126
6 Asymmetric Quantum Codes
trace Hermitian inner product. In [37], asymmetric quantum MDS codes obtained
from generalized Reed–Solomon (GRS) codes were constructed. In the papers [94,
95], we constructed families of AQECCs by expanding GRS codes and by applying
product codes, respectively.
In this chapter, we construct families of asymmetric quantum codes derived from
(classical) BCH codes (Sect. 6.3), Reed–Solomon and generalized Reed–Solomon
codes (Sect. 6.5), tensor product codes (Sect. 6.6). In Sect. 6.7, we generalize to asymmetric quantum codes the known results which are valid to quantum codes. More
precisely, we show how to construct AQECCs by applying to methods of puncturing,
extending, expanding, direct sum and by the technique of (u|u + v) construction.
In particular, several families of AQECCs are obtained by employing these results
to generalized Reed–Muller codes, character codes, BCH, quadratic residue and
affine-invariant codes.
6.1 Preliminaries
As always, p is a prime, q is a prime power, F q is the finite field with q elements
and α ∈ F q m is a primitive nth root of unity.
Recall that the trace map (see Definition 3.5.3) tr q m /q : F q m −→ F q is defined as
tr q m /q (a) :=
m−1
i=0
a
q
i . As it is usual in group theory, we write H ≤ G to denote that
H is a subgroup of a group (G, ∗). The center of G is denoted by Z (G). If S ≤ G,
then we write C G (S) meaning the centralizer of S in G. Further, we write S Z(G)
to denote the subgroup generated by S and the center Z (G) (these notations are
essentially the same to that of [80]).
We write wt(C) to denote the minimum weight of a code C, and by d(C) its
minimum distance. Sometimes we abuse the notation by writing C = [n, k, d] q . If
C is an [n, k, d] q linear code, recall that its Euclidean dual (see Definition 4.2.1) is
defined as
C
⊥
= {y ∈ F
n
q | y · x = 0, ∀ x ∈ C}.
If C is an [n, k, d] q 2 code over F q 2 , recall that its Hermitian dual (see Definition 4.2.3)
is defined by
C
⊥ h = {y ∈ F
n
q 2 | y
q
· x = 0, ∀ x ∈ C},
where y
q
= (y
q
1 , . . . , y
q
n ) denotes the conjugate of the vector y = (y 1 , . . . , y n ).
If C ≤ F
2n
q is an additive code, then we write swt(C) to denote the symplectic
weight (see Definition 3.5.7) of C and C
⊥ s to denote the trace-symplectic dual of C
(see Definition 6.2.1).
6 Asymmetric Quantum Codes
trace Hermitian inner product. In [37], asymmetric quantum MDS codes obtained
from generalized Reed–Solomon (GRS) codes were constructed. In the papers [94,
95], we constructed families of AQECCs by expanding GRS codes and by applying
product codes, respectively.
In this chapter, we construct families of asymmetric quantum codes derived from
(classical) BCH codes (Sect. 6.3), Reed–Solomon and generalized Reed–Solomon
codes (Sect. 6.5), tensor product codes (Sect. 6.6). In Sect. 6.7, we generalize to asymmetric quantum codes the known results which are valid to quantum codes. More
precisely, we show how to construct AQECCs by applying to methods of puncturing,
extending, expanding, direct sum and by the technique of (u|u + v) construction.
In particular, several families of AQECCs are obtained by employing these results
to generalized Reed–Muller codes, character codes, BCH, quadratic residue and
affine-invariant codes.
6.1 Preliminaries
As always, p is a prime, q is a prime power, F q is the finite field with q elements
and α ∈ F q m is a primitive nth root of unity.
Recall that the trace map (see Definition 3.5.3) tr q m /q : F q m −→ F q is defined as
tr q m /q (a) :=
m−1
i=0
a
q
i . As it is usual in group theory, we write H ≤ G to denote that
H is a subgroup of a group (G, ∗). The center of G is denoted by Z (G). If S ≤ G,
then we write C G (S) meaning the centralizer of S in G. Further, we write S Z(G)
to denote the subgroup generated by S and the center Z (G) (these notations are
essentially the same to that of [80]).
We write wt(C) to denote the minimum weight of a code C, and by d(C) its
minimum distance. Sometimes we abuse the notation by writing C = [n, k, d] q . If
C is an [n, k, d] q linear code, recall that its Euclidean dual (see Definition 4.2.1) is
defined as
C
⊥
= {y ∈ F
n
q | y · x = 0, ∀ x ∈ C}.
If C is an [n, k, d] q 2 code over F q 2 , recall that its Hermitian dual (see Definition 4.2.3)
is defined by
C
⊥ h = {y ∈ F
n
q 2 | y
q
· x = 0, ∀ x ∈ C},
where y
q
= (y
q
1 , . . . , y
q
n ) denotes the conjugate of the vector y = (y 1 , . . . , y n ).
If C ≤ F
2n
q is an additive code, then we write swt(C) to denote the symplectic
weight (see Definition 3.5.7) of C and C
⊥ s to denote the trace-symplectic dual of C
(see Definition 6.2.1).