Bounds on the minimum distance of additive quantum codes
Bounds on [[75,55]]2
lower bound: | 5 |
upper bound: | 6 |
Construction
Construction of a [[75,55,5]] quantum code:
[1]: [[122, 104, 5]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[74, 56, 5]] quantum code over GF(2^2)
Shortening of [1] at { 1, 3, 16, 20, 21, 23, 24, 25, 28, 29, 32, 33, 36, 37, 39, 40, 42, 44, 46, 52, 57, 58, 60, 62, 69, 71, 73, 74, 75, 76, 78, 79, 81, 82, 86, 87, 91, 96, 101, 102, 103, 104, 105, 114, 117, 119, 120, 121 }
[3]: [[74, 55, 5]] quantum code over GF(2^2)
Subcode of [2]
[4]: [[75, 55, 5]] quantum code over GF(2^2)
ExtendCode [3] by 1
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 1 0 1 0 1 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 0 1 1 1 0 1 1 0 0 0 0 1 1 1 0 0 1 1 0 1 1 1 0 1 0 1 0 1 1 1 0|0 1 0 1 1 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 1 1 1 1 1 1 0 0 0 0 1 0 1 0 1 1 1 0 0 0 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 1 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 1 0 0 1 1 0 1 1 1 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0|1 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 1 0 0 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 0 1 1 0 1 1 0 1 1 0 1 0 0 1 1 1 1 0 0 0 1 1 0 1 1 1 0 1 0 1 1 1 1 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 1 1 0 1 1 0 1 0 1 0 1 0 1 1 0 0 1 0|0 0 0 0 1 0 1 1 0 0 1 1 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 0 1 0 0 1 1 1 1 1 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1 1 1 0]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 0 1 0 0 0 1 0 0 1 1 1 0 0 0 1 0 0 1 1 1 1 0 1 0 1 0 0 1 0 1 1 0 1 0 0|1 1 0 1 0 1 1 0 1 0 0 1 0 1 0 1 0 0 1 1 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 1 1 1 0 0 0 1 1 0 1 0 0 0 0 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 1 1 0 0 1 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 1 0 1 1 1 1 0 1 0 1 1 1 1 0 0 0 1 0 1 0 1 1 1 1 1 0 0 0 0|1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0 1 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 1 0 1 1 0 0 0 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 0 0 0 0 1 1 0]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 1 1 1 1 0 1 0 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 1 0 1 0 0 1 1 1 1 0 0 1 0 0 0|1 1 1 1 1 1 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 0 1 0 1 1 1 1 1 0 1 1 0 1 1 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 0 1 0 0 1 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 0 0 0 0 1 0 1 0 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 1 1 0 1 1 0 0 0 0 1 0 0 0 0|1 1 1 0 1 1 0 0 1 1 1 1 0 1 1 0 0 1 0 0 1 0 1 1 1 1 0 1 1 0 0 1 0 0 1 1 0 0 0 0 1 1 0 1 0 1 0 0 0 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 0 1 1 1 0 1 0 1 1 0 0 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 0 1 0 0 1 0|1 1 1 1 0 1 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 0 0 1 0 1 1 1 1 1 1 0 0 1 1 0 0 0 0 1 0 0 1 0 1 1 0 1 1 0 1 1 0 0 0 1 0 1 0 1 0 0 0 1 0 1 0 0 1 1 1 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 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 1 1 0 1 1 1 1 0 0 1 1 1 1 1 0 1 1 1 1 1 0 0 0 1 1 1 1 1 0 1 0 1 0 0|1 0 1 0 1 0 0 1 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 1 1 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 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 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 0 0 0 0 0 0 0 0 0 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 1 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 1 1 1 0 1 0 1 1 0 1 1 0 1 1 0 1 0 0 0 1 1 0 0 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0|1 1 1 1 0 0 1 0 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 1 1 1 0 1 0 0 1 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 1 0 0 0 0 0 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 1 1 1 0 0 1 0 0 0 0 1 1 1 1 0 1 0 1 1 1 0 0 0 1 0 0 0 1 1 1 1 1 1 1 1 0 0 1 1 0 1 1 1 1 1 0 1 0 1 0 0 0|0 0 1 1 1 1 1 1 1 1 1 0 1 0 0 1 0 0 0 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 0 1 0 1 1 1 1 0 1 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1 0 0 1 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 1 1 1 1 1 1 1 1 0 1 0 1 0 0 1 0 0 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 1 0 1 0|1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 0 0 1 1 1 1 0 0 0 1 1 0 0 0 1 0 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 0 1 0 0 1 0 0 1 1 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 1 1 0 1 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 1 0 0 1 1 0 1 1 0 0 1 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 0 0 0 1 0 1 0 0 0 1 1 1 0 1 0 1 0 0 0 1 0|0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 1 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 1 1 0 0 0 1 0 0 0 1 1 0 0 1 0 0 1 1 0 1 1 1 0 0 0 0 0 0 1 1 0 1 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 0 1 1 1 1 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 0 1 1 1 1 0 1 1 1 1 0 1 0|1 1 0 0 1 1 1 0 0 0 0 0 1 0 1 0 1 1 0 0 0 1 0 1 1 1 1 1 0 0 0 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 1 0 0 0 0 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 1 0 0 0 1 0 0 0 1 0 1 1 1 0 0 0 0 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 1 0 0 0 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0|0 0 0 1 1 0 0 0 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 1 0 0 1 1 0 1 0 1 0 1 0 0 1 1 1 0 1 0 1 1 1 1 1 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 1 0 0 1 1 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 0 1 1 1 0 1 0 1 1 0 1 1 1 0 0 0 0 1 0 0 1 1 1 1 0 1 0 1 1 1 0 0 1 0 1 0 1 0 1 1 0 0 1 0 1 0 0|1 1 1 0 1 1 1 0 1 0 1 0 1 0 1 0 0 1 1 1 0 0 1 1 1 0 0 0 1 0 1 0 1 1 1 1 0 0 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 1 1 1 0 1 0 0 1 1 1 1 1 1 1 1 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 0 1 1 1 0 1 1 1 0 0 0 0 0 0 1 1 1 0 1 1 0 0 1 1 1 0 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 0 1 0 0 1 1 0 1 1 0 1 0|1 1 1 1 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 1 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 1 1 0 0 1 1 0 0 1 1 0 1 1 0 1 0 1 0 0 1 1 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 0 1 0 0 0 1 1 1 0 0 1 0 0 0 0 1 0|1 0 0 1 1 0 1 1 1 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 0 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 0 1 0 1 1 0 1 1 1 0 1 0 0 1 1 1 1 1 1 1 1 1 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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]
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