Bounds on the minimum distance of additive quantum codes
Bounds on [[40,18]]2
lower bound: | 6 |
upper bound: | 8 |
Construction
Construction of a [[40,18,6]] quantum code:
[1]: [[70, 48, 5]] quantum code over GF(2^2)
quasicyclic code of length 70 stacked to height 2 with 4 generating polynomials
[2]: [[40, 18, 6]] quantum code over GF(2^2)
Shortening of [1] at { 1, 2, 6, 8, 9, 13, 15, 16, 20, 22, 23, 27, 29, 30, 34, 37, 38, 42, 44, 45, 49, 51, 52, 56, 58, 59, 63, 65, 66, 70 }
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 1 1 0 1 0 0 0 0|0 1 1 0 0 0 0 1 1 1 0 0 1 0 0 1 1 0 1 0 1 1 1 1 1 0 1 0 0 0 0 1 1 0 0 1 1 1 1 0]
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 0 0 0 0|0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1]
[0 0 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 1 1 0 0 0 1 0 0 1 0 1 1|0 0 0 0 0 0 1 0 1 0 1 0 0 0 1 1 0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 1 0 0 0 0 0 1 1 1]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 1 0 1 1 1 1 0|0 1 1 0 0 1 0 0 1 1 0 0 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 0 1 0 0 0 0 1 1]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 1 1 0|1 1 0 0 0 1 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0]
[0 0 0 0 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 1 0 0 0 0 0 0 1 1 1|1 1 1 0 0 1 0 1 1 0 0 1 0 0 0 0 0 1 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 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 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 0 1 0 1|0 1 0 1 1 1 1 0 0 1 0 1 1 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0 1 0]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 1 1 0 1 0|0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1 1 1 1 0 0 0 0 0 1 0 1 0 1 1 1 0 0 0 0]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0|1 1 1 1 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 0 1 1 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 0 1 0]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1 0|1 0 1 1 1 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 1 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 1 1 1|0 1 0 1 1 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 0 1 0 1 0]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 1 1 0 0|1 1 0 0 0 1 1 0 1 0 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 1 1 1 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 1 1 1 0|0 1 1 0 0 0 1 1 0 1 0 1 0 0 1 1 0 1 0 1 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 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 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0|1 1 1 1 0 1 0 0 1 0 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 1 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1|1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 1 0 0 0 1 1 0 1 1 0 0 0 1 0 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 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 1 1 0 0|1 1 1 0 1 0 1 1 1 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 1 1 1 1 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 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1|0 1 0 0 0 1 0 1 0 0 1 1 1 0 1 0 1 0 0 0 1 0 1 0 1 0 1 1 1 1 0 0 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 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0 1 1 0|1 1 0 1 1 0 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 1 1 1 0 0 0 0 1 0 0 1 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 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1|0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 0 1 1 0 1 1 1 0 1 1 0 0 0 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 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 1|1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 1 0 1 1 1 0 1 1 0 0 0 0 1 0 1 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 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0|0 1 1 0 1 0 0 1 0 0 1 0 1 1 0 0 0 0 0 1 0 1 1 1 0 1 1 1 1 1 0 1 1 1 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 1 0 1 0 1 0 1 0 0|0 1 0 1 0 0 1 1 1 0 1 0 1 0 0 0 0 1 0 0 1 0 1 1 1 1 0 0 1 0 1 1 0 1 1 0 1 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