Bounds on the minimum distance of additive quantum codes
Bounds on [[76,64]]2
lower bound: | 4 |
upper bound: | 4 |
Construction
Construction of a [[76,64,4]] quantum code:
[1]: [[126, 114, 4]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[76, 64, 4]] quantum code over GF(2^2)
Shortening of [1] at { 2, 4, 5, 7, 11, 14, 19, 20, 27, 32, 34, 35, 36, 41, 42, 43, 47, 52, 54, 55, 56, 59, 62, 63, 68, 72, 75, 76, 81, 84, 85, 86, 90, 92, 102, 105, 106, 107, 109, 110, 111, 113, 114, 117, 118, 120, 122, 123, 124, 126 }
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 1 1 1 0 0 1 0 1 1 0 1 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0 1 1 0 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 1 0|0 1 0 0 1 1 0 0 1 0 1 0 0 0 0 1 0 1 1 0 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 1 1 1 0 0 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0]
[0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 0 1 0 0 1 1 0 0 0 1 0 1 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 1 1 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 0 1|0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 0 1 0 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 1 1 1 0 1 0 1 1 0 1 1 1 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 1 1 0 1 1 1 0]
[0 0 1 0 0 0 0 0 1 0 1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 1 1 1 0 0 1 1 0 0 0 1 0 1 1 1 0 1 1 1 0 1 0 1 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0|0 0 1 0 0 0 0 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 0 1 1 0 0 1 1 0 0 0 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 0 1 1 0 1 1 1 0 1 0 1 0 0 0 1 1 0 1 0 1]
[0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 0 1 1 1 0 0 1 0 1 0 0 1 0 0 1 1 0 0 1 0 1 1 0 0 0 0 0 1 0 0 1 0 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 0 0 1 1 0|0 1 0 0 0 1 1 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 1 1 1 0 0 1 1 1 1 1 0 0 1 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1]
[0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 0 1 0 1 0 0 1 0 1 0 0 1 1 1 0 0 1 1 1 1 1 1 1 0 0 1|0 1 0 1 0 0 0 1 1 0 0 1 1 1 1 1 1 1 0 1 1 1 0 1 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 1 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 0 0]
[0 0 0 0 0 1 0 0 0 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 0 1 1 1 1 1 1 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 1 0 1 0 1 1 0 0 0 1 1 0 0 1 1 0 0 1 1 1 0|0 1 0 0 1 1 0 0 1 1 1 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 0 1 1 1 0 0 0 0 0 1 0 1 1 0 0 1 0 1 1 0 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 0]
[0 0 0 0 0 0 1 0 1 0 1 1 0 0 1 0 0 1 1 0 0 0 1 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 0 0 0 0 1 0 1 0 0 1 1 1 0 0 0|0 0 1 0 1 0 0 1 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 1 1 1 0 0 1 0 1 1 0 0 0 0 1 1 1 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0]
[0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 1 1 0 0 1 1 0 1 1 0 1 1 0 0 1 1 1 0 0 1 0 0 1 0 0 1 1 0 1 0 0 1 0 1 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 1 0 0 1|0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 1 0 0 0 1 0 1 0 1 0 0 0 1 0 0 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 1 0 0 1 1 1 0 1 1 0 0 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 0 0 1 0 1 1 1 1 1 0 0 1 0 0 1 0|0 0 0 0 1 1 0 0 0 0 0 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 0 0 0 1 1 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 1 1 1 0 0 1 0 1 1 0 1 1 0 1 1 1 1 1 1 0 0|0 1 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 0 1 1 1 1 1 0 1 1 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0|0 0 0 1 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 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 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0|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 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 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 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