120
5 Quantum Code Constructions
Proof Since C 1 and C 3 are cyclic codes, the sum code C 1 + C 3 = {c 1 + c 3 |c 1 ∈
C 1 and c 3 ∈ C 3 } is also cyclic. We know that C 1 ⊂ C 1 + C 3 , so (C 1 + C 3 )
⊥
⊂ C
⊥
1 .
As C 1 contains its dual code, it follows that (C 1 + C 3 )
⊥
⊂ C
⊥
1 ⊂ C 1 ⊂ C 1 + C 3 .
Thus the sum C 1 + C 3 is also a dual-containing cyclic code.
Let g (x) = gcd(g 2 (x), g 4 (x)). Since g 2 (x)|g 1 (x) and g 4 (x)|g 3 (x), it follows
that g (x) divides g(x), which implies that the inclusion C 1 + C 3 ⊂ C 2 + C 4
is true, where C 2 + C 4 is also a cyclic code. From the fact that deg(g (x)) <
deg(g(x)), one has C 1 + C 3 C 2 + C 4 . Applying Theorem 5.5.1 to the codes
C 1 + C 3 and C 2 + C 4 , we can construct a QSC with parameters (a l , a r ) − [[n +
a l + a r , n − 2 deg(gcd(g 1 (x), g 3 (x)))]], where a l + a r < deg(gcd(g 1 (x), g 3 (x))) −
deg(gcd(g 2 (x), g 4 (x))), and corrects up to at least
d−1
2
phase errors and up to at
least
d
∗ −1
2
bit errors.
The second theorem is obtained by considering the intersection of cyclic codes.
Theorem 5.5.3 Let n ≥ 3 be an integer such that gcd(n, 2) = 1 and consider that
m = ord n (2). Let C 1 be an [n, k 1 , d 1 ] self-orthogonal cyclic code. Assume also that
C 2 and C 3 are two cyclic codes with parameters [n, k 2 , d 2 ] and [n, k 3 , d 3 ], respectively, such that {0} C
⊥
3 C 1 ∩ C 2 . Then for any pair of nonnegative integers
(a l , a r ) satisfying a l + a r < n − deg(g 3 (x)) − deg(lcm(g 1 (x), g 2 (x))), there exists
an
(a l , a r ) − [[n + a l + a r , 2 deg(lcm(g 1 (x), g 2 (x))) − n]]
QSC that corrects up to at least
d−1
2
phase errors and up to at least
d 3 −1
2
bit
errors, where d is the minimum distance of (C 1 ∩ C 2 )
⊥ , and g i (x) is the generator
polynomial of C i , i = 1, 2, 3.
Proof Since C 1 and C 2 are cyclic code, it follows that C 1 ∩ C 2 is cyclic, which
implies that its dual code (C 1 ∩ C 2 )
⊥ is also cyclic. As C 1 ∩ C 2 ⊂ C 1 , the inclusion C
⊥
1 ⊂ (C 1 ∩ C 2 )
⊥ holds. Since C 1 is self-orthogonal, we have C 1 ∩ C 2 ⊂ C 1 ⊂
C
⊥
1 ⊂ (C 1 ∩ C 2 )
⊥ , i.e., C 1 ∩ C 2 is self-orthogonal, which implies that (C 1 ∩ C 2 )
⊥
is dual-containing cyclic code. As C
⊥
3 C 1 ∩ C 2 , it follows that (C 1 ∩ C 2 )
⊥
C 3 .
The dimension of the QSC is equal to 2 deg(lcm(g 1 (x), g 2 (x))) − n and a l + a r <
n − deg(g 3 (x)) − deg(lcm(g 1 (x), g 2 (x))). Applying Theorem 5.5.1 to the codes
(C 1 ∩ C 2 )
⊥ and C 3 , for any pair of nonnegative integers (a l , a r ) satisfying a l + a r <
n − deg(g 3 (x)) − deg(lcm(g 1 (x), g 2 (x))), there exists an QSC with the specified
parameters.
5.5.2 Synchronizable Codes from BCH Codes
The results presented here are concerned with constructions of QSCs derived from
BCH codes.
