Bounds on the minimum distance of additive quantum codes
Bounds on [[123,114]]2
lower bound: | 3 |
upper bound: | 3 |
Construction
Construction of a [[123,114,3]] quantum code:
[1]: [[168, 159, 3]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[123, 114, 3]] quantum code over GF(2^2)
Shortening of [1] at { 4, 10, 11, 15, 22, 23, 25, 26, 29, 30, 31, 37, 38, 49, 69, 84, 85, 86, 87, 89, 90, 94, 96, 108, 109, 111, 117, 120, 123, 127, 134, 135, 138, 139, 141, 146, 148, 149, 152, 153, 154, 156, 162, 167, 168 }
stabilizer matrix:
[1 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 1 1 1 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 0 1 0 1 0 1 1 1 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 0 1 1 1 1 1 0 0 0 0|0 0 1 1 1 0 1 0 1 1 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 1 1 1 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 1 1 0 1 1 0 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 0 0 0]
[0 1 0 0 1 0 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 0 1 1 1 1 0 0 1 0 0 0 1 0 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 1 0 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0|0 1 1 0 1 1 0 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 0 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 0 0 0 1 1 1 1 0 0 1 0 0 0 1 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1 1 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 0 1 1 0 0 0 1 0 1 1 1 0 0 0 0 0 1 0]
[0 0 1 0 0 1 1 0 0 1 1 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 1 1 0 0 1 1 1 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 1 1 0 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0|0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 1 0 1 1 1 0 1 0 1 1 1 1 1 0 0 0 1 1 0 0 1 1 0 1 1 1 1 0 0 1 0 1 1 1 1 1 1]
[0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 0 1 1 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 1 1 1 1 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 0 1 1 0 1 1 0 1 1 1 1 1 1 0 1 0 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 1 1 1 1 1 0 0|0 1 0 0 1 0 1 0 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 1 1 0 0 1 1 1 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 1 1 0 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0]
[0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 1 0 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 0 1 1 1 1 0 0 1|1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 1 1 1 1 0 1 0 0 0 0 1 1 1 0 0 0 1 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 1 0 0 1 0 1 0 0 1 0 1 1 1 0 1 0 1 1 0 0 1 1 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 1 1]
[0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 1 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 1 1 1 1 1 0 1 1 1 1 1 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 0 1 1 0 0 0 1 0 1 1 1 0 0 1 1 1 0 0 1 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0|1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 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 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 1 0 0 1 1 1 1 0 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 1 0 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0|0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 1 1 0 1 1 0 1 1 1 1 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 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 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 1 0 1 1 0 0 0 0 1 1 1 1 1 0 1 0 1 1 1 0 1 0 1 1 1 1 1|0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 1 1 0 1 1 0 1 1 1 1 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|0 0 0 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]
last modified: 2008-08-05
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