Bounds on the minimum distance of additive quantum codes
Bounds on [[35,0]]2
| lower bound: | 11 |
| upper bound: | 13 |
Construction
Construction of a [[35,0,11]] quantum code:
[1]: [[36, 0, 12]] self-dual quantum code over GF(2^2)
quasicyclic code of length 36 stacked to height 2 with 4 generating polynomials
[2]: [[35, 1, 11]] quantum code over GF(2^2)
Shortening of the stabilizer code of [1] at 36
[3]: [[35, 0, 11]] self-dual quantum code over GF(2^2)
Subcode of [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|1 1 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 1 0 0 1 1 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0|1 1 1 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 1 0 0 1 1 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0|0 1 1 1 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 1 0 0 1 1 0 0 0 1 1 0 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|1 0 1 1 1 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 1 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|1 1 0 1 1 1 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 1 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1 0 0 1 1 0 0 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 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 1 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 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 0 1 1 1 1 1 0 0 1 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 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 0 1 1 1 1 1 0 0 1 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 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 0 1 1 1 1 1 0 0 1 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 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 0 1 1 1 1 1 0 0 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 1 1 1 0 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 1 0 1 1 1 1 1 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 1 1 0 1 1 1 0 1 1 1 1 1 0 0 0 1 1 0 1 1 0 1 1 1 1 1 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 0 0 0 0 0 1 1 1 0 1 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 1 0 1 1 1 1 1]
[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 0 0 0 0 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 0 1 1 0 1 1 0 1 1 1 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|1 1 1 0 0 0 0 0 0 0 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 1 0 0 1 1 0 0 0 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|1 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 1 0 0 1 1 0 0 0 1 1 0 1 1 0 1]
[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|1 1 0 1 1 0 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1 1 1 0 0 0 0 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|1 1 1 0 1 1 0 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1 1 1 0 0 0 0 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|1 1 1 1 0 1 1 0 1 1 0 0 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0|1 1 1 1 1 0 1 1 0 1 1 0 0 0 1 1 0 0 1 0 1 0 1 0 1 1 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 1 0 0 0 0 0 0 0 0 0 0 0 0|0 1 1 1 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 1 0 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 1 0 0 0 0 0 0 0 0 0 0 0|0 0 1 1 1 1 1 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 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 1 0 0 0 0 0 0 0 0 0 0|1 0 0 1 1 1 1 1 0 1 1 0 1 1 0 0 0 1 0 1 1 1 0 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 1 0 0 0 0 0 0 0 0 0|1 1 0 0 1 1 1 1 1 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 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 1 0 0 0 0 0 0 0 0|0 1 1 0 0 1 1 1 1 1 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 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 1 0 0 0 0 0 0 0|0 0 1 1 0 0 1 1 1 1 1 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 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 1 0 0 0 0 0 0|0 0 0 1 1 0 0 1 1 1 1 1 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 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 1 0 0 0 0 0|1 0 0 0 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 0 0 0 0 0 1 1 1 0 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 1 0 0 0 0|1 1 0 0 0 1 1 0 0 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 1 1 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 0 0 0 0 0 0 0 0 0 0 1 0 0 0|0 1 1 0 0 0 1 1 0 0 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 1 1 1 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 0 0 0 0 0 1 0 0|1 0 1 1 0 0 0 1 1 0 0 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0|1 1 0 1 1 0 0 0 1 1 0 0 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1|0 1 1 0 1 1 0 0 0 1 1 0 0 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 1 0]
last modified: 2005-07-05
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