5.3 BCH Codes—Part III
93
Table 5.9 Code comparison
Our CSS codes—Construction I
CSS codes in [4]
[[n, n − 2m(c − 2) − m/2 − 1, d ≥ c]] q
[[n
, k
, d
]] q
m = 2
[[15, 9, d ≥ 3]] 4
[[15, 7, d
≥ 3]] 4
[[15, 5, d ≥ 4]] 4
[[15, 3, d
≥ 4]] 4
[[24, 18, d ≥ 3]] 5
[[24, 16, d
≥ 3]] 7
[[24, 14, d ≥ 4]] 5
[[24, 12, d
≥ 4]] 7
[[24, 10, d ≥ 5]] 5
[[24, 8, d
≥ 5]] 7
[[63, 57, d ≥ 3]] 8
[[63, 55, d
≥ 3]] 8
[[63, 53, d ≥ 4]] 8
[[63, 51, d
≥ 4]] 8
[[63, 49, d ≥ 5]] 8
[[63, 47, d
≥ 5]] 8
[[63, 45, d ≥ 6]] 8
[[63, 43, d
≥ 6]] 8
[[63, 41, d ≥ 7]] 8
[[63, 39, d
≥ 7]] 8
[[63, 37, d ≥ 8]] 8
[[63, 35, d
≥ 8]] 8
m = 4
[[255, 244, d ≥ 3]] 4
[[255, 239, d
≥ 3]] 4
[[255, 236, d ≥ 4]] 4
[[255, 231, d
≥ 4]] 4
[[624, 613, d ≥ 3]] 5
[[624, 608, d
≥ 3]] 5
[[624, 605, d ≥ 4]] 5
[[624, 600, d
≥ 4]] 5
[[624, 597, d ≥ 5]] 5
[[624, 592, d
≥ 5]] 5
The first construction generates quantum codes with parameters
(i) [[n, n − 4(c − 2) − 2, d ≥ c]] q , where n = q
4
− 1, and 3 ≤ c ≤ q
2 ;
The second one produces codes with parameters
(ii) [[n, n − 2mc − 2, d ≥ c + 2]] q , for all 1 ≤ c ≤ q
2
− 2;
(iii) [[n, n − 2m(q
2
− 1) − 2, d ≥ q
2
+ 2]] q ;
(iv) [[n, n − 2m(c − 1) − 2, d ≥ c + 2]] q , for all q
2
+ 1 ≤ c ≤ 2q
2
+ 2;
(v) [[n, n − 4m(q
2
− 1) − 2, d ≥ 2q
2
+ 2]] q , for all q
2
+ 1 ≤ c ≤ 2q
2
+ 2,
where n = q
2m
− 1, q ≥ 4 is a prime power, and m = ord n (q
2
) ≥ 3.
The third one generates families of quantum codes with parameters
(vi) [[n, n − m(2c − 1) − 2, d ≥ c + 2]] q , for all 1 ≤ c ≤ q − 2;
(vii) [[n, n − m(2q − 3) − 2, d ≥ q + 1]] q ;
(viii) [[n, n − m(2q − 1) − 1, d ≥ q + 3]] q ;
(ix) [[n, n − m(2c − 4) − 2, d ≥ c + 2]] q ;
(x) [[n, n − m(4q − 8) − 2, d ≥ 2q]] q ;
(xi) [[n, n − m(4q − 5) − 2, d ≥ 2q + 2]] q , where q + 1 < c < 2q − 2;
where n = q
m
− 1, q ≥ 4 is a prime power, and m = ord n (q) ≥ 3.