Bounds on the minimum distance of additive quantum codes
Bounds on [[71,47]]2
lower bound: | 6 |
upper bound: | 8 |
Construction
Construction of a [[71,47,6]] quantum code:
[1]: [[128, 105, 6]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[70, 47, 6]] quantum code over GF(2^2)
Shortening of [1] at { 1, 2, 7, 8, 9, 10, 13, 14, 15, 21, 22, 24, 25, 26, 27, 29, 30, 33, 34, 37, 38, 42, 43, 45, 51, 53, 56, 57, 61, 63, 65, 67, 76, 79, 80, 82, 84, 86, 89, 92, 93, 94, 95, 97, 98, 99, 102, 103, 107, 108, 109, 110, 112, 116, 119, 121, 125, 126 }
[3]: [[71, 47, 6]] 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 1 0 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 1 0 1 1 0 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 1 0 0 1 1 0|0 0 0 0 0 1 0 0 1 0 0 1 1 0 1 0 0 1 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 1 0 1 0 0 1 1 1 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 1 0 0 0 1 0 1 1 0 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 0|0 0 0 0 0 1 0 1 0 0 0 1 1 1 1 0 0 1 0 0 0 1 1 1 0 1 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 1 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 1 1 0 0 0 1 0 1 1 1 1 1 1 1 0 1 0 0 1 1 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 0 1 0 0 1 1 1 1 0 0 1 0 0 0 0|0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 1 0 0 1 1 1 0 1 1 0 1 1 0 0 0 1 0 1 1 1 0 1 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 1 0 0 0 1 1 0 1 0 0]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 0 0 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 1 1 1 1 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 1 1 0 0 1 1 1 0 0]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 0 0 1 1 1 1 0 0 1 1 0 0 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 0 0 0 0 0 0 0|0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 0]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 1 1 0 0 0 0 1 0 1 1 1 1 0 1 0 1 1 0 1 1 1 0 1 0 0 0 0 0 0 0|0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 1 1 1 0 1 0 0 1 0 0 0 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 1 1 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 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 0 0 1 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 0 0 0 1 0 0 1 1 1 1 0 1 1 0 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 1 1 0 0 1 1 1 0 1 0 0 0 1 1 1 0 1 0]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 0 1 1 1 0 1 0 1 0 0 0 1 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0|0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0 0 0 1 0 1 0 1 1 1 0 1 0 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 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 1 0 1 0 0 0 0 1 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 0 1 1 0 0 0 1 0 0|0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 1 1 1 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 1 0 1 1 0 1 0 1 0 1 1 1 0 0 1 1 0 0 1 1 1 1 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 0 0|0 0 0 0 0 0 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 0 0 1 0 1 0 0 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 1 0 0 0 0 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 0 0 0 1 1 0 1 1 0 0 1 0 0 0 0 1 0 1 1 1 0 1 1 0 1 0 1 1 0 0 0 1 0 1 1 1 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 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 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 0 0 0 1 0 0 1 1 1 1 0 1 1 0 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 1 1 0 0 1 1 1 0 1 0 0 0 1 1 1 0 1 0|0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 0 0 0 1 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 1 1 1 0 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 1 0|0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 1 1 0 0 0 0 1 0 1 1 1 1 0 1 0 1 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 1 0 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 0 0 1 0 1 0 1 1 1 0 0 0 0 1 0 0 0 1 1 0 1 0 1 1 1 1 0|0 0 0 0 0 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 0 1 0 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 1 0 1 0 0 1 1 1 0 0 1 1 0 0 1 1 1 1 0 0 0 1 1 0 0 0 0 0 1 1 0 1 1 1 0 0 1 0 1 0 1 0 1 0 1 1 1 1 0 1 0 0 0 1 1 1 0|0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 1 0 0 1 0 1 1 0 0 0 0 1 1 1 1 0 0 1 1 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 1 1 1 0 1 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 1 1 1 1 1 0 0 1 1 0 0 0 1 0 1 1 0 1 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 1 1 1 1 0 1 1 1 1 1 1 0 0 1 0 0 0 0 1 1 0 1 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 0 1 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 1 1 1 0 1 0 1 1 0 1 1 0 1 0 1 0 0 0 1 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 0 1 1 1 0 1 0 1 1 0 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 1 0 0 0 1 0 1 0 0 1 1 1 0 0 0 1 1 1 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 0 1 0 1 0 0 1 0 1 1 0 0 0 1 0 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 1 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 1 1 1 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 1 1 0 1 0 1 0 1 0 1 1 1 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 1 1 1 0 1 1 0 1 1 0 0 1 1 1 0 1 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 1 0 0 0 1 0 0 1 0 0 1 1 1 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 1 1 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 1 0 1 0 0 1 0 0 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 0 0 1 0 0 0 1 1 1 1 1 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
(codes@codetables.de).
Last change: 23.10.2014