138
6 Asymmetric Quantum Codes
6.5.1 Review of RS and GRS Codes
We review basics concepts on Reed–Solomon (RS) and generalized Reed–Solomon
(GRS) codes. The reader can consult [67, 114] for more details concerning RS and
GRS codes. Let us recall the definition of such codes.
Let q be a prime power and n be a positive integer with gcd(q, n) = 1. Recall
that a cyclic code C (see Definition 4.4.4) of length n over F q is a BCH code with
designed distance δ if
Table 6.1 Quantum code comparison
New asymmetric codes
Asymmetric codes shown in [3]
[[n, k, d z ≥ d/d x ≥ (d − c)]] q
[[n, k ∗ , d z ∗ /d x ∗ ]] q
[[26, 16, d z ≥ 4/d x ≥ 3]] 3
[[26, 14, d z ∗ ≥ 4/d x ∗ ≥ 3]] 3
[[26, 15, d z ≥ 5/d x ≥ 3]] 3
[[26, 11, d z ∗ ≥ 5/d x ∗ ≥ 3]] 3
[[26, 13, d z ≥ 5/d x ≥ 4]] 3
[[26, 11, d z ∗ ≥ 5/d x ∗ ≥ 4]] 3
[[80, 58, d z ≥ 6/d x ≥ 5]] 3
[[80, 52, d z ∗ ≥ 6/d x ∗ ≥ 5]] 3
[[80, 54, d z ≥ 8/d x ≥ 5]] 3
[[80, 48, d z ∗ ≥ 8/d x ∗ ≥ 5]] 3
[[242, 210, d z ≥ 8/d x ≥ 5]] 3
[[242, 202, d z ∗ ≥ 8/d x ∗ ≥ 5]] 3
[[242, 205, d z ≥ 8/d x ≥ 6]] 3
[[242, 197, d z ∗ ≥ 8/d x ∗ ≥ 6]] 3
[[728, 690, d z ≥ 8/d x ≥ 5]] 3
[[728, 680, d z ∗ ≥ 8/d x ∗ ≥ 5]] 3
[[728, 684, d z ≥ 8/d x ≥ 6]] 3
[[728, 674, d z ∗ ≥ 8/d x ∗ ≥ 6]] 3
[[728, 691, d z ≥ 7/d x ≥ 5]] 3
[[728, 686, d z ∗ ≥ 7/d x ∗ ≥ 5]] 3
[[124, 110, d z ≥ 5/d x ≥ 3]] 5
[[124, 106, d z ∗ ≥ 5/d x ∗ ≥ 3]] 5
[[124, 107, d z ≥ 5/d x ≥ 4]] 5
[[124, 103, d z ∗ ≥ 5/d x ∗ ≥ 4]] 5
[[124, 86, d z ≥ 10/d x ≥ 8]] 5
[[124, 82, d z ∗ ≥ 10/d x ∗ ≥ 8]] 5
[[124, 87, d z ≥ 11/d x ≥ 7]] 5
[[124, 85, d z ∗ ≥ 11/d x ∗ ≥ 7]] 5
[[124, 86, d z ≥ 12/d x ≥ 7]] 5
[[124, 82, d z ∗ ≥ 12/d x ∗ ≥ 7]] 5
[[124, 83, d z ≥ 12/d x ≥ 8]] 5
[[124, 79, d z ∗ ≥ 12/d x ∗ ≥ 8]] 5
[[124, 80, d z ≥ 12/d x ≥ 9]] 5
[[124, 76, d z ∗ ≥ 12/d x ∗ ≥ 9]] 5
[[124, 77, d z ≥ 12/d x ≥ 10]] 5
[[124, 73, d z ∗ ≥ 12/d x ∗ ≥ 10]] 5
[[624, 606, d z ≥ 5/d x ≥ 3]] 5
[[624, 600, d z ∗ ≥ 5/d x ∗ ≥ 3]] 5
[[624, 602, d z ≥ 5/d x ≥ 4]] 5
[[624, 596, d z ∗ ≥ 5/d x ∗ ≥ 4]] 5
[[624, 574, d z ≥ 10/d x ≥ 8]] 5
[[624, 568, d z ∗ ≥ 10/d x ∗ ≥ 8]] 5
[[624, 575, d z ≥ 11/d x ≥ 7]] 5
[[624, 572, d z ∗ ≥ 11/d x ∗ ≥ 7]] 5
[[624, 574, d z ≥ 12/d x ≥ 7]] 5
[[624, 568, d z ∗ ≥ 12/d x ∗ ≥ 7]] 5
[[624, 562, d z ≥ 12/d x ≥ 10]] 5
[[624, 556, d z ∗ ≥ 12/d x ∗ ≥ 10]] 5
[[342, 322, d z ≥ 7/d x ≥ 3]] 7
[[342, 318, d z ∗ ≥ 7/d x ∗ ≥ 3]] 7
[[342, 316, d z ≥ 7/d x ≥ 5]] 7
[[342, 312, d z ∗ ≥ 7/d x ∗ ≥ 5]] 7
[[342, 292, d z ≥ 12/d x ≥ 10]] 7
[[342, 288, d z ∗ ≥ 12/d x ∗ ≥ 10]] 7
[[342, 284, d z ≥ 15/d x ≥ 10]] 7
[[342, 282, d z ∗ ≥ 15/d x ∗ ≥ 10]] 7
[[342, 286, d z ≥ 16/d x ≥ 9]] 7
[[342, 282, d z ∗ ≥ 16/d x ∗ ≥ 9]] 7
6 Asymmetric Quantum Codes
6.5.1 Review of RS and GRS Codes
We review basics concepts on Reed–Solomon (RS) and generalized Reed–Solomon
(GRS) codes. The reader can consult [67, 114] for more details concerning RS and
GRS codes. Let us recall the definition of such codes.
