Bounds on the minimum distance of additive quantum codes
Bounds on [[56,33]]2
lower bound: | 6 |
upper bound: | 8 |
Construction
Construction of a [[56,33,6]] quantum code:
[1]: [[128, 105, 6]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[56, 33, 6]] quantum code over GF(2^2)
Shortening of [1] at { 1, 2, 4, 7, 8, 9, 12, 14, 15, 16, 19, 20, 21, 22, 23, 24, 26, 27, 30, 32, 33, 34, 39, 40, 41, 42, 43, 44, 46, 47, 48, 50, 51, 52, 53, 54, 57, 61, 62, 67, 69, 71, 73, 74, 76, 77, 79, 80, 81, 82, 83, 84, 85, 86, 91, 93, 95, 97, 100, 102, 103, 105, 106, 108, 112, 114, 115, 117, 121, 122, 123, 127 }
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 1 0|0 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 0 1 0 1 0 1 1 1 1 0 1 0 0 1 0 1 1]
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 1 1 1 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0|0 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 1 1 1 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 0]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 0 0 0 1 1 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 0 1 1 0 1 0 1|0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 1 0 1 0 0 1 1 1 0 1 0 1 0 0 0 1 1 1 0 0 1 0 1 1 0]
[0 0 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 1 0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 1 1 0|0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 1 1 0 1 0 0 1 0 0 0 0 1 1 1 0 0 0 1 0 1 1 1 0 1 0 1 1 0 1]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1 1 1 0 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 1|0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 1 1 0 0 1 0 0 1 1 0 1 1 1 1]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 1 1 0 0 1|0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 1 1 0 1]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1 0 1 0|0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 0 0 1 1 1 1 1 1 0 1]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 1 1 0|0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 1 0 0|0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 1 0 0 1 1 1 0 1 1 0 0 0 1 0 0 0 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 1 0 0 1 1 0 1 0 1 1 1 1 1 0 1 1 1 1 0 1 1 0 0 1 0 1 1 1 0 1 1 1 1 1 0 0 1 0 0 1 1|0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 0 1 0 1 1 0 0 1 0]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 1 1 1 1 1 0 0 1 1 1 0 0 1|0 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 1 1 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 0 1 0 1 0 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 1 0 0 0 0 1 1 0 0 0 1 1 0 0 1 1 0 1 0 1 1 0 0 1 0 0 0 1 0 0 1 1 0 1 1 1 0 0 0 1 1 0 0 1 0|0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 1 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 0 1 1 1 0 1 1 1 1 1 0 1 1 1 0 1 0 1 1 0|0 0 0 0 1 0 0 1 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 1 0 1 1 0 0 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 1 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 0 0 0 1 0 0 1 0 1 1 1 0 1 0 0|0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 1 1 1 1 1 1 1 1 0 0 0 0 1 0 0 1 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 0 1 1 0 1 1 0 0 1 0 1 0 0 1 1 0 1 0 1 1 1 0 1 1 0 0 0 0 1 0 0|0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 1 1 1 1 1 1 1 1 0 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 0 0 0 0 0 0 0 0 0 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 0 1 1 0 1 1 1 1 0 0 1 1 0 1 0 1 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 1 1 1 1 0 0 0 0 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 0 0 0 0 0 0 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 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 1 1 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 1 1 1 0 0 0 1 1 0 1 0 0 1 0 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 1 0 1 0 1 0 1 1 1 0 1 1 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 0 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 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 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 0 1 1 1 1 0 1 0 1 1 1 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 1 1 0 0 1 1 0 1 1 1 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 1 1 1 0 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 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 1 0 1 1 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 1 0 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 1 1 0 0 1 1 1 0 1 0 1 0 0 1 1 1 1 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