198
7 Constructions of QCCs
Table 7.8 Some new codes
MDS QCCs
QCCs
[(q, q − 2m, 1; 1, d f = m + 2)] q
[(2q 2 , 2q 2 + q − 2m, 1; 1, d f ≥
m − q + 2)] q 2
1 ≤ m ≤ (q − 2)/2
q − 2 ≤ m ≤ 2q − 2 and q = 2 t with t ≥ 3
[(4, 2, 1; 1, d f = 3)] 4
—
[(7, 5, 1; 1, d f = 3)] 7
—
[(7, 3, 1; 1, d f = 4)] 7
—
[(8, 6, 1; 1, d f = 3)] 8
[(128, 118, 1; 1, d f ≥ 3)] 8
[(8, 2, 1; 1, d f = 5)] 8
[(128, 114, 1; 1, d f ≥ 5)] 8
[(32, 30, 1; 1, d f = 3)] 32
[(2048, 2014, 1; 1, d f ≥ 3)] 32
[(32, 2, 1; 1, d f = 17)] 32
[(2048, 1986, 1; 1, d f ≥ 17)] 32
[(64, 62, 1; 1, d f = 3)] 64
[(8192, 8126, 1; 1, d f ≥ 3)] 64
[(64, 2, 1; 1, d f = 33)] 64
[(8192, 8066, 1; 1, d f ≥ 33)] 64
[(128; 126; 1; 1; d f = 3)] 128
[(32768; 32638; 1; 1; d f ≥ 3)] 128
[(128; 2; 1; 1; d f = 65)] 128
[(256, 254, 1; 1, d f = 3)] 256
[(256, 2, 1; 1, d f = 129)] 256
[(512; 510; 1; 1; d f = 3)] 512
[(512; 2; 1; 1; d f = 257)] 512
[(1024, 1022, 1; 1, d f = 3)] 1024
[(1024, 2, 1; 1, d f = 513)] 1024
have different probabilities. The parameters of an AQCC are denoted by [(n, k, μ
∗
; γ,
[d z ] f /[d x ] f )] q , where n is the frame size, k is the number of logical qudits per frame,
m is the memory, [d z ] f and [d x ] f are the free distance corresponding to phase-shift
and qudit-flip errors, respectively, and γ is the degree of the code.
In this subsection, we present the first families of AQQCs exhibited in the literature
[102]. Our asymmetric quantum convolutional codes have parameters given in the
following:
• [(n, 2i − 4, μ
∗
; 6, [d z ] f /[d x ] f )] q ,
where q = 2
t , t ≥ 4, n = q + 1, (d z ) f ≥ n − 2i − 1 and (d x ) f ≥ 3, for all 3 ≤
i ≤
q
2
− 1;
• [(n, 2i − 2t − 2, μ
∗
; 6, [d z ] f ≥ n − 2i − 1/[d x ] f ≥ 2t + 3)] q ,
where q = 2
l , l ≥ 4, n = q + 1, t integer with 1 ≤ t ≤ i − 2, 3 ≤ i ≤
q
2
;
• [(n, 2i − 2t, μ
∗
; 4, [d z ] f /[d x ] f )] q ,
where (d z ) f ≥ n − 2i − 1 and (d x ) f ≥ 2t + 3, q = 2
l , l ≥ 4, n = q + 1, t integer
with 1 ≤ t ≤ i − 1, 2 ≤ i ≤
q
2
;
• [(n, 2i − 2t − 2, μ
∗
; 6, [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 − 2, where 3 ≤ i ≤
n
2
− 1;
7 Constructions of QCCs
Table 7.8 Some new codes
MDS QCCs
QCCs
[(q, q − 2m, 1; 1, d f = m + 2)] q
[(2q 2 , 2q 2 + q − 2m, 1; 1, d f ≥
m − q + 2)] q 2
1 ≤ m ≤ (q − 2)/2
q − 2 ≤ m ≤ 2q − 2 and q = 2 t with t ≥ 3
[(4, 2, 1; 1, d f = 3)] 4
—
[(7, 5, 1; 1, d f = 3)] 7
—
[(7, 3, 1; 1, d f = 4)] 7
—
[(8, 6, 1; 1, d f = 3)] 8
[(128, 118, 1; 1, d f ≥ 3)] 8
[(8, 2, 1; 1, d f = 5)] 8
[(128, 114, 1; 1, d f ≥ 5)] 8
[(32, 30, 1; 1, d f = 3)] 32
[(2048, 2014, 1; 1, d f ≥ 3)] 32
[(32, 2, 1; 1, d f = 17)] 32
[(2048, 1986, 1; 1, d f ≥ 17)] 32
[(64, 62, 1; 1, d f = 3)] 64
[(8192, 8126, 1; 1, d f ≥ 3)] 64
[(64, 2, 1; 1, d f = 33)] 64
[(8192, 8066, 1; 1, d f ≥ 33)] 64
[(128; 126; 1; 1; d f = 3)] 128
[(32768; 32638; 1; 1; d f ≥ 3)] 128
[(128; 2; 1; 1; d f = 65)] 128
[(256, 254, 1; 1, d f = 3)] 256
[(256, 2, 1; 1, d f = 129)] 256
[(512; 510; 1; 1; d f = 3)] 512
[(512; 2; 1; 1; d f = 257)] 512
[(1024, 1022, 1; 1, d f = 3)] 1024
[(1024, 2, 1; 1, d f = 513)] 1024
have different probabilities. The parameters of an AQCC are denoted by [(n, k, μ
∗
; γ,
[d z ] f /[d x ] f )] q , where n is the frame size, k is the number of logical qudits per frame,
m is the memory, [d z ] f and [d x ] f are the free distance corresponding to phase-shift
and qudit-flip errors, respectively, and γ is the degree of the code.
In this subsection, we present the first families of AQQCs exhibited in the literature
[102]. Our asymmetric quantum convolutional codes have parameters given in the
following:
• [(n, 2i − 4, μ
∗
; 6, [d z ] f /[d x ] f )] q ,
where q = 2
t , t ≥ 4, n = q + 1, (d z ) f ≥ n − 2i − 1 and (d x ) f ≥ 3, for all 3 ≤
i ≤
q
2
− 1;
• [(n, 2i − 2t − 2, μ
∗
; 6, [d z ] f ≥ n − 2i − 1/[d x ] f ≥ 2t + 3)] q ,
where q = 2
l , l ≥ 4, n = q + 1, t integer with 1 ≤ t ≤ i − 2, 3 ≤ i ≤
q
2
;
• [(n, 2i − 2t, μ
∗
; 4, [d z ] f /[d x ] f )] q ,
where (d z ) f ≥ n − 2i − 1 and (d x ) f ≥ 2t + 3, q = 2
l , l ≥ 4, n = q + 1, t integer
with 1 ≤ t ≤ i − 1, 2 ≤ i ≤
q
2
;
• [(n, 2i − 2t − 2, μ
∗
; 6, [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 − 2, where 3 ≤ i ≤
n
2
− 1;