6.7 New Codes from Old
157
6.7.6.4 Construction IV—QR Codes
Here, we construct families of AQECCs derived from quadratic residue (QR) codes
[67, 114]. A family of quantum codes derived from classical QR codes was constructed in [80, Theorems 40 and 41].
Definition 6.7.1 Let p be an odd prime not dividing q, where q is a prime power
that is a square modulo p. Let Q be the set of nonzero squares modulo p and let C
consisting of non-squares modulo p. The quadratic residue codes Q, Q
, C and C
are
cyclic codes with generator polynomials q(x), (x − 1)q(x), c(x), (x − 1)c(x),
respectively, where
q(x) =
r ∈Q
(x − α
r
), c(x) =
s∈C
(x − α
s
)
have coefficients from F q , and α is a primitive pth root of unity belonging to some
extension field of F q .
The codes Q and C have the same parameters [ p, (p + 1)/2, d 1 ] q , where (d 1 )
2
≥
p. Analogously, the codes Q
and C
also have the same parameters [ p, (p − 1)/2, d 2 ] q ,
where (d 2 )
2
≥ p.
In the next result, we construct families of AQECCs by expanding quadratic
residue codes.
Theorem 6.7.14 Let p be a prime number of the form p ≡ 1 mod 4, and let q =
p
t
∗ (t ≥ 1) be a power of a prime that is not divisible by p. If q is a quadratic
residue modulo p, then there exists an [[t p, t, d z /d x ]] p ∗ asymmetric quantum errorcorrecting code, where d z and d x satisfy d z ≥
√ p and d x ≥
√ p.
Proof Let us consider the codes Q, Q
and C given above. Since p = 4k + 1, we have
Q
= C
⊥ , hence, C
⊥
⊂ Q. The codes Q and C
⊥ have parameters, respectively, given
by [ p, (p + 1)/2, d 1 ] q , with (d 1 )
2
≥ p and [ p, (p − 1)/2, d 2 ] q , where (d 2 )
2
≥ p.
Proceeding similarly as in the proof of Theorem 6.7.2 we have an [[t p, t, d z /d x ]] p ∗
AQECC, where d z and d x satisfy d z ≥
√ p and d x ≥
√ p.
Theorem 6.7.15 Let p be a prime of the form p ≡ 3 mod 4, and let q = p
t
∗ (t ≥ 1)
be a power of a prime that is not divisible by p. If q is a quadratic residue modulo p,
then there exists an [[t p, t, d z /d x ]] p ∗ AQECC, where d z ≥ d, d x ≥ d and d satisfies
d
2
− d + 1 ≥ p.
Proof Since p = 4k − 1, the dual code Q
⊥ of the code Q is equal to Q
⊥
=
Q
, so Q
⊥
⊂ Q. The codes Q and Q
⊥ have parameters [ p, (p + 1)/2, d] q and
[ p, (p − 1)/2, d
≥ d] q , respectively, and the minimum distance is bounded by
d
2
− d + 1 ≥ p (see, for instance, the proof of Theorem 40 in [80]). Applying Theorem 6.7.3, we get an [[t p, t, d z /d x ]] p ∗ code, where d z ≥ d, d x ≥ d and
d
2
− d + 1 ≥ p.
157
6.7.6.4 Construction IV—QR Codes
Here, we construct families of AQECCs derived from quadratic residue (QR) codes
[67, 114]. A family of quantum codes derived from classical QR codes was constructed in [80, Theorems 40 and 41].
Definition 6.7.1 Let p be an odd prime not dividing q, where q is a prime power
that is a square modulo p. Let Q be the set of nonzero squares modulo p and let C
consisting of non-squares modulo p. The quadratic residue codes Q, Q
, C and C
are
cyclic codes with generator polynomials q(x), (x − 1)q(x), c(x), (x − 1)c(x),
respectively, where
q(x) =
r ∈Q
(x − α
r
), c(x) =
s∈C
(x − α
s
)
have coefficients from F q , and α is a primitive pth root of unity belonging to some
extension field of F q .
The codes Q and C have the same parameters [ p, (p + 1)/2, d 1 ] q , where (d 1 )
2
≥
p. Analogously, the codes Q
and C
also have the same parameters [ p, (p − 1)/2, d 2 ] q ,
where (d 2 )
2
≥ p.
In the next result, we construct families of AQECCs by expanding quadratic
residue codes.
Theorem 6.7.14 Let p be a prime number of the form p ≡ 1 mod 4, and let q =
p
t
∗ (t ≥ 1) be a power of a prime that is not divisible by p. If q is a quadratic
residue modulo p, then there exists an [[t p, t, d z /d x ]] p ∗ asymmetric quantum errorcorrecting code, where d z and d x satisfy d z ≥
√ p and d x ≥
√ p.
Proof Let us consider the codes Q, Q
and C given above. Since p = 4k + 1, we have
Q
= C
⊥ , hence, C
⊥
⊂ Q. The codes Q and C
⊥ have parameters, respectively, given
by [ p, (p + 1)/2, d 1 ] q , with (d 1 )
2
≥ p and [ p, (p − 1)/2, d 2 ] q , where (d 2 )
2
≥ p.
Proceeding similarly as in the proof of Theorem 6.7.2 we have an [[t p, t, d z /d x ]] p ∗
AQECC, where d z and d x satisfy d z ≥
√ p and d x ≥
√ p.
Theorem 6.7.15 Let p be a prime of the form p ≡ 3 mod 4, and let q = p
t
∗ (t ≥ 1)
be a power of a prime that is not divisible by p. If q is a quadratic residue modulo p,
then there exists an [[t p, t, d z /d x ]] p ∗ AQECC, where d z ≥ d, d x ≥ d and d satisfies
d
2
− d + 1 ≥ p.
Proof Since p = 4k − 1, the dual code Q
⊥ of the code Q is equal to Q
⊥
=
Q
, so Q
⊥
⊂ Q. The codes Q and Q
⊥ have parameters [ p, (p + 1)/2, d] q and
[ p, (p − 1)/2, d
≥ d] q , respectively, and the minimum distance is bounded by
d
2
− d + 1 ≥ p (see, for instance, the proof of Theorem 40 in [80]). Applying Theorem 6.7.3, we get an [[t p, t, d z /d x ]] p ∗ code, where d z ≥ d, d x ≥ d and
d
2
− d + 1 ≥ p.