Bounds on the minimum distance of additive quantum codes
Bounds on [[107,89]]2
lower bound: | 4 |
upper bound: | 6 |
Construction
Construction of a [[107,89,4]] quantum code:
[1]: [[122, 104, 5]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[106, 88, 5]] quantum code over GF(2^2)
Shortening of [1] at { 20, 22, 23, 27, 28, 29, 34, 39, 45, 46, 48, 50, 55, 63, 87, 99 }
[3]: [[105, 89, 4]] quantum code over GF(2^2)
Shortening of the stabilizer code of [2] at 106
[4]: [[107, 89, 4]] 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 1 0 0 0 0 1 1 0 1 1 1 1 1 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 1 0 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 0 1 1 1 0 1 0 1 0 0 0 1 1 0 0 1 0 1 0 0|1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 0 0 1 1 1 0 0 0 0 1 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 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 0 1 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 1 1 0 1 1 0 1 1 0 1 1 0 1 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0|1 1 1 0 1 0 0 0 0 1 1 1 1 0 1 0 0 1 0 1 1 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 0 1 1 1 1 1 1 1 0 0 1 0 0 0 1 0 1 0 0 0 0 1 0 1 1 1 0 1 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 1 1 1 0 1 1 0 0 0 1 1 0 1 1 1 0 1 0 0 0 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 1 1 1 0 1 0 1 1 0 1 0 1 1 0 0 1 0 0 0 1 1 1 1 0 1 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 0 1 1 1 0 1 0 0 0|1 0 0 1 0 1 1 0 0 1 1 1 1 1 0 1 1 0 1 1 0 1 1 0 0 1 0 0 1 0 1 1 0 0 1 1 0 1 1 0 1 1 1 0 1 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 1 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 0 1 0 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 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 1 0 1 0 1 0 1 1 0 0 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 1 1 1 1 1 0 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 1 0 1 0 1 1 0 0 1 0 1 1 1 0 0 1 1 0 0 0|0 0 0 1 1 0 1 1 1 0 1 0 1 0 0 1 1 0 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 1 1 0 0 1 1 0 1 1 1 0 0 1 0 1 1 0 0 0 1 0 0]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 0 1 0 0 1 0 0 0 1 1 0 1 1 0 1 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 0 0|0 0 1 1 0 0 0 1 1 0 1 1 1 1 0 1 0 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 1 1 0 0]
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[0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 1 1 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 0 1 1 1 0 0 0 1 1 1 1 1 0 0 0 1 0 0 0 1 0 1 0 0 0|1 1 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 1 0 0 0 0 1 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 1 0 0 1 1 0 1 0 1 0 0 0 1 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 1 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 0 0 1 0 1 1 1 1 0 1 0 0 0 1 0 0 1 1 0 1 1 1 0 1 0 0 0 0 0 0|1 0 0 1 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 0 0 0 0 1 0 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 0 1 0 1 1 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 0 0 1 1 0 1 1 0 1 0 1 1 0 0]
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[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 1 1 0 1 0 1 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 0 1 1 1 1 1 1 1 0 1 1 0 1 0 1 0 0|0 1 0 0 1 1 1 0 0 0 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 0 1 0 0 0 1 1 1 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 1 1 1 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 0 0 1 1 1 0 1 0 1 0 0 0 1 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 1 1 0 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 1 0 1 1 0 1 0 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 0 0 0 0 0|1 0 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 0 1 1 1 1 1 1 1 0 0 1 1 0 0 1 1 0 1 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 1 0 1 1 1 1 0 0 0 0 1 0 1 1 1 0 1 1 0 1 1 1 1 0 1 1 0 0 0]
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[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 0 1 1 0 1 0 1 0 1 1 1 0 0 0 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 0 1 1 0 0 1 1 1 1 1 1 1 1 0 1 0 1 0 0 1 1 1 1 0 0 0 1 0 1 0 0|0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 1 0 1 1 0 0 0 0 0 0 1 1 1 0 1 1 1 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 1 1 1 0 1 1 0 1 0 1 1 1 1 1 1 1 0 1 0 0 0 1 0 0 1 1 0 1 1 1 1 1 0 1 0 0 0 0 1 1 1 0 1 1 0 1 0 0 1 1 1 1 1 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 1 1 0 0 1 1 1 1 1 1 0 0 1 0 1 0 0 1 0 1 1 1 0 0 0 0 0 1 0 1 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 1 1 1 1 0 1 1 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 1 1 1 0 1 1 1 1 0 0|0 0 1 0 1 1 0 0 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 0 1 0 0 1 0 1 1 0 0 1 1 0 0 0 0 0 1 1 0 0 0 1 0 1 1 0 0 1 0 1 0 1 0 1 0 0 1 0 0 0 1 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 0]
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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