Bounds on the minimum distance of additive quantum codes
Bounds on [[80,56]]2
lower bound: | 6 |
upper bound: | 8 |
Construction
Construction of a [[80,56,6]] quantum code:
[1]: [[128, 105, 6]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[80, 57, 6]] quantum code over GF(2^2)
Shortening of [1] at { 5, 7, 8, 13, 14, 20, 21, 25, 31, 33, 35, 39, 41, 42, 47, 48, 49, 50, 52, 54, 60, 61, 69, 70, 71, 72, 73, 74, 81, 85, 88, 91, 95, 97, 102, 104, 107, 108, 112, 114, 115, 117, 120, 121, 123, 124, 125, 128 }
[3]: [[80, 56, 6]] 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 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 0 0 1 0 1 1 1 0 1 0 1 0 1 1 1 0 1 0 1 1 1 1 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 1 0 0 0 1|0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 1 1 1 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 0 0 1 1 1 1 1]
[0 1 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 1 1 1 1 0 0 1 0 0 1 1 0 1 0 1 1 1 1 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 1 1 0 0 1 0 0 0 0 1 1 0 1 1|0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 1 1 1 0 1 1 0 0 0 0 1 1 1 1 0 0 1 0 0 0 1 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 0 1 1 1 1 0 1 0]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 0 1 0 0 0 0 1 0 0 1 1 1 0 1 1 0 1 1 0 0 0 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1|0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 1 1 0 1 0 0 0 1 1 1 1 1 0 1 0 1 1 0 1 1 0 1 1 1 0 1 0 1 1 0 1 0 1 1 1 1 1 0 0 1 0 1 1 1 1 0 1 0 1 0 1 1 1 0 1 1 1 0 1 1 0 0 1 0 1]
[0 0 0 1 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 1 0 0 1 1 0 1 1 0 0 1 0 1 1 1 1 1 1 1 0 0 0 0 1 0 1 1 1 1 1 1 0 0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 1 0 0 0|0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 0 1 0 1 0 1 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 1 1 0 0 1 1 1 1 1 1 0 0 1 0]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 0 1 1 0 1 1 0 0 1 1 1 1 1 0 1 0 1 0 0 0 0 1 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1|0 0 0 0 0 0 1 0 1 1 0 0 1 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 1 1 1 1 1 0 1 1 1 0 0 1 0]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 1 0 0 1 1 1 0 0 1 0 1 1 0 1 1 1 0 0 1 1 1 1 0 0 1 0 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 1 1 0 0 1 0 0 0|0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 1 0 0 1 1 1 0 1 1 0 1 0 1 1 0 1 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 1 0 1 1 0 1 1 1 0 0 1 1 0 0 0 0 0 1 1 0 1]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 1 0 1 0 1 0 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 1 0 0 0 1 0 1|0 0 0 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 1 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 1 0 1 1 1 1 0 0 0 0 0 1 0 1 1 0 0 1 1 0 1 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1 1 1 1 1]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 1 1 0 1 1 1 1 1 0 0 0 1 0 1 0 0 1 1 0 0 1 0 0 1 1 1 0 1 1 1 0|0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1 1 1 1 0 0 1 0 1 1 1 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 0 1]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 1 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 0 0 1 1 1 0 0 1 1 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 1 1 1 0 0 0 1 0 1 0 1|0 0 0 0 0 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 0 1 0 0 1 0 1 0 0 1 0 0 0 1 1 0 0 0 0 1 0 1 0 1 1 0 0 0 0 0 1 0 1 0 1 1 0 1 0 0 0 1]
[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 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 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 1 1 0 0 0 0 1 0 1 1 0 1]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 0 1 0 0 0 0 0 1 0 1 0 0 1 1 0 1 0 0 1 1 0|0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 1 0 0 0 0 1 0 1 0 0 1 1 1 1 0 1 0 1 1 1 0 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 1 0 0 1 1 1 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 0 1 1 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 1 1 0 0 1 0 1 1 1 1 0 1 1|0 0 0 0 0 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 1 1 0 1 0 0 0 0 1 0 0 0 1 1 1 0 0 1 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 0 1 0 0 1 1 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 0 1 0 0 1 0 0 1 1 1 0 0 1 0 0 1 1 1 1 0 1 0 1 1 1 0 0 0 1 1 1 1 1 1 0 0 1 1 0 1 1 1 0 1 1 1 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0|0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 0 1 0 0 0 1 0 1 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 1 0 1 0 0 0 0 1 1 0 0 0 0 1 1 1 0 0 1 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 0 1 0|0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 1 1 1 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 1 0 0 1 0 1 0 1 1 0 0 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 1 0 1 0 1 1 1 0 1 0 1 1 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 0 1 1|0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 1 1 1 1 1 0 1 1 1 0 1 1 0 1 0 0 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 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 1 0 0 0 0 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 1 0 1 1 0 1 1 1 0 1 1 0 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 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 0 0 0 0 0 0 0 0 0 0|0 1 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 1 1 0 0 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 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 1 0 0 0 0 0 0 0 1 1 1 0 0 1 1 1 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 1 1 1 1 1 0 1 1 1 0 0 1 1 1 1 0 0 0 0 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 0 0 0 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 1 1 1 0 1 0 0 0 1 0 1 0 1 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 1 0 0 1 1 1 1 0 1 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1]
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[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 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