128
6 Asymmetric Quantum Codes
wt(e) = #{i : 1 ≤ i ≤ n|(a i , b i ) = (0, 0)}.
Definition 6.2.1 The trace-symplectic form of two vectors (a|b), (a
∗
|b
∗
) ∈ F
2n
q is
defined by
(a|b)|(a
∗
|b
∗
) s = tr q/ p (b · a
∗
− b
∗
· a).
In the sequence, we define formally the concept of asymmetric quantum errorcorrecting codes.
Definition 6.2.2 An AQECC with parameters ((n, K , d z /d x )) q is an K -dimensional
subspace of the Hilbert space C
q
n . The code corrects all qudit-flip errors up to
d x −1
2
and all phase-shift errors up to
d z −1
2
. An ((n, q
k
, d z /d x )) q code is denoted by
[[n, k, d z /d x ]] q .
Since the last two decades, many authors have focused the attention in the construction of good quantum codes [4, 11, 17, 25, 28, 30, 55, 56, 62, 80, 81, 89, 105,
108, 113, 138, 148, 153, 155, 156, 162]. On the other hand, some authors have also
presented constructions of asymmetric quantum codes with good parameters [3, 62,
70, 139, 140, 149].
In the same manner that the CSS construction works as an important tool in the
construction of quantum codes, it also works in the case of asymmetric errors.
Lemma 6.2.1 ([25, 80, 121]) (CSS construction) Let C 1 and C 2 denote two classical
linear codes with parameters [n, k 1 , d 1 ] q and [n, k 2 , d 2 ] q , respectively. Assume that
C 2 ⊂ C 1 . Then there exists an AQECC with parameters [[n, K = k 1 − k 2 , d z /d x ]] q ,
where d x =wt(C
⊥
2 \C
⊥
1 )} and d z = wt(C 1 \C 2 ). The resulting code is said pure if, in
the above construction, d x = d(C
⊥
2 ) and d z = d(C 1 ).
6.3 Asymmetric BCH Codes
In this section, we deal with constructions of families of nonbinary asymmetric
quantum BCH codes. These quantum codes have good parameters and they can be
applied in quantum systems where the asymmetry between qudit-flip and phase-shift
errors is large.
More precisely, we construct several families of asymmetric q-ary (q is an odd
prime power) quantum BCH codes by means of the CSS construction (Lemma 6.2.1)
applied to two distinct q-ary classical BCH codes. To do this, we construct subclasses
of (classical) BCH codes with great dimension, also computing lower bounds to
the corresponding minimum distances d z and d x by applying the well-known BCH
bound. The proposed families have parameters better than the ones available in the
literature.
Although in [105] families of quantum codes were constructed by applying similar technique, the parameters (dimension and minimum distances) of the proposed
6 Asymmetric Quantum Codes
wt(e) = #{i : 1 ≤ i ≤ n|(a i , b i ) = (0, 0)}.
Definition 6.2.1 The trace-symplectic form of two vectors (a|b), (a
∗
|b
∗
) ∈ F
2n
q is
defined by
(a|b)|(a
∗
|b
∗
) s = tr q/ p (b · a
∗
− b
∗
· a).
In the sequence, we define formally the concept of asymmetric quantum errorcorrecting codes.
Definition 6.2.2 An AQECC with parameters ((n, K , d z /d x )) q is an K -dimensional
subspace of the Hilbert space C
q
n . The code corrects all qudit-flip errors up to
d x −1
2
and all phase-shift errors up to
d z −1
2
. An ((n, q
k
, d z /d x )) q code is denoted by
[[n, k, d z /d x ]] q .
Since the last two decades, many authors have focused the attention in the construction of good quantum codes [4, 11, 17, 25, 28, 30, 55, 56, 62, 80, 81, 89, 105,
108, 113, 138, 148, 153, 155, 156, 162]. On the other hand, some authors have also
presented constructions of asymmetric quantum codes with good parameters [3, 62,
70, 139, 140, 149].
In the same manner that the CSS construction works as an important tool in the
construction of quantum codes, it also works in the case of asymmetric errors.
Lemma 6.2.1 ([25, 80, 121]) (CSS construction) Let C 1 and C 2 denote two classical
linear codes with parameters [n, k 1 , d 1 ] q and [n, k 2 , d 2 ] q , respectively. Assume that
C 2 ⊂ C 1 . Then there exists an AQECC with parameters [[n, K = k 1 − k 2 , d z /d x ]] q ,
where d x =wt(C
⊥
2 \C
⊥
1 )} and d z = wt(C 1 \C 2 ). The resulting code is said pure if, in
the above construction, d x = d(C
⊥
2 ) and d z = d(C 1 ).
6.3 Asymmetric BCH Codes
In this section, we deal with constructions of families of nonbinary asymmetric
quantum BCH codes. These quantum codes have good parameters and they can be
applied in quantum systems where the asymmetry between qudit-flip and phase-shift
errors is large.
More precisely, we construct several families of asymmetric q-ary (q is an odd
prime power) quantum BCH codes by means of the CSS construction (Lemma 6.2.1)
applied to two distinct q-ary classical BCH codes. To do this, we construct subclasses
of (classical) BCH codes with great dimension, also computing lower bounds to
the corresponding minimum distances d z and d x by applying the well-known BCH
bound. The proposed families have parameters better than the ones available in the
literature.
Although in [105] families of quantum codes were constructed by applying similar technique, the parameters (dimension and minimum distances) of the proposed