Bounds on the minimum distance of additive quantum codes
Bounds on [[57,37]]2
lower bound: | 5 |
upper bound: | 7 |
Construction
Construction of a [[57,37,5]] quantum code:
[1]: [[93, 73, 5]] quantum code over GF(2^2)
quasicyclic code of length 93 stacked to height 2 with 6 generating polynomials
[2]: [[57, 37, 5]] quantum code over GF(2^2)
Shortening of [1] at { 1, 4, 11, 14, 16, 18, 19, 20, 21, 24, 28, 31, 36, 40, 41, 46, 48, 53, 54, 55, 56, 58, 60, 62, 63, 64, 67, 71, 74, 75, 78, 85, 88, 90, 92, 93 }
stabilizer matrix:
[1 0 0 0 0 0 0 0 1 0 0 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 1 0 0 1 1 0 1 1 0 1 1 0 0 0 1 0 1|0 0 0 0 1 1 1 0 0 1 0 1 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1]
[0 1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 1 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 0 1 1 0 1 0 1 0 1 1 1|1 0 0 0 1 1 1 1 1 0 1 1 0 0 1 1 1 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 0 1 0 1 0 0 1 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0 0 0 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 1 0 0 1 1 0 1 1 1 0 1 1 1 0 1 1 0 0 1|0 1 0 0 1 0 1 0 0 0 0 1 1 0 0 1 0 1 1 1 0 0 1 1 0 1 0 0 1 0 0 1 1 1 1 0 0 1 1 0 1 1 1 1 1 0 1 0 0 1 1 0 0 0 0 0 1]
[0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 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 1 0 1 1 1 1 0 1 0 0 0 1 0 0 0 1 0|1 0 0 0 1 0 1 1 0 1 0 0 0 0 1 0 1 0 1 0 1 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 1 0 1 1 1 1 1 0]
[0 0 0 0 1 0 0 0 0 0 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 1 1 0 1 0 1 1 1 1 0 1 0 1 0 1 1 1 0 0|0 1 1 0 0 0 0 0 1 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 0 1 0 0 0 0 0 1 1 1 1 1]
[0 0 0 0 0 1 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 1 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0|0 0 0 1 1 1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 0 1 1 0 1 0 1 0 1 1 0 0 1 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 0 1 1 0 1 1 0 0 1]
[0 0 0 0 0 0 1 0 1 0 0 0 0 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 1 1 0 1 1 0 1 0 0 0 0 0 0 0 1 1 0 1|1 0 1 0 0 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 0 1 0 0 1 0 0 1 1 0 0 1 0 0 0 0 1 1]
[0 0 0 0 0 0 0 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 1 1 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0|1 0 0 1 0 1 0 0 0 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 0 0 0 0 1 1 0]
[0 0 0 0 0 0 0 0 0 1 0 0 1 0 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 1 1 1 1 0 1 0 1 1 0 1 0 1 1 0 1 1 0|1 0 0 0 0 0 1 1 0 0 1 0 1 1 1 0 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 1 0 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0]
[0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 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 0 1 0 0 1 0 0 0 1 0 0 1|0 0 0 0 1 0 1 1 1 0 0 1 0 1 0 0 0 1 1 1 1 1 0 1 1 1 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 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 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1 0 1 1 1|0 1 0 0 0 1 1 0 1 1 0 0 1 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 0 0 1 1 0 1 1 0 0 1 1 0 1 1 0 0 0 1 0 1 1 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 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 0 0|0 1 0 0 0 1 1 1 1 0 1 1 0 1 1 1 0 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 1 1 1 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 1 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 0 1 0 1 0 0 0 0 1 1 0 1 1 0 0 1 1 0|1 0 0 1 1 0 1 1 0 0 0 1 1 1 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 1 1 1 1 0 1 0 1 1 0 0 0 0 1 1 1 1 0 0 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 0 1 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0|1 1 0 0 1 1 0 1 1 0 0 0 1 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 0 0 1 0 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 1 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 1 0 0 0 0 0 0 1 1 0 1 1 1 0 0 1 0 0 0 0 0 0 1 1 1 1 0 1 0 1 1 0 1 1|0 1 0 1 1 1 0 0 0 0 1 1 0 1 1 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 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 1 0 1 0 0 0 1 0 1 1 0 1 0 0 1 1 1 0 1 0 0 1 1 0 1 0 1 1 1 0 0 1 1|0 1 1 0 1 0 0 0 1 1 1 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 0 1 0 0 1 1 0 0 1 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 0 0 0 1 1 0 0 0 0 1 1 1 1 1 1 0 1 0 0 1 1 1 1 0 0 1 0 1 0 1 1 1 1 0 1|0 1 1 1 0 0 0 0 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 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 1 0 0 1 1 1 0 0 1 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 0|0 1 1 1 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 1 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 1 0 0 1 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 1 0 1 1 1 1 0 0 1 0 1 1 0 0 1 0 0 1 1 1 0 0 1 0 0 1 1 1 0|1 0 0 0 0 1 0 1 0 0 1 0 1 1 0 0 0 0 0 1 0 1 0 1 0 1 0 1 1 0 1 1 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 1 0 0 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 0 0 0 1 0 0 1 1 1 1 0 1 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 0 0 0 0|1 1 1 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 1 1 0 1 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 0 0 0 1 0 0 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