Bounds on the minimum distance of additive quantum codes
Bounds on [[72,40]]2
lower bound: | 7 |
upper bound: | 10 |
Construction
Construction of a [[72,40,7]] quantum code:
[1]: [[128, 96, 7]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[72, 40, 7]] quantum code over GF(2^2)
Shortening of [1] at { 1, 3, 7, 9, 11, 15, 17, 19, 23, 24, 32, 34, 38, 39, 40, 42, 44, 47, 48, 50, 53, 57, 58, 60, 61, 65, 68, 70, 71, 74, 78, 79, 80, 81, 82, 89, 90, 92, 95, 96, 97, 100, 102, 104, 109, 110, 111, 112, 113, 117, 118, 119, 122, 123, 125, 128 }
stabilizer matrix:
[1 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 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 0 1 1 0 1 1 0 1 0 0 0 1 0 0 0 0 0 1 0|0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 1 1 0 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 1 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 1 1 0 1 0 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 0 0 0 0 0 0 1 0 0 1 1 0 1 0 1 0 0 0 1|0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 1 0 1 0]
[0 0 1 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 0 1 0 0 1 1 1 0 0 0 0 0 0 1 1 0 0 1 1 1 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 1 0 0 1 0 1|0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 1 1 0 1 1 0 1 1 0 1 1 0 1 0 1 0 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 1 1 0 0|0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 0 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 1 1 1 0 1 0 0 1 1 0 1 0 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0|0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 1 1 1 1 0 0 1 0 1 1 1 1 1 1 1 0 0 0 1 0 0 1 1 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 1 1 0 0 0]
[0 0 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 0 1 0 1 1 1 0 0 0 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 0 0 1 1 1 1 0 1 1 1 1 1 1 0 1 0 0 0|0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 1 0 0 0 1 0 1 1 1 0 0 1 1 1 0 0 1 0 1 0 1 0 1 1 1 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 1 0 1 0 0 0 1 0 0 1 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1|0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 0 0 1 1 0 0 1 1 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 1 1 0 1 1 1 0 0 1]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 1 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 1 1|0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 0 0 1 0 0 0 0 0 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 1 1 1 0 1 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 0 1 1 1 1 0 1 1 0 0 0 0 1 1 1 0 1 0 1 0 1 0 0 0 1 0 1 1 1 0 1 0 1 0 0 0 0|0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 1 1 1 0 0 1 0 0 0 0 1 1 0 1 1 1 1 1 1 0 1 1 0 0 1 0 1 1 0 0 1 0 0 0 1 0 1 1 1 0]
[0 0 0 0 0 0 0 0 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 1 0 1 1 0 0 0 1 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 0 1 0|0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 1 1 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 1 1 0 0 0 0 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0|0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 0|0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 1 1 0 0 0 1 1 1 1 0 0 0 1 1 1 0 1 0 1 0 1 1 1 1 0 0 1 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 0 0|0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1 0 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0 1 0 1|0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 1 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 0 1 0 0 0 0 1 0 0 0 1 1 1 1 1 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 1 0 0 0 1 0 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0|0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 1 1 0 1 0 1 0 1 1 1 1 1 1 1 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 0 0 0 0 0 0 1 0 1 1 1 0 1 1 1 0 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 0 1 1 0 1 1 1|0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 1 0 1 0 1 1 0 0 1 1 0 0 1 1 1 1 0 1 1 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 0 1 1 1 0 1 1 0 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1|0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 1 0 1 0 0 0 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 1 0 1 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 1 0 1 1 1 1 1 0 0 0 0 1 0 1 0 0 0 1 1 1 0 1 0 0 0 0 1 0 1 0 1 0 1 1 1 0 0|0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 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 1 1 0 1 1 0 1 0 1 0 0 1 1 0 1 0 0 1 1 1 1 1 0 0 0 1 0 1 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 0 0 1 1|0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 1 0 0 0 0 1 1 1 1 1 0 1 1 0 0 1 0 1 1 1 1 0 0 0 0 1 1 1 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 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 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 0 0|0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 1 1 0 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 1 0 1 1 0 1 1 0 0 0 1 1 1 1 1 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 1 0 0 0 0 0 0 1 1 0 1 1 1 1 1 0 1 0 1 1 0 1 1 1 0 0 0 1 0 0 1 1 1 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 0 0 0|0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 1 0 1 0 1 0 1 1 1 0 0 0 1 1 1 1 0 1 0 0 0 0 1 0 1 1 1 0 1 1 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 1 0 0 1 1 1 1 0 1 0 1 1 1 1 1 0 0 0 1 0 0 1 0 0 1 1 1 0 1 1 1 0 1 1 0 0 1 1 0 0 1 0|0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 0 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 1 0 0 0 1 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 0 1 1]
[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 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 0 1 1 1 1 0 1 1 0 1 0 0 0 1 1 0 0|0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 1 1 0 0 1 1 0 0 1 1 0 1 1 1 0 1 0 1 0 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 1 0 1 1 1 1 1 0 1 1 1 0 0 1 0 1 0 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 1 1 1 1 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 1 1 0 1 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1|0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 1 1 1 0 0 1 0 0 0 0 1 0 1 0 0 1 1 1 1 0 1 1 0 0 0 1 0 1 0 1 0 0 0 1 0 1 0 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|1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 1 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 1 0 1 1 1 0 0 0 0 1 1 1 0 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 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 1 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 1 0 0 1 1 1 0 1 1 1 1 0 1 1 0 0 0 1 0 1 1 1 0 0 1 1 1 1 1 1 0 0 1 0 1 1 1 0 1 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 1 0 0 0 0 0 1 1 1 1 1 1 0 1 0 1 1 0 1 0 0 1 1 1 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 0 0 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 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 1 1 0 0 1 1 1 1 1 0 1 1 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 1 1 0 1 0 1 0 0 1 1 0 1 1 1 0 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 1 0 0 0 0 0 1 1 1 1 0 1 1 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 0 1 0 1 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1 0 0 0 1 0 0 0 0 1 1 1 1 1 0 0 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 1 0 0 1 0 0 0 0 1 0 1 1 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 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 1 0 1 1 1 0 1 1 1 1 0 0 1 1 0 0 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 1 1 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 0 0 0 0 1 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 1 1 1 1 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 0 1 0 1 1 1 0 0 1 0 0 1 1 0 1 0 1 1 1 0 1 1 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 1 1 0 1 1 1 0 1 0]
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
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Last change: 23.10.2014