Bounds on the minimum distance of additive quantum codes
Bounds on [[48,33]]2
lower bound: | 4 |
upper bound: | 5 |
Construction
Construction of a [[48,33,4]] quantum code:
[1]: [[65, 51, 4]] quantum code over GF(2^2)
quasicyclic code of length 65 stacked to height 2 with 10 generating polynomials
[2]: [[48, 34, 4]] quantum code over GF(2^2)
Shortening of [1] at { 5, 6, 9, 10, 11, 12, 15, 17, 25, 31, 32, 33, 36, 39, 44, 46, 54 }
[3]: [[48, 33, 4]] quantum code over GF(2^2)
Subcode of [2]
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 1 0 1 0 0 0 1 0|1 0 1 1 1 0 0 1 1 1 1 0 1 1 1 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 1 1 0 0 1 1 1 1 1 1 1 1 1]
[0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 1 0 1 0 0 0 1 0 1 1 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0|0 1 1 1 1 0 0 1 1 1 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 1 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 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 1 1 0|1 1 0 0 0 1 0 0 1 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 1 0 1 0 0 1 0 1 0 1 0 0 0 0 0 0 1 1 1 1 1]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 1|0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 1]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0 0 0|0 1 0 0 0 1 0 1 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 0 1 0 0 1 1 1 0 0 0 0 0 0 1 1 0]
[0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 1 0 0 0 1 0 1 0 0 0 1 0|1 1 1 1 0 1 1 1 1 0 0 1 1 1 1 1 0 1 1 1 1 0 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1]
[0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1 0 0 0 0 0 1 1 1 1 1 0 1 1 0 1 1 1 0|0 0 1 0 1 0 0 1 1 1 0 0 1 0 0 1 0 1 1 1 1 1 1 0 0 0 1 1 0 1 1 0 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 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 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 1 0 0 0 0]
[0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0|0 0 0 1 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 0 1]
[0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 1 1 0 1 1 0 0 1 0 0 0 1 0 0|1 1 1 0 1 1 1 0 0 1 0 1 0 0 0 1 1 1 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 1 1 0 0 0 1 1 0 1 1|0 1 0 1 1 0 0 0 0 0 1 0 0 0 1 0 1 1 1 0 0 1 1 1 0 0 1 0 1 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 1 1 0 1 1 1 1 1 0 0 1 1|1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 1 1 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1|0 1 0 1 0 0 0 1 1 0 0 0 1 1 1 1 1 0 0 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0 1 0 0 1 1 1 1 1 0 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 0 1 1 0 0 0 1 0 1 0 1 0 1 1|0 1 0 1 1 1 0 1 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 0 1 1 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 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 0 0 1 0 1 0 1 0 1|1 0 0 1 1 1 1 1 1 0 1 1 0 0 0 1 1 0 0 1 1 1 1 0 0 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 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