Bounds on the minimum distance of additive quantum codes
Bounds on [[63,35]]2
lower bound: | 7 |
upper bound: | 10 |
Construction
Construction of a [[63,35,7]] quantum code:
[1]: [[63, 35, 7]] quantum code over GF(2^2)
cyclic code of length 63 with generating polynomial x^61 + x^60 + x^59 + x^57 + w^2*x^56 + x^55 + x^54 + w*x^51 + x^50 + w^2*x^48 + w*x^47 + x^44 + w*x^43 + w*x^42 + w^2*x^40 + x^39 + w^2*x^38 + w*x^36 + w^2*x^34 + x^33 + w^2*x^29 + w*x^27 + w^2*x^26 + w^2*x^25 + x^24 + w^2*x^23 + w*x^22 + w*x^21 + x^18 + w^2*x^17 + w*x^16 + w*x^14 + 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 1 0 1 0 0 0 0 1 1 0 1 0 0 1 1 1 1 0 0 0 1 0 0 0 1 1 1 0 0 1 1 0 0 1 0|1 1 0 1 1 0 1 1 1 1 0 1 0 0 1 1 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 1 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 1 0 1 0 0 0 0 1 1 0 1 0 0 1 1 1 1 0 0 0 1 0 0 0 1 1 1 0 0 1 1 0 0 1|1 1 1 0 1 1 0 1 1 1 1 0 1 0 0 1 1 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 1 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 1 0 0 0 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 0|1 0 1 0 1 1 0 1 0 0 1 0 0 1 1 1 0 0 1 1 0 1 1 1 1 1 0 1 0 1 1 1 0 0 1 0 0 0 0 1 0 1 1 1 0 1 1 1 1 0 1 1 0 0 0 1 1 0 0 1 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 1 0 0 0 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1|0 1 0 1 0 1 1 0 1 0 0 1 0 0 1 1 1 0 0 1 1 0 1 1 1 1 1 0 1 0 1 1 1 0 0 1 0 0 0 0 1 0 1 1 1 0 1 1 1 1 0 1 1 0 0 0 1 1 0 0 1 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 1 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1 1 0 1|1 1 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 0 0 0 1 0 1 1 0 1 1 0 0 0 1 1 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 1 1 1 0 0 0 0 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 0 1 1 1 0 0 1 1 1 0 1 0 0|0 0 1 0 0 0 1 1 1 0 0 1 1 1 1 0 1 1 0 0 0 0 1 1 1 0 0 1 1 1 1 1 0 0 1 0 0 1 0 1 1 0 0 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 0 1 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 1 1 1 0 0 0 0 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 0 1 1 1 0 0 1 1 1 0 1 0|0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 1 0 1 1 0 0 0 0 1 1 1 0 0 1 1 1 1 1 0 0 1 0 0 1 0 1 1 0 0 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 0 1 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 1 1 1 0 0 0 0 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 0 1 1 1 0 0 1 1 1 0 1|1 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 1 0 1 1 0 0 0 0 1 1 1 0 0 1 1 1 1 1 0 0 1 0 0 1 0 1 1 0 0 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 0 1]
[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 1 0 1 1 1 1 0 1 1 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 0 1 1 1 1 1 1 0 0|0 0 0 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1 1 1 0 1 1 0 0 0 0 1 1 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 1 0 1 1 1 1 0 1 1 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 0 1 1 1 1 1 1 0|1 0 0 0 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1 1 1 0 1 1 0 0 0 0 1 1]
[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 1 0 1 1 1 1 0 1 1 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 0 1 1 1 1 1 1|1 1 0 0 0 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1 1 1 0 1 1 0 0 0 0 1]
[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 1 0 1 1 0 1 1 0 0 0 0 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1|0 0 1 1 1 0 0 0 0 0 1 0 0 1 1 1 1 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 0 0 1 0 0 0 1 1 0 1 0 1 1 0 1 0 0 0 1 0 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 1 1 1 1 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 1 1 1 0 0 0 1 1 0 0 0 1 0 0|0 1 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 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 1 1 1 1 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 1 1 1 0 0 0 1 1 0 0 0 1 0|1 0 1 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 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 1 1 1 1 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 1 1 1 0 0 0 1 1 0 0 0 1|1 1 0 1 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1]
[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 1 0 1 1 1 1 1 0 1 1 1 1 0 0 1 0 0 1 0 1 1 1 0 1 0 0 0 0 0 1 0 1 0 1 0|0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 1 1 0 0 0 0 1 0 1 1 0 1 0 1 1 0 0 1 0 1 0 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 1 1 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 1 0 1 1 1 1 1 0 1 1 1 1 0 0 1 0 0 1 0 1 1 1 0 1 0 0 0 0 0 1 0 1 0 1|1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 1 1 0 0 0 0 1 0 1 1 0 1 0 1 1 0 0 1 0 1 0 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 1 1]
[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 0 1 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0|0 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 0 0 1 1 0 1 1 0 1 1 1 1 0 0 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 0 0 0 0 1 0 1 1 1 0 0 1 1 0 0 0 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 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 1 0 1 0 0 1 1 1 0 0|0 0 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 0 0 1 1 0 1 1 0 1 1 1 1 0 0 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 0 0 0 0 1 0 1 1 1 0 0 1 1 0 0 0 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 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 1 0 1 0 0 1 1 1 0|1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 0 0 1 1 0 1 1 0 1 1 1 1 0 0 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 0 0 0 0 1 0 1 1 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 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 1 0 1 0 0 1 1 1|0 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 0 0 1 1 0 1 1 0 1 1 1 1 0 0 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 0 0 0 0 1 0 1 1 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 1 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 1 0 0 0 1 1 0 0 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 0 1|1 1 1 1 1 0 1 0 1 0 1 0 0 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 1 0 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 1 0 0 0 0 0 1 1 1 1 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1 1 0 1 0 0 0 0 1 0|0 0 1 0 0 1 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 0 0 0 1 0 1 1 0 1 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 1 0 0 0 0 0 1 1 1 1 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1 1 0 1 0 0 0 0 1|0 0 0 1 0 0 1 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 0 0 0 1 0 1 1 0 1 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 1 0 0 0 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 0 0 0 1 0|1 1 0 1 0 0 1 0 0 1 1 1 0 0 1 1 0 1 1 1 1 1 0 1 0 1 1 1 0 0 1 0 0 0 0 1 0 1 1 1 0 1 1 1 1 0 1 1 0 0 0 1 1 0 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 1 0 0 0 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 0 0 0 1|0 1 1 0 1 0 0 1 0 0 1 1 1 0 0 1 1 0 1 1 1 1 1 0 1 0 1 1 1 0 0 1 0 0 0 0 1 0 1 1 1 0 1 1 1 1 0 1 1 0 0 0 1 1 0 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 1 0 1 0 0 0 0 1 1 0 1 0 0 1 1 1 1 0 0 0 1 0 0 0 1 1 1 0 0 1 1 0 0 1 0 1 0|0 1 1 0 1 1 1 1 0 1 0 0 1 1 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 1 0 0 0 0 0 1 1 1 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 1 0 1 0 0 0 0 1 1 0 1 0 0 1 1 1 1 0 0 0 1 0 0 0 1 1 1 0 0 1 1 0 0 1 0 1|1 0 1 1 0 1 1 1 1 0 1 0 0 1 1 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 1 0 1 1]
last modified: 2008-02-18
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
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Last change: 23.10.2014