Bounds on the minimum distance of additive quantum codes
Bounds on [[45,31]]2
lower bound: | 4 |
upper bound: | 5 |
Construction
Construction of a [[45,31,4]] 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]: [[45, 31, 4]] quantum code over GF(2^2)
Shortening of [1] at { 4, 6, 10, 13, 22, 24, 29, 30, 31, 38, 42, 44, 49, 52, 53, 55, 57, 62, 64, 65 }
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 1 0 0 1 0 1 0 0 0 0 1 1 0 1 0|0 0 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 0 0 0 0 1 1 1 0 1 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 1 1 1 1]
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 1 1 1 0 0 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 1 1|1 1 0 1 0 1 1 0 1 1 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 1 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 1 1 1 1 0 0 0 1 1 0 1|0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 0 0 1 1 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 0 1 0 1 1 1 0 0|0 1 0 0 1 1 0 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 1 0 1 0 1 1 0 1 0 0 0 1 0 1 1 1 0 1 0 0|0 1 1 0 1 1 1 0 1 1 1 0 0 0 1 1 1 1 0 0 1 1 0 0 0 0 0 1 0 1 1 0 1 0 1 1 0 0 1 0 0 1 1 1 1]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 1 1 1 0 0 1 1 0 0 0 0 1 1 0 1 0 1 1 0|0 1 1 1 1 0 1 1 0 1 1 1 1 1 0 1 1 1 0 1 0 0 0 0 0 0 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 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 1 1 0 1 0 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 0 0 0|0 0 1 0 0 0 0 1 1 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1 0 1 1 1 0 1 1 0 1 1 1]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 1 0 1|0 1 1 0 1 1 1 1 0 0 1 1 0 1 0 1 0 0 1 1 1 0 0 0 1 1 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 1 0 0 1]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 1 1 0 1 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 1 0 0|0 0 1 1 1 0 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1 1 0]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 1 1 1 0 1 0 0 0 1 1 1 0 1 0 1 1 1 0 0|1 0 1 1 0 0 1 0 1 0 1 0 0 1 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 1 0 1 1 0 1 0 0 1 0 1 0 0]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 0 0 1 1 1 0 1 1 0 0 0 1 1 0 0 1 1 1 0 0 1 1 1 1 1 1|0 0 1 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 1 1 0 0 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 0 0 0 1 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 1 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 1 1|0 0 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 1 1 1 1 1 0 1 1 1 0 0 1 1 0 0 0 1 1 1 1 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 1 1 0 0 1 1 1 1|1 0 0 1 0 1 0 1 1 1 0 1 1 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 1 0 1 0 0 0 1 1 0 1 1 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 0 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 0 0|1 1 0 1 0 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 0 1 1 1 0 1 1 1 1 0 0 1 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