1 and C 2 be two narrow-sense BCH codes of length q
m/2
< n ≤
q
m
− 1 over F q with designed distances δ 1 and δ 2 in the range 2 ≤ δ 1 , δ 2 ≤ δ max =
min{{nq
m/2
/(q
m
− 1), n} and δ 1 < δ
⊥
2 ≤ δ 2 < δ
⊥
1 . Assume also that S 1 ∪ . . . ∪
S δ 1 −1 = S 1 ∪ . . . ∪ S δ 2 −1 , where S i denotes a q-coset. Then there exists an [[tn, t (n −
m(δ 1 − 1)(1 − 1/q) − m(δ 2 − 1)(1 − 1/q)), d
∗
z /d
∗
x ]] p AQECC, where d
∗
z =
wt (C 2 \C
⊥
1 ) ≥ δ 2 and d
∗
x = wt (C 1 \C
⊥
2 ) ≥ δ 1 .
Proof It suffices to apply Theorem 6.7.2 in those codes shown in [3, Theorem 8].
Remark 6.7.5 Note that one can obtain more families of AQECCs by applying
Theorem 6.7.3 in the existing families shown in [89]. Moreover, by expanding generalized Reed–Solomon (GRS) codes we have [94, Theorem 7.1] as a particular case
of Theorem 6.7.2.