Bounds on the minimum distance of additive quantum codes
Bounds on [[81,68]]2
lower bound: | 4 |
upper bound: | 4 |
Construction
Construction of a [[81,68,4]] quantum code:
[1]: [[126, 114, 4]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[80, 68, 4]] quantum code over GF(2^2)
Shortening of [1] at { 1, 2, 3, 4, 8, 10, 11, 14, 16, 19, 20, 23, 24, 25, 30, 31, 33, 42, 44, 47, 49, 51, 52, 55, 57, 58, 61, 63, 68, 74, 80, 84, 85, 87, 92, 98, 99, 109, 113, 114, 116, 117, 118, 119, 120, 124 }
[3]: [[81, 68, 4]] quantum code over GF(2^2)
ExtendCode [2] by 1
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 1 1 1 1 0 0 1 1 1 1 1 1 0 0 1 0 0 0 0 1 1 0 0 0 0 0|0 0 0 1 0 0 0 0 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 0 0 0 1 0 0 1 1 0 0]
[0 1 0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 1 1 0 0 0 1 1 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 0 0|0 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 1 0 0 0 0 1 1 0 1 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 0 1 0 1 0 1 0 0 1 1 1 0 1 1 1 1 1 1 0]
[0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 1 0 1 1 1 1 0 0 1 1 1 1 1 0 0 0 0 1 0 1 0 1 1 0 0 0 0 0 0|0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 1 0 1 0 1 1 1 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0]
[0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 1 0 1 0 0 1 1 0 0|0 0 1 1 1 1 1 1 1 0 1 1 0 0 1 0 1 1 1 1 1 0 1 1 1 1 1 0 0 1 0 1 0 1 1 1 1 1 1 1 1 0 1 0 0 0 1 0 0 1 1 1 0 0 0 1 0 1 0 1 1 0 0 1 0 0 1 1 0 0 1 1 1 0 0 0 0 0 0 1 0]
[0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 1 0 1 1 1 1 0 1 0 0 0 1 1 0 0 1 1 1 0 1 1 1 1 1 0 0 0 1 0 0 0 1 0 1 0 0 1 1 0 0|0 0 1 1 1 1 1 0 1 0 1 0 0 0 1 1 0 1 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 1 1 0 1 0 0 1 0 0 0 0 1 0 1 0 0 1 1 1 0 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 1 0 0 1 0 1 0]
[0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 1 1 1 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 0 0|0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 1 1 0 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 1 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 0 0 1 0 0 0 1 1 1 1 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 0 0|0 1 1 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 0 1 0 1 1 1 1 0 0 0 0 1 0 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 1 1 0 0 1 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 0 1 0 0 0 0]
[0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 0 0 1 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 0 1 0|0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 1 1 0]
[0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 1 1 1 0 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 0 1 1 0 1 0 0 1 0 1 0 0 0 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 0 0 0 1 0 1 0 1 1 1 0 1 0|0 1 0 1 0 1 0 1 1 1 0 1 1 1 0 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 1 1 1 0 0 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 1 0 0 1 0 1 1 1 0]
[0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 1 0 0 1 0 1 0 0 1 1 0 1 0 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 1 0 0 0|0 0 1 0 0 1 0 1 0 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 1 0 1 1 1 1 0 1 0 1 1 1 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 1 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0|0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 1 1 1 1 1 1 0 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 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 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 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 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