Bounds on the minimum distance of additive quantum codes
Bounds on [[83,52]]2
lower bound: | 6 |
upper bound: | 10 |
Construction
Construction of a [[83,52,6]] quantum code:
[1]: [[85, 53, 7]] quantum code over GF(2^2)
cyclic code of length 85 with generating polynomial w*x^84 + w*x^83 + x^81 + w*x^80 + w^2*x^79 + x^78 + x^77 + w^2*x^76 + x^75 + w^2*x^74 + w^2*x^73 + x^72 + w^2*x^71 + x^70 + w*x^69 + w^2*x^68 + w*x^66 + w^2*x^64 + x^63 + x^62 + x^61 + x^60 + w^2*x^59 + w^2*x^58 + w^2*x^57 + w*x^55 + w^2*x^54 + x^53 + w*x^52 + x^51 + w*x^50 + w*x^49 + x^48 + x^47 + x^45 + x^43 + x^42 + x^41 + w^2*x^39 + w*x^38 + x^37 + w*x^36 + x^35 + x^34 + w*x^33 + w*x^32 + x^30 + w*x^28 + w^2*x^27 + x^26 + w*x^25 + w*x^24 + w^2*x^23 + w*x^22 + x^20 + x^19 + x^18 + w*x^17 + x^16 + 1
[2]: [[83, 55, 6]] quantum code over GF(2^2)
Shortening of the stabilizer code of [1] at { 1, 8 }
[3]: [[83, 52, 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 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 1 1 1 1 1 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1|1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 0 1 0 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 1 1 1 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 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 1 0 0 1 1 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 1 0 0 0 1 1 1 1 0 1 0 0 1 1 0 1 0|0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 0 1 0 0 1 0 0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 0 1 0 0 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 1 0 1 1 1 1 0 0 1 1 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 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 1 0 1 1 0 0 1 1 0 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 1 1 1 0 0 1 1 0 1 1 0|1 0 1 1 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 1 1 1 0 1 0 0 1 1 1 1 0 1 1 0 0 1 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 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 0 0 0 1 1 1 1 1 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 1 0 0|1 0 0 0 1 0 1 1 1 1 0 1 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 0 1 1 1 1 0 1 0 1 0 1 0 1 0 0 0 1 1 1 0 0 1 1 1 0 0 1 1 0 0 1 0 0 0 1 0 1 0 1 1 1 1 0 0 0 0 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 1 0 0 1 0 1 0 0 0 1 1 1 0 0 1 1 1 0 0 0 0 1 0 1 0 1 1 1 1 0 1 1 0 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 1 0 1 0 0 1 0|1 1 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0 1 1 0 1 1 0 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 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 1 1 0 1 1 1 1 0 0 1 1 1 0 0 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0|0 0 1 0 0 0 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 1 1 1 1 0 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 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 0 0 0 0 1 0 0 0 0 1 1 1 0 0 1 0 0 1 1 0 0 1 0 0 1 1 0 0 0 0 1 1 0 0 1 0 1 1 1 1 1 0 0 1 0 0 1 0 1 1 0 0 0 1 1 1|0 1 0 1 0 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 0 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 0 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 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 1 0 1 1 0 0 1 1 1 1 0 0 0 1 1 0 1 1 0 1 1 0 1 1 1 0 1 1 1 1 0 0 0 0 1|0 0 0 0 1 0 1 1 1 1 1 0 0 1 0 0 1 0 1 0 1 1 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 0 0 0 0 1 1 0 1 0 1 1 0 0 1 0 0 0 0 1 0 1 1 0 0 1 1 0 0 1 0 0 0 0 1 0 1 1 1 0 1 1 0 0 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 1 1 0 0 0 0 1 1 1 0 1 0 1 0 1 1 0 0 1 0 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 0 0 0 1 0 1 0 1 0 1|1 1 0 0 1 0 1 0 0 1 1 0 1 0 1 0 1 0 0 1 1 1 1 0 1 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 1 1 1 0 1 0 1 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 0 1 1 1 1 1 0 1 0 1 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 0 0 0 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 1 1 0 1 0 0 0 1 1 0 1 1 0 1 0 0 0 1|0 1 1 0 1 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 0 1 1 1 1 0 1 1 0 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 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 1 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 1 1 1 0 0 1 1 0 0 0 1 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 0 1 1 1 1 1 1 0 0 0|0 1 0 0 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 0 1 0 0 0 0 1 0 1 0 1 1 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 0 0 1 0 1 0 0 0 0 0 1 0 1 1 0 1 1 0 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 0 0 0 0 0 1 0 0 1 0 0 1 0 1 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 0 1 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 0 1 1 0 0 1 1 0|1 0 0 0 1 0 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 0 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 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 0 0 1 1 0 1 1 0 1|1 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 1 1 0 0 1 1 0 0 1 0 1 0 1 1 0 1 1 1 0 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 0 0 0 0 0 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 1 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 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 0 1 0 1 1 1 1 0 0 0 0 1 1|1 1 0 0 0 1 1 1 0 1 1 0 1 1 0 1 0 0 1 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0 1 1 0 0 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 0 1 0 1 0 0 1 1 0 1 1 0 1 0 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 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 0 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 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 1 1 1 1 1 0 0 1 1 0 0 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 1 0 0 1 1 1 1]
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[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 1 1 0 1 1 0 0 1 1 0 1 0 1 0 1 0 0 1 1 0 1 1 1 1 0 1 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0|0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 0 0 0 0 1 1 0 0 1 1 1 1 1 0 1 0 1 1 1 1 1 1 0 0 0 0 0 1 1 0 0 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 1 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 0 1 0 1 1 0 1 0 0 1 0 0 1 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 0 1|0 0 1 1 1 0 0 1 1 0 1 1 1 1 1 0 1 0 0 0 0 1 1 0 1 1 1 0 0 0 1 0 1 0 0 1 1 1 1 1 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 1]
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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 1 0 0 1 1 1 0 0 1 1 1 0 1 0 0 1 0 1 0 0 1 1 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 1 1 1 0 0 0 0 1 1 1 1 0 1 0 1 1 1 0|0 1 1 0 1 1 1 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 1 1 0 1 1 0 1 1 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 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 1 0 1 0 1 0 1 1 1 1 1 0 0 0 1 0 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 0 1 0 0 1 0 0 0 1 1 0 1|1 0 0 1 0 0 0 1 0 1 1 1 1 1 0 1 1 0 0 1 1 1 1 0 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 1 1 0 0 0 0 1 0 0 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 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 1 0 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 0|1 0 0 0 1 1 0 1 0 0 0 0 0 1 0 1 0 1 0 1 0 0 1 0 1 1 1 0 0 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 0 0 1 1 1 1 1 1 1 1 1 0 0 0 1 0 0 0 1 1 1 0 1 1 1 1 0 0 0 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 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 1 0 1 0 1 1 0 0 0 0|0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 1 1 1 1 1 0 0 1 0 0 0 1 1 1 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 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 0 0 0 0 0 0 0 0 0 0 0 0 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 1 1 0 0 0 1 0 1 1 1 1]
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