Bounds on the minimum distance of additive quantum codes
Bounds on [[84,66]]2
lower bound: | 5 |
upper bound: | 6 |
Construction
Construction of a [[84,66,5]] quantum code:
[1]: [[122, 104, 5]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[84, 66, 5]] quantum code over GF(2^2)
Shortening of [1] at { 4, 69, 72, 74, 75, 76, 77, 78, 79, 80, 83, 84, 85, 87, 89, 92, 95, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109, 110, 111, 112, 113, 114, 116, 117, 118, 119, 120, 122 }
stabilizer matrix:
[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 1 1 1 1 1 1 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 0 1 1 0 0 1 0 0 1 0 1 1 1 1 1 0 1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 1 1|1 0 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 1 0 1 0 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 1 1 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 1 1 1 0 0 1 1 0 1 1 1 0 0 0 1 0 1 0 0 0 1 1 0 0 1 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 0 1 1 1 1 1 0 0 1 1 1|0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 0 1 0 1 0 0 1 1 1 1 1 1 0 0 1 0 0 0 1 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1 0 1 1 1 1 0 1 1 1 0 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 1 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 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 1 0 1 0 0 1 0 1 1 1 0 1 1 0 0 1 0 1 0 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 1|1 1 0 0 1 1 0 1 1 1 0 0 0 1 0 1 1 0 1 1 0 0 1 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 0 0 1 1 1 1 0 1 1 1 0 1 1 0 1 1 0 1 0 0 0 1 0 0 1 1 1 0 1 0 1 1 0 1 0 0 0 1 1 1 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 1 0 0 0 1 0 1 0 1 0 1 1 1 1 1 1 1 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 0 1 1 1 1 1 1 0 1 0 1 0 0 1 1 1|0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 1 0 1 1 0 0 1 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 1 1 1 1]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 1 0 0 0 1 0 0 1 1 1 0 0 0 1 0 0 1 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 1 0 1 1 0 1 1 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1|0 0 0 1 0 0 1 0 1 1 1 0 0 1 0 1 1 1 0 0 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1 0 0 1 1 1 1 0 0 1 1 0 0 0 1 0 1 1 1 0 1 0 1 1 1 1 1 1 1 1 0]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 1 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 1 0 1 0 0 0 1 1 0 1 0 0 1 0 0 0 0 1 0 0 1|0 0 0 0 0 1 1 0 1 1 0 1 0 0 1 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 0 1 1 0 0 1 1 0 0 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 0 0 1 1 0 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 1 0 0 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 0 1 0 0 0 1 1 0 1 0 1 0 0 1 0 0 1 1|1 1 0 0 0 1 0 0 1 1 1 0 0 0 0 1 0 1 1 1 0 0 0 0 0 1 0 1 1 0 0 1 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 1 0 1 0 1 0 0 1 1 0 1 0]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 0 1 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 0|1 1 1 1 0 1 0 0 1 0 1 0 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 0 0 0 1 0 1 1 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 1 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 0 1 0 1 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0|0 0 0 0 1 0 1 0 0 1 0 0 0 1 1 1 0 0 1 0 1 1 1 1 1 0 1 0 1 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 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 1 0 0 0 0 1 1 0 0 0 0 1]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 0 0 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 0 1 1 0 1 1 1 1 1 0 0|0 0 1 0 1 1 0 1 0 0 0 0 1 1 0 1 1 0 0 1 1 1 0 0 1 0 1 0 1 0 0 1 0 0 1 1 0 1 1 1 0 1 0 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 1 1 1 1 1 0 0 1 0 1 0 1 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 1 1 0 1 0 0 0 0 0 1 1 0 1 1 0 1 1 1 1 0 0 1 0 1 1 0 1 1 1 0 0 1 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 1 0 0 1 0|1 0 1 0 0 0 1 0 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 0 0 0 1 0 0 0 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 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 1 1 1 1 0 0 1 1 1 0 0 1 0 1 1 0 1 1 0 1 1 1 0 1 1 1 1 0 0 1 0 0 0 0 1 1 0|1 1 1 0 0 1 1 0 1 0 0 1 0 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 1 1 0 1 0 1 0 1 0 1 1 0 1 1 0 0 1 0 0 1 0 1 0 0 1 1 1 1 0 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 1 1 1 0 1 1 1 0 1 1 1 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 1 0 1 1 1 0 0 0 0 1 1 0 0 1 0 0 0|1 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 0 0 1 1 0 1 1 0 0 1 1 0 1 0 1 0 1 1 0 0 0 1 0 0 1 1 1 0 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 1 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 1 1 0 0 0 0 1 1 1 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 0 0 0 1 1 0 0 1 1 0 0 1 1 1 1 0 1 0 0 0 1|1 1 1 0 0 1 1 1 1 0 1 0 0 1 1 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 1 0 1 0 0 1 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 1 1 0 0 1 0 0 1 1 0 1 0 1 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 1|0 1 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 1 0 0 0 0 0 1 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 0 1 0 1 1 1 1 1 0 0 0 1 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 0 0 1 1 1 1 1 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 1 0 1 0 1 1 1 0 1 0 1 1 0 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 1 1 0 1 0 1 1 1|0 1 0 1 0 1 1 0 0 1 1 0 1 1 0 1 1 0 1 0 1 0 1 0 1 1 1 1 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 0 1 0 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 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 1 0 0 1 1 0 1 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 1 1 0 0 1 1 0 0 0 1 1 1 1 1 0 1|1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 1 0 1 0 1 0 1 0 1 0 0 1 1 1 1 0 0 1 0 0 0 1 0 0 0 0 0 1 1 1 1 0 1 1 1 0 1 1 1 1 0 1 1 0 0 1 0 0 1 1 0 0 0 1 1 1 1 1 0 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 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 1 0 0 0 0 0 0 1 1 0 1 1 1 1 1 0 1 0 1 1 1|0 1 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 1 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 1 0 1 0 1 0 1 1 1 1]
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