Bounds on the minimum distance of additive quantum codes
Bounds on [[57,34]]2
lower bound: | 5 |
upper bound: | 8 |
Construction
Construction of a [[57,34,5]] quantum code:
[1]: [[87, 59, 6]] quantum code over GF(2^2)
quasicyclic code of length 87 with 2 generating polynomials
[2]: [[84, 62, 5]] quantum code over GF(2^2)
Shortening of the stabilizer code of [1] at { 3, 24, 30 }
[3]: [[56, 34, 5]] quantum code over GF(2^2)
Shortening of [2] at { 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55 }
[4]: [[57, 34, 5]] quantum code over GF(2^2)
ExtendCode [3] by 1
stabilizer matrix:
[1 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 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0|1 1 0 0 0 0 0 1 1 1 1 0 1 1 0 1 0 1 1 0 1 0 1 0 1 1 1 0 1 0 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0]
[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 1 0 0 1 0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0 1 0|0 0 0 0 1 0 1 1 1 1 0 0 1 1 1 1 0 1 0 0 0 0 1 1 0 1 1 1 0 0 1 0 0 1 1 0 0 1 1 1 0 1 0 0 0 1 0 1 1 1 0 1 1 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 1 0 0 0 0 1 0 1 0 0 0 1 1 0 1 0 0 0 1 1 1 1 1 0 1 0 0 0 0 1 0 0|1 0 1 1 0 1 0 1 0 0 1 0 1 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 1 1 0 1 1 1 1 1 1 1 0 0 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 0 1 0 0 1 0 1 0 0 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 0 0 0 0 1 0 0 1 0 1 0|1 1 1 0 0 1 0 0 0 0 1 1 1 0 0 1 1 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 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 1 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 0 0 1 1 0|0 0 1 0 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 1 0 0 1 1 1 1 0]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 0|0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 1 0 1 0 0 0 1 1 1 0 1 1 0 0 0 1 1 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 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 0 0 1 1 0 0 1 0 0 0 1 1 1 0 0|1 1 0 1 1 1 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 1 1 0 1 1 0 0 0 0 0 0 1 0 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 1 0 0 0 1 1 1 0 1 1 1 1 1 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0|1 0 0 1 1 0 1 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0 1 1 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 1 1 0]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 0 0 1 0 1 1 1 1 0 1 0 0 1 1 1 1 1 1 0 0|0 1 0 1 0 0 0 1 1 1 0 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 0 0 0 1 0 1 0 0 1 1 1 0 1 0 1 0 0 1 0 1 1 1 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 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 0|1 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 1 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 0 1 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 1 0 0 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 1 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0 0|0 1 1 0 1 0 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 0 1 1 1 1 0 0 0 1 1 1 1 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 1 1 1 1 0 0 0 0 1 0 1 0 0 0 0 1 1 0 1 0 1 1 0 0 0 1 0 0 0 0 0 0|1 1 0 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 1 0 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 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 0 0 1 1 1 0 1 1 0 0 1 0 1 0 1 0 1 0 0 0 1 1 1 0 0 1 0 0 1 0|0 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 1 1 1 0 1 1 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 1 1 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0|1 1 1 1 0 1 1 0 1 1 0 1 1 0 0 0 0 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0|1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 1 1 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 1 0 0 0 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 0|1 0 1 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 0 0 1 1 1 0 0 0 1 0 1 1 0 0 0 0 1 1 0 1 0 1 0 0 0 1 1 1 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 1 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1 1 1 0 1 0 0 1 1 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0|0 1 1 1 0 0 0 0 1 0 1 1 0 0 1 0 1 0 1 0 0 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 0 0 1 1 0 0 0 1 1 0 0 0 1 0 0 1 0 1 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 0 1 1 1 1 1 0 0 1 1 0 0 1 0 0 0 1 1 1 0 1 0 0 1 1 0 1 1 0 0|0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 0 1 1 1 1 1 0 1 0 1 1 0 1 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 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 1 0 0 0 0 1 1 0 1 0 0 0 0 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 0 0 0 1 0 0 0 1 0 1 0|0 0 0 0 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 1 1 0 1 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 0 1 0 1 0 1 0 0 1 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 1 0 0 1 0 1 0 1 1 1 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1 0|1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 1 0 0 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 0 0 0 0 0 0 1 1 0 1 0 0 1 1 1 1 1 1 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 1 1 0 0 1 0 1 0|1 0 0 1 1 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 1 0 0 1 1 1 0 0 0 0 1 0 1 0 1 0 1 1 1 0 0 0 0 1 0 1 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 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 0 0 1 1 0 1 0 0 1 1 0|0 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 0 1 0 1 0 0 1 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 1 1 1 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 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]
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
(codes@codetables.de).
Last change: 23.10.2014