Bounds on the minimum distance of additive quantum codes
Bounds on [[82,51]]2
lower bound: | 6 |
upper bound: | 10 |
Construction
Construction of a [[82,51,6]] quantum code:
[1]: [[85, 49, 8]] quantum code over GF(2^2)
cyclic code of length 85 with generating polynomial w^2*x^83 + x^82 + x^79 + w^2*x^76 + w*x^75 + w^2*x^74 + w^2*x^73 + w*x^71 + w*x^70 + x^69 + w^2*x^68 + w*x^67 + x^66 + w^2*x^64 + w^2*x^63 + w*x^62 + w*x^61 + w^2*x^59 + x^58 + x^57 + w^2*x^56 + w*x^55 + x^54 + x^52 + w^2*x^50 + w^2*x^49 + x^48 + x^45 + x^44 + w*x^42 + w^2*x^41 + w*x^35 + w^2*x^34 + w^2*x^33 + w*x^31 + x^30 + w^2*x^29 + x^27 + x^26 + x^24 + x^23 + w^2*x^22 + x^20 + x^19 + x^18 + 1
[2]: [[82, 52, 6]] quantum code over GF(2^2)
Shortening of the stabilizer code of [1] at { 5, 7, 80 }
[3]: [[82, 51, 6]] 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 1 1 1 1 0 1 0 1 1 0 1 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 0 0 1 1 0 1 0 1 1 1 1 1 0 0 0 1 1 0 1 1 1 1|1 0 0 0 1 0 1 1 1 1 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 0 1 1 1 0 0 0 1 1 1 1 0 1 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 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 1 1 0 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 1 0 1 1 0|0 1 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 1 0 1 0 1 0 1 0 0 0 1 1 1 0 1 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 1 1 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 0 0 1 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0|1 1 0 0 0 1 1 1 0 0 1 0 0 1 1 1 0 1 1 0 0 0 0 0 0 1 1 0 1 1 1 1 0 0 1 0 0 1 0 1 0 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 1 1 1 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 0 0 0 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 1 0 0 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 0 1|0 1 0 0 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 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 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 1 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0|1 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 1 1 1 0 0 1 0 1 0 1 1 1 1 0 1 1 1 0 0 1 0 1 1 0 0 0 1 0 1 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 1 0 1 1 0 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 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 1 0 1 0 0 0 0 1 1 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 1 1 0 1 1 0 0 1 0 1 1 1 1 0 1|1 1 1 0 1 0 0 0 1 0 0 0 0 1 0 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 0 1 0 0 0 0 1 1 1 1 0 1 1 1 1 0 0 1 0 0 1 0 1 1 1 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 0 1 0 0 1 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 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 1 1 0 1 0 0 0 1 0 1 1 1 0 1 0 1 0 1 0|1 1 0 0 1 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 0 1 0 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 0 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 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 1 0|1 0 1 0 1 1 1 0 1 0 1 1 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 1 1 0 1 1 0 1 1 1 1 1 1 0 1 1 0 0 0 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 0 1 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 1 0 1 0 0 0 1 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 0 0 0 0 1 1 0 1 0 1 0 1 0 1 1 1 0 0 1 1 0 0 0 0 1 0 0|0 1 1 0 0 1 1 0 1 1 1 0 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 1 0 1 1 1 1 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 1 0 1 1 0 0 1 0 0 0 0 1 1 1 0 0 1 1 0 1 1 0 1 0 1 0 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 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 1 0 1 0 1 0 1 1 1 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 1 0 0 0 1 0 1 0 0 1 1 1|0 1 1 0 1 0 0 1 1 1 1 1 0 0 1 0 1 1 0 0 0 0 0 1 0 1 0 1 1 1 0 1 1 0 1 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 0 1 1 1 0 0 0 1 1 1 1 1 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 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 0 1 1 0 0 0 1 1 1 0 1 1 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 0 0 0 1 1 0 1 1 0 0 1 0|0 0 1 0 1 1 0 0 0 0 0 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 0 1 0 0 1 1 0 0 1 1 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0|1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 1 1 1 1 0 1 1 0 0 0 1 1 1 0 0 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 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 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 1 0 0 0 1|0 0 1 0 1 0 1 1 1 1 0 0 0 1 0 0 0 0 1 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 