Bounds on the minimum distance of additive quantum codes
Bounds on [[60,36]]2
lower bound: | 6 |
upper bound: | 8 |
Construction
Construction of a [[60,36,6]] quantum code:
[1]: [[128, 105, 6]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[60, 37, 6]] quantum code over GF(2^2)
Shortening of [1] at { 2, 3, 4, 5, 7, 8, 9, 10, 14, 18, 19, 23, 24, 25, 26, 27, 31, 33, 34, 35, 36, 40, 42, 43, 45, 47, 49, 50, 52, 53, 54, 55, 57, 59, 62, 64, 65, 66, 67, 68, 69, 71, 75, 78, 80, 81, 84, 85, 87, 88, 89, 90, 93, 95, 96, 100, 101, 102, 105, 108, 110, 115, 116, 118, 121, 124, 126, 128 }
[3]: [[60, 36, 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 0 0 1 0 0 0 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 1 1 1 0 0 1 0 0 0 0 1 1 1|0 0 0 0 0 0 0 0 1 1 0 1 0 1 1 1 1 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 0 0 1 0 0 1 1 1 1 1 1 1 1 0 0 1 0 1 0 0 0 1]
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 1 1 0 1 1 1 1 0 1 1 1 1 0 0 1 0 0 1 1 0 0 1 1 1 1 1 1 1 0 0 0 1 0 1 0|0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 0 1 0 0 0 1 1 1 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 0 0 0 0 1 0 1 1 0]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 1 0 1 1 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 1 0 1 1 1 0 0 1 0 1 1 1|0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 1 1 1 1 0 0 1 0 0 1 0 0]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 1 1 1 1 0|0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 1 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 0 0 1 0 0 0 0 1]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 1 0 1 0 1 1 0 1 0 1 0|0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0 0 0 0 1 1 0 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 1 0 1 1 0 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 1 1 1 0 0 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 1 1 1 1 1 1|0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 1 0 0 0 1 1 1 1 0 1 0 0 0 0 1 1 1 1 0 1 1 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 0 1 1 0 0 1 1]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 1 1 0 0 1 0 1 0 0 0 1 1 1 1 0 1 0 0 0 1 1 1 0 0 0|0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 0 1 0 1 1 0 0 1 1 1 0 0 1 0 0 0 1 0 0 0 0 1]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 0 0 1 0 0 1 1 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 0 1 1 1|0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 0 1 0 0 0 0 1 0 1 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 1 1 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 0 1 0 0 1 1 1 0 1 0 0 1 0 0|0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 0 1 0 0 0 1 1 1 1 1 0 1 0 0 0 0 0 1]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 0 0 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 1 0 0 1 0 1 0 0 1 0|0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 0 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 1 1]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 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 0 0 1 1 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 1 0 1 1 0 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 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 1 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 1 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0|0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 1 1 1 1 0 0 1 1 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 1 0 0 0 1 0 1 0 1 1 0 1 1 1 0 0 1 0|0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 1 1 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 1 0 0 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 1 1 0 0 1 1 1 0 0 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 1 0 0 1 1 0 0 1 0|0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 1 1 0 1 0 0 0 1 1 1 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 1 1 0 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 1 0|0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 1 0 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 1 1 1 0 1 1 0 1 0 0 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|1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 1 0 1 0 0 1 0 1 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 1 0 0 0 0 0 0 1 1 1 1 1 0 1 1 0 1 1 0 1 0 1 1 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 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 1 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 0 0 1 0 1 0 0 1 1 1 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1 1 1 0 0 0 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 1 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 0 0 1 0 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1 0 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 1 0 0 0 0 1 0 1 0 0 0 1 1 0 1 0 0 0 1 1 0 0 1 1 1 0 1 1 1 1 1 1 0 1 0 0 1 1 0 0 1 1 1 1 0 1 1 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 0 1 0 0 1 0 0 1 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 1 0 1 1 1 0 0 1 1 0 0 1 0 1 0 1 0 0 0 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 1 0 1 1 1 0 0 1 1 1 1 0 0 0 1 1 1 1 1 0 0 0 0 1 0 1 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1 0 0 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 1 1 1 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 1 0 0 0 0 1 1 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 1 0 0 0 0 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