Bounds on the minimum distance of additive quantum codes
Bounds on [[88,62]]2
lower bound: | 6 |
upper bound: | 8 |
Construction
Construction of a [[88,62,6]] quantum code:
[1]: [[85, 65, 5]] quantum code over GF(2^2)
cyclic code of length 85 with generating polynomial x^84 + w^2*x^83 + x^82 + x^81 + x^80 + w^2*x^78 + w^2*x^77 + x^76 + w^2*x^75 + w^2*x^73 + w*x^72 + w^2*x^71 + w*x^69 + w^2*x^68 + w*x^65 + x^64 + x^63 + x^62 + w^2*x^61 + w*x^60 + w*x^59 + x^58 + w*x^57 + w^2*x^56 + w^2*x^55 + x^53 + x^52 + w^2*x^51 + x^49 + w*x^48 + w*x^47 + w*x^46 + x^45 + x^44 + x^43 + x^42 + w*x^41 + w^2*x^40 + x^38 + x^37 + x^34 + w^2*x^33 + w^2*x^32 + w*x^31 + x^30 + w^2*x^29 + w*x^28 + x^27 + w*x^26 + x^25 + x^24 + x^23 + w*x^22 + w*x^20 + w^2*x^19 + w^2*x^18 + x^17 + x^16 + w^2*x^15 + w*x^14 + x^13 + x^12 + x^10 + 1
[2]: [[86, 64, 6]] quantum code over GF(2^2)
standard lengthening of [1]
[3]: [[86, 62, 6]] quantum code over GF(2^2)
Subcode of [2]
[4]: [[88, 62, 6]] quantum code over GF(2^2)
ExtendCode [3] by 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 1 1 1 0 0 1 1 0 0 1 0 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 0 1 0 1 0 1 1 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0 0|0 1 1 0 1 1 1 0 1 1 1 0 0 0 1 1 0 1 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 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 0 0 0 0 0 0 0 1 1 1 1 0 1 0 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 0 1 0 0 0 1 1 1 1 1 0 1 1 1 0 1 0 0 1 1 1 0 1 1 0 1 0 0 0 0 0|0 1 1 1 0 1 0 0 1 0 0 0 1 1 0 1 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 1 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 1 0 0 0 1 0 1 1 0 1 1 1 1 0 0 1 1 1 0 1 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 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 1 1 0 0 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 0 1 0 0 0 1 0 1 1 1 0 0 1 1 1 0 0 0|0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 1 0 0 1 0 1 0 0 0 0 1 1 1 0 1 1 1 0 0 0 0 1 1 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 1 0 1 1 1 0 0 1 0 0 1 0 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 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 1 1 1 1 0 0 1 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0|0 1 1 0 0 1 0 1 0 1 0 0 1 1 1 1 1 1 1 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 1 1 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 0 1 1 0 1 0 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 1 1 0 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 1 0 0 0 0 1 1 1 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 1 1 1 1 1 1 1 1 0 0 1 0 0|0 0 0 1 1 1 1 1 1 0 1 1 1 0 0 0 0 1 1 1 1 0 1 0 0 1 0 0 0 1 1 1 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 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 0 0 1 1 0 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 1 0 0 0 0 1 1 1 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 1 1 0 0 1 1 1 0 1 0 0|0 0 0 0 1 1 1 1 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 1 0 0 1 0 0 0 1 1 1 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 0 1 0 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 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 1 0 0 0 0 1 1 1 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 0 0|0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 0 0 0 0 1 1 1 1 0 1 0 0 1 0 0 0 1 1 1 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 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 1 1 1 1 1 1 0 1 0 1 1 0 1 0 1 1 1 1 1 0 1 1 0 1 0 0 0 0 1 1 1 0 1 1 1 1 0 0 0 1 1 0 1 1 1 1 1 1 0 1 1 0 0 1 1 1 0 1 1 1 1 1 0 0 0 0|0 0 0 1 0 0 1 0 1 1 1 1 1 0 1 1 1 0 0 0 1 0 0 1 1 1 1 0 1 1 0 1 0 0 1 1 1 1 1 0 0 1 1 0 1 0 0 1 0 1 0 0 1 1 1 0 1 0 1 0 0 1 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 1 1 1 1 1 0 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 1 1 1 1 0 1 0 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 0 1 0 0 0 1 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 0 0 0 0|0 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 1 1 0 1 0 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 1 1 1 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 1 1 1 1 0 1 0 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 0 1 0 0 0 1 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 0 0 0|0 0 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 1 1 0 1 0 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 1 1 1 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 1 1 0 1 1 0 1 1 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 1 0 1 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 1 1 1 1 0 0|0 1 1 1 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 1 1 1 0 0 1 0 0 0 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 0 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 1 1 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 1 1 0 1 1 0 1 1 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 1 0 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 0 0 1 1 0 0|0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 1 0 1 1 1 0 1 1 0 1 0 1 1 0 0 0 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 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 1 1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0|0 1 1 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 1 1 1 1 0 1 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 1 1 1 1 0 1 0 0 1 1 0 1 0 1 0 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 1 0 1 0 0 1 1 0 0 1 1 1 0 1 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0|0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 0 1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 1 1 0 1 1 0 0 0 1 0 0 1 0 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 1 0 0 0 0 0 0 1 0 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 1 0 0 1 0 1 1 0 1 1 0 1 1 0 1 0 0 0 0 1 0 0 0 1 1 1 1 0 0 0 1 1 1 0 1 0 0 0 1 1 1 1 1 1 0 1 1 0 0|0 1 1 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 1 0 0 1 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 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 1 1 0 1 1 1 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 1 0 1 0 1 0 0 0 1 0 1 0 0 1 1 0 0 0 0 0 1 0 0 0 0|0 1 1 1 0 0 1 1 1 0 0 1 1 0 1 0 1 1 1 1 1 0 1 1 1 1 0 1 0 0 1 0 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 0 1 0 0 1 1 1 0 1 0 0 1 0 0 1 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 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 1 1 0 0 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 1 0 1 1 0 1 0 1 0 0 0 0 0|0 0 0 1 0 1 0 0 1 1 0 0 1 0 1 0 1 1 1 1 0 0 0 1 0 0 0 1 1 1 1 0 0 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 0 1 1 0 1 0 1 0 0 1 0 0 0 1 0 0 0 1 1 0 1 1 0 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 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 1 1 0 0 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 0 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0|0 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 1 1 1 0 1 1 1 0 0 0 0 1 1 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 1 0 1 1 1 0 0 1 0 0 1 0 0 0 1 1 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 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 1 1 0 0 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0|0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 0 1 1 1 1 0 0 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 0 1 1 0 1 0 1 0 0 1 0 0 0 1 0 0 0 1 1 0 1 1 0 1 0 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 1 0 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 0|0 0 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 1 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 0 0 0 0 1 1 1 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 0 0 1 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 1 0 0 0 1 0 1 1 1 1 0 1 0 1 0 0 0 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0|0 1 0 1 1 0 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 1 1 1 0 0 0 1 0 1 0 1 1 1 1 1 0 0 0 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 1 0 1 1 1 1 1 1 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 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
(codes@codetables.de).
Last change: 23.10.2014