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5 Quantum Code Constructions
Theorem 5.5.5 Let n ≥ 3 be an integer such that gcd(n, 2) = 1 and assume that
2
m/2
< n ≤ 2
m
− 1, where m = ord n (2). Take integers a, b, e and f such that 2 ≤
e < a < b < f < min{{n2
m/2
/(2
m
− 1) − 1, n − 1, κ},
where
κ =
n
2 m −1
(2
m/2
− 1) and a, b, e, f ≡ 0 mod 2. Then, for any pair of nonnegative integers
(a l , a r ) satisfying a l + a r < m(t − w), there exists an (a l , a r ) − [[n + a l + a r , n −
2m(t + 1)]] QSC that corrects up to at least
d−1
2
phase errors and up to at least
d
∗ −1
2
bit errors, where d ≥ b + 1, d
∗
≥ e + 1, t = (b − 1)/2 and w = (e − 1)/2.
Proof Let C 1 be the binary BCH code of length n generated by
C 1 = =M 1 (x)M 3 (x) · · · M b (x),
where b = 2t + 1 and t is a nonnegative integer. Let C 2 be the binary BCH code of
length n generated by
C 2 = =M 1 (x)M 3 (x) · · · M a (x),
where a = 2u + 1 and u ≥ 1.
From construction, C 1 ⊂ C 2 ; by Proposition 5.5.1 it follows that C 1 contains its
dual code. Further, consider the binary BCH codes of length n generated by
C 3 = =M 1 (x)M 3 (x) · · · M f (x)
and
C 4 = =M 1 (x)M 3 (x) · · · M e (x),
where f = 2v + 1 with v ≥ 1, and e = 2w + 1 with w ≥ 0. From construction,
C 3 C 4 . It then follows that C 1 + C 3 C 2 + C 4 . Since e < a and b < f , from
Lemma 5.5.1 and a straightforward computation, the codes C 2 + C 4 and C 1 + C 3
have dimensions K 2 = n − m(w + 1) and K 1 = n − m(t + 1), respectively, which
implies that the dimension of the corresponding QSC is equal to K = n − 2m(t + 1)
and K 2 − K 1 = m(t − w). Since C 1 is dual-containing, C 1 + C 3 is also dualcontaining. From the BCH bound, the minimum distance d 13 of C 1 + C 3 satisfies
d 13 ≥ b + 1 and the minimum distance d 24 of C 2 + C 4 satisfies d 24 ≥ e + 1. Applying Theorem 5.5.2 to C 1 + C 3 and C 2 + C 4 we get the result.
5.5.3 Synchronizable Codes from Product Codes
For the reader convenience, we recall some basic concepts on product codes. Let us
consider C 1 and C 2 be two linear codes with parameters [n 1 , k 1 , d 1 ] and [n 2 , k 2 , d 2 ],
respectively, both over F q . Suppose G
(1) and G
(2) are the generator matrices of C 1
and C 2 , respectively. Then the product code C 1 ⊗ C 2 is a linear code with parameters
5 Quantum Code Constructions
Theorem 5.5.5 Let n ≥ 3 be an integer such that gcd(n, 2) = 1 and assume that
2
m/2
< n ≤ 2
m
− 1, where m = ord n (2). Take integers a, b, e and f such that 2 ≤
e < a < b < f < min{{n2
m/2
/(2
m
− 1) − 1, n − 1, κ},
where
κ =
n
2 m −1
(2
m/2
− 1) and a, b, e, f ≡ 0 mod 2. Then, for any pair of nonnegative integers
(a l , a r ) satisfying a l + a r < m(t − w), there exists an (a l , a r ) − [[n + a l + a r , n −
2m(t + 1)]] QSC that corrects up to at least
d−1
2
phase errors and up to at least
d
∗ −1
2
bit errors, where d ≥ b + 1, d
∗
≥ e + 1, t = (b − 1)/2 and w = (e − 1)/2.
Proof Let C 1 be the binary BCH code of length n generated by
C 1 = =M 1 (x)M 3 (x) · · · M b (x),
where b = 2t + 1 and t is a nonnegative integer. Let C 2 be the binary BCH code of
length n generated by
C 2 = =M 1 (x)M 3 (x) · · · M a (x),
where a = 2u + 1 and u ≥ 1.
From construction, C 1 ⊂ C 2 ; by Proposition 5.5.1 it follows that C 1 contains its
dual code. Further, consider the binary BCH codes of length n generated by
C 3 = =M 1 (x)M 3 (x) · · · M f (x)
and
C 4 = =M 1 (x)M 3 (x) · · · M e (x),
where f = 2v + 1 with v ≥ 1, and e = 2w + 1 with w ≥ 0. From construction,
C 3 C 4 . It then follows that C 1 + C 3 C 2 + C 4 . Since e < a and b < f , from
Lemma 5.5.1 and a straightforward computation, the codes C 2 + C 4 and C 1 + C 3
have dimensions K 2 = n − m(w + 1) and K 1 = n − m(t + 1), respectively, which
implies that the dimension of the corresponding QSC is equal to K = n − 2m(t + 1)
and K 2 − K 1 = m(t − w). Since C 1 is dual-containing, C 1 + C 3 is also dualcontaining. From the BCH bound, the minimum distance d 13 of C 1 + C 3 satisfies
d 13 ≥ b + 1 and the minimum distance d 24 of C 2 + C 4 satisfies d 24 ≥ e + 1. Applying Theorem 5.5.2 to C 1 + C 3 and C 2 + C 4 we get the result.
5.5.3 Synchronizable Codes from Product Codes
For the reader convenience, we recall some basic concepts on product codes. Let us
consider C 1 and C 2 be two linear codes with parameters [n 1 , k 1 , d 1 ] and [n 2 , k 2 , d 2 ],
respectively, both over F q . Suppose G
(1) and G
(2) are the generator matrices of C 1
and C 2 , respectively. Then the product code C 1 ⊗ C 2 is a linear code with parameters