Bounds on the minimum distance of additive quantum codes
Bounds on [[75,62]]2
lower bound: | 4 |
upper bound: | 4 |
Construction
Construction of a [[75,62,4]] quantum code:
[1]: [[126, 114, 4]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[74, 62, 4]] quantum code over GF(2^2)
Shortening of [1] at { 1, 2, 3, 5, 7, 9, 10, 12, 13, 14, 16, 17, 18, 23, 24, 30, 33, 37, 38, 41, 44, 47, 48, 50, 51, 52, 54, 64, 67, 68, 69, 71, 73, 74, 76, 86, 89, 91, 93, 94, 95, 98, 99, 101, 104, 105, 106, 108, 110, 112, 118, 124 }
[3]: [[75, 62, 4]] quantum code over GF(2^2)
ExtendCode [2] by 1
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 0 0 1 1 1 1 0 0 1 0 0 0 0 1 1 0 0 1 1 0 1 0 1 0 0 1 0 0 1 1 0 0 1 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 0 1 0 0|0 1 0 0 0 1 0 0 1 0 0 1 0 1 1 1 0 1 0 0 0 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 1 1 0 1 0 1 1 0 0 1 0 0]
[0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 1 0 1 1 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 1 1 0 0 1 0 0 0|0 0 1 0 1 1 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 1 1 0 1 0 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 1 1 1 0 1 0 0 1 0 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 1 1 1 0 0 1 0|0 1 0 1 1 1 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 0 0 0 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 1 1 0 1 1 1 1 1 0 0 1 1 1 1 0 1 1 0 1 1 1 0 1 0 0 1 0 1 1 1 1 1 0|0 0 0 1 1 1 0 1 0 1 1 1 0 1 1 1 1 1 1 0 0 1 0 0 0 1 0 0 0 1 1 1 1 1 1 0 0 1 0 0 0 0 1 1 1 0 0 0 1 1 1 0 1 1 0 0 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0 1 1 0]
[0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 1 1 0 1 1 1 0 1 1 0 1 0 0 0 1 1 0 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 0 1 1 1 1 0 0 0 1 1 0 1 1 0 1 0 1 1 0 1 1 1 1 0 0|0 1 0 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 0 0 1 0 0 1 1 0 1 1 1 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0]
[0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0|0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 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 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 1 1 0 0 1 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 1 0 1 1 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 1 0 1 0 1 1 0 1 0|0 1 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 0 1 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 0 0 1 1 1 0 1 0 0 0]
[0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 1 1 0 1 1 1 0|0 0 0 1 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 1 0 1 0 1 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 1 1 0 1 0]
[0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 1 0 0 0 1 0 1 0 1 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 0|0 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 0 1 0 1 1 0 1 1 0 0 0 1 1 0 0]
[0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 1 0 1 0 0 0 1 0 1 0 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 1 0 1 1 1 0 1 1 1 0 1 0|0 0 1 0 1 0 1 0 1 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1 0 1 1 0 0 1 1 0 1 0 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 1 0 0 1 1 0 0 1 1 1 1 1 1 0 1 0 1 0 1 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 1 1 0 1 1 0 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 0|0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 1 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 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