Bounds on the minimum distance of additive quantum codes
Bounds on [[92,60]]2
lower bound: | 7 |
upper bound: | 10 |
Construction
Construction of a [[92,60,7]] quantum code:
[1]: [[91, 60, 7]] quantum code over GF(2^2)
cyclic code of length 91 with generating polynomial w^2*x^89 + w*x^88 + w*x^87 + x^86 + x^84 + w*x^83 + w*x^82 + x^81 + w^2*x^80 + w^2*x^79 + x^78 + w^2*x^75 + w*x^74 + x^73 + w*x^71 + w*x^70 + w*x^68 + w^2*x^67 + w*x^66 + w^2*x^65 + w^2*x^63 + w^2*x^62 + w*x^61 + w*x^60 + w^2*x^59 + w^2*x^58 + x^57 + w*x^56 + x^55 + w^2*x^54 + w*x^53 + w*x^52 + x^51 + w^2*x^50 + w*x^49 + w*x^47 + w^2*x^46 + w*x^44 + w^2*x^43 + w*x^42 + x^41 + w*x^40 + w^2*x^38 + w*x^37 + x^35 + w*x^34 + w^2*x^33 + w*x^31 + w^2*x^30 + w^2*x^29 + w^2*x^28 + x^26 + w^2*x^25 + x^23 + w*x^22 + w*x^21 + w*x^20 + x^19 + x^17 + w*x^16 + x^8
[2]: [[92, 60, 7]] quantum code over GF(2^2)
ExtendCode [1] by 1
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 1 0 1 1 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 1 0 0 1 0 1 1 1 1 0 0 1 0 1 1 1 0 0 1 0 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 0 1 0 1 1 0|1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 0 1 0 0 1 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 1 0 1 1 1 1 1 1 1 0 0 1 0 1 1 0 0 0 1 1 1 0 0 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 1 1 1 0 1 0 0 0 0 1 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 0 1 0 1 1 1 0 1 0 0 1 1 1 0 0 0 1 1 0 0 1 0 1 1 1 0 0|0 1 0 1 0 1 1 1 0 0 0 1 0 0 0 1 1 0 1 1 0 1 1 0 1 0 0 1 0 1 0 1 1 0 1 1 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 1 1 0 0 1 1 0 1 0 1 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 1 0 1 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 1 1 1 0 1 0 0 0 0 1 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 0 1 0 1 1 1 0 1 0 0 1 1 1 0 0 0 1 1 0 0 1 0 1 1 1 0|0 0 1 0 1 0 1 1 1 0 0 0 1 0 0 0 1 1 0 1 1 0 1 1 0 1 0 0 1 0 1 0 1 1 0 1 1 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 1 1 0 0 1 1 0 1 0 1 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 1 0 1 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 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0|0 0 0 0 1 1 1 1 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 0 1 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 1 0 1 1 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 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 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0|1 0 0 0 0 1 1 1 1 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 0 1 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 1 0 1 1 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 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 0 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0|0 1 0 0 0 0 1 1 1 1 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 0 1 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 1 0 1 1 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 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 0 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0 0|1 0 1 0 0 0 0 1 1 1 1 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 0 1 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 1 0 1 1 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 0 1 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 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0|0 1 0 1 0 0 0 0 1 1 1 1 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 0 1 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 1 0 1 1 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 0 1 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 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 0|0 0 1 0 1 0 0 0 0 1 1 1 1 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 0 1 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 1 0 1 1 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 0 1 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 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0|0 0 0 1 0 1 0 0 0 0 1 1 1 1 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 0 1 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 1 0 1 1 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 0 1 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 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0|0 0 0 0 1 0 1 0 0 0 0 1 1 1 1 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 0 1 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 1 0 1 1 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 0 1 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 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0|0 0 0 0 0 1 0 1 0 0 0 0 1 1 1 1 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 0 1 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 1 0 1 1 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 0 1 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 1 0 1 1 0 0 0 0 0 0 0 1 1 0 0 1 0 1 1 1 0 0 1 0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 0 1 1 0 0 0 1 0 1 1 1 1 1 1 0 0 0 0 1 0 0 1 0|0 0 0 1 1 0 0 0 1 0 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 1 0 0 0 1 0 1 1 0 1 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 0 0 1 1 0 0 0 0]
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[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 1 1 0 0 0 1 0 0 0 1 1 1 0 1 0 0 0 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 0 1 0 1 1 0 0 0 0 1 0 1 1 0 0 1 1 0 0 1 1 1 0 1 1 0 0 0|0 1 0 1 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 0 0 1 1 1 1 0 1 1 0 0 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 1 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 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 1 1 0 0 0 1 1 0 0 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 1 1 1 1 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 1 1 1 0|0 1 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 1 0 1 1 1 1 1 0 0 1 1 0 0 0 1 1 1 1 1 0 1 1 0 0 0 1 1 0 0 1 1 0 1 0 1 0 0 1 1 1 1 0 1 0 0 0 0 0 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 0 0 0 1 0 0 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 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 1 1 1 1 1 1 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0|1 0 0 1 1 1 1 0 1 1 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 1 1 0 1 0 1 0 0 0 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 1 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 0 0 1 0 0 1 1 1 0 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 0 1 1 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 0 1 0 1 1 0|1 1 1 0 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 0 1 1 1 0 1 1 1 1 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 1 1 1 1 1 0 1 0 0 1 0 0 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 1 1 1 0 1 1 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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]
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.
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Markus Grassl
(codes@codetables.de).
Last change: 23.10.2014