Bounds on the minimum distance of additive quantum codes
Bounds on [[76,55]]2
lower bound: | 5 |
upper bound: | 6 |
Construction
Construction of a [[76,55,5]] 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]: [[76, 55, 5]] quantum code over GF(2^2)
Shortening of [2] at { 2, 7, 8, 11, 14, 16, 20, 21, 22, 24, 27, 28, 33, 35, 38, 41, 45, 46, 47, 48, 53, 54, 56, 58, 60, 61, 62, 64, 66, 67, 70, 73, 75, 78, 81, 86, 92, 93, 94, 100, 103, 104, 108, 111, 113, 114, 120, 121, 123, 124, 127 }
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 0 1 1 1 0 0 1 1 0 1 1 1|0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 1 1]
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 1 1 0 1 1 0 0 1 1 1 0|0 0 0 0 0 0 1 0 0 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 1 1 1 0 0 1 0 0 0 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 0 1 1 1 0 1]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 0 0 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 0 0 0 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 0 1 0 1 0 0 0 1|0 0 0 0 0 0 0 1 0 0 1 0 1 1 1 1 0 1 1 0 0 1 0 1 0 1 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 1 1 1 0]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 1 1 0 0 0 0|0 0 0 0 0 0 1 1 0 1 1 1 1 1 0 0 0 1 1 1 0 1 0 1 0 1 0 1 1 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 1]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 0 1 0 1 0 0 0 1 0 1 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 0 1 1 0 1 1 1 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 1 0 1|0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 1 1 1 0 0 1 0 1 0 0 0 0 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 0 0 0 1 1 0 1 0 0 1 0 1]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 1 1 0 1 0 1 1 0 1 1 0 0 1 1 0 0 1 1 1 0 1 0|0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 1 1 1 1 1 1 1 0 0 1 1 1 0 1 0 0 1 1 1 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 1 1 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 1 0 0 1 0 1 1 0 1 1 1 0 0 1 1 1 1 0 1 1 1 1 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 0 1 0 0 1 1 1 1 1 1 0 0 1 1 1 1 0|0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 0 0 1 1 1 1 1 1 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0 1 0 1 0 0 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 1 1 1 0 1 1 0 1 0 0 1 1 1 0 0 0 1 1]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 1 0 1 0 0 1 1 0 1 0 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 1 1 0 1|0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 1 0 0 0 1 1 1 0 1 0 0 0 1 1 1 0 0 1 0 0 1 0 0 1 0 1 1 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 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 1 0 0 1 0 1 0 1 0 0 0 1 1 0 1 1 0 1 1 0 1 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 0 1 0|0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 0 1 0 0 1 1 0 0 1 1 1 1 0 1 1 1 1 0 1 1 0 0 0 1 1 1 0 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 1 1 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 1 0 1 0 1 0 1 0 0 1 1 0 0 0 0 0|0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 1 1]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 1 1 0 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 0 0 0 1 0 1 0 1 0 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 0 0 0 0|0 0 0 0 0 0 1 0 0 1 1 0 1 1 0 0 1 1 1 0 1 1 1 0 0 0 0 1 1 0 1 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 1 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 1 0 1 1 0 1]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 1 0 0 0 1 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 0 1 0 1 0 1|0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 1 0 1 1 1 0 0 0 0 0 1 1 0 1 0 0 1 0 0 1 1 0 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 1 1 0 1 0 1 1 1 1 0 1 1 0 1 1 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 1 0 1 0 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 1 0 1 1 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 1 1 0 0 1 0 0 1 0 1|0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 0 1 0 0 1 1 0 0 1 1 1 1 0 1 1 1 1 0 1 1 0 0 0 1 1 1 0 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 1 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 0 1 0 0 0 1 1 0|0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 1 1 1 1 1 1 1 0 0 1 1 1 0 1 0 0 1 1 1 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 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|1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 1 1 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 1 0 1 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 1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 1 0 0 0 1 0 1 1 1 1 0 0 1 0 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 1 0 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 1 1 1 0 1 1 1 1 0 0 0 1 0 0 0 0 1 0 1 1 0 1 0 1 1 1 0 1 0 0 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 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 1 0 0 0 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1 0 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 0 0 0 1 0 0 1 0 1 1 1 1 0 1 1 0 1 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 1 0 0 1 0 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 1 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 1 0 1 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 1 0 1 0 0 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 1 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 0 1 1 0 1 1 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 1 0 0 0 1 0 1 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 1 1 1 0 1 0 1 1 1 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 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