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5 Quantum Code Constructions
5.1.5 Construction V
In this subsection, we show the existence of (classical) cyclic codes whose defining
set consists of only one cyclotomic coset containing at least two consecutive integers.
This fact allows us to construct quantum codes with good parameters.
Lemma 5.1.6 in the following is a particular case of the CSS construction.
Lemma 5.1.6 ([4, Lemma 17]) If there exists a classical linear [n, k, d] q code C
such that C
⊥
⊂ C, then there exists an [[n, 2k − n, ≥ d]] q stabilizer code that is
pure to d.
Theorem 5.1.10 establishes conditions for the existence of a cyclic code whose
defining set consists of only one q-coset containing at least two consecutive integers.
This fact produces conditions to construct quantum codes with good parameters (in
the sense of the QSB).
Theorem 5.1.10 Let q ≥ 3 be a prime power and n > m be a positive integer such
that gcd(q, n) = 1 and gcd(q
a i − 1, n) = 1 for every i = 1, 2, . . . , r , where m =
ord n (q) ≥ r + 2 and 1 ≤ r, a 1 , a 2 , . . . , a r < m are integers. If n| gcd(t 2 , . . . , t r ),
where t j = [( j − ( j − 1)q
a j )(q
a j − 1)
−1
− (q
a 1 − 1)
−1
] for every j = 2, . . . , r (the
operations are performed modulo n), then there exists an [n, n − m
∗
, d ≥ r + 2] q
cyclic code, where m
∗ is the cardinality of the q-coset containing r + 1 consecutive
integers.
Proof We will investigate the following system of congruences:
xq
a 1 ≡ (x + 1) mod n
(x + 1)q
a 2 ≡ (x + 2) mod n
(x + 2)q
a 3 ≡ (x + 3) mod n
. . .
(x + r − 1)q
a r ≡ (x + r ) mod n,
where 1 ≤ r, a 1 , a 2 , . . . , a r < m. Since gcd(q
a i − 1, n) = 1 for every i = 1, 2, . . . , r ,
it follows that such system is equivalent to
x ≡ (q
a 1 − 1)
−1
mod n
x ≡ (2 − q
a 2 )(q
a 2 − 1)
−1
mod n
x ≡ (3 − 2q
a 3 )(q
a 3 − 1)
−1
mod n
. . .
x ≡ [r − (r − 1)q
a r ](q
a r − 1)
−1
mod n,
5 Quantum Code Constructions
5.1.5 Construction V
In this subsection, we show the existence of (classical) cyclic codes whose defining
set consists of only one cyclotomic coset containing at least two consecutive integers.
This fact allows us to construct quantum codes with good parameters.
Lemma 5.1.6 in the following is a particular case of the CSS construction.
Lemma 5.1.6 ([4, Lemma 17]) If there exists a classical linear [n, k, d] q code C
such that C
⊥
⊂ C, then there exists an [[n, 2k − n, ≥ d]] q stabilizer code that is
pure to d.
Theorem 5.1.10 establishes conditions for the existence of a cyclic code whose
defining set consists of only one q-coset containing at least two consecutive integers.
This fact produces conditions to construct quantum codes with good parameters (in
the sense of the QSB).
Theorem 5.1.10 Let q ≥ 3 be a prime power and n > m be a positive integer such
that gcd(q, n) = 1 and gcd(q
a i − 1, n) = 1 for every i = 1, 2, . . . , r , where m =
ord n (q) ≥ r + 2 and 1 ≤ r, a 1 , a 2 , . . . , a r < m are integers. If n| gcd(t 2 , . . . , t r ),
where t j = [( j − ( j − 1)q
a j )(q
a j − 1)
−1
− (q
a 1 − 1)
−1
] for every j = 2, . . . , r (the
operations are performed modulo n), then there exists an [n, n − m
∗
, d ≥ r + 2] q
cyclic code, where m
∗ is the cardinality of the q-coset containing r + 1 consecutive
integers.
Proof We will investigate the following system of congruences:
xq
a 1 ≡ (x + 1) mod n
(x + 1)q
a 2 ≡ (x + 2) mod n
(x + 2)q
a 3 ≡ (x + 3) mod n
. . .
(x + r − 1)q
a r ≡ (x + r ) mod n,
where 1 ≤ r, a 1 , a 2 , . . . , a r < m. Since gcd(q
a i − 1, n) = 1 for every i = 1, 2, . . . , r ,
it follows that such system is equivalent to
x ≡ (q
a 1 − 1)
−1
mod n
x ≡ (2 − q
a 2 )(q
a 2 − 1)
−1
mod n
x ≡ (3 − 2q
a 3 )(q
a 3 − 1)
−1
mod n
. . .
x ≡ [r − (r − 1)q
a r ](q
a r − 1)
−1
mod n,