6.5 Reed–Solomon and GRS Codes
143
Table 6.2 Quantum MDS
Our AQECCs
Codes in [3, Theorem 8]
[[N , K , d z ≥ d/d x ≥ (d − c)]] q
[[N
, K
, d z /d x ]] q
[[14, 4, d z ≥ 4/d x ≥ 3]] 3
[[14, 2, d z ≥ 2/d x ≥ 2]] 5
[[16, 2, d z ≥ 5/d x ≥ 4]] 3
[[16, 0, d z ≥ 5/d x ≥ 2]] 5
[[16, 4, d z ≥ 5/d x ≥ 3]] 3
[[16, 0, d z ≥ 5/d x ≥ 2]] 5
[[18, 2, d z ≥ 7/d x ≥ 3]] 5
[[18, 0, d z ≥ 3/d x ≥ 2]] 5
[[22, 10, d z ≥ 5/d x ≥ 3]] 5
[[22, 2, d z ≥ 4/d x ≥ 2]] 5
[[22, 8, d z ≥ 5/d x ≥ 4]] 5
[[22, 2, d z ≥ 4/d x ≥ 2]] 5
[[22, 8, d z ≥ 6/d x ≥ 3]] 5
[[22, 2, d z ≥ 4/d x ≥ 2]] 5
[[24, 12, d z ≥ 5/d x ≥ 3]] 5
[[24, 12, d z ≥ 5/d x ≥ 3]] 5
[[48, 18, d z ≥ 12/d x ≥ 5]] 5
[[48, 0, d z ≥ 12/d x ≥ 4]] 5
[[124, 88, d z ≥ 8/d x ≥ 3]] 5
[[124, 100, d z ≥ 8/d x ≥ 3]] 5
[[124, 64, d z ≥ 12/d x ≥ 5]] 5
[[124, 85, d z ≥ 12/d x ≥ 5]] 5
[[78, 66, d z ≥ 3/d x ≥ 3]] 5
[[78, 62, d z ≥ 3/d x ≥ 3]] 5
[[156, 128, d z ≥ 6/d x ≥ 3]] 5
[[156, 132, d z ≥ 6/d x ≥ 3]] 5
[[312, 252, d z ≥ 12/d x ≥ 5]] 5
[[312, 260, d z ≥ 12/d x ≥ 5]] 5
[[372, 318, d z ≥ 15/d x ≥ 5]] 5
[[372, 276, d z ≥ 15/d x ≥ 5]] 5
[[372, 282, d z ≥ 25/d x ≥ 7]] 5
[[372, 222, d z ≥ 25/d x ≥ 7]] 5
[[30, 18, d z ≥ 5/d x ≥ 3]] 7
[[30, 6, d z ≥ 5/d x ≥ 3]] 7
[[30, 8, d z ≥ 10/d x ≥ 3]] 7
[[30, 2, d z ≥ 6/d x ≥ 3]] 7
[[96, 70, d z ≥ 10/d x ≥ 5]] 7
[[96, 48, d z ≥ 10/d x ≥ 5]] 7
[[96, 48, d z ≥ 21/d x ≥ 5]] 7
[[96, 8, d z ≥ 21/d x ≥ 5]] 7
length n, and for fixed values of d z and d x , the better code is the one which achieves
greater values of the number of qudits than the other.
In Table 6.2, [[N , K , d z ≥ d/d x ≥ (d − c)]] q denotes the parameters of our codes,
where N = mn, K = m(2k − n + c) and k = n − d + 1. On the other hand, we
denote [[N
, K
, d z /d x ]] q meaning the parameters of the codes shown in [3, Theorem
8].
As we can see in Table 6.2, our AQECCs have parameters comparable with the
ones available in [3, Theorem 8]. Since our construction and that construction showed
in such paper are quite different, there exist cases where our codes are better than
the ones shown in the cited article and reciprocally.
6.6 Tensor Product Codes
This section is devoted to construct AQECCs derived from product codes. Although
the parameters of our codes are not so good, these codes have great asymmetry.
