94
5 Quantum Code Constructions
Table 5.10 Code comparison
Our CSS codes—Construction II
CSS Codes in [4]
[[q m − 1, q m − 2m − 3, d ≥ 3]] q
[[n
, k
, d
]] q
[[q m − 1, q m − 4m − 3, d ≥ 4]] q
[[n
, k
, d
]] q
[[q m − 1, q m − 6m − 3, d ≥ 5]] q
[[n
, k
, d
]] q
m = 2
[[80, 74, d ≥ 3]] 9
[[80, 72, d
≥ 3]] 9
[[80, 70, d ≥ 4]] 9
[[80, 68, d
≥ 4]] 9
[[80, 66, d ≥ 5]] 9
[[80, 66, d
≥ 5]] 9
m = 3
[[124, 116, d ≥ 3]] 5
[[124, 112, d
≥ 3]] 5
[[124, 110, d ≥ 4]] 5
[[124, 106, d
≥ 4]] 5
[[124, 104, d ≥ 5]] 5
[[124, 100, d
≥ 5]] 5
[[342, 334, d ≥ 3]] 7
[[342, 330, d
≥ 3]] 7
[[342, 328, d ≥ 4]] 7
[[342, 324, d
≥ 4]] 7
[[342, 322, d ≥ 5]] 7
[[342, 318, d
≥ 5]] 7
m = 4
[[624, 614, d ≥ 3]] 5
[[624, 608, d
≥ 3]] 5
[[624, 606, d ≥ 4]] 5
[[624, 600, d
≥ 4]] 5
[[624, 598, d ≥ 5]] 5
[[624, 592, d
≥ 5]] 5
We need to recall three useful lemmas from [4].
Lemma 5.3.1 ([4, Lemma 1]) Assume that gcd(q, n) = 1. A cyclic code of length n
over F q with defining set Z contains its Euclidean dual code if and only if Z ∩ Z
−1
=
∅, where Z
−1
= {−z mod n|z ∈ Z }.
Lemma 5.3.2 Assume that gcd(q, n) = 1. A cyclic code of length n over F q 2 with
defining set Z contains its Hermitian dual code if and only if Z ∩ Z
−q
= ∅, where
Z
−q
= {−qz mod n|z ∈ Z }.
Lemma 5.3.3 (Hermitian Construction) If there exists a classical linear [n, k, d] q 2
code D such that D
⊥ H ⊂ D, then there exists an [[n, 2k − n, ≥ d]] q stabilizer code
that is pure to d. If the minimum distance d
⊥ H of D
⊥ H exceeds d, then the stabilizer
code is pure and has minimum distance d.
We utilize the notation C [a] to denote the cyclotomic coset containing a, where a
is not necessarily the smallest number in C [a] .
5 Quantum Code Constructions
Table 5.10 Code comparison
Our CSS codes—Construction II
CSS Codes in [4]
[[q m − 1, q m − 2m − 3, d ≥ 3]] q
[[n
, k
, d
]] q
[[q m − 1, q m − 4m − 3, d ≥ 4]] q
[[n
, k
, d
]] q
[[q m − 1, q m − 6m − 3, d ≥ 5]] q
[[n
, k
, d
]] q
m = 2
[[80, 74, d ≥ 3]] 9
[[80, 72, d
≥ 3]] 9
[[80, 70, d ≥ 4]] 9
[[80, 68, d
≥ 4]] 9
[[80, 66, d ≥ 5]] 9
[[80, 66, d
≥ 5]] 9
m = 3
[[124, 116, d ≥ 3]] 5
[[124, 112, d
≥ 3]] 5
[[124, 110, d ≥ 4]] 5
[[124, 106, d
≥ 4]] 5
[[124, 104, d ≥ 5]] 5
[[124, 100, d
≥ 5]] 5
[[342, 334, d ≥ 3]] 7
[[342, 330, d
≥ 3]] 7
[[342, 328, d ≥ 4]] 7
[[342, 324, d
≥ 4]] 7
[[342, 322, d ≥ 5]] 7
[[342, 318, d
≥ 5]] 7
m = 4
[[624, 614, d ≥ 3]] 5
[[624, 608, d
≥ 3]] 5
[[624, 606, d ≥ 4]] 5
[[624, 600, d
≥ 4]] 5
[[624, 598, d ≥ 5]] 5
[[624, 592, d
≥ 5]] 5
We need to recall three useful lemmas from [4].
Lemma 5.3.1 ([4, Lemma 1]) Assume that gcd(q, n) = 1. A cyclic code of length n
over F q with defining set Z contains its Euclidean dual code if and only if Z ∩ Z
−1
=
∅, where Z
−1
= {−z mod n|z ∈ Z }.
Lemma 5.3.2 Assume that gcd(q, n) = 1. A cyclic code of length n over F q 2 with
defining set Z contains its Hermitian dual code if and only if Z ∩ Z
−q
= ∅, where
Z
−q
= {−qz mod n|z ∈ Z }.
Lemma 5.3.3 (Hermitian Construction) If there exists a classical linear [n, k, d] q 2
code D such that D
⊥ H ⊂ D, then there exists an [[n, 2k − n, ≥ d]] q stabilizer code
that is pure to d. If the minimum distance d
⊥ H of D
⊥ H exceeds d, then the stabilizer
code is pure and has minimum distance d.
We utilize the notation C [a] to denote the cyclotomic coset containing a, where a
is not necessarily the smallest number in C [a] .