Let q be a prime power and n be a positive integer with gcd(q, n) = 1. Recall
that a cyclic code C (see Definition 4.4.4) of length n over F q is a BCH code with
designed distance δ if
Table 6.1 Quantum code comparison
New asymmetric codes
Asymmetric codes shown in [3]
[[n, k, d z ≥ d/d x ≥ (d − c)]] q
[[n, k ∗ , d z ∗ /d x ∗ ]] q
[[26, 16, d z ≥ 4/d x ≥ 3]] 3
[[26, 14, d z ∗ ≥ 4/d x ∗ ≥ 3]] 3
[[26, 15, d z ≥ 5/d x ≥ 3]] 3
[[26, 11, d z ∗ ≥ 5/d x ∗ ≥ 3]] 3
[[26, 13, d z ≥ 5/d x ≥ 4]] 3
[[26, 11, d z ∗ ≥ 5/d x ∗ ≥ 4]] 3
[[80, 58, d z ≥ 6/d x ≥ 5]] 3
[[80, 52, d z ∗ ≥ 6/d x ∗ ≥ 5]] 3
[[80, 54, d z ≥ 8/d x ≥ 5]] 3
[[80, 48, d z ∗ ≥ 8/d x ∗ ≥ 5]] 3
[[242, 210, d z ≥ 8/d x ≥ 5]] 3
[[242, 202, d z ∗ ≥ 8/d x ∗ ≥ 5]] 3
[[242, 205, d z ≥ 8/d x ≥ 6]] 3
[[242, 197, d z ∗ ≥ 8/d x ∗ ≥ 6]] 3
[[728, 690, d z ≥ 8/d x ≥ 5]] 3
[[728, 680, d z ∗ ≥ 8/d x ∗ ≥ 5]] 3
[[728, 684, d z ≥ 8/d x ≥ 6]] 3
[[728, 674, d z ∗ ≥ 8/d x ∗ ≥ 6]] 3
[[728, 691, d z ≥ 7/d x ≥ 5]] 3
[[728, 686, d z ∗ ≥ 7/d x ∗ ≥ 5]] 3
[[124, 110, d z ≥ 5/d x ≥ 3]] 5
[[124, 106, d z ∗ ≥ 5/d x ∗ ≥ 3]] 5
[[124, 107, d z ≥ 5/d x ≥ 4]] 5
[[124, 103, d z ∗ ≥ 5/d x ∗ ≥ 4]] 5
[[124, 86, d z ≥ 10/d x ≥ 8]] 5
[[124, 82, d z ∗ ≥ 10/d x ∗ ≥ 8]] 5
[[124, 87, d z ≥ 11/d x ≥ 7]] 5
[[124, 85, d z ∗ ≥ 11/d x ∗ ≥ 7]] 5
[[124, 86, d z ≥ 12/d x ≥ 7]] 5
[[124, 82, d z ∗ ≥ 12/d x ∗ ≥ 7]] 5
[[124, 83, d z ≥ 12/d x ≥ 8]] 5
[[124, 79, d z ∗ ≥ 12/d x ∗ ≥ 8]] 5
[[124, 80, d z ≥ 12/d x ≥ 9]] 5
[[124, 76, d z ∗ ≥ 12/d x ∗ ≥ 9]] 5
[[124, 77, d z ≥ 12/d x ≥ 10]] 5
[[124, 73, d z ∗ ≥ 12/d x ∗ ≥ 10]] 5
[[624, 606, d z ≥ 5/d x ≥ 3]] 5
[[624, 600, d z ∗ ≥ 5/d x ∗ ≥ 3]] 5
[[624, 602, d z ≥ 5/d x ≥ 4]] 5
[[624, 596, d z ∗ ≥ 5/d x ∗ ≥ 4]] 5
[[624, 574, d z ≥ 10/d x ≥ 8]] 5
[[624, 568, d z ∗ ≥ 10/d x ∗ ≥ 8]] 5
[[624, 575, d z ≥ 11/d x ≥ 7]] 5
[[624, 572, d z ∗ ≥ 11/d x ∗ ≥ 7]] 5
[[624, 574, d z ≥ 12/d x ≥ 7]] 5
[[624, 568, d z ∗ ≥ 12/d x ∗ ≥ 7]] 5
[[624, 562, d z ≥ 12/d x ≥ 10]] 5
[[624, 556, d z ∗ ≥ 12/d x ∗ ≥ 10]] 5
[[342, 322, d z ≥ 7/d x ≥ 3]] 7
[[342, 318, d z ∗ ≥ 7/d x ∗ ≥ 3]] 7
[[342, 316, d z ≥ 7/d x ≥ 5]] 7
[[342, 312, d z ∗ ≥ 7/d x ∗ ≥ 5]] 7
[[342, 292, d z ≥ 12/d x ≥ 10]] 7
[[342, 288, d z ∗ ≥ 12/d x ∗ ≥ 10]] 7
[[342, 284, d z ≥ 15/d x ≥ 10]] 7
[[342, 282, d z ∗ ≥ 15/d x ∗ ≥ 10]] 7
[[342, 286, d z ≥ 16/d x ≥ 9]] 7
[[342, 282, d z ∗ ≥ 16/d x ∗ ≥ 9]] 7