Bounds on the minimum distance of additive quantum codes
Bounds on [[87,78]]2
lower bound: | 3 |
upper bound: | 3 |
Construction
Construction of a [[87,78,3]] quantum code:
[1]: [[168, 159, 3]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[87, 78, 3]] quantum code over GF(2^2)
Shortening of [1] at { 1, 2, 3, 4, 7, 8, 9, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 33, 35, 37, 39, 44, 47, 48, 50, 51, 52, 56, 57, 58, 60, 63, 64, 65, 70, 71, 72, 73, 74, 79, 80, 83, 98, 99, 124, 126, 127, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 144, 145, 146, 147, 151, 152, 153, 154, 156, 157, 158, 159, 160, 164, 166, 167, 168 }
stabilizer matrix:
[1 0 0 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 1 1 0 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 1|0 0 0 0 0 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 1 0 1 1 0 1 0 1 1 0 1 1 1 1 0 1 1 0 0 0 1 0 0 1 0 1 0 1 0 1 1 0 0 1 0]
[0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 1 1 0 0 0 1 1 0 1 1 0 1 1 0 1 0 0 0 1 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1|1 0 1 1 0 1 0 1 0 1 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 1 1 0 0 1 0 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 1 0 1 0]
[0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 1 1 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 1 1 1 0 0 1 0 1 1 0 1 0 0 0 1 0 0 0 1 1 1 0 1 1 1 0 1 0 0 1 0 1|0 0 0 0 1 0 1 1 0 1 1 0 1 0 1 1 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 1 1 1 0 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 0 0 0 1 1 0 1 1 1 0 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0]
[0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 1 1 0 0 0 1 0 1 1 0 1 0 1 1 1 0 0 0 0 0 1 1 0 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 0 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 0 0 1 1|0 1 1 1 0 1 1 0 0 1 1 0 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 1 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0 0 0 1 0 1 0 0]
[0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 0 0 1 1 1 1 1 1 0 0 1 0 1 0 0 0 1 1 0 1 1 1 0 1 0 0 1 1 0 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 0 1|1 0 1 0 1 0 0 1 0 1 1 0 0 0 1 0 1 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 0 0 1 0 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 0 0 1 1 1]
[0 0 0 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 0 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1|1 1 1 0 1 0 1 1 1 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 0 1 1 1 0 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 0 1 1 1 1 0 0 1 0 0 1 0 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 0 0 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 1 1 1 0 1 1 0 1 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0|0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 1 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 1 1 0 1 0 1 1 1 1 0 1 0 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 1 0 1 1 0 0 1 1 1 1 1|0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 1 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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]
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