Bounds on the minimum distance of additive quantum codes
Bounds on [[47,0]]2
lower bound: | 13 |
upper bound: | 17 |
Construction
Construction of a [[47,0,13]] quantum code:
[1]: [[48, 0, 14]] self-dual quantum code over GF(2^2)
cyclic code of length 48 with generating polynomial w*x^47 + w^2*x^44 + w^2*x^43 + x^42 + w^2*x^40 + w^2*x^39 + x^38 + w^2*x^37 + x^35 + w^2*x^34 + x^31 + w^2*x^30 + w^2*x^29 + w*x^27 + w*x^25 + w*x^24 + w*x^23 + 1
[2]: [[47, 1]] quantum code over GF(2^2)
Shortening of the stabilizer code of [1] at 48
[3]: [[47, 0, 13]] self-dual quantum code over GF(2^2)
Subcode of [2]
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 0 0 0 0 0 0 0 0 0 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 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0|1 1 0 1 1 1 0 1 0 0 0 0 1 0 0 1 1 0 1 1 1 0 0 1 0 0 1 1 1 0 1 1 0 0 1 0 0 0 0 1 0 1 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 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0|0 1 0 1 1 1 0 1 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 0 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 0 0 0 0 1 0 0 0 0 0 0 0 0|0 0 0 1 1 1 0 1 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 0 1 0 0 1 1 1 0 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 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 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0|1 0 1 0 1 1 0 1 0 0 0 0 1 0 0 1 1 1 0 0 1 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 0 0 1 0 1 1 1 1 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 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 1 0 1 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 1 1 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 1 1 1 1 0 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 0 0 0 0 0 1 0 0 0 0 0 0 0 0|0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 1 0 0 1 1 1 1 1 0 0 1 1 1 1 1 0 0 0 0 1 0 0 1 0 0 1 1 1 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 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 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0|1 0 1 0 1 0 1 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 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 1 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 1 1 0 0 1 1 0 0 1 1 1 0 0 1 1 1 1 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 0 0 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 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 0 0|0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 1 1 0 1 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 0 1 1 1 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 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 1 0 0 1 1 0 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 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 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|1 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 0 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 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 1 1 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 1 1 0 0 1 1 1 0 0 1 1 1 0 1 0 1 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 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|1 1 1 0 0 0 0 0 0 0 1 1 1 0 1 1 0 0 1 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 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 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 1 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 1 1 1 0 0 1 1 1 0 1 1 1 0 1 1 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 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 1 0 0 1 0 0 1 1 1 1 1 1 0 1 1 0 0 1 1 1 0 0 1 0 0 0 0 1 0 1 0 0 1 1 0 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|1 1 1 0 0 1 1 1 0 1 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 1 1 0 1 0 1 0 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 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 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 0 1 1 1 1 0 1 1 1 1 0 0 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 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0|1 1 1 0 0 1 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0 0 1 1 1 0 1 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 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0|0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 1 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 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0|0 0 0 1 0 0 1 1 0 1 1 1 0 0 1 0 0 1 1 1 0 1 1 0 0 1 0 0 0 0 1 0 1 1 1 0 1 1 0 1 1 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 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0|0 0 1 1 1 0 1 0 0 0 1 1 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 1 1 1 0 1 1 1 1 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|1 1 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 1 1 0 0 1 1 1 1 0 1 1 1 0 1 1 1 0 1 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 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0|1 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0|0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1 1 1 0 0 0 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 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0|0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 1 0 1 1 1 0 1 1 1 0 0 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 1 0 0 0 0 0 0 0 0 0 0 0|1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 0 1 0 0 0 0 0 0 0 0|1 0 1 0 1 0 1 0 0 1 1 1 1 0 1 1 1 1 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 1 0 0 0 0 0 0 0 0|0 1 1 0 0 1 1 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1 1 0 0 1 1 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 0 0 0 0 0 0 0 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 0 1 1 1 0 0 0 0 1 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 1 0 0 0 0 1 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 1 0 0 0 0 0 0|0 1 1 1 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 1 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 1 0 0 0 0 0|0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 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 1 0 0 0 0|0 0 0 1 1 1 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 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 1 0 0 0|0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 1 0 0 1 1 1 0 0 0 0 0 0 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 1 0 0|0 0 0 0 0 1 1 1 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 1 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 1 0|0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 0 0 1 1 1 1 0 1 1 1 1 0 0 1 0 1 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 0 0 0 0 0 0 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 0 0 1 1 0 0 1 1 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1 1 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 1 1 1 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0]
last modified: 2009-02-08
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.
This page is maintained by
Markus Grassl
(codes@codetables.de).
Last change: 23.10.2014