Bounds on the minimum distance of additive quantum codes
Bounds on [[60,41]]2
lower bound: | 5 |
upper bound: | 6 |
Construction
Construction of a [[60,41,5]] quantum code:
[1]: [[122, 104, 5]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[60, 42, 5]] quantum code over GF(2^2)
Shortening of [1] at { 7, 11, 13, 15, 17, 18, 19, 20, 21, 22, 24, 27, 29, 30, 33, 34, 38, 40, 41, 46, 47, 48, 50, 51, 52, 54, 56, 58, 59, 60, 62, 63, 64, 67, 68, 69, 70, 72, 76, 78, 79, 80, 81, 82, 87, 88, 89, 90, 91, 92, 93, 97, 102, 104, 105, 106, 107, 114, 115, 116, 117, 120 }
[3]: [[60, 41, 5]] 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 1 1 1 0 1 1 0 1 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 0 1 0 0 1 0|0 1 1 0 0 1 1 1 0 1 1 1 1 1 0 1 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 0 0 0 0 1 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 1 0 1 1 0 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 1 1 0 1 1 1 0 0 1 0 0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 0 1 1 1 1 1 0 1 1 0 0 0 1|0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 1 0 1 0 1 0 0 1 1 1 0 1 0 0 1 1 0 0 0 0 1 1 1 1 0 1 1 1 0 0 1 1 1 0 1 0 0 0 0 0 1 1]
[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 0 0 1 1 1 0 0 0 0 1 1 0 0 1 1 1 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0 0 1|1 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 1 1 0 1 1 0 1 1 0 1 1 1 0 1 1 0 1 0 1 0 0 0 1 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 1 0 0 0 0 1 1 0 0 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 0 1 1 0 0 1 0 1 0 0 1 0 1 1 1 1 0 0|0 0 1 1 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 1 1 0 0 1 0 1 1 1 1 1 0 1 0 1 0 1 1 0 1 0 0 0 0 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 1 0 1 0 1 1 1 1 0 1 0 1 0 0 0 0 1 0 0 1 1 0 1 1 0 0 1 0|0 0 0 0 1 1 1 1 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 1 0 1 0 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 0 1 0 1 0 1 1 1 1 0 1 1 0 0 1 1 0 0 0|1 1 0 1 1 0 1 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 1 0 1 0 0 0 0 0 1 1 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 0 0 0 0 1 1 0 1 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 1 0 0 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 1 1 1 1 1 0|1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 1 1 0 0 0 0 1 1 1 1 1 0 0]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 0 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 0 1 0 1 1 0 1 0 0 0|1 1 0 0 0 1 0 1 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1 0 0 1 1 0 0 1 0 0 1 0 1 1 1 0 0 1 1 1 1 0 0 1 1 1 0 0 1 0 0 1 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 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 1 0 1 1 0 1 0 0 1 0 1 1 1 0 1|0 0 1 1 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 0 0 1 0 1 0 1 0 1 1 0 1 1]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1 0 0 0|1 0 1 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 1 1 0 0 1 0 1 1 1 0 1 0 1 0 1 1 1 1 1 1 1 0 0 1 1 1 0 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 1 0 0 1 1 1 0 1 1 1 0 0 1 0 0 0 1 0 1 0 1 1 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0|0 0 1 1 1 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 1 0 1 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0 1 1 1 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 1 1 1 0 1 1 1 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 1 1 0 0 0|0 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 1 0 1 0 1 1 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 1 0 0 1 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0|0 0 0 1 0 1 0 1 1 1 0 0 0 1 1 1 0 1 1 0 0 0 1 1 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 0 0 0 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 1 0 1 0 0 1 1 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 1|1 0 0 1 1 1 0 1 1 1 0 1 1 1 1 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 0 1 1 1 0 0 0 0 1 0 0 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 1 0 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 1 1 0 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1|1 0 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 1 0 1 1 1 1 1 0 1 1 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 0 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 1 1 1 1 0 0 1 0 1 0 1 1 1 0 0 1 1 0 0 0 1 0 1 1 0 0 1 1 0 0 0 0 1 0 0 1|1 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 1 0 0 0 1 0 0 1 1 1 0 0 0 1 1 1 1 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 0 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 1 1 0 1 1 1 1 0 1 1 0 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 0|1 1 1 1 0 0 0 0 0 1 1 1 0 1 0 0 1 1 0 1 0 1 1 1 1 0 0 1 1 1 1 0 1 1 0 0 0 0 1 1 1 1 1 1 0 1 0 0 0 1 0 0 1 0 0 1 1 0 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 1 0 1 1 1 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 0|0 1 1 1 0 0 0 0 1 0 1 0 0 1 1 1 0 0 0 1 0 1 1 0 1 1 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 1 1 0 1 1 1]
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