5.5 Quantum Synchronizable Codes
123
Table 5.18 Some QSCs from Theorem 5.5.4
[[n + a l + a r , n − 2m(t + 1)]]
d, d ∗ , a l + a r
(a l , a r ) − [[63 + a l + a r , 27]]
d ≥ 6, d ∗ ≥ 2, 0 ≤ a i + a r < 12
(a l , a r ) − [[85 + a l + a r , 53]]
d ≥ 4, d ∗ ≥ 2, 0 ≤ a i + a r < 8
(a l , a r ) − [[127 + a l + a r , 29]]
d ≥ 14, d ∗ ≥ 2, 0 ≤ a i + a r < 42
(a l , a r ) − [[127 + a l + a r , 29]]
d ≥ 14, d ∗ ≥ 4, 0 ≤ a i + a r < 35
(a l , a r ) − [[127 + a l + a r , 29]]
d ≥ 14, d ∗ ≥ 6, 0 ≤ a i + a r < 28
(a l , a r ) − [[127 + a l + a r , 29]]
d ≥ 14, d ∗ ≥ 8, 0 ≤ a i + a r < 21
(a l , a r ) − [[127 + a l + a r , 29]]
d ≥ 14, d ∗ ≥ 10, 0 ≤ a i + a r < 14
(a l , a r ) − [[1365 + a l + a r , 1125]]
d ≥ 20, d ∗ ≥ 2, 0 ≤ a i + a r < 108
(a l , a r ) − [[1365 + a l + a r , 1125]]
d ≥ 20, d ∗ ≥ 4, 0 ≤ a i + a r < 96
(a l , a r ) − [[1365 + a l + a r , 1125]]
d ≥ 20, d ∗ ≥ 6, 0 ≤ a i + a r < 84
(a l , a r ) − [[1365 + a l + a r , 1125]]
d ≥ 20, d ∗ ≥ 8, 0 ≤ a i + a r < 72
(a l , a r ) − [[1365 + a l + a r , 1125]]
d ≥ 20, d ∗ ≥ 18, 0 ≤ a i + a r < 12
(a l , a r ) − [[1365 + a l + a r , 1245]]
d ≥ 10, d ∗ ≥ 2, 0 ≤ a i + a r < 48
(a l , a r ) − [[1365 + a l + a r , 1245]]
d ≥ 10, d ∗ ≥ 4, 0 ≤ a i + a r < 36
(a l , a r ) − [[1365 + a l + a r , 1245]]
d ≥ 10, d ∗ ≥ 6, 0 ≤ a i + a r < 24
(a l , a r ) − [[1365 + a l + a r , 1245]]
d ≥ 10, d ∗ ≥ 8, 0 ≤ a i + a r < 12
[n 1 n 2 , k 1 k 2 , d 1 d 2 ] code over F q generated by the Kronecker product matrix G
(1)
⊗
G
(2) (see Definition 1.7.14) defined as
G
(1)
⊗ G
(2)
=
⎡
⎢
⎢
⎢
⎢
⎣
g
(1)
11 G
(2)
g
(1)
12 G
(2)
· · · g
(1)
1n 1
G
(2)
g
(1)
21 G
(2)
g
(1)
22 G
(2)
· · · g
(1)
2n 1
G
(2)
. . .
. . .
. . .
. . .
g
(1)
k 1 1 G
(2)
g
(1)
k 1 2 G
(2)
· · · g
(1)
k 1 n 1
G
(2)
⎤
⎥
⎥
⎥
⎥
⎦
We can now prove the following result.
Theorem 5.5.6 Let n and n
∗ be two positive odd integers such that gcd(n, n
∗
) = 1.
Let C 1 be an [n, k 1 , d 1 ] self-orthogonal cyclic code and let C 2 be an [n, k 2 , d 2 ] cyclic
code, both over F 2 . Assume also that C 3 and C 4 are two cyclic codes with parameters
[n
∗
, k 3 , d 3 ] and [n
∗
, k 4 , d 4 ], respectively, both over F 2 , such that (C 1 ⊗ C 3 )
⊥
C 2 ⊗ C 4 . Then for any pair of nonnegative integers (a l , a r ) satisfying a l + a r <
k 1 k 3 + k 2 k 4 − nn
∗ , there exists an (a l , a r ) − [[nn
∗
+ a l + a r , nn
∗
− 2k 1 k 3 ]] QSC
that corrects up to at least
d−1
2
phase errors and up to at least
d 2 d 4 −1
2
bit errors,
where d is the minimum distance of (C 1 ⊗ C 3 )
⊥ , which satisfies d ≥ d 2 d 4 .
