6.7 New Codes from Old
155
where α
⊥
= m(q − 1) − 1 − α.
We utilize the properties of GRM codes in order to obtain more asymmetric
quantum error-correcting codes:
Theorem 6.7.10 Let 0 ≤ α 1 ≤ α 2 < m(q − 1) and assume that q = p
t is a prime
power, where t ≥ 1. Then there exists an p-ary asymmetric quantum GRM code with
parameters [[tq
m
, t[k(α 2 ) − k(α 1 )], d z /d x ]] p , where d z ≥ d(α 2 ), d x ≥ d(α
⊥
1 ), k(α 2 )
and k(α 1 ) are given in Eq. (6.1), d(α 2 ) is shown in Eq. (6.2) and d(α
⊥
1 ) = (a + 1)q
b ,
where α 1 + 1 = (q − 1)b + a and 0 ≤ a ≤ q − 1.
Proof We first note that since the inequality α 1 ≤ α 2 holds, then the inclusion
R q (α 1 , m) ⊂ R q (α 2 , m) also holds. The codes β(R q (α 1 , m)) and β(R q (α 2 , m))
have parameters [tq
m
, tk(α 1 ), d(α 1 )] p and [tq
m
, tk(α 2 ), d(α 2 )] p , respectively, where
k(α 1 ) and k(α 2 ) are computed according to Eq. (6.1) and d(α 1 ), d(α 2 ) are computed
by applying Eq. (6.2). We know that the parameter α
⊥
1 of the dual code [R q (α 1 , m)]
⊥
= R q (α
⊥
1 , m) is equal to α
⊥
1 = m(q − 1) − 1 − α 1 ; thus, the minimum distance of
[R q (α 1 , m)]
⊥ is d(α
⊥
1 ) = (a + 1)q
b , where α 1 + 1 = (q − 1)b + a and 0 ≤ a ≤
q − 1. Hence, the code [β(R q (α 1 , m))]
⊥ has minimum distance greater than or equal
to d(α
⊥
1 ). Applying Theorem 6.7.3, we obtain an [[tq
m
, t[k(α 2 ) − k(α 1 )], d z /d x ]] p
asymmetric stabilizer code, where d z ≥ d(α 2 ) and d x ≥ d(α
⊥
1 ). The proof is complete.
6.7.6.2 Construction II—Character Codes
The class of (classical) character codes was introduced by Ding et al. [33].
Before proceeding further, we need to define some concepts on character codes.
Let us consider the commutative additive group G = Z
m
2 , where m ≥ 1, and a finite
field F q of odd characteristic. Recall that the code C q (r, m) = C X , where X ⊂ Z
m
2
consists of all elements with Hamming weight greater than r , has parameters [2
m
,
s m (r ), 2
m−r
] q (see [33, Theorem 6]), where s m (r ) =
r
i=0
m
i
. The Euclidean dual
code [C q (r, m)]
⊥ of C q (r, m) is equivalent to C q (m − r − 1, m) (see [33, Theorem
8]) and, consequently, it has parameters [2
m
, s m (m − r − 1), 2
r +1
] q .
In the following, we utilize the code expansion applied to character codes to
generate AQECCs, as establishes the next result.
Theorem 6.7.11 Assume that 0 ≤ r 1 < r 2 ≤ m and let q = p
t be a power of an odd
prime p, where t ≥ 1. Then there exists an [[t2
m
, t[k(r 2 ) − k(r 1 )], d z /d x ]] p AQECC,
where k(r ) =
r
i=0
m
i
, d z ≥ 2
m−r 2 and d x ≥ 2
r 1 +1 .
Proof It is easy to see that the inclusion C q (r 1 , m) ⊂ C q (r 2 , m) holds. The dual code
[C q (r 1 , m)]
⊥ is equivalent to the code C q (m − r 1 − 1, m). Applying Theorem 6.7.3,
we get an [[t2
m
, t (k(r 2 ) − k(r 1 )), d z /d x ]] p AQECC, where t, k(r 1 ), k(r 2 ), d x and d z
are specified in the hypotheses.
