Bounds on the minimum distance of additive quantum codes
Bounds on [[55,39]]2
lower bound: | 4 |
upper bound: | 6 |
Construction
Construction of a [[55,39,4]] quantum code:
[1]: [[85, 69, 4]] quantum code over GF(2^2)
quasicyclic code of length 85 with 4 generating polynomials
[2]: [[55, 39, 4]] quantum code over GF(2^2)
Shortening of [1] at { 4, 7, 8, 9, 12, 18, 21, 23, 27, 28, 29, 33, 35, 36, 39, 48, 51, 56, 58, 59, 60, 62, 63, 64, 66, 71, 73, 75, 81, 82 }
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 1 1 0 1 0 0|1 1 1 1 1 0 0 1 0 0 0 0 1 1 1 1 1 1 0 0 1 0 0 1 1 1 0 1 0 0 1 0 0 0 1 0 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0]
[0 1 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 1 1 0 1 0 0 1 0 1 1 1 0 1 1 0 0 1 0 0 1 0 1 0 0 0 1 1 0|0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 0 1 1 0 1 1 1 1 1 0 1 0 0 0 1 1 0 1 1 0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 1 1 1 1 1 0 0 1|0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 0 1 0 0 0 0 0 0 1 1 1]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 0 0 1 1 0 1 0 1 0 0 1 0 0 1 1 0 1 0|1 1 1 1 1 1 1 1 0 0 0 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 0 1 0 0 1 0]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 0 0 1 0 1 1 1 0 1 0 1 1 1 1 1 0|1 1 1 0 0 1 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 1 1 0 1 1 0 0 1 1 1]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 1 1 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 1 1 1|1 1 0 1 1 0 0 1 0 1 1 1 0 1 1 1 1 1 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 0 0 0 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0|0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 1 1 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 0 1 1 0 0 1 1 0 1 1 1 1 1|1 0 1 0 1 0 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 1 0 1 1 1 0 1 0 1 1 0 1 1 0 0 1 1 1 0 0 1 0 0 1 1 1 0 0 1 1 0]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 0 1 1 1 0 0 1 0 0|1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 0 0 1 1 0 1 0 1 1 0 0 1 0 0 1 1 1 1]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 1 1 1 1 1 0 0 1 1 1 0 0 0 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0|0 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 1 0 0 1 0 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 1 1 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 0 1 0 1 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 1 0 1 1 0 0|0 1 0 1 0 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 0 1|0 0 0 0 1 1 0 0 0 1 0 1 1 1 1 1 1 1 1 0 0 1 0 0 1 1 1 1 0 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 0 0 1 1 0 1 0 0 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 1 1 1 1 0 0 1 0 1 0 1 0 0 0 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 0 0|1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 0 1 0 0 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 0 0 1 0 1 0 1 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 1 1 0 0 1 0 1 0 0 0 1 1 0 0 1 1 1 0 0 1 0 1 0 1 0 1 1 0 0 0 0 0 0 0|1 0 1 1 0 1 0 1 1 0 1 0 1 1 1 1 1 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 1]
[0 0 0 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 0 1 0 0 1 1 0 0 1 1 0 1 1 0 0 1 0 0 1 0 0 0 1 0 1 1|1 0 0 0 0 1 1 0 1 0 1 0 0 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 1 0 1 0 1 1 1 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1 0 1 1 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 1 1|1 1 0 0 0 1 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 0 0 1 1 1 1 0 0 1 0 0 1 1 1 1 1 0 1 1 1]
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