Bounds on the minimum distance of additive quantum codes
Bounds on [[77,54]]2
lower bound: | 5 |
upper bound: | 7 |
Construction
Construction of a [[77,54,5]] quantum code:
[1]: [[128, 105, 6]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[78, 55, 6]] quantum code over GF(2^2)
Shortening of [1] at { 1, 2, 4, 5, 7, 8, 15, 16, 18, 19, 21, 22, 23, 34, 35, 36, 37, 40, 43, 48, 49, 51, 53, 58, 60, 66, 71, 73, 74, 75, 78, 82, 83, 86, 87, 90, 92, 95, 97, 99, 102, 103, 104, 109, 111, 115, 116, 119, 121, 124 }
[3]: [[78, 54, 6]] quantum code over GF(2^2)
Subcode of [2]
[4]: [[77, 54, 5]] quantum code over GF(2^2)
Puncturing of [3] at { 78 }
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 0 1 1 0 1 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 1 0 0 0 0 1 1 0|0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 1 1 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1]
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 0 0 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 0 1 1 0 1 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 0 0 1 0|0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 0 1 1 0 0 1 0]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 0 1 1 1 0 0 1 0 0 0 1 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 1 1 1 0 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 1 0 1 0|0 0 0 0 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 0 1 0 1 0 1 1 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 0 1 0 0 1 0 1 0 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 0 0 0 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 1 0 0 0 1 1 0 1 0 0 1 1 0 0 0 1 1 0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0|0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 0 1 0 1 1 1]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 1 0 1 1 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1 0 0 1 1 1 0 0 0 0 0 0 1 1 0|0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 0 1 0 0 0 1 1 1 0 0 0 1 0 1 0 0 0 0 0 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 1 1 1]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 0 0 0 1 0 0 0 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 1 1 0 0 0 1|0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 1 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 0 1|0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 0 0 1 1 0 0 1 0 1 0 1 0 1 1 1 1]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 0 1 0 0 1 0 1 1 0 0 1 0 0 1 1 1 0 1|0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 0 1 0 1 0 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 0 1 0 1 0 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 1 0 0 0 1 1 0 1 1 0 0 0 1 0 1 0 1 1 0 1 1]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 1 0 0 1 1 1 1 0 0 1 1 1 0 1 1 1 0 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0 1 1 0 0 1 0|0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 1 0 0 0 1 1 0 1 0 1 0 1 0 1 1 0 1 1 0 0 1 0 0 1 1 0 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 0 0 0 1 0 0 0 1]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 1 1 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0|0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 0 0 1 1 1 1 0 1 0 0 1 0 0 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 1 0 0 0 1 0 1 0 0 0|0 0 0 0 0 1 0 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 0 1 0 0 1 1 1 1 0 0 0 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 1 1 1 0 1 0 1 0 0 1 1 0 1 0 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 1 1 1 1 0 1 0 0 1 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 1 1 1 0 1 1 0 1 0 0 0 1 1 0 1 0 1 0 0 1 1 0 0 1 0 1 0 1 1 1 1 0 1|0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 0 0 0 0 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 0 0 0 1 0 1 1 0 0|0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 0 0 0 0 1 1 1 1 1 0 1 1 0 0 0 1 1 1 0 1 0 0 0 1 1 1 1 0 0 0 0 1 1 0 1 0 0 0 0 0]
[0 0 0 0 0 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 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 1 1 1 0 1 0 0 1 1 1 1 0 1 0 0 1 0 0 1 0 1 1 0 1 1 0|0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 0 0 0 1 1 0 0 0 0 1 1 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 1 0 1 1 1 0 1 0 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 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 1 0 0 1 0 1 1 1 1 0 1 1 1 0 0 0 0 1 0 1 1 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 1 1 0 0|0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 1 0 1 0 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 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 1 1 0 1 1 1 1 1|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 1 1 1 0 1 1 0 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 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 0 0 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 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 0 0|0 1 0 0 0 1 0 0 0 0 1 0 1 1 0 1 1 1 1 0 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 1 0 1 1 1 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 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 1 0 0 1 1 0 1 0 1 1 1 1 0 0 0 0 1 1 1 0 0 1 0 1 1 0 1 0 1 1 0 1 1 1 0 0 1 0 0 0 1 1 0 1 1 0 1 1 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 1 0 1 0 0 1 0 0 0 1 0 1 1 0 0 1 1 1 0 1 1 1 1 0 1 0 1 1 0 0 0 0 1 0 0 1 1 1 1 0 1 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0 1 1 1 1 1 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 1 0 0 1 0 1 1 1 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 0 1 0 1 0 0 1 0 1 1 1 0 1 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 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 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 0 0 1 0 1 1 0 0 1 1 0 1 1 1 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 0 0 0 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 0|0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 0 1 1 0 1 1 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 0 0 0 0 1 0 1 0 0 1 1 0 1 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 1 1 1 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