143
Table 6.2 Quantum MDS
Our AQECCs
Codes in [3, Theorem 8]
[[N , K , d z ≥ d/d x ≥ (d − c)]] q
[[N
, K
, d z /d x ]] q
[[14, 4, d z ≥ 4/d x ≥ 3]] 3
[[14, 2, d z ≥ 2/d x ≥ 2]] 5
[[16, 2, d z ≥ 5/d x ≥ 4]] 3
[[16, 0, d z ≥ 5/d x ≥ 2]] 5
[[16, 4, d z ≥ 5/d x ≥ 3]] 3
[[16, 0, d z ≥ 5/d x ≥ 2]] 5
[[18, 2, d z ≥ 7/d x ≥ 3]] 5
[[18, 0, d z ≥ 3/d x ≥ 2]] 5
[[22, 10, d z ≥ 5/d x ≥ 3]] 5
[[22, 2, d z ≥ 4/d x ≥ 2]] 5
[[22, 8, d z ≥ 5/d x ≥ 4]] 5
[[22, 2, d z ≥ 4/d x ≥ 2]] 5
[[22, 8, d z ≥ 6/d x ≥ 3]] 5
[[22, 2, d z ≥ 4/d x ≥ 2]] 5
[[24, 12, d z ≥ 5/d x ≥ 3]] 5
[[24, 12, d z ≥ 5/d x ≥ 3]] 5
[[48, 18, d z ≥ 12/d x ≥ 5]] 5
[[48, 0, d z ≥ 12/d x ≥ 4]] 5
[[124, 88, d z ≥ 8/d x ≥ 3]] 5
[[124, 100, d z ≥ 8/d x ≥ 3]] 5
[[124, 64, d z ≥ 12/d x ≥ 5]] 5
[[124, 85, d z ≥ 12/d x ≥ 5]] 5
[[78, 66, d z ≥ 3/d x ≥ 3]] 5
[[78, 62, d z ≥ 3/d x ≥ 3]] 5
[[156, 128, d z ≥ 6/d x ≥ 3]] 5
[[156, 132, d z ≥ 6/d x ≥ 3]] 5
[[312, 252, d z ≥ 12/d x ≥ 5]] 5
[[312, 260, d z ≥ 12/d x ≥ 5]] 5
[[372, 318, d z ≥ 15/d x ≥ 5]] 5
[[372, 276, d z ≥ 15/d x ≥ 5]] 5
[[372, 282, d z ≥ 25/d x ≥ 7]] 5
[[372, 222, d z ≥ 25/d x ≥ 7]] 5
[[30, 18, d z ≥ 5/d x ≥ 3]] 7
[[30, 6, d z ≥ 5/d x ≥ 3]] 7
[[30, 8, d z ≥ 10/d x ≥ 3]] 7
[[30, 2, d z ≥ 6/d x ≥ 3]] 7
[[96, 70, d z ≥ 10/d x ≥ 5]] 7
[[96, 48, d z ≥ 10/d x ≥ 5]] 7
[[96, 48, d z ≥ 21/d x ≥ 5]] 7
[[96, 8, d z ≥ 21/d x ≥ 5]] 7
length n, and for fixed values of d z and d x , the better code is the one which achieves
greater values of the number of qudits than the other.
In Table 6.2, [[N , K , d z ≥ d/d x ≥ (d − c)]] q denotes the parameters of our codes,
where N = mn, K = m(2k − n + c) and k = n − d + 1. On the other hand, we
denote [[N
, K
, d z /d x ]] q meaning the parameters of the codes shown in [3, Theorem
8].
As we can see in Table 6.2, our AQECCs have parameters comparable with the
ones available in [3, Theorem 8]. Since our construction and that construction showed
in such paper are quite different, there exist cases where our codes are better than
the ones shown in the cited article and reciprocally.
6.6 Tensor Product Codes
This section is devoted to construct AQECCs derived from product codes. Although
the parameters of our codes are not so good, these codes have great asymmetry.