5 Quantum Code Constructions
Proof Since C 1 and C 3 are cyclic codes, the sum code C 1 + C 3 = {c 1 + c 3 |c 1 ∈
C 1 and c 3 ∈ C 3 } is also cyclic. We know that C 1 ⊂ C 1 + C 3 , so (C 1 + C 3 )
⊥
⊂ C
⊥
1 .
As C 1 contains its dual code, it follows that (C 1 + C 3 )
⊥
⊂ C
⊥
1 ⊂ C 1 ⊂ C 1 + C 3 .
Thus the sum C 1 + C 3 is also a dual-containing cyclic code.
Let g (x) = gcd(g 2 (x), g 4 (x)). Since g 2 (x)|g 1 (x) and g 4 (x)|g 3 (x), it follows
that g (x) divides g(x), which implies that the inclusion C 1 + C 3 ⊂ C 2 + C 4
is true, where C 2 + C 4 is also a cyclic code. From the fact that deg(g (x)) <
deg(g(x)), one has C 1 + C 3 C 2 + C 4 . Applying Theorem 5.5.1 to the codes
C 1 + C 3 and C 2 + C 4 , we can construct a QSC with parameters (a l , a r ) − [[n +
a l + a r , n − 2 deg(gcd(g 1 (x), g 3 (x)))]], where a l + a r < deg(gcd(g 1 (x), g 3 (x))) −
deg(gcd(g 2 (x), g 4 (x))), and corrects up to at least
d−1
2
phase errors and up to at
least
d
∗ −1
2
bit errors.
The second theorem is obtained by considering the intersection of cyclic codes.
Theorem 5.5.3 Let n ≥ 3 be an integer such that gcd(n, 2) = 1 and consider that
m = ord n (2). Let C 1 be an [n, k 1 , d 1 ] self-orthogonal cyclic code. Assume also that
C 2 and C 3 are two cyclic codes with parameters [n, k 2 , d 2 ] and [n, k 3 , d 3 ], respectively, such that {0} C
⊥
3 C 1 ∩ C 2 . Then for any pair of nonnegative integers
(a l , a r ) satisfying a l + a r < n − deg(g 3 (x)) − deg(lcm(g 1 (x), g 2 (x))), there exists
an
(a l , a r ) − [[n + a l + a r , 2 deg(lcm(g 1 (x), g 2 (x))) − n]]
QSC that corrects up to at least
d−1
2
phase errors and up to at least
d 3 −1
2
bit
errors, where d is the minimum distance of (C 1 ∩ C 2 )
⊥ , and g i (x) is the generator
polynomial of C i , i = 1, 2, 3.
Proof Since C 1 and C 2 are cyclic code, it follows that C 1 ∩ C 2 is cyclic, which
implies that its dual code (C 1 ∩ C 2 )
⊥ is also cyclic. As C 1 ∩ C 2 ⊂ C 1 , the inclusion C
⊥
1 ⊂ (C 1 ∩ C 2 )
⊥ holds. Since C 1 is self-orthogonal, we have C 1 ∩ C 2 ⊂ C 1 ⊂
C
⊥
1 ⊂ (C 1 ∩ C 2 )
⊥ , i.e., C 1 ∩ C 2 is self-orthogonal, which implies that (C 1 ∩ C 2 )
⊥
is dual-containing cyclic code. As C
⊥
3 C 1 ∩ C 2 , it follows that (C 1 ∩ C 2 )
⊥
C 3 .
The dimension of the QSC is equal to 2 deg(lcm(g 1 (x), g 2 (x))) − n and a l + a r <
n − deg(g 3 (x)) − deg(lcm(g 1 (x), g 2 (x))). Applying Theorem 5.5.1 to the codes
(C 1 ∩ C 2 )
⊥ and C 3 , for any pair of nonnegative integers (a l , a r ) satisfying a l + a r <
n − deg(g 3 (x)) − deg(lcm(g 1 (x), g 2 (x))), there exists an QSC with the specified
parameters.
5.5.2 Synchronizable Codes from BCH Codes
The results presented here are concerned with constructions of QSCs derived from
BCH codes.