155
where α
⊥
= m(q − 1) − 1 − α.
We utilize the properties of GRM codes in order to obtain more asymmetric
quantum error-correcting codes:
Theorem 6.7.10 Let 0 ≤ α 1 ≤ α 2 < m(q − 1) and assume that q = p
t is a prime
power, where t ≥ 1. Then there exists an p-ary asymmetric quantum GRM code with
parameters [[tq
m
, t[k(α 2 ) − k(α 1 )], d z /d x ]] p , where d z ≥ d(α 2 ), d x ≥ d(α
⊥
1 ), k(α 2 )
and k(α 1 ) are given in Eq. (6.1), d(α 2 ) is shown in Eq. (6.2) and d(α
⊥
1 ) = (a + 1)q
b ,
where α 1 + 1 = (q − 1)b + a and 0 ≤ a ≤ q − 1.
Proof We first note that since the inequality α 1 ≤ α 2 holds, then the inclusion
R q (α 1 , m) ⊂ R q (α 2 , m) also holds. The codes β(R q (α 1 , m)) and β(R q (α 2 , m))
have parameters [tq
m
, tk(α 1 ), d(α 1 )] p and [tq
m
, tk(α 2 ), d(α 2 )] p , respectively, where
k(α 1 ) and k(α 2 ) are computed according to Eq. (6.1) and d(α 1 ), d(α 2 ) are computed
by applying Eq. (6.2). We know that the parameter α
⊥
1 of the dual code [R q (α 1 , m)]
⊥
= R q (α
⊥
1 , m) is equal to α
⊥
1 = m(q − 1) − 1 − α 1 ; thus, the minimum distance of
[R q (α 1 , m)]
⊥ is d(α
⊥
1 ) = (a + 1)q
b , where α 1 + 1 = (q − 1)b + a and 0 ≤ a ≤
q − 1. Hence, the code [β(R q (α 1 , m))]
⊥ has minimum distance greater than or equal
to d(α
⊥
1 ). Applying Theorem 6.7.3, we obtain an [[tq
m
, t[k(α 2 ) − k(α 1 )], d z /d x ]] p
asymmetric stabilizer code, where d z ≥ d(α 2 ) and d x ≥ d(α
⊥
1 ). The proof is complete.
6.7.6.2 Construction II—Character Codes
The class of (classical) character codes was introduced by Ding et al. [33].
Before proceeding further, we need to define some concepts on character codes.
Let us consider the commutative additive group G = Z
m
2 , where m ≥ 1, and a finite
field F q of odd characteristic. Recall that the code C q (r, m) = C X , where X ⊂ Z
m
2
consists of all elements with Hamming weight greater than r , has parameters [2
m
,
s m (r ), 2
m−r
] q (see [33, Theorem 6]), where s m (r ) =
r
i=0
m
i
. The Euclidean dual
code [C q (r, m)]
⊥ of C q (r, m) is equivalent to C q (m − r − 1, m) (see [33, Theorem
8]) and, consequently, it has parameters [2
m
, s m (m − r − 1), 2
r +1
] q .
In the following, we utilize the code expansion applied to character codes to
generate AQECCs, as establishes the next result.
Theorem 6.7.11 Assume that 0 ≤ r 1 < r 2 ≤ m and let q = p
t be a power of an odd
prime p, where t ≥ 1. Then there exists an [[t2
m
, t[k(r 2 ) − k(r 1 )], d z /d x ]] p AQECC,
where k(r ) =
r
i=0
m
i
, d z ≥ 2
m−r 2 and d x ≥ 2
r 1 +1 .
Proof It is easy to see that the inclusion C q (r 1 , m) ⊂ C q (r 2 , m) holds. The dual code
[C q (r 1 , m)]
⊥ is equivalent to the code C q (m − r 1 − 1, m). Applying Theorem 6.7.3,
we get an [[t2
m
, t (k(r 2 ) − k(r 1 )), d z /d x ]] p AQECC, where t, k(r 1 ), k(r 2 ), d x and d z
are specified in the hypotheses.