Bounds on the minimum distance of additive quantum codes
Bounds on [[58,20]]2
lower bound: | 9 |
upper bound: | 14 |
Construction
Construction of a [[58,20,9]] quantum code:
[1]: [[55, 20, 9]] quantum code over GF(2^2)
cyclic code of length 55 with generating polynomial x^54 + w*x^52 + w^2*x^51 + x^50 + w*x^48 + w*x^46 + w^2*x^45 + w^2*x^44 + x^43 + w^2*x^42 + w^2*x^40 + x^39 + w*x^37 + w^2*x^36 + w^2*x^35 + x^34 + x^33 + w^2*x^32 + w*x^30 + x^29 + x^28 + w*x^27 + x^26 + w*x^25 + x^24 + w*x^23 + x^22 + x^21 + w*x^8
[2]: [[58, 20, 9]] quantum code over GF(2^2)
ExtendCode [1] by 3
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 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 1 1 0 1 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 0 0 0 1 1 0 0 1 1 1 1 1 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 1 0 1 0 0 1 1 1 0 0 0 0 1 1 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 0 1 1 0 1 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 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 1 1 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 1 0 1 1 1 1 0 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 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 1 1 0 1 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 0 0 0 1 1 0 0 1 1 1 1 1 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 1 0 1 0 0 1 1 1 0 0 0 0 1 1 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 0 1 1 0 1 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 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 1 1 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 1 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 1 1 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 1 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 1 1 0 1 1 0 1 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 1 0 1 0 0 1 1 1 0 0 0 0 1 1 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 0 1 1 0 1 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 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 1 1 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 1 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 1 1 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 1 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 1 1 0 1 1 0 1 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 1 0 1 1 0 0 1 1 1 1 1 0 1 1 1 0 0 1 0 0 0 0 0 1 1 0 1 1 1 0 0 1 0 0 1 0 1 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 1 1 0 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 0 1 0 0 0 1 0 1 1 0 1 1 1 1 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 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 1 1 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 1 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 1 1 0 1 1 0 1 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 1 1 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 1 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 0 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 1 1 1 1 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 1 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 1 1 1 0 1 0 0 1 0 1 1 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 1 0 0 1 1 0 1 1 1 1 0 1 1 1 0 1 1 1 1 1 0 0 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 0 0 0 1 1 0 1 0 1 0 1 1 0 0 1 1 1 1 1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 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 1 0 1 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 1 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 1 0 0 1 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1 1 0 1 0 0 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 1 0 0 1 1 0 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 0 1 1 0 1 1 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 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 0 1 1 1 1 1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 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 1 0 1 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 1 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 1 0 0 1 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1 1 0 1 0 0 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 1 0 0 1 1 0 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 0 1 1 0 1 1 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 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 0 1 1 1 1 1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 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 1 0 1 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 1 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 1 0 1 1 0 1 0 0 1 0 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 1 0 0 1 1 1 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 1 1 0 1 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 1 1 1 0 1 0 1 0 1 1 1 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 1 1 0 1 0 1 0 1 1 0 0 1 1 1 1 1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 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 1 0 1 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 1 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 1 1 1 0 0 1 1 0 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 0 1 0 0 0 0 1 0 1 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 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 1 0 1 1 1 0 1 0 0 1 1 1 1 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 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 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 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 1 1 0 0 1 0 0 0 0 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 1 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 0 1 1 0 0 1 1 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 1 0 0 0 1 0 0 0 1 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 0 1 0 0 0 1 1 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 1 0 1 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 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 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 1 1 0 0 1 0 0 0 0 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 1 0 1 1 1 1 1 0 0 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 0 1 0 1 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 1 0 0 1 1 1 0 1 1 0 1 1 0 1 0 0 1 0 1 1 1 1 1 1 1 0 0 1 1 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 1 1 1 1 0 0 1 1 1 1 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1 1 1 0 0 0|0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 0 1 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 0 1 1 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 1 1 1 0 1 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 1 0 0 1 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 1 1 1 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 1 0 1 1 0 0 1 1 1 1 1 0 1 0 1 0 1 1 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 1 0 0 0|0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 1 1 0 1 0 0 0 0 1 0 0 1 1 1 0 0 1 0 1 1 0 1 0 1 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 1 1 0 1 0 1 1 0 1 1 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 0 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 1 1 0 1 1 1 0 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 0 1 1 1 1 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 1 0 0 0 1 0 1 1 0 0 1 1 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0|0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 1 1 0 0 0 1 1 1 1 1 0 1 1 0 1 1 1 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 1 1 0 1 0 1 1 0 1 1 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 0 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 1 1 0 1 1 1 0 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 0 1 1 1 1 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 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 1 0 0 0 1 0 1 1 1 1 1 0 0 1 0 0 1 0 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 1 1 0 1 1 0 0 0 1 0 1 1 1 1 0 0 1 0 0 1 1 0 1 1 1 0 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 1 0 0 0 0 1 0 1 1 1 0 1 0 1 1 0 1 1 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 0 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 1 1 0 1 1 1 0 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 0 1 1 1 1 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 1 0 1 1 0 0 0 1 1 0 1 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 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 1 0 1 1 0 1 0 1 0 0 0 1 1 0 0 1 0 1 1 0 0 0 1 0 1 0 1 0 0 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 0 0 0 0 0 1 0 1 0 1 0 1 0 1 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 1 0 0 0 0 0 1 1 0 1 0 1 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 1 1 0 1 1 1 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 1 1 1 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 0 0 1 1 0 1 1 1 1 0 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 1 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 0 0 1 1 0 1 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 0 0 0 1 1 0 0 1 1 1 1 1 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 1 0 1 0 0 1 1 1 0 0 0 0 1 1 1 1 1 0 1 1 0 1 1 1 0 1 0 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 0 0 0 0 0 1 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 1 1 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
last modified: 2022-08-02
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