Bounds on the minimum distance of additive quantum codes
Bounds on [[60,35]]2
lower bound: | 6 |
upper bound: | 8 |
Construction
Construction of a [[60,35,6]] quantum code:
[1]: [[128, 105, 6]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[58, 35, 6]] quantum code over GF(2^2)
Shortening of [1] at { 3, 4, 5, 6, 11, 12, 16, 17, 18, 19, 20, 23, 28, 31, 32, 35, 36, 39, 40, 41, 44, 46, 47, 48, 49, 50, 52, 54, 55, 58, 59, 60, 62, 64, 66, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 81, 83, 85, 87, 88, 90, 91, 96, 100, 101, 104, 105, 106, 111, 113, 114, 115, 117, 118, 120, 122, 123, 124, 127, 128 }
[3]: [[60, 35, 6]] quantum code over GF(2^2)
ExtendCode [2] by 2
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 0 1 0 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0|0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 0 1 1 1 1 1 0 0 1 1 1 1 0 1 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 1 1 1 1 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 1 0 1 0 1 1 1 0 0|0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1 0 0 1 0 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 1 1 0 0 0 1 1 0 1 0 1 0 0 0 0|0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 1 1 1 1 1 0 0 1 1 0 1 1 1 1 1 0 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 1 0 1 1 1 0 0 1 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 1 0 1 1 1 0 1 1 0 0 1 0 1 1 0 1 0 1 1 0 0 0 0|0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 1 1 1 1 1 1 1 0 0 0 0 1 0 0 1 1 0 1 0 1 1 1 0 1 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 1 1 0 1 0 0 1 0 0 0 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 1 0 0|0 0 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 0 1 0 1 1 1 0 0]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 1 1 0 0|0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 1 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 1 1 0 1 1 1 0 1 0 1 1 1 1 1 0 0 1 0 1 0 1 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 0 0 0|0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 1 0 1 0 1 1 1 0 0 1 0 1 0 0 1 0 0 1 1 1 1 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 1 1 0 1 0 0]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 1 1 0 0 1 1 1 1 1 1 0 0 1 0 1 0 0 1 0 1 1 0 1 1 1 1 1 1 1 0 0|0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 0 0 1 1 1 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 0 0 1 0 0 1 1 1 0 0]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 1 1 0 0 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 1 0 0 0|0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 0 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 0 0 1 0 0 1 1 1 0 1 1 0 1 0 1 1 0 1 0 1 1 1 0 0]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 0 0 0 1 1 0 0 1 0 0 1 0 1 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 1 0 0 0 0 0|0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 0 1 0 0 1 0 1 1 1 0 1 1 1 1 1 0 1 1 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 0 1 1 1 1 1 0 0 1 1 1 1 0 1 0 0 0 0|0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 1 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 0 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 0 1 1 0 0|0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 1 1 0 1 1 1 1 1 1 1 1 0 0 1 0 1 1 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0|0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 1 1 0 1 0 0 1 0 0 0 1 1 0 1 1 0 0 0 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 0 1 0 1 0 0|0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 1 1 1 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 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 0 0 1 1 0 1 0 1 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0 1 0 0 1 1 1 0 0|0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 1 0 1 0 1 1 1 0 0 1 0 1 0 0 1 0 0 1 1 1 1 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 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 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 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|1 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 1 0 0 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 1 0 0 0 0 0 1 1 1 1 0 0 1 0 0 1 0 0 1 0 1 0 0 1 1 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 1 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 1 0 0 0 0 1 0 1 0 0 1 1 0 1 1 1 1 0 1 0 0 0 1 0 1 0 1 1 1 1 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 1 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 1 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 1 1 1 1 1 1 0 1 1 0 1 0 1 0 0 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 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 1 0 0 0 1 1 0 0 1 0 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 1 1 0 1 0 0 1 1 1 0 1 1 0 1 0 1 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 1 0 0 0 0 1 0 0 1 0 1 0 1 1 0 1 1 1 1 0 1 0 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 1 1 1 1 1 1 0 1 0 1 1 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 0 0|0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 0 0 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 1 1 0 1 1 0 1 0 0 1 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 0|0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 0 1 0 1 0 0 0 1 1 0 0 1 1 1 0 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