Bounds on the minimum distance of additive quantum codes
Bounds on [[53,34]]2
lower bound: | 5 |
upper bound: | 6 |
Construction
Construction of a [[53,34,5]] quantum code:
[1]: [[122, 104, 5]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[52, 34, 5]] quantum code over GF(2^2)
Shortening of [1] at { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 24, 25, 26, 30, 31, 32, 33, 36, 37, 40, 41, 42, 43, 44, 47, 49, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 64, 65, 66, 67, 68, 74, 77, 80, 82, 84, 86, 87, 88, 94, 97, 98, 99, 100, 101, 102, 103, 110, 111, 115 }
[3]: [[53, 34, 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 0 0 0 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 0|1 1 0 1 1 1 1 1 0 0 1 1 1 1 1 0 1 1 1 1 1 1 0 1 0 0 1 0 0 1 0 1 0 0 1 1 1 1 0 0 1 0 1 1 1 0 0 1 0 1 1 1 0]
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 1 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 1 0 0 1 0 1 0 0 0|0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 0 1 0 1 1 1 0 1 0 1 1 1 0 1 1 1 1 0 0 0 1 1 0 1 1 0 1 1 0 0 0 1 0 1 0 0]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 0 0 0 1 0|0 1 0 1 1 1 1 0 0 0 0 1 0 1 0 1 0 1 0 0 1 1 1 0 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 1 1 0 1 1 0 0 1 0]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 1 1 1 0 1 1 0 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0|0 0 1 1 0 1 0 0 0 1 1 1 1 0 1 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 0 0 1 0 1 1 1 0 1 0 1 1 0 0 1 0 1 0 0 0 1 0 0]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0|1 0 1 0 1 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 1 0 1 1 1 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 1 0 0 1 0 0 1 0 1 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 1 0|1 1 1 1 1 0 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 0 1 0 1 1 0 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 1 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 0 1 1 1 1 0 0|1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 0 1 1 1 0 0 0 0 1 1 1 0 0 0 1 0 1 1 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 1 1 0 1 0 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 1 0 1 1 0 1 1 0 1 1 0|0 1 1 1 0 0 0 1 0 1 0 1 0 1 1 0 0 0 1 0 1 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 0 0]
[0 0 0 0 0 0 0 0 1 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 1 1 0 1 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0|0 1 1 1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 0 1 1 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 0 1 0 1 1 0 1 1 1 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 1 1 1 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 1 0 0 1 0 1 0|0 1 1 1 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 0 0 1 0 0 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 0 1 0 0]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 1 1 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 0 0 1 0 1 1 0|0 0 0 0 1 0 0 1 1 1 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 1 0 1 1 1 1 1 0 0 0 0 0 1 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 1 1 1 1 1 1 0 1 1 1 1 0 0 1 0 0 0 0 1 0 1 1 1 1 1 1 0 0 1 0|1 1 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 1 1 0 1 0 0 0 0 1 1 1 0 1 0 1 1 1 0 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 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 0 0 1 0 1 1 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0 0|1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 1 1 1 0 0 1 0 1 0 0 1 1 1 0 1 1 0 0 1 0 1 0 0 0 1 1 1 1 0 1 0 0 0 1 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 0 1 0 0 1 0 1 0 0 0 0 1 0 0 0 0|1 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 1 1 0 1 0 0 0 0 1 1 1 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 1 0 0 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 1 0 0 1 0 0|0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 1 1 0 1 0 0 1 1 1 0 1 0 1 0 1 0 1 1 0 0 1 0 1 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 0|0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 1 1 1 0 1 1 1 1 0 1 0 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1 1 1 0 0 1 1 0 0 0 0 0|0 1 0 1 0 1 0 0 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 1 0 1 0 0 1 1 1 1 0 1 0 0 1 1 0 1 0 0 0 1 0 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 1 1 1 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 1 0|1 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 0 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 0 0 0 0 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]
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