Bounds on the minimum distance of additive quantum codes
Bounds on [[47,16]]2
lower bound: | 8 |
upper bound: | 11 |
Construction
Construction of a [[47,16,8]] quantum code:
[1]: [[45, 16, 8]] quantum code over GF(2^2)
cyclic code of length 45 with generating polynomial x^44 + x^43 + x^42 + x^41 + x^40 + w*x^39 + w*x^38 + w^2*x^36 + w*x^35 + w*x^34 + w*x^33 + x^32 + w^2*x^30 + w*x^29 + x^28 + x^27 + w*x^26 + w^2*x^25 + w*x^23 + w^2*x^22 + w^2*x^21 + w*x^17 + w^2*x^14
[2]: [[47, 16, 8]] quantum code over GF(2^2)
ExtendCode [1] by 2
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 1 1 0 0 1 1 0 0 0 1 0 1 0 1 0 0|0 0 1 0 0 0 0 1 1 1 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 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 0 1 1 1 0 0 1 0 1 0 1 0 0 1 1 1 1 1 0 0|0 1 0 0 1 1 1 0 1 1 1 0 1 1 0 0 1 0 0 0 0 1 1 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 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 0 0 0 1 1 0 0 1 0 1 1 0 0 1 0 1 1 0 1 0 0 0|0 0 0 0 0 1 1 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 0 1 1 1 0 0 1 0 0 1 0 0 1 1 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 1 0 0 1 0 1 1 0 0 1 0 1 1 0 1 0 0|0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 0 1 1 1 0 0 1 0 0 1 0 0 1 1 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 1 0 1 1 0 1 0 0 0 0 0 1 1 0 0|0 1 0 1 1 1 1 1 1 0 1 1 1 0 0 0 1 0 1 0 1 0 1 0 0 1 1 1 0 1 1 1 1 0 0 1 0 0 0 1 0 1 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 1 1 1 1 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0|0 1 1 1 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 1 0 1 0 1 1 1 1 0 1 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 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 1 0 1 0 0 0|0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 1 0 1 0 1 1 1 1 0 1 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 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 1 0 1 0 0|0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 1 0 1 0 1 1 1 1 0 1 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 0 0 0 1 0 1 0 0 1 1 0 0 0 0 0 1 1 1 1 1 0 0|0 0 1 0 1 1 1 1 1 1 0 1 1 0 1 0 0 1 1 0 1 1 1 0 0 1 1 1 1 1 1 0 1 1 0 1 1 1 0 1 1 0 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 1 1 1 0 1 0 1 0 1 0 0 0 1 1 0 1 0 0 0|0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 0 0 1 0 0 1 1 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 1 1 1 0 1 0 1 0 1 0 0 0 1 1 0 1 0 0|0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 0 0 1 0 0 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 0 1 1 0 1 1 0 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 1 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0|0 0 0 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 1 1 0 0 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 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0|0 0 1 0 1 0 0 0 0 0 1 0 1 1 0 1 0 1 1 0 1 0 1 0 1 0 1 1 1 1 0 1 0 0 0 0 1 1 0 0 0 1 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 0 0 0 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0|0 1 1 0 1 0 1 1 1 1 1 0 1 0 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 1 1 1 0 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 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0|0 1 0 0 1 0 1 0 0 0 0 0 1 0 1 1 0 1 0 1 1 0 1 0 1 0 1 0 1 1 1 1 0 1 0 0 0 0 1 1 0 0 0 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 0 0 1 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 0 0|0 1 1 1 1 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 1 1 0 1 1 0 1 0 1 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 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 0|0 1 0 0 0 0 1 0 0 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 0 0 1 0 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 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 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 0 0 0 0 1 0 0 1 0 1 1 0 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 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 1 0 0|0 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0|0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 1 1 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 0 0 0 0 0 0 0 1 1 0 1 1 0 0 1 0 0 1 0 1 0 0 0 0 0 0|0 1 0 1 1 0 0 1 0 1 1 1 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 1 0 1 1 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 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 1 0 0 1 0 1 0 0 0 0 0|0 0 1 0 1 1 0 0 1 0 1 1 1 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 1 0 1 1 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 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 1 0 0 1 0 1 0 0 0 0|0 1 1 0 1 0 0 1 1 0 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 1 1 1 0 0 1 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 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 1 0 0 1 0 1 0 0 0|0 1 0 0 1 0 1 1 0 0 1 0 1 1 1 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 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 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 1 0 0 1 0 1 0 0|0 0 1 0 0 1 0 1 1 0 0 1 0 1 1 1 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 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 1 0 0 1 0 1 1 1 1 1 1 0 1 0 0 0 0 1 1 1 0 0|0 1 0 0 1 1 0 0 1 1 0 1 0 1 1 0 0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 1 0 1 0 0 0 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 1 0 1 1 1 0 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0|0 1 1 1 1 0 0 0 0 1 1 1 0 1 1 0 1 0 1 1 1 0 1 1 1 1 1 0 0 1 0 1 1 1 0 1 0 1 1 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 0 0 0 0 0 1 0 1 1 1 0 0 1 1 0 0 0 1 0 1 0 1 1 0 0|0 1 0 0 0 0 1 1 1 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 1 0 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 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0]
last modified: 2022-08-02
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