Bounds on the minimum distance of additive quantum codes
Bounds on [[70,51]]2
lower bound: | 5 |
upper bound: | 6 |
Construction
Construction of a [[70,51,5]] quantum code:
[1]: [[122, 104, 5]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[70, 52, 5]] quantum code over GF(2^2)
Shortening of [1] at { 3, 5, 6, 7, 10, 11, 12, 17, 23, 26, 27, 28, 29, 32, 35, 37, 38, 41, 43, 44, 45, 49, 53, 56, 57, 58, 60, 63, 65, 70, 72, 74, 75, 77, 80, 89, 90, 91, 92, 93, 96, 98, 102, 105, 106, 108, 109, 111, 115, 119, 120, 121 }
[3]: [[70, 51, 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 1 0 0 0 1 1 1 0 1 0 0 1 0 1 0 0 0 0 0 1 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 1 1 1 1 1 0 1 0 0 1 1 0 0 0 0 0|0 1 0 1 0 0 0 1 1 1 1 1 0 1 0 0 0 1 1 1 0 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 1 0 0 1 1 1 1 1 0 0 1 0 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 1 0 1 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 1 0 1 1 0 1|1 0 1 0 1 1 0 0 1 0 1 0 1 1 1 1 1 1 0 1 0 0 0 1 0 1 0 1 1 1 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 1 0 0 1 0 1 0 1]
[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 0 0 0 1 0 0 0 1 0 1 0 1 1 1 1 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 1 1 1 0 1 0 0 0 1 1|0 1 0 0 0 0 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 1 1 0 0 0 0 1 0 1 1 1 1 1 0 1 0 1 0 1 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 1 1 0 0 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0|0 1 1 0 1 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0|0 1 0 0 1 1 0 0 1 1 1 1 0 0 0 1 1 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 1 0 1 0 0 1 0 1 1 0 0 0 1 0 1 0 0 1 1 1 0 1 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 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 0 1 1|0 0 1 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 1 0 1 0 0 1 1 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 0 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 1 1 0 0 1 1 0 1 0 0 0 1 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 0 0 0 0 1 0 1 0|0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 0 1 0 0 1 0 1 1 0 1 0 1 1 1 1 1 1 1 1 0 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 1 1 1 1 0 1 1]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1|0 0 1 0 1 1 1 1 1 1 0 0 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 0 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 0 0 1 1 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 1 1 0 0 1 0 0 1 0]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 1 1 0 0 0 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1|0 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 0 1 1 0 1 1 0 0 1 1 1 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 1]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 0 1 0|1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 1 0 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0 1 1 1 1 1 0 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 1 1 1 1 1 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 1 0 1 0 0 1 1 0 1 1 0 1|0 0 1 0 1 0 1 0 1 1 0 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 1 0 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1|0 1 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 0 1 0 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 0 1 1 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 0 0 0 1 1 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 1 1 1 0 0 0 0 0 1 0 1 1 1 0 1 1 1 1 0 0 0 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 0 0 0|0 0 1 1 0 0 0 1 1 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 1 1 1 0 0 1 1 1 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 1 1 1 0 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 1 0 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 0 1 0 0 0 1 1 0 0 1 1 1 1 1 0|1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 1 1 0 0 1 0 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 1 0 0 0 0 0 1 0 1 1 0 1 0 1 1]
[0 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 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 1 1 1 0 1 0 1 0 1 1 0 1 1 1 0 0 1 1 1 0|1 0 1 1 1 0 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 0 1 0 1 1 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 1 1 0 1 1 1 0 0 0 0 1 1 0 0 1 0 0 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 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 0 1 1 1 0 0 1 0 1 0 0 0 0 0 0 0 1|1 0 1 0 1 0 1 0 1 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 1 1 0 0 1 1 1 1 0 1 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 1 1 0 1 0 0 1 1 0 0 1 1 1 0 0 1 0 0 0 1 0 1 1 1 1 0 0 1 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0|1 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 0 1 1 0 0 1 0 1 0 0 1 1 1 0 0 1 1 0 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 1 0 1 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 0 1 1 0 1 1 0 0 0 1 0 0 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 1 1 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 1 1 1 0|1 1 1 1 1 1 1 1 0 1 0 0 0 1 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 0 1 0 1 1 1 1 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