Bounds on the minimum distance of additive quantum codes
Bounds on [[99,76]]2
lower bound: | 5 |
upper bound: | 7 |
Construction
Construction of a [[99,76,5]] quantum code:
[1]: [[128, 105, 6]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[100, 77, 6]] quantum code over GF(2^2)
Shortening of [1] at { 6, 10, 58, 60, 61, 63, 66, 71, 74, 79, 82, 90, 93, 95, 96, 98, 99, 100, 101, 102, 103, 104, 106, 110, 111, 116, 118, 125 }
[3]: [[100, 76, 6]] quantum code over GF(2^2)
Subcode of [2]
[4]: [[99, 76, 5]] quantum code over GF(2^2)
Puncturing of [3] at { 100 }
stabilizer matrix:
[1 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 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 1 1 0 1 0 1 0 0 1 0 1 0 0 1 1 1 0 0 0 0 1 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 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 1 1 0 1 0 0 1 1 0 0 1 1 1 1 1 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 1 0 1 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 1 0 1 1 0 1 0 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 1 1 1 1 0 0 0 1 0 0 0 0 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 1 1 0 0 1 0 1 1 0 0 0 1|0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 0 1 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 1 1 0 0 1 1 0 1 1 0 1 0 1 0 0 1 1 1 0 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 1 1 1 0 1 1 0 0 0 1 0 0 1 0 1]
[0 0 1 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 0 0 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 0 1 0 1 0 1 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 0 0 0 1 1 1 1 0 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 1 0 0 0 1 0 1 0|0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 0 1 1 1 1 0 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 0 1 0 1 0 0 1 0 0 1 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 1 1 1 0 1 1 0 0 0 0 0 1 1 1 1 1 0 1 0 0 1 1 0 0 1 1 1 0 1 1 1 0 1 1 0 1 0 0 0 1 0 0 1 1 0 0|0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 1 0 0 0 1 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 0 1 1 1 1 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 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 1 0 1 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 0 0 0 1 0 1 1 0 0 1 1 1 0 0 1 1 0 1 1 1 0 1 1 1 1 1 1 0 1 0 1 1 0 0 1 0 0 0 1 0 1 0|0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 1 1 1 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 0]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 1 1 0 0 1 1 0 1 1 1 0 0 0 1 1 1 0 0 1 0|0 0 0 0 0 0 1 0 1 0 1 1 0 1 0 0 0 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 1 0 1 1 1 0 0 1 0 1 0 1 1 1 1 1 1 1 0 0 1 0 1 0 1 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 0 1 1 1 1 1 1 0 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1|0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 0 1 1 1 0 1 0 1 1 0 0 1 1 0 0 1 1 1 1 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 0 1 0 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0]
[0 0 0 0 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 1 0 0 1 0 1 0 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 1 0 0 1 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 0 1 0 1 0 1 1 1 1 1 0 1 0 1 0 0 0 1 0 1 0 1 0 1 1 1 0 1|0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 1 0 1 1 1 0 0 1 0 1 0 1 1 1 1 1 1 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 1 0 1 1 1 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 0 1 1]
[0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 1 1 1 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 1 0 0 0 1 0 1 1 1 1 0 0 1 0 0 0 1 1 0 1 0 1 0 1 1 0 0 1 1 1|0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 1 1 1 1 1 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1]
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[0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 0 0 1 1 1 0 1 1 1 0 0 1 1 0 0 0 1 1 1 0 1 0 1|0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 1 1 0 1 0 0 1 0 1 1 0 0 1 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0]
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[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 0 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 0 1 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 1 1 1 0 0 0 0 0 1 0 0 1 0|0 0 0 0 0 0 1 1 0 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 1 0 1 0 0 1 1 1 0 0 1 0 0 0 1 1 0 0 1 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 1 0 0 0 1 1 1 1 0 0 1 0 0 0 1 0 1 1 0 0 1 1 1 0 1 0 1 0 0 1 1 1 1 1 0 1 0 0 0 1 0 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 0 0 1 0 1 1 0 0 0 0 1 0 0 0 1 1 1 1 1 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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Markus Grassl
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Last change: 23.10.2014