7.7 AQCCs
199
• [(n, 2i − 2t, μ
∗
; 4, [d z ] f /[d x ] f )] q ,
where q = p
l , p is an odd prime, l ≥ 2, n = q + 1, (d z ) f ≥ n − 2i and (d x ) f ≥
2t + 2, for all 1 ≤ t ≤ i − 1, with 2 ≤ i ≤
n
2
− 1;
• [(q − 1, i − t − 1, μ
∗
; 3, [d z ] f /[d x ] f )] q ,
where q ≥ 8 is a prime power (d z ) f ≥ q − i − 1 and (d x ) f ≥ t + 2, for all 1 ≤
t ≤ i − 2, where 3 ≤ i ≤ q − 3;
• [(q − 1, i − t, μ
∗
; 2, [d z ] f /[d x ] f )] q ,
where (d z ) f ≥ q − i − 1, (d x ) f ≥ t + 2, for all 1 ≤ t ≤ i − 1, where 2 ≤ i ≤
q − 3;
• [(n, n − t − k − 2, μ
∗
; 3, [d z ] f /[d x ] f )] q ,
where (d z ) f ≥ t + 2 and (d x ) f ≥ k + 1, where q ≥ 5 is a prime power, k ≥ 1
and n are integers such that 5 ≤ n ≤ q and k ≤ n − 4 and t is an integer with
1 ≤ t ≤ n − k − 2;
• [(n, n − t − k − 1, μ
∗
; 2, [d z ] f /[d x ] f )] q ,
where q ≥ 5 is a prime power, k ≥ 1, n ≥ 5 are integers such that n ≤ q, k ≤
n − 4, 1 ≤ t ≤ n − k − 1, (d z ) f ≥ t + 2 and (d x ) f ≥ k + 1.
For propaedeutic purposes we divide into three different types of constructions.
Construction I, shown in Sect. 7.7.1, is a general construction of AQCCs, i.e., we do
not consider any specific class of codes. In Construction II (Sect. 7.7.2) we utilize the
class of BCH codes to derive families of AQCCs and in Construction III (Sect. 7.7.3)
we show how to derive families of AQCCs from Reed–Solomon (RS) and generalized
Reed–Solomon (GRS) codes.
7.7.1 Construction I—General Construction
Theorem 7.7.1 establishes the first construction method.
Theorem 7.7.1 (General Construction) Let q be a prime power and n be a positive
integer. Then there exists asymmetric quantum convolutional codes with parameters
[(n, rk H 0 , μ
∗
; γ 1 + γ 2 , (d z ) f /(d x ) f )] q ,
where γ 1 = μ(rk H μ + rk H
μ ) +
μ−1
i=1
(μ − i)[rk H
(μ−i) − rk H
(μ−i+1) ], γ 2 = μ(rk
H
μ ) +
μ−1
i=1
(μ − i)[rk H
(μ−i) − rk H
(μ−i+1) ], (d x ) f ≥ (d 1 ) f ≥ d
⊥ and (d z ) f ≥ (d 2 )
⊥
f ,
where (d 1 ) f , d
⊥
, (d 2 )
⊥
f and the matrices H 0 , H
0 , H 1 , H
1 , . . . , H μ , H
μ , are constructed below.
Proof Consider a set of m < n linearly independent (LI) vectors v i ∈ F
n
q , i =
1, 2, . . . , m. Let
199
• [(n, 2i − 2t, μ
∗
; 4, [d z ] f /[d x ] f )] q ,
where q = p
l , p is an odd prime, l ≥ 2, n = q + 1, (d z ) f ≥ n − 2i and (d x ) f ≥
2t + 2, for all 1 ≤ t ≤ i − 1, with 2 ≤ i ≤
n
2
− 1;
• [(q − 1, i − t − 1, μ
∗
; 3, [d z ] f /[d x ] f )] q ,
where q ≥ 8 is a prime power (d z ) f ≥ q − i − 1 and (d x ) f ≥ t + 2, for all 1 ≤
t ≤ i − 2, where 3 ≤ i ≤ q − 3;
• [(q − 1, i − t, μ
∗
; 2, [d z ] f /[d x ] f )] q ,
where (d z ) f ≥ q − i − 1, (d x ) f ≥ t + 2, for all 1 ≤ t ≤ i − 1, where 2 ≤ i ≤
q − 3;
• [(n, n − t − k − 2, μ
∗
; 3, [d z ] f /[d x ] f )] q ,
where (d z ) f ≥ t + 2 and (d x ) f ≥ k + 1, where q ≥ 5 is a prime power, k ≥ 1
and n are integers such that 5 ≤ n ≤ q and k ≤ n − 4 and t is an integer with
1 ≤ t ≤ n − k − 2;
• [(n, n − t − k − 1, μ
∗
; 2, [d z ] f /[d x ] f )] q ,
where q ≥ 5 is a prime power, k ≥ 1, n ≥ 5 are integers such that n ≤ q, k ≤
n − 4, 1 ≤ t ≤ n − k − 1, (d z ) f ≥ t + 2 and (d x ) f ≥ k + 1.
For propaedeutic purposes we divide into three different types of constructions.
Construction I, shown in Sect. 7.7.1, is a general construction of AQCCs, i.e., we do
not consider any specific class of codes. In Construction II (Sect. 7.7.2) we utilize the
class of BCH codes to derive families of AQCCs and in Construction III (Sect. 7.7.3)
we show how to derive families of AQCCs from Reed–Solomon (RS) and generalized
Reed–Solomon (GRS) codes.
7.7.1 Construction I—General Construction
Theorem 7.7.1 establishes the first construction method.
Theorem 7.7.1 (General Construction) Let q be a prime power and n be a positive
integer. Then there exists asymmetric quantum convolutional codes with parameters
[(n, rk H 0 , μ
∗
; γ 1 + γ 2 , (d z ) f /(d x ) f )] q ,
where γ 1 = μ(rk H μ + rk H
μ ) +
μ−1
i=1
(μ − i)[rk H
(μ−i) − rk H
(μ−i+1) ], γ 2 = μ(rk
H
μ ) +
μ−1
i=1
(μ − i)[rk H
(μ−i) − rk H
(μ−i+1) ], (d x ) f ≥ (d 1 ) f ≥ d
⊥ and (d z ) f ≥ (d 2 )
⊥
f ,
where (d 1 ) f , d
⊥
, (d 2 )
⊥
f and the matrices H 0 , H
0 , H 1 , H
1 , . . . , H μ , H
μ , are constructed below.
Proof Consider a set of m < n linearly independent (LI) vectors v i ∈ F
n
q , i =
1, 2, . . . , m. Let