Bounds on the minimum distance of additive quantum codes
Bounds on [[50,28]]2
lower bound: | 6 |
upper bound: | 8 |
Construction
Construction of a [[50,28,6]] quantum code:
[1]: [[64, 44, 6]] quantum code over GF(2^2)
quantum twisted code of length 64 with interval [ 1, 2, 3, 4 ] and parameter kappa 2
[2]: [[48, 28, 6]] quantum code over GF(2^2)
Shortening of [1] at { 13, 22, 23, 32, 34, 39, 43, 47, 48, 51, 53, 54, 56, 60, 62, 63 }
[3]: [[50, 28, 6]] quantum code over GF(2^2)
ExtendCode [2] by 2
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 1 0 0 1 0 0 1 1 1 1 0 1 1 0 0 0 1 1 1 0 0|0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 0 0 0 1 0 1 1 1 0 0 1 0 1 0 0 1 0 0 0 1 1 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 0 0 0 1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 1 1 0 1 0 1 0 1 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 1 1 0 1 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0|0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 1 0 1 1 1 0 1 1 0 1 1 0 1 0 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 0 0]
[0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 0 0|0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 0 0 1 1 1 0 1 1 0 0 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 0 0]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 1 0 0 1 1 0 1 1 1 0 1 0 1 0 0 1 0 0 0 1 0 0|0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 1 1 1 1 0 1 0 1 0 0 0 0 1 0 0]
[0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 0 0 1 0 1 0 0 0 1 1 1 1 0 1 1 0 0|0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0]
[0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 0 1 0 1 0 0 0 1 1 0 0|0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 0 0 1 1 1 1 0 1 0 0 1 1 0 1 1 1 0 0 1 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 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0|0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 1 1 0 1 0 0 0 0 0 1 0 1 1 0 1 0 0 0 1 0 1 1 0 0 0 0]
[0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 1 0 0 1 0 0 1 1 1 0 0|0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 1 1 0 0 1 0 0]
[0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0|0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 0 0 1 0 1 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 1 0 0 1 0 0 0 0 1 0 1 1 0 0|0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 1 1 1 0 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 0 1 1 1 0 1 0 1 1 0 1 0 1 0 0 1 0 1 1 1 0 0 1 1 0 1 0 1 0 1 0 0|0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 1 1 0 1 1 1 1 1 0 1 0 0 1 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 1 1 0 0 1 0 1 0 0 1 0 0|0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 1 1 0 1 1 1 1 1 0 1 1 1 1 0 1 1 0 0 1 1 0 1 0 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 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 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|1 0 0 0 0 0 0 1 1 0 1 0 0 1 1 1 0 0 1 1 0 0 1 0 0 0 1 0 0 1 0 1 1 0 1 0 1 1 1 1 0 1 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 1 0 0 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 0 0 1 0 0 0 1 1 0 1 0 1 0 0 1 0 1 1 1 1 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 0 0 0 0 0 0 0 0 0|0 0 1 0 0 0 0 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0 1 0 1 0 1 0 0 1 0 0 1 1 1 0 1 1 0 1 1 1 1 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 0 0 0 0 0 0 0 0 0 0|0 0 0 1 0 0 0 1 0 1 0 0 0 1 1 0 0 1 0 0 1 0 1 1 0 1 1 1 0 0 1 1 1 1 0 1 0 0 0 1 1 1 0 1 1 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 0 0 0 0 0 0 0 0|0 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 1 1 0 0 1 1 1 1 0 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 0 0 0 0 0 0 0 0 0 0 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 0 1 0 1 0 1 0 1 1 0 0 1 1 1 1 1 1 0 1 0 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 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 1 1 1 1 0 0 1 1 0 0 1 0 0 0 1 0 1 0 0 1 1 0 1 0 0 1 1 0 0 1 1 0 1 1 1 1 1 0 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