154
6 Asymmetric Quantum Codes
C 4 ), (C 1 |C 1 + C 3 ) and [(C 2 |C 2 + C 4 )]
⊥ , one obtains an [[2n, (k 1 + k 3 ) − (k 2 +
k 4 ), d z /d x ]] q = [[2n, k
∗
+ k
, d z /d x ]] q asymmetric stabilizer code, where d z ≥ min
{2d 1 , d 3 } and d x ≥ min{2d
⊥
4 , d
⊥
2 }, as required.
As an alternative proof, we can also write the codewords of [(C 2 |C 2 + C 4 )]
⊥ in
the form {(u + v, −v)|u ∈ C
⊥
2 , v ∈ C
⊥
4 }, and because the Hamming weights of v
and −v are the same, the latter code is equivalent to {(u + v, v)|u ∈ C
⊥
2 , v ∈ C
⊥
4 },
and the result follows.
6.7.6 Code Constructions
In this subsection, we only apply the asymmetric quantum code expansion shown
in Sect. 6.7.1 in order to generate families of AQECCs, although it is clear that all
construction techniques proposed in Sect. 6.7.1–6.7.5 can be also applied. In the
sequence, we construct AQECCs derived from generalized Reed–Muller (GRM),
character codes, BCH, quadratic residue (QR) and affine-invariant codes, respectively.
Remark 6.7.4 It is important to observe that in all results presented in the following,
we expand the codes defined over F q (where q = p
t , t ≥ 1 and p prime) with respect
to the prime field F p . However, it is clear that the method also holds if the expansion
is performed over any subfield of the field F q .
6.7.6.1 Construction I—Generalized RM Codes
The first family of AQECCs derived from binary Reed–Muller (RM) codes were
constructed in [140, Lemma 4.1]. In this subsection, we present a construction of
AQECCs derived from generalized Reed–Muller (GRM) codes [114, 129].
The GRM code R q (α, m) over F q of order α, 0 ≤ α < q(m − 1), has parameters
[q
m
, k(α), d(α)] q , where
k(α) =
m
i=0
(−1)
i
m
i
m + α − iq
α − iq
(6.1)
and
d(α) = (t + 1)q
u
,
(6.2)
where m(q − 1) − α = (q − 1)u + t and 0 ≤ t < q − 1. The dual of a GRM code
R q (α, m) is also a GRM code given by
[R q (α, m)]
⊥
= R q (α
⊥
, m),
6 Asymmetric Quantum Codes
C 4 ), (C 1 |C 1 + C 3 ) and [(C 2 |C 2 + C 4 )]
⊥ , one obtains an [[2n, (k 1 + k 3 ) − (k 2 +
k 4 ), d z /d x ]] q = [[2n, k
∗
+ k
, d z /d x ]] q asymmetric stabilizer code, where d z ≥ min
{2d 1 , d 3 } and d x ≥ min{2d
⊥
4 , d
⊥
2 }, as required.
As an alternative proof, we can also write the codewords of [(C 2 |C 2 + C 4 )]
⊥ in
the form {(u + v, −v)|u ∈ C
⊥
2 , v ∈ C
⊥
4 }, and because the Hamming weights of v
and −v are the same, the latter code is equivalent to {(u + v, v)|u ∈ C
⊥
2 , v ∈ C
⊥
4 },
and the result follows.
6.7.6 Code Constructions
In this subsection, we only apply the asymmetric quantum code expansion shown
in Sect. 6.7.1 in order to generate families of AQECCs, although it is clear that all
construction techniques proposed in Sect. 6.7.1–6.7.5 can be also applied. In the
sequence, we construct AQECCs derived from generalized Reed–Muller (GRM),
character codes, BCH, quadratic residue (QR) and affine-invariant codes, respectively.
Remark 6.7.4 It is important to observe that in all results presented in the following,
we expand the codes defined over F q (where q = p
t , t ≥ 1 and p prime) with respect
to the prime field F p . However, it is clear that the method also holds if the expansion
is performed over any subfield of the field F q .
6.7.6.1 Construction I—Generalized RM Codes
The first family of AQECCs derived from binary Reed–Muller (RM) codes were
constructed in [140, Lemma 4.1]. In this subsection, we present a construction of
AQECCs derived from generalized Reed–Muller (GRM) codes [114, 129].
The GRM code R q (α, m) over F q of order α, 0 ≤ α < q(m − 1), has parameters
[q
m
, k(α), d(α)] q , where
k(α) =
m
i=0
(−1)
i
m
i
m + α − iq
α − iq
(6.1)
and
d(α) = (t + 1)q
u
,
(6.2)
where m(q − 1) − α = (q − 1)u + t and 0 ≤ t < q − 1. The dual of a GRM code
R q (α, m) is also a GRM code given by
[R q (α, m)]
⊥
= R q (α
⊥
, m),