Bounds on the minimum distance of additive quantum codes
Bounds on [[96,81]]2
lower bound: | 4 |
upper bound: | 4 |
Construction
Construction of a [[96,81,4]] quantum code:
[1]: [[126, 114, 4]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[93, 81, 4]] quantum code over GF(2^2)
Shortening of [1] at { 2, 12, 13, 18, 20, 29, 47, 61, 80, 83, 94, 95, 97, 99, 100, 101, 103, 106, 107, 109, 110, 111, 113, 115, 116, 117, 118, 119, 121, 123, 124, 125, 126 }
[3]: [[96, 81, 4]] quantum code over GF(2^2)
ExtendCode [2] by 3
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 0 0 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 0 1 1 1 1 1 0 0 1 1 1 0 0 0 1 0 0 0|0 1 1 0 1 0 0 0 1 0 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 0 1 0 1 0 0 1 0 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 0 1 1 0 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 1 1 0 1 1 1 1 0 0 1 1 1 1 1 0 0 1 1 0 1 1 0 0 0]
[0 1 0 0 0 0 0 0 1 1 1 0 0 1 0 1 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 0 1 0 0 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 1 1 1 1 1 0 1 0 1 0 1 0 1 1 1 0 1 0 0 0|0 0 1 0 1 0 1 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0 1 1 1 1 1 0 0 0 1 0 1 1 1 1 1 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 1 1 0 0 0 1 0 1 0 1 0 1 1 1 0 1 1 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 0 1 0 0 0 1 1 0 0 0 0 0|0 0 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 1 1 1 1 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 1 0 0 0|0 1 0 1 1 0 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 1 1 0 1 1 1 1 0 1 0 1 1 1 1 0 1 1 1 0 0 0 0 0]
[0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 1 0 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 0 1 1 0 1 0 1 0 0 0 0|0 1 0 1 1 0 1 0 1 0 0 1 1 1 1 1 1 0 0 0 1 1 0 0 0 1 0 0 1 0 1 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 0 0 0 0 0 0]
[0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 0 1 0 0 1 0 1 1 0 0 0 1 1 0 1 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 1 1 1 0 0 1 0 1 1 0 0 0|0 1 1 1 0 0 1 0 1 1 0 1 0 1 1 1 1 0 0 0 0 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 1 0 0 0 0 1 0 1 1 0 0 0 0 1 1 1 0 1 0 1 0 1 0 1 1 1 1 0 0 1 1 1 0 0 0 1 1 0 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0|0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0]
[0 0 0 0 0 0 0 1 1 1 1 1 0 1 0 1 1 1 0 0 0 0 1 1 0 0 1 1 1 1 1 0 0 0 1 0 0 1 1 1 0 1 1 1 0 0 0 0 1 0 0 1 0 0 0 1 0 1 0 1 1 1 1 1 0 1 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 0|0 0 1 1 0 1 0 0 1 0 1 0 0 1 0 0 0 1 1 1 1 0 0 1 0 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 1 0 1 1 1 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 0 1 1 1 0 1 1 0 0 1 0 1 0 0 1 1 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0|0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 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 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 0 1 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 0 1 0 0 0 1 0 1 1 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 0 1 0 0 0 1 1 0 1 0 1 0 0 1 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 0 0 0|0 0 0 1 1 0 1 0 0 1 1 1 0 0 1 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 0 1 0 0 0 0 0 1 1 0 1 1 1 0 1 1 1 1 0 1 1 0 1 1 1 0 0 1 0 0 0 0 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 1 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0|0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 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 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 0 0 0 0 0 0 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 0 0 0 0 0 0 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 0 0 0 0 0 0 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 0 1 1 1 1 1 1 1 1 1 1 1 1 0 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 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 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