Bounds on the minimum distance of additive quantum codes
Bounds on [[88,67]]2
lower bound: | 5 |
upper bound: | 6 |
Construction
Construction of a [[88,67,5]] quantum code:
[1]: [[128, 105, 6]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[127, 106, 5]] quantum code over GF(2^2)
Shortening of the stabilizer code of [1] at 128
[3]: [[88, 67, 5]] quantum code over GF(2^2)
Shortening of [2] at { 4, 9, 10, 14, 16, 20, 23, 24, 27, 33, 44, 46, 50, 53, 55, 57, 59, 61, 65, 66, 68, 70, 71, 75, 81, 83, 85, 86, 93, 95, 103, 104, 105, 106, 107, 118, 120, 121, 125 }
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 1 1 1 0 1 0 1 0 0 1 1 0 1 0 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 1 0 1 0 1 0 0 1 0 0 1 1 0 0|0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 1 0 0 1 1 1 1 0 1 0 1 1 1 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 1]
[0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 1 1 1 0 0 0 1 1 1 1 0 1 1 0 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 1 1 1 1 0 1 0|0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 0 1 0 0 1 0 0 0 1 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 1 0 1 1 0 1 1 0 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 0 0 1 1 0 0 0 1 1 1 0 1 0 0 0 1 1 1 0]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 1 1 0 1 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 1 0 0 0 1 0 1 1 1 1 1 0 1 1|0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 0 0 1 1 0 0 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 1 1 0 0 1 0 0 1 1 0 1 1 1|0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 0 1 1 0 1 1 0 1 0 0 1 1]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 0 0 0 1 0 0 1 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 1 0 1 0 0 1 1 0 0 1 1 1 1 1 0 0 1 0 1 1 1 1 0 1 1 0 0 0 1 0 0|0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 0 1 1 0 1 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 1 0 1 1 1 1 1 0 0 0 0 1 1 0 0 0 0 0 1 1]
[0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 0 1 0 0 1 0 0 1 1 1 0 1 1 0 1 0 1 0 1 1 1 1 0 1 1 1 1 0 0 1 1 0 1 0 1 0 1 1 1 1 1 1 1 0 0 1 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0|0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 1 0 0 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 0 0 1 1 1 0 0 0 0 1 0 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 1 0 1 0 1 1 1 1 1 1 0 1 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0|0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 0 1 0 0 1 0 0 0 1 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 1 0 1 1 0 1 1 0 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 0 0 1 1 0 0 0 1 1 1 0 1 0 0 0 1 1 1 0]
[0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 1 1 0 0 0 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 1 1 0 0 1|0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 0 0 1 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 1 1 1 0 1 0 0 0 0 1 0 0 1 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 0 1 0 1 0 1 0 1 0]
[0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 1 1 1 1 0 1 1 1 0 1 0 1 1 1 0 0 0 1 0 1 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 0 1 0 0 1 1 1 0 1 1 0 1 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 1 1 0 1|0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 1 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 1 1 0 1 1 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 1]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 0 1 1 0 0 1 1 1 0 0 1 1 1 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 1 0 1 1 0 0 1 1 0 0 0 1 0|0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 0 0 1 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 1 0 0 1 0 1 1 0 1 0 0 0 0 1 1 1 0 0 1 1 0 1 0 1 1 1 1 0 1 0 1 0 1 0 1 0]
[0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 0 1 1 0 1 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 0 1 1 0 1 0 0 0 1 0 1 1 0 1 0 1 1 1 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 0 1 1 1 0 1 1 0 0 1|0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 1 1 1 0 0 1 1 1 1 1 1 0 0 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 1 1 1 0 1 1 0 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 1 1 0 0 1 0 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 1 1 0 0 0 0 1 0 0 1 0 1 0 1 1 0 1 1 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 1 0 1 1 0 1 0|0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 1 1 0 1 0 0 1 0 0 1 0 0 0 1 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 1 0 0 1 1 0 1 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 1 0 1 1 0 0 0 0 0 0 0 1 0 1 1 0 1 0 1 0 0 1 1 1 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 1 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 0 0 1 1 0 1 1 1 0 1 1 1 0 1 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 1 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 1 1 0 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 1 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1 0 1 1 0 0 0 0 1 1 1 1 0 0 1 0 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 0 1|0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 1 1 1 0 0 0 0 1 0 1 0 0 1 1 1 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 1 0 0 1 0 1 0 1 0 0 0 1 0 1 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 0 1 1 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 1 1 0 1 1 0 1 1 0 1 1 0 0 1 0 1 0 0 1 0 1 1 0 0 1 1 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 0 0 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 1 0 1 0 0 0 1 1 0 1 1 1 0 1 1 1 1 0 0 1 1 0 1 0 0 1 1 1 0 1 1 0 0 1 0 1 0 1 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 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 0 0 0 0 0 1 0 1 0 0 1 1 1 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 1 0 1 1 1 0 1 1 0 0 1 0 1 0 1 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 0 0 0 0 0 0 0 0|0 0 0 1 0 0 0 0 1 0 0 1 1 1 0 1 0 1 1 1 0 0 1 1 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 1 1 0 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 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 1 0 0 1 0 0 0 1 1 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 1 1 0 0 0 1 1 1 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 1 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 0 1 1 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 1 0 1 1 0 0 1 0 1 0 0 1 0 1 1 0 0 1 1 0 0 1 0 1 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 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 0 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 0 1 1 1 1 0 0 0 1 1 0 0 1 1 0 0 1 1 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 1 0 1 1 0 0 1 0 1 0 1 0 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