Bounds on the minimum distance of additive quantum codes
Bounds on [[38,13]]2
lower bound: | 7 |
upper bound: | 9 |
Construction
Construction of a [[38,13,7]] quantum code:
[1]: [[35, 13, 7]] quantum code over GF(2^2)
cyclic code of length 35 with generating polynomials [
x^34 + x^33 + w*x^32 + x^31 + x^30 + w^2*x^29 + w*x^27 + w*x^26 + x^25 + w^2*x^24 + w*x^23 + x^21 + w^2*x^20 + w*x^19 + w^2*x^18 + w^2*x^17 + w^2*x^15 + w*x^14 + w^2*x^13 + w^2*x^12 + x^11 + 1,
w*x^34 + w*x^33 + w^2*x^32 + w*x^31 + w*x^30 + x^29 + w^2*x^27 + w^2*x^26 + w*x^25 + x^24 + w^2*x^23 + w*x^21 + x^20 + w^2*x^19 + x^18 + x^17 + x^15 + w^2*x^14 + x^13 + x^12 + w*x^11 + w
]
[2]: [[38, 13, 7]] quantum code over GF(2^2)
ExtendCode [1] by 3
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 1 0 1 1 0 0 1 1 0 0 0 1 1 1 0 1 1 0 0 0|0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 1 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 1 0 0 0 0 0|1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 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 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 0|0 1 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 1 1 0 0 1 0 1 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 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 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0|0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 1 1 0 0 1 0 1 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 1 0 1 1 0 1 0 1 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 1 1 0 1 1 0 0 1 1 1 0 0 1 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 1 1 0 1 1 0 0 1 1 1 0 0 1 1 0 0 0 0|0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0]
[0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 1 0 1 1 0 1 0 1 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 1 1 0 1 1 0 0 1 1 1 0 0 1 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 1 1 0 1 1 0 0 1 1 1 0 0 1 1 0 0 0|0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 0 0 1 0 0 0 1 1 0 0 0 0 0 0]
[0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 1 1 0 1 0 1 1 0 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0|0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 1 0 1 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 1 0 1 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0|0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 1 1 0 1 0 1 1 0 1 1 0 0 0 0 1 0 1 0 1 0 0 0|0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 1 0 1 1 1 0 1 0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 1 0 1 1 1 0 1 0 0 0 0 0 0 1 0 0 0|0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0]
[0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0|0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 0 0 0|0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 1 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 1 1 0 1 0 1 0 1 0 0 0|0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 1 0 0 0 0 1 1 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 1 0 0 0 0 1 1 0 0 0 0|0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 0 1 1 0 0 0]
[0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 1 1 0 0 1 1 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 1 1 1 0 0 1 1 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0|0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 1 0 1 0 0 1 1 0 1 0 1 0 0 1 1 0 1 1 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 1 0 1 1 0 0 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0|0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 1 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 1 0 0 0 0 0 0|0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 0 1 0 1 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 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 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 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]
last modified: 2006-04-07
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 necessarily 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: 10.06.2024