Bounds on the minimum distance of additive quantum codes
Bounds on [[78,57]]2
lower bound: | 5 |
upper bound: | 6 |
Construction
Construction of a [[78,57,5]] quantum code:
[1]: [[128, 105, 6]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[127, 106, 5]] quantum code over GF(2^2)
Shortening of the stabilizer code of [1] at 128
[3]: [[78, 57, 5]] quantum code over GF(2^2)
Shortening of [2] at { 3, 4, 5, 6, 8, 9, 11, 17, 18, 23, 24, 25, 29, 31, 33, 34, 35, 40, 42, 49, 53, 54, 61, 64, 66, 68, 71, 72, 74, 76, 85, 87, 88, 89, 92, 95, 96, 97, 98, 100, 103, 105, 107, 108, 109, 112, 113, 118, 125 }
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 1 0 1 1 0 1 0 0 0 0 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 0 1 1 0 1 0 0|0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 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 1 0 0 1 0 0 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 1 0 1 1 1 1 0 0 0 0 0 0 1 1 0 1 1 1 0 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1|0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 1 1 1 0 0 0 1 0]
[0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 1 1 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 1 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 1 1 1|0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 1 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 0 1 0 0 1 1 0 0 1 0 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 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0 0 1 1 1 0 1 0 0 0 1 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 0 1|0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 1 1 0 0 1 1 1 0 1 1 1 1 0 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 0 1 0 0]
[0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 1 0 1 1 1 1 0 0 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 0 1 1 1 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0|0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 0 1 0 1 0 0 1 1 1 1 0 1 1 0 1 1 0 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 0 0 0 1 0 0 0 1 1 1 1 1 0 1 1 0 1 1 1 0 0 0 0 1 1|0 0 0 0 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 1 1 0 1 1 1 0 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 1 1 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 1 1]
[0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 1 1 0 0 1 0 0 0 1 0 0 1 1 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0|0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 1 0 1 1 0 1 1 0 0 0 0 1 1 1 0 0 0 0 1 0 1 1 0 0 0 1 0 1 0 0 1]
[0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 0 0 0 0 0 0 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 1 1 0 0 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 0 0 0 0 0 1 0 1 1|0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 1 1 1 0 0 1 0 1 1 0 0 1 0 1 1 0 1 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 1 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 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 1 1 0 1 1 0 1 1 0 0 0 1 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0|0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 0 1 0 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 0 1 0 1 0 0 1 1 0 0]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 0 0 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 1 1 1 1 1 1 1 0 1 0 1 0 1 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 0 1 0 1 0 0 1 1 1 1 0 1 1 0 1 1 0 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 1 1 0 1 0 1 1 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1 0 1 1 1 1 0 0 0|0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 1 1 1 0 0 1 1 0 1 1 0 0 0 1 1 1 0 0 0 0 0 1 1 1 0 1 1 0 0 1 0 1 0 0 1 1 1 0 1 0 1 0 0 1 0 1 0 1]
[0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 1 0 0 1 1 0 1 1 0 1 0 1 0 1 1 1 0 1 1 1 1 1 0 0 1 1 0 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 0 1 1 1 1 1 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 0 0 1 0 1 0 1 0 0 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 1 1 1 1 1 1 0 0 1 1 0 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 0 1 0 1 1 1 0 0 0 1 0 0 1 0 1 1 0 0 0 0 1 1 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1|0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 1 1 1 1 1 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1 1 1 1 0 1 1 1 1 1 0 0 0 1 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 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 1 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1|0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 0 1 0 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 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 0 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 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 1 1 0 1 1 1 0 0 1 1 0 0 1 1 1 0 1 0 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|0 1 0 0 0 0 1 0 1 0 1 1 1 1 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 0 0 0 1 0 1 1 1 1 0 1 0 1 0 0 0 1 1 1 0 1 1 0 1 0 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 0 0 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 1 0 1 0 1 1 0 0 0 1 1 1 0 1 0 1 1 1 1 1 1 1 1 0 0 1 0 0 1 1 1 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 1 0 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 1 0 0 1 0 1 0 0 1 1 0 0 1 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 1 0 1 1 0 0 1 1 0 1 1 1 1 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 0 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 0 0 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 0 1 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 0 1 0 0 1 1 1 1 1 0 0 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 0 0 0 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 1 0 1 0 0 1 0 1 0 1 1 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 1 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 0 0 0 0|0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 1 0 1 0 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 1 0 0 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 0 0 1 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