1 1 0 0 1 0 0 1 1 1 0 1 1 1 1 0 1 0 1 0 1 1 1 1 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 1 0 0 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 0 0 0 0 0 0 0 0 0 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 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 1 0 0 1 0 1 1 1 0 0 0 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 1 0 0 1 1 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 1 1|0 1 0 1 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 1 0 1 1 1 0 1 0 1 0 0 1 0 0 1 1 0 0 1 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 0 0 1 1 0 0 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 0 0 1 0 1 1 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 0 0 0 0 0|0 0 0 0 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 0 0 1 0 0 1 1 1 0 0 1 1 0 1 1 1 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 1 1 0 0 1 1 1 0 1 1 1 0 0 0 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 1 0 0 1 0 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1|0 0 1 1 1 0 1 1 1 0 0 0 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 1 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 1 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 0 0 0 0 0 1 0 1 1 1 0 1 0 1 0 0 1 1 1 1 0 1 1 0 0 1 1 0 0 1 1 1 1 0 1 1 0 0 1 0 1 0 1 1 0 1 0 1 1 1 1 0 0 1|1 1 0 1 1 1 0 1 0 0 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 0 1 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 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 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 1 1 1 1 0 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 1 0 0 1 1 1|0 0 1 0 0 0 1 1 0 1 1 0 1 0 1 0 0 0 0 1 0 1 0 0 1 1 1 0 1 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 1 1 0 1 1 0 1 1 1 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 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 1 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 0 0 0 0 1 1 1 1 0 0 1 1|1 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 0 0 1 1 0 1 0 1 0 0 1 1 1 1 1 1 0 1 0 1 1 0 1 1 1 0 0 1 0 0 1 0 0 0 0 0 1 1 0 1 0 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0]
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[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 0 0 1 1 1 1 1 0 0 1 1 1 0 0 1 0 1 1 0 0 1 1 1 0 0 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 1 1 1 1 1 1 1 0|0 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 1 1 0 0 1 1 0 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 1 1 1 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 0 0 0 0 0 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0|1 1 0 1 0 1 1 1 0 0 1 1 0 1 1 1 1 1 1 0 0 0 0 1 1 0 0 1 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 0 0 1 1 1 1 1 0 1 1 0 0 1 0 1 1 1 0 0 0 0 0 1 1 1 0 1 0 1 1 1 1 0 1 0 1 0 1 0]
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[0 0 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 1 1 0 1 0 1 1 0 0 1 1 0 1 0 1 0 0 1 0 0 0 1 1 0 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 1 0 1 1 1 0 1|0 0 1 0 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 0 0 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 0 0 0 0 1 0 0 0 1 1 1 0 1 1 1 0 1 1 0 1 0 1 0 1 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 1 0 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 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 1 1 1 0 0 0 1|1 1 1 0 0 0 1 1 0 1 1 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 1 1 0 1 1 1 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 0 1 1 1 0 1 1 1 1 0 1 0 0 1 1 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 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 1 1 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 1 0 0 0|1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 0 1 1 1 0 1 0 1 1 1 1 0 1 1 1 1 1 0 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 1 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 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 0 1 1 1 1 1 0 1 0 1 0 0 1 1 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 1 1|0 1 1 0 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 1 0 1 1 0 0 1 0 1 0 0 0 0 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 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 1 1 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 1 0|0 0 1 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 0 1 1 1 0 1 0 1 1 1 1 0 1 1 1 1 1 0 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 1 1 1 0 1 0 0 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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Last change: 23.10.2014