Proof As gcd(n, n
∗
) = 1, it follows that the product code C 2 ⊗ C 4 ( consequently
C 1 ⊗ C 3 and (C 1 ⊗ C 3 )
⊥ ), is also cyclic (see [114, Theorem 1, p. 570]). The elements
of the code C 1 ⊗ C 3 are linear combinations of vectors v
(1)
i ⊗ w
(3)
j , where v
(1)
i
∈ C 1
123
Table 5.18 Some QSCs from Theorem 5.5.4
[[n + a l + a r , n − 2m(t + 1)]]
d, d ∗ , a l + a r
(a l , a r ) − [[63 + a l + a r , 27]]
d ≥ 6, d ∗ ≥ 2, 0 ≤ a i + a r < 12
(a l , a r ) − [[85 + a l + a r , 53]]
d ≥ 4, d ∗ ≥ 2, 0 ≤ a i + a r < 8
(a l , a r ) − [[127 + a l + a r , 29]]
d ≥ 14, d ∗ ≥ 2, 0 ≤ a i + a r < 42
(a l , a r ) − [[127 + a l + a r , 29]]
d ≥ 14, d ∗ ≥ 4, 0 ≤ a i + a r < 35
(a l , a r ) − [[127 + a l + a r , 29]]
d ≥ 14, d ∗ ≥ 6, 0 ≤ a i + a r < 28
(a l , a r ) − [[127 + a l + a r , 29]]
d ≥ 14, d ∗ ≥ 8, 0 ≤ a i + a r < 21
(a l , a r ) − [[127 + a l + a r , 29]]
d ≥ 14, d ∗ ≥ 10, 0 ≤ a i + a r < 14
(a l , a r ) − [[1365 + a l + a r , 1125]]
d ≥ 20, d ∗ ≥ 2, 0 ≤ a i + a r < 108
(a l , a r ) − [[1365 + a l + a r , 1125]]
d ≥ 20, d ∗ ≥ 4, 0 ≤ a i + a r < 96
(a l , a r ) − [[1365 + a l + a r , 1125]]
d ≥ 20, d ∗ ≥ 6, 0 ≤ a i + a r < 84
(a l , a r ) − [[1365 + a l + a r , 1125]]
d ≥ 20, d ∗ ≥ 8, 0 ≤ a i + a r < 72
(a l , a r ) − [[1365 + a l + a r , 1125]]
d ≥ 20, d ∗ ≥ 18, 0 ≤ a i + a r < 12
(a l , a r ) − [[1365 + a l + a r , 1245]]
d ≥ 10, d ∗ ≥ 2, 0 ≤ a i + a r < 48
(a l , a r ) − [[1365 + a l + a r , 1245]]
d ≥ 10, d ∗ ≥ 4, 0 ≤ a i + a r < 36
(a l , a r ) − [[1365 + a l + a r , 1245]]
d ≥ 10, d ∗ ≥ 6, 0 ≤ a i + a r < 24
(a l , a r ) − [[1365 + a l + a r , 1245]]
d ≥ 10, d ∗ ≥ 8, 0 ≤ a i + a r < 12
[n 1 n 2 , k 1 k 2 , d 1 d 2 ] code over F q generated by the Kronecker product matrix G
(1)
⊗
G
(2) (see Definition 1.7.14) defined as
G
(1)
⊗ G
(2)
=
⎡
⎢
⎢
⎢
⎢
⎣
g
(1)
11 G
(2)
g
(1)
12 G
(2)
· · · g
(1)
1n 1
G
(2)
g
(1)
21 G
(2)
g
(1)
22 G
(2)
· · · g
(1)
2n 1
G
(2)
. . .
. . .
. . .
. . .
g
(1)
k 1 1 G
(2)
g
(1)
k 1 2 G
(2)
· · · g
(1)
k 1 n 1
G
(2)
⎤
⎥
⎥
⎥
⎥
⎦
We can now prove the following result.
Theorem 5.5.6 Let n and n
∗ be two positive odd integers such that gcd(n, n
∗
) = 1.
Let C 1 be an [n, k 1 , d 1 ] self-orthogonal cyclic code and let C 2 be an [n, k 2 , d 2 ] cyclic
code, both over F 2 . Assume also that C 3 and C 4 are two cyclic codes with parameters
[n
∗
, k 3 , d 3 ] and [n
∗
, k 4 , d 4 ], respectively, both over F 2 , such that (C 1 ⊗ C 3 )
⊥
C 2 ⊗ C 4 . Then for any pair of nonnegative integers (a l , a r ) satisfying a l + a r <
k 1 k 3 + k 2 k 4 − nn
∗ , there exists an (a l , a r ) − [[nn
∗
+ a l + a r , nn
∗
− 2k 1 k 3 ]] QSC
that corrects up to at least
d−1
2
phase errors and up to at least
d 2 d 4 −1
2
bit errors,
where d is the minimum distance of (C 1 ⊗ C 3 )
⊥ , which satisfies d ≥ d 2 d 4 .
Proof As gcd(n, n
∗
) = 1, it follows that the product code C 2 ⊗ C 4 ( consequently
C 1 ⊗ C 3 and (C 1 ⊗ C 3 )
⊥ ), is also cyclic (see [114, Theorem 1, p. 570]). The elements
of the code C 1 ⊗ C 3 are linear combinations of vectors v
(1)
i ⊗ w
(3)
j , where v
(1)
i
∈ C 1