Bounds on the minimum distance of additive quantum codes
Bounds on [[37,21]]2
lower bound: | 4 |
upper bound: | 6 |
Construction
Construction of a [[37,21,4]] quantum code:
[1]: [[31, 21, 4]] quantum code over GF(2^2)
cyclic code of length 31 with generating polynomial x^30 + x^29 + w*x^26 + w^2*x^25 + w^2*x^24 + w^2*x^23 + x^21 + w*x^20 + w*x^18 + w^2*x^17 + x^16 + w*x^15 + x^14 + w*x^13 + x^11 + w^2*x^10 + x^9 + x^8 + w^2*x^7 + x^6 + w*x^5 + 1
[2]: [[32, 21, 4]] quantum code over GF(2^2)
ExtendCode [1] by 1
[3]: [[35, 21, 4]] quantum code over GF(2^2)
ExtendCode [2] by 3
[4]: [[37, 21, 4]] quantum code over GF(2^2)
ExtendCode [3] by 2
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0|0 1 1 0 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 1 0 0 1 1 0 1 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1 1 1 1 1 0 1 1 1 0 1 0 0 0 0 0 0 0 0|1 1 0 1 0 1 1 0 0 0 0 1 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1 1 1 1 1 0 1 1 1 0 1 0 0 0 0 0 0 0|1 1 1 0 1 0 1 1 0 0 0 0 1 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1 1 1 1 1 0 1 1 1 0 1 0 0 0 0 0 0|1 1 1 1 0 1 0 1 1 0 0 0 0 1 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0]
[0 0 0 0 1 0 0 0 0 0 1 0 1 1 1 0 1 0 0 1 1 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0|0 0 0 1 1 1 1 0 1 1 0 1 1 1 1 1 0 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 0 0 0 0 1 1 1 1 1 0 0 1 1 0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0|0 1 1 0 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 0 1 0 0 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 1 1 1 0 0 1 1 0 0 0 0 0 0 0|1 1 0 1 0 0 0 1 1 0 1 0 0 1 0 0 0 1 0 1 1 0 1 1 1 1 0 0 0 1 1 0 0 0 0 0 0]
[0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 1 1 1 0 0 1 1 0 0 0 0 0 0|1 1 1 0 1 0 0 0 1 1 0 1 0 0 1 0 0 0 1 0 1 1 0 1 1 1 1 0 0 0 1 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0|1 0 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 0 0 0 0 0 0|1 1 0 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 1 0 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 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 0 0 0 0]
[0 0 0 0 0 0 0 0 0 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 0 0]
[0 0 0 0 0 0 0 0 0 0 0 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: 2006-04-07
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 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