Bounds on the minimum distance of additive quantum codes
Bounds on [[56,8]]2
lower bound: | 12 |
upper bound: | 17 |
Construction
Construction of a [[56,8,12]] quantum code:
[1]: [[54, 8, 12]] quantum code over GF(2^2)
Code found by Petr Lisonek and Vijaykumar Singh
Construction from a stored generator matrix
[2]: [[56, 8, 12]] quantum code over GF(2^2)
ExtendCode [1] by 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 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0|0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 1 1 0 1 0 1 1 1 0 1 1 0 1 1 0 0 1 0 1 0 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 1 0 0 0 0 0|0 1 0 1 0 0 1 0 0 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 0 1 0 1 1 0 1 1 0 0 0 0 1 0 0 0 1 1 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0|0 0 1 1 0 1 0 1 0 1 0 1 1 1 0 1 1 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0|1 0 0 1 1 0 1 0 1 0 1 0 1 1 1 0 1 1 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 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 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 0 0|0 1 0 0 1 1 0 1 0 1 0 1 0 1 1 1 0 1 1 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 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 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 0|0 0 1 0 0 1 1 0 1 0 1 0 1 0 1 1 1 0 1 1 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0|0 1 0 1 0 1 0 0 0 1 0 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 0 1 1 0 0 1 0 1 0 0 0 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0|1 1 0 1 1 0 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 1 1 0 0 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0|1 0 1 1 0 1 1 0 1 0 0 1 1 0 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 1 0 0 1 0 0 1 0 1 1 1 1 0 1 1 0 1 1 1 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0|0 1 1 0 1 1 0 1 1 0 0 1 0 1 1 1 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 0 0 1 1 0 1 1 0 1 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0|0 0 0 1 1 1 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 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 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 0 0 0|1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 0 1 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 0 0 1 1 1 1 0 1 1 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 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 0 0|0 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 1 1 0 1 1 0 1 0 1 0 0 0 0 0 0 1 1 0 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 1 1 0 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 1 0 0 1 1 1 1 1 0 0 0 0 0|0 1 1 0 0 1 0 0 1 0 0 1 1 1 0 0 1 0 0 0 0 1 1 1 1 0 1 0 1 0 1 1 1 0 1 1 0 1 1 0 1 0 1 0 1 0 1 1 1 1 1 1 1 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 1 1 0 0 0 0 1 0 0 0 0 0 0|0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 1 0 1 0 0 1 1 0 0 0 0 0 1 0 0 0 1 0 1 1 1 0 1 1 1 0 0 0 0 0 0]
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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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 0|1 0 0 1 0 1 1 1 1 0 0 0 0 1 0 0 1 0 1 1 0 1 1 1 0 0 0 0 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 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 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 1 1 0 0 1 0 0 0 0 0 0 0|1 1 0 1 0 1 1 1 1 0 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 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 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 1 1 0 0 1 0 0 0 0 0 0|1 1 1 0 1 0 1 1 1 1 0 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 0 0 0 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 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 1 1 0 0 1 0 0 0 0 0|1 1 1 1 0 1 0 1 1 1 1 0 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 0 0 0 0 1 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 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 1 0 0 1 0 0 0 1 0 0 0 0 0|1 0 1 1 1 1 0 1 1 1 1 0 0 1 1 0 1 1 0 1 1 0 1 0 0 0 0 1 1 1 0 1 1 0 0 1 0 1 0 0 0 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 0 0 0|1 0 1 0 1 1 1 1 0 0 1 1 1 0 0 1 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 0 0 0 1 1 1 1 1 1 1 1 0 0 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0|1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 1 1 1 0 1 1 1 1 0 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 1 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 0 0 0 0 0|1 0 1 0 0 0 1 0 1 1 1 0 0 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 1 1 0 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 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 1 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 0 0 0 0|1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 0 0 1 1 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 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0|1 0 0 0 0 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 1 0 1 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 1 0 0 0 1 1 0 1 0 0 1 0 0 0 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0|1 1 0 0 0 1 1 1 1 1 0 0 1 0 1 1 0 0 0 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 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 0 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 1 1 0 0 0 0 0 0 0|0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 1 1 0 1 0 1 1 1 0 1 1 0 1 1 0 0 1 0 1 0 1 1 1 1 1 0 1 1 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 0 0 0 0 0 0|1 0 1 0 1 0 1 0 1 0 0 1 1 1 1 1 0 1 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 1 0 1 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 1 0 0 0 1 1 0 1 1 0 0 0 0 0|0 0 1 1 1 0 0 0 1 1 1 1 1 0 0 1 0 1 1 0 0 0 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 1 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 0 0 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 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 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 0 0 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 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 0 0 0 0 0 0 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 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 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 0 0 0 0 0 0 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 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]
last modified: 2013-11-13
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