Bounds on the minimum distance of additive quantum codes
Bounds on [[55,21]]2
lower bound: | 8 |
upper bound: | 12 |
Construction
Construction of a [[55,21,8]] quantum code:
[1]: [[55, 21, 8]] quantum code over GF(2^2)
cyclic code of length 55 with generating polynomial w^2*x^49 + x^48 + x^47 + x^46 + w*x^44 + w^2*x^43 + w*x^41 + w*x^37 + x^35 + w*x^34 + w*x^32 + w^2*x^31 + x^30 + w^2*x^27 + w*x^25 + x^24 + x^22 + 1
stabilizer matrix:
[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 1 0 0 1 0 1 0 0 1 1 1 0 1 1 1 0 1 1 0 1 1|0 1 1 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 0 0 0 1 0 0 1 1 0 0 0 1 1 1 1 0 1 0 0 1 0 1 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 0 1 1 0 0 1 1 0 1 1 0|0 1 0 1 0 0 0 1 0 1 1 1 1 0 1 0 1 0 1 1 0 1 0 0 0 0 1 1 0 1 0 1 0 0 1 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 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 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 0 1 1 0 0 1 1 0 1 1|0 0 1 0 1 0 0 0 1 0 1 1 1 1 0 1 0 1 0 1 1 0 1 0 0 0 0 1 1 0 1 0 1 0 0 1 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 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 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 1 0 0 1 0 0 0 1 0 1 1 0|0 1 1 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 1 1 0 1 0 1 1 1 0 0 0 1 1 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 1 0 0 1 0 0 0 1 0 1 1|1 0 1 1 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 1 1 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 1 1 1 0|1 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 1 0 1 0 1 0]
[0 0 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 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 1 1 1|0 1 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 1 0 1 0 1]
[0 0 0 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 0 0 0 1 0 1 1 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0|1 1 0 0 1 1 1 0 1 0 0 1 1 0 0 0 1 1 0 0 1 0 0 1 0 1 1 1 1 1 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 0 0 1 1 0 0 1 0]
[0 0 0 0 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 0 0 0 1 0 1 1 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0|0 1 1 0 0 1 1 1 0 1 0 0 1 1 0 0 0 1 1 0 0 1 0 0 1 0 1 1 1 1 1 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 0 0 1 1 0 0 1]
[0 0 0 0 0 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 0 0 0 1 0 1 1 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1|1 0 1 1 0 0 1 1 1 0 1 0 0 1 1 0 0 0 1 1 0 0 1 0 0 1 0 1 1 1 1 1 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 0 0 1 1 0 0]
[0 0 0 0 0 0 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 0 0 0 0 1 1 1 0 0 1 0 0 1 0 1 1 1 0 0|0 0 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 1 1 1 0 1 1 0 0 0 1 1 1 1 1 1 0 1 1 1 0]
[0 0 0 0 0 0 0 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 0 0 0 0 1 1 1 0 0 1 0 0 1 0 1 1 1 0|0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 1 1 1 0 1 1 0 0 0 1 1 1 1 1 1 0 1 1 1]
[0 0 0 0 0 0 0 0 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 0 0 0 0 1 1 1 0 0 1 0 0 1 0 1 1 1|1 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 1 1 1 0 1 1 0 0 0 1 1 1 1 1 1 0 1 1]
[0 0 0 0 0 0 0 0 0 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 1 0 0 0 0 1 0 0 1 0 0 1 1 1 0 0 1 0 0 0 0|1 0 1 0 0 1 1 0 1 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 1 0 0 1 1 0 0 1 1 1 1 0 1 0 1]
[0 0 0 0 0 0 0 0 0 0 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 1 0 0 0 0 1 0 0 1 0 0 1 1 1 0 0 1 0 0 0|1 1 0 1 0 0 1 1 0 1 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 1 0 0 1 1 0 0 1 1 1 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 1 1 1 0 0 1 0 0|0 1 1 0 1 0 0 1 1 0 1 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 1 0 0 1 1 0 0 1 1 1 1 0 1]
[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 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 1 1 1 0 0 1 0|1 0 1 1 0 1 0 0 1 1 0 1 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 1 0 0 1 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 1 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 0 0 1 1 1 0 0 1|0 1 0 1 1 0 1 0 0 1 1 0 1 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 1 0 0 1 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 1 1|1 1 0 0 1 1 0 0 1 0 0 1 1 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 0 1 0 1 0 1 0 1 1 1 0 0 1 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 1 0 0 0 1 0 1 1 1 1 0 0 0|1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 0 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 1 0 0 0 1 0 1 1 1 1 0 0|1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 0 0 1 1 1 1 1 1 1 0 0 0 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 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 1 0 0 0 1 0 1 1 1 1 0|1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0 1 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 1 0 0 0 1 0 1 1 1 1|1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 0 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 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 0 1 1 0 0|1 0 0 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 1 1 1 0 1 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 1 0 0 0 0 0 1 1 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 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 0 1 1 0|1 1 0 0 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 1 1 1 0 1 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 1 0 0 0 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 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 0 1 1|1 1 1 0 0 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 1 1 1 0 1 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 1 0 0 0 0 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 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 0 1 0|0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 1 0 0 0 0 0 1 1 0 1 0 1 0 0 1 1 1 0 0 1 0 0 0 0 1 1 1 1 0 0 1 0 0 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 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 0 1|1 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 1 0 0 0 0 0 1 1 0 1 0 1 0 0 1 1 1 0 0 1 0 0 0 0 1 1 1 1 0 0 1 0 0 0 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 1 0 0 0 0 0 1 0 1 1 0 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1|1 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 1 1 1 0 0 0 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 1 0 0 0 1 0 1 0 0 1 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 1 0 0 0 0 1 1 0 0 1 1 1 0 0 1 1 0 0 0 0 0 1 1 1 1 0|1 0 1 1 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 1 0 1 1 1 1 1 0 1 0 1 0 0 0 1 1 0 1 0 0 0 1 0 0 1 0 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 1 0 0 0 0 1 1 0 0 1 1 1 0 0 1 1 0 0 0 0 0 1 1 1 1|1 1 0 1 1 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 1 0 1 1 1 1 1 0 1 0 1 0 0 0 1 1 0 1 0 0 0 1 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 1 0 0 1 0 1 0 0 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0 0|0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 0 0 0 1 0 0 1 1 0 0 0 1 1 1 1 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 0 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 1 0 0 1 0 1 0 0 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0|1 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 0 0 0 1 0 0 1 1 0 0 0 1 1 1 1 0 1 0 0 1 0 1 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 1 0 1 1 0 1 1 1|1 1 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 0 0 0 1 0 0 1 1 0 0 0 1 1 1 1 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0]
last modified: 2008-02-14
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