Bounds on the minimum distance of additive quantum codes
Bounds on [[50,29]]2
lower bound: | 5 |
upper bound: | 7 |
Construction
Construction of a [[50,29,5]] quantum code:
[1]: [[93, 73, 5]] quantum code over GF(2^2)
quasicyclic code of length 93 stacked to height 2 with 6 generating polynomials
[2]: [[49, 29, 5]] quantum code over GF(2^2)
Shortening of [1] at { 2, 4, 6, 9, 10, 12, 16, 17, 18, 20, 21, 22, 23, 25, 29, 31, 32, 33, 35, 37, 38, 42, 43, 47, 49, 53, 57, 62, 63, 64, 65, 66, 68, 72, 74, 76, 78, 80, 83, 84, 86, 90, 91, 92 }
[3]: [[50, 29, 5]] quantum code over GF(2^2)
ExtendCode [2] by 1
stabilizer matrix:
[1 0 0 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 1 1 0 0 1 0 1 1 1 1 0 1 0 0|0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 1 0 1 0 0 0 1 0 0]
[0 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 1 1 0 1 1 0 0 0 1 1 1 1 1 0 1 0|1 0 1 1 1 0 0 1 0 0 1 1 0 0 1 1 1 0 1 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 1 1 1 0 0 1 1 0 1 0]
[0 0 1 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 1 1 0 0 1 1 1 0 1 1 1 1 1 0 0 0|1 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 1 1 0 0 0 1 1 0 0]
[0 0 0 1 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 1 0 1 1 0 1 0 0 0 1 0 0 1 0|0 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 0 1 1 0 1 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 0 1 1 1 0 0 0 1 0 0]
[0 0 0 0 1 0 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 0 0 1 0 1 0 1 0 1 1 1 0 1 1 1 0 1 0|0 0 0 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 1 0 1 1 1 1 0 1 0 1 0 0 1 1 1 0 1 0]
[0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 1 0 0 0 1 0 0|0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 1 1 0 0 0 1 0 1 0 1 1 1 1 0 0 1 1 1 0 1 0 1 0 0 0 0 0 0 0 1 0]
[0 0 0 0 0 0 1 0 0 0 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 1 1 0 0 0 1 0 1 0 1 0 0 0 1 0|0 0 1 1 1 1 1 0 1 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 1 0 0]
[0 0 0 0 0 0 0 1 0 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 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0|0 0 1 1 0 0 1 0 1 1 1 0 0 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 1 0 0 0 1 1 0]
[0 0 0 0 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 1 1 0 1 1 1 1 0 1 0 0 1 0 0|0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 0 1 0 0 0 1 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 1 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 1 0 0 0 1 0 1 1 0 1 1 0 1 1 0|0 0 0 0 1 0 0 0 1 0 0 0 1 1 0 1 1 1 1 0 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 0 0 1 0 1 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 1 0 0 1 0 1 1 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0|0 1 1 1 1 0 1 0 0 0 0 1 0 1 1 1 0 1 1 1 0 1 1 1 0 0 0 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 1 1 0 1 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 1 0 1 1 1 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1 1 0 0|0 1 1 0 1 1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 0 0 0 0 1 1 0 0 1 1 1 1 0 1 1 0 1 0 0 1 1 0 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 1 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 1 1 1 0 1 0 0 0 1 0 0 0 1 0 0 1 0|1 0 1 1 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0|1 1 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 1 1 0 1 0 0 0 0 1 0 1 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 1 0 0 0 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 0 0 1 0 0 0 1 0 1 0|0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 0 0 1 0 1 0 0 1 1 1 1 1 0 1 0 0 0 1 1 1 0 0 0 0 1 1 1 0 0 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 1 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0|1 1 0 1 1 1 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 0 0 1 1 0 1 1 1 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 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0|0 1 0 1 1 1 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 0 0 0 0 0 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 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 1 0 0|0 1 1 1 1 0 1 0 1 1 0 1 0 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1 1 1 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 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 1 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 0 1 1 0 0 0|0 0 0 1 1 0 1 1 1 1 0 0 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 0 1 0 1 1 0 1 0 1 1 1 0 1 0 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 1 0 1 0 1 1 1 1 0 0 0 1 1 0 1 0 0 1 1 0 1 0 0 0|0 0 1 0 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 0 0 1 1 0 1 1 1 0 0 1 0 0 0 1 1 0 0 0 1 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 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]
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