Bounds on the minimum distance of additive quantum codes
Bounds on [[59,38]]2
lower bound: | 5 |
upper bound: | 7 |
Construction
Construction of a [[59,38,5]] 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]: [[58, 38, 5]] quantum code over GF(2^2)
Shortening of [1] at { 1, 5, 6, 10, 12, 16, 20, 25, 26, 27, 29, 31, 34, 36, 42, 44, 46, 47, 51, 54, 56, 59, 61, 63, 68, 69, 70, 72, 74, 75, 79, 80, 84, 86, 90 }
[3]: [[59, 38, 5]] quantum code over GF(2^2)
ExtendCode [2] by 1
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 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 1 0 1 0 1 1 1 1 1 1 1 1 0 1 1 1 0 0|1 0 1 0 1 0 0 0 0 0 1 1 1 0 0 1 0 0 1 1 1 0 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 0 0 1 1 1 0 0]
[0 1 0 0 0 0 0 0 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 0 0 0 1 1 0 1 0 1 1 0 0 0 1 0 0 0 0 0 1 1 0|0 1 0 0 0 0 0 0 1 0 1 1 1 1 1 0 0 0 1 0 1 1 1 1 0 0 1 0 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 0 0 0 0 1 0 1 1 1 1 0]
[0 0 1 0 0 0 0 0 0 0 0 0 1 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 1 0 0 0 0 0 1 0 0 1 0 1 1 1 0 1 0 0 1 0|1 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 0 1 0 0 0 0 0 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 0 1 0 0]
[0 0 0 1 0 0 0 0 0 0 1 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 1 1 0 1 1 1 1 1 0 1 0 0 1 0 0 0 1 0 0|0 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0 1 1 1 0 1 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0]
[0 0 0 0 1 0 0 0 0 0 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 1 0 0 1 1 1 1 0 0 0 1 0 0 0 1 1 0 1 1 0|1 0 0 0 1 0 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 1 1 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 1 0]
[0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 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 1 1 1 0 1 1 1 1 0 1 1 0 0 1 1 0 0 1 0|0 1 0 1 0 0 0 0 0 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 1 0 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0 1 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 1 1 0 1 1 0 0 0 1 0 1 0 0 0|0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 0 1 1 0 1 1 1 0 0 0 1 1 1 1 1 0 1 1 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0]
[0 0 0 0 0 0 0 1 0 0 1 0 1 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 0 1 1 1 1 0 1 1 0 1 1 1 1 1 0 1 1 1 0 0 0|1 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 1 0 0 0]
[0 0 0 0 0 0 0 0 1 0 1 1 1 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 1 0 1 0 0 0 1 1 1 1 1 0 0 1 1 1 0 0 0|1 1 1 0 0 0 0 1 0 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 1 1 1 0 0 0 0 1 1 0 0 1 1 1 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 1 1 0 1 0]
[0 0 0 0 0 0 0 0 0 1 0 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 1 1 0 0 1 0 1 1 0 1 0 1 1 0 0 1 1 0 0 0|1 0 1 0 0 0 1 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 0 1 0 1 0 1 1 1 1 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 0 0 1 1 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 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 0|0 1 1 0 1 1 1 1 0 0 0 0 1 0 1 1 1 1 1 0 0 0 0 1 0 1 0 0 0 1 1 0 1 1 0 0 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0 1 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 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 0 0 1 1 0|0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 1 1 1 1 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 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 1 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 0 0 1 1 0 1 1 1 1 1 1 1 0 0 0 0|0 1 1 1 0 0 0 0 1 1 0 0 1 1 0 0 1 0 0 1 1 1 0 0 0 1 0 1 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 0 0 0 1 1 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 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0|0 1 1 1 1 0 0 0 1 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 0 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 0 0 1 1 1 0 0 1 0 0 1 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 0 0 1 0 0 0 0 0 1 0 1 0 1 1 1 0 1 1 0 1 1 0 0 0 1 1 1 0 1 1 0 0 0 1 1 1 0 0|0 1 0 1 0 0 1 1 0 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 1 0 0 1 0 1 0 1 0 0 1 0 0 0 1 1 0 0 1 1 0 1 0 1 1 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 1 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 1 0 0 1 1 1 1 1 0 0 0 1 1 0 1 1 0 0|0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 1 0 1 1 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 1 0 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 1 0 0|1 0 0 1 0 1 1 0 0 0 1 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 1 0 1 1 0 0 1 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 1 0 0 1 1 1 0 1 0 0 1 1 1 0 1 1 0 1 0 1 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 0 0 1 1 0 1 0 1 0 0 1 0 1 1 1 1 0 0 1 0 0 1 1 1 1 0 1 1 0 1 1 1 0 0 0 1 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 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 1 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 0 0 0 1 0|0 1 1 0 0 0 1 0 1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 0 1 0 1 0 1 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 0 1 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 1 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0|1 1 0 1 0 0 0 1 1 0 1 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 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 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]
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