Bounds on the minimum distance of additive quantum codes
Bounds on [[37,18]]2
lower bound: | 5 |
upper bound: | 7 |
Construction
Construction of a [[37,18,5]] quantum code:
[1]: [[35, 20, 5]] quantum code over GF(2^2)
cyclic code of length 35 with generating polynomial x^33 + x^30 + x^29 + w*x^28 + w*x^27 + x^26 + x^25 + w^2*x^24 + x^23 + x^22 + w*x^20 + w*x^19 + w^2*x^17 + w^2*x^16 + w*x^14 + w*x^11 + w^2*x^10 + w*x^9 + w*x^8 + 1
[2]: [[35, 18, 5]] quantum code over GF(2^2)
Subcode of [1]
[3]: [[37, 18, 5]] quantum code over GF(2^2)
ExtendCode [2] by 2
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 1 1 1 1 0 0 1 1 0 0 1 0 0 0|0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 1 1 0 1 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 1 1 0 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 1 0 0 0 0 0|1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 1 1 1 0 0 1 0 0 1 0 1 1 1 0 1 0 0]
[0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 1 1 1 1 0 0 1 1 0 0 1 0 0|0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 1 1 0 1 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 1 0 1 0 1 1 0 0 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0|0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 1 1 1 1 1 0 0]
[0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 1 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 0 0 0|0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 0 0 1 1 0 1 1 1 0 1 0 1 1 1 1 1 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 1 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 0|0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0]
[0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 1 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 0 0|0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 0 0 1 1 0 1 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 0 1 1 0 0|0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 1 0 0]
[0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 1 1 0 1 0 0|0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 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 1 0 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 1 1 0 0 0 1 1 0 0|0 0 0 0 1 0 0 0 1 1 1 1 0 0 1 0 0 1 1 1 0 0 0 1 1 1 0 0 1 0 1 0 1 0 1 0 0]
[0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 0 0 0 0|0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 1 1 0 0 0 0 0 0|0 0 0 0 0 1 0 0 0 1 0 1 1 1 0 1 1 0 1 0 0 1 1 0 1 0 1 1 1 0 1 1 0 1 1 0 0]
[0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 0 0 0|0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 0|0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 0 1 1 0 1 0 0 1 1 0 1 0 0 0 0 0 0 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 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 1 1 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 1 1 1 1 0 0 1 1 0 0 1 0 1 0 0|0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 1 1 0 1 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 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 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]
last modified: 2006-04-06
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