Bounds on the minimum distance of additive quantum codes
Bounds on [[90,67]]2
lower bound: | 6 |
upper bound: | 7 |
Construction
Construction of a [[90,67,6]] quantum code:
[1]: [[128, 105, 6]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[90, 67, 6]] quantum code over GF(2^2)
Shortening of [1] at { 37, 58, 62, 65, 70, 73, 74, 78, 79, 80, 81, 82, 83, 84, 85, 88, 91, 92, 93, 94, 96, 97, 100, 101, 103, 104, 107, 109, 111, 114, 115, 116, 118, 119, 123, 126, 127, 128 }
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 0 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 1 0 0 0 1 1 0 0 1 1 1|0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 1 0 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 0 1 0 0 0 1 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 1 0 1 1 1 0 0 1 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0 1 0 1 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 1 1 0 1 1 0 0 0 0 1 0 0 1 0 0 0 1 0 1 0 0 0 1 1 0|0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 0 1 1 1 1 0 1 0 0 0 0 1 1 0 1 1 1 0 0 1 1 1 0 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 0 1 1]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1 1 0 0 0 1 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 1 1 0 0 1 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 0 0|0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 0 1 0 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 0 0 1 0 1 1 1 1 1 1 1 0 1 1]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1 1 0 0 0 1 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 1 1 1 0 1 1 1 0 0 0 0 1 0 1 1 1 1 0 1 0 1 1 0 0 1 0 0 1 1 0 0 1 0 0 1 0|0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 0 0 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 0 0 1 0 1 1|0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 1 1 0 0 1 0 0 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 1 0 0 0 0 1 1 1 1 1 1 0]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 1 0 1|0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 0 1 0 1 0 1 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 1 0 1 1 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 0 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 1 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 0|0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 1 1 1 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 1 0 1 0 1 0 0 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 0 0 1 1 0 0 0 1 1 1 1 1 1 0 0 1 0]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 0 0 0 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 0 1 1 1 0 0 0 1 1 0 1 0 1 0 1 1 1 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 0 0 1 0 1|0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 1 0 1 1 0 1 1 1 0 1 0 1 1 0 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 1 0 1 1 0 1]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 0 1 1 1 0 1 0 1 1 0 1 0 1 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0|0 0 0 0 0 0 0 0 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 1 1 1 0 1 0 0 1 1 0 1 0 1 0 0 0 1 1 1 1 0 0 1 0 1 0 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 1 0 0 0 1 0 0 1 1 1 0 0]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 1 1 0 0 1 0|0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 0 1 1 1 0 1 0 1 1 0 1 0 1 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 0 0 0 1 1 1 1 1 1 0|0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 1 1 1 0 1 1 1 1 1 0 1 0 0 1 0 0 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 0 1 1 0 0 1 1 1 0 0 0 1 0]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 1 1 0 0 0 0 0 1 1 0 0 1 1 1 0 0 1 1 0 1 1 0 0 0 1 0 1 0 0 0 0 1 0 1 1 0 0 0 1 1 0 0 0 0 0 1 0 1 1 0 1 0 1 0 0 0 0 1 0 1 1 0 1 1 0 0 1 0 1 0|0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 1 0 0 1 1 1 0 0 0 1 1 1 0 0 1 0 0 1 0 0 1 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1 1 0 0 0 1 1 1 1 1 0 0 1 1 0 0 0 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 0 0 0|0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 1 0 0 0 1 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 0 1 1 1 1 1|0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 1 1 1 1 1 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 1 1 1 0 1 1 0 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 0 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 0 1 0 1 0 1 1 0 0 1 0 1 0|0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 1 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 0 1 0 0 0 1 0 0 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 1 1 0 0 0 0 1 1 0 1 1 1 1 1 0 0 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 1 0 1 1 1 0 0 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|0 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 1 1 0 0 0 1 1 0 1 1 1 0 1 0 0 1 1 0 0 0 1 0 1 0 1 0 0 0 0 1 1 1 1 1 0 0 1 0 1 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 0 0 1 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 1 1 1 1 0 0 1 0 0 0 1 0 1 1 0 0 1 1 0 1 0 1 0 0 1 1 1 1 1 0 1 0 0 0 0 1 1 0 0 1 0 1 0 0 1 0 1 1 0 1 1 0 0 1 0 1 1 1 0 0 1 0 0 1 1 1 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 1 1 1 1 0 0 1 0 0 0 1 0 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 1 1 0 1 0 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 1 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 0 1 1 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 1 1 1 1 0 0 1 0 0 0 1 0 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 1 1 0 1 0 0 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 1 0 0 1 1 0 1 1 1 0 1 0 1 1 1 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 1 1 1 1 0 0 1 0 0 0 1 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1 1 1 1 0 1 0 0 0 1 1 0 0 1 0 0 1 0 1 1 1 0 1 0 1 0 0 0 1 1 1 1 0 1 0 0 0 1 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 1 1 1 1 0 0 1 0 0 0 1 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1 1 1 1 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 1 1 1 1 1 1 1 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 1 1 1 1 0 0 1 0 1 1 1 1 0 1 1 1 0 1 0 1 1 1 0 0 1]
last modified: 2006-04-03
Notes
- All codes establishing the lower bounds where constructed using MAGMA.
- Most upper bounds on qubit codes for n≤100 are based on a MAGMA program by Eric Rains.
- For n>100, the upper bounds on qubit codes are weak (and not even monotone in k).
- Some additional information can be found in the book by Nebe, Rains, and Sloane.
- My apologies to all authors that have contributed codes to this table for not giving specific credits.
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Markus Grassl
(codes@codetables.de).
Last change: 23.10.2014