Bounds on the minimum distance of additive quantum codes
Bounds on [[49,30]]2
lower bound: | 5 |
upper bound: | 6 |
Construction
Construction of a [[49,30,5]] quantum code:
[1]: [[122, 104, 5]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[48, 30, 5]] quantum code over GF(2^2)
Shortening of [1] at { 4, 5, 6, 8, 11, 13, 14, 17, 20, 22, 23, 25, 26, 27, 30, 31, 32, 33, 34, 35, 38, 39, 40, 41, 42, 43, 44, 47, 49, 50, 51, 54, 55, 56, 57, 58, 59, 60, 62, 64, 65, 66, 67, 68, 69, 70, 73, 74, 75, 79, 80, 81, 83, 84, 85, 87, 91, 92, 93, 95, 99, 102, 103, 104, 105, 107, 111, 112, 114, 115, 117, 120, 121, 122 }
[3]: [[49, 30, 5]] quantum code over GF(2^2)
ExtendCode [2] by 1
stabilizer matrix:
[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 1 0 0 1 1 0 1 0 1 0 0 0 0 1 1 1 0 1 0 1 1 0 0|0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 1 1 0 0 1 0 0 1 0 1 0 0 0 1 1 1 1 1 1 0 1 1 0 1 0 0 0 0 0 0 1 0 1 0]
[0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 1 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0|1 0 0 1 1 1 1 0 1 1 0 0 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 1 0 1 1 0 1 0 0 1 1 0 0 1 1 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 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 1 0 1 1 0 0 1 1 1 0 0 0 0 0 1 0|0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 1 1 0 0 1 0 0 1 0 1 0 0 0]
[0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 1 0 1 0 1 1 1 0 1 0 1 1 0 1 0 0 0|1 1 1 0 1 0 1 1 0 1 0 1 0 0 0 1 0 1 0 1 0 1 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 1 0 1 1 1 0 1 0|1 1 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 0 1 1 1 1 0]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 1 0 1 0 0 1 1 1 0 0 0 1 1 0 1 1 1 0 1 1 0 0 0|0 1 1 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 0 0 1 0 1 0 0 0 0 0 1 0 1 1 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0]
[0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 1 0 0 1 0 1 0 1 0 1 1 1 1 0 1 1 1 1 1 0|0 1 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1 1 0 0 1 0 1 1 1 0 1 0 0 1 0 1 1 1 1 0]
[0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 0 0 1 0 0 0 1 1 1 0|0 0 1 1 1 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 1 0 0 0 1 1 1 1 0 1 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1 1 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 0 0 0 1 0 0 1 0 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0|1 1 0 0 1 0 0 0 0 0 1 1 1 1 1 0 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 0 1 1 1 0 0 0 1 1 1 1 1 1 0 1 1 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 0 0 0 1 0 1 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0|1 1 0 1 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0 1 0 1 1 0 0 0 0 1 0 0 1 0 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 1 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 0 0 1 0 0 0 1 1 1 0|0 1 0 1 0 1 1 0 1 1 0 1 1 1 0 1 0 1 1 0 0 0 1 0 1 0 1 0 0 1 1 1 1 1 0 1 1 0 1 1 0 1 1 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 0 1 0 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 1 0 0 1 0 1 0 1 0|0 0 0 0 1 1 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 1 0 1 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 1 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 1 0 1 1 0|0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 1 1 0 0 1 1 1 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 0 1 0 1 0 0|1 1 1 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 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 1 0 0 1 0 0 0 1 0 1 1 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0|0 0 1 1 0 0 0 1 0 0 1 0 1 1 1 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 1 1 0 0 0 0 1 1 1 0 0 1 1 0 0 0 1 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 1 0 1 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 0|0 1 0 0 0 1 1 1 1 1 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 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 0 1 1 1 1 1 0 1 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 0|1 1 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 1 1 0 0 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 0 0 0 0 1 0 1 1 0 1 1 1 1 0 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 0 0|0 0 0 1 1 1 0 0 1 1 1 1 1 1 0 0 1 0 0 0 0 0 1 0 1 1 1 1 1 0 1 1 0 0 1 0 1 0 0 0 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 0 0 0 0 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]
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