Bounds on the minimum distance of additive quantum codes
Bounds on [[61,30]]2
lower bound: | 7 |
upper bound: | 11 |
Construction
Construction of a [[61,30,7]] quantum code:
[1]: [[58, 30, 7]] quantum code over GF(2^2)
QuasiCyclicCode of length 58 with generating polynomials: w^2*x^28 + x^27 + w^2*x^25 + w*x^24 + w*x^23 + x^22 + w*x^21 + w*x^19 + w*x^18 + x^15 + w*x^14 + x^13 + x^12 + w*x^10 + x^8 + x^7 + w*x^6 + w*x^5 + w*x^4 + 1, x^28 + w*x^27 + w*x^26 + x^24 + w*x^23 + x^22 + w^2*x^21 + w*x^19 + w^2*x^18 + x^17 + x^15 + w^2*x^14 + w^2*x^13 + w*x^12 + w^2*x^11 + w*x^10 + w^2*x^9 + x^6 + w*x^3 + w^2*x^2 + w^2*x + w
[2]: [[61, 30, 7]] quantum code over GF(2^2)
ExtendCode [1] by 3
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1 1 1 1 1 0 1 0 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 0 0 1 1 0 0 1 0 1 1 1 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 1 0 0 1 1 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 0 0 1 1 0 0 1 0 1 1 1 0 1 0 0 0|1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 0 0 0 1 0 1 0 1 1 1 1 1 1 1 0 1 1 0 0 1 1 1 0 1 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 1 0 0 1 1 0 1 0 1 0 0 1 1 0 0 1 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 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 0 1 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 1 0 0 1 0 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 0 1 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 1 0 0 1 0 0 1 0 0 1 0 0 0|0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 0 1 1 0 1 1 1 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 1 1 0 1 1 0 1 1 0 0 1 1 0 1 1 1 0 0 1 1 0 0 1 0 0 1 0 0 0 1 0 1 0 1 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 0 1 1 1 1 0 0 1 0 1 0 1 0 0 1 1 1 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 1 0 0 1 0 0 1 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 1 0 1 0 0 1 1 1 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 1 0 0 1 0 0 1 0 0 0 0 0|0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 0 1 1 0 1 1 1 1 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 0 0 1 0 0 1 1 0 0 0 0 1 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 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 1 1 1 1 0 1 0 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 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 0 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 1 0 0 0|0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 1 1 1 0 0 1 1 1 1 0 0 0 0 1 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 1 1 0 0 1 0 1 1 1 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 1 0 0 0 1 0 0 0|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 1 0 1 0 1 1 0 0 1 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 1 1 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 1 0 1 0 1 1 0 0 1 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 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 0 1 0 1 1 1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 0 0 0]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 1 1 1 0 1 0 0 1 0 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 1 0 1 1 1 0 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 1 0 1 0 0 1 1 0 1 1 1 1 0 1 0 1 0 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 0 1 1 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 1 0 0 1 1 0 1 1 1 1 0 1 0 1 0 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 0 1 1 1 1 1 0 1 0 0 0|0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 0 1 1 1 1 0 1 1 0 0 1 1 1 0 1 1 0 1 1 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1 1 1 0 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 1 0 1 1 0 1 0 1 1 0 0 1 0 0 1 1 0 1 1 1 0 1 1 0 1 0 0 0 1 0 0 1 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 1 0 1 1 0 1 0 1 1 0 0 1 0 0 1 1 0 1 1 1 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0|0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 0 1 1 1 0 0 0 1 0 0 0 1 1 1 1 1 0 0 0 0 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 0 1 1 0 0 0 0]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 1 0 0 0 1 0 1 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 1 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 1 1 0 1 0 1 0 1 1 0 0 0 1 0 1 1 1 1 1 1 0 0 1 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 1 1 0 1 0 1 0 1 1 0 0 0 1 0 1 1 1 1 1 1 0 0 1 0 1 0 0 0|0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 1 0 1 0 1 0 0 1 1 1 1 1 1 1 0 1 0 1 1 1 0 1 0 0 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 0 0 0 0 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 1 1 0 0 0 1 1 0 1 1 1 1 0 0 0 1 0 0 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 1 1 1 1 0 1 1 0 0 1 0 1 0 0 0 1 0 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 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 1 1 1 1 0 1 1 0 0 1 0 1 0 0 0 1 0 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 0 0 0 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 1 1 0 1 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 1 1 1 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 1 1 1 1 1 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 1 1 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 1 1 0 0 1 1 1 1 0 0 0 1 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 1 1 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 1 1 0 0 1 1 1 1 0 0 0 1 1 1 0 0 1 0 0 0|0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 1 0 1 1 0 0 1 0 0 1 0 0 0 0 1 0 0 1 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 1 1 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 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 1 0 1 1 1 0 0 1 0 1 0 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 0 1 1 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 1 0 1 1 1 0 0 1 0 1 0 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 0 1 1 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 1 0 0 0 1 0 1 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 0 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 1 1 1 0 0 1 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 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 0 1 0 0 1 0 1 0 1 0 0 1 1 1 1 0 0 1 0 0 1 1 1 0 1 1 0 0 0 0 1 0 0 0 1 1 1 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 1 0 0 1 0 1 0 1 0 0 1 1 1 1 0 0 1 0 0 1 1 1 0 1 1 0 0 0 0 1 0 0 0 1 1 1 0 1 1 1 0 1 0 0 0 0 0|0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 1 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 1 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 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 0 1 0 0 1 0 1 0 1 0 0 1 1 1 1 0 0 1 0 0 1 1 1 0 1 1 0 0 0 0 1 0 0 0 1 1 1 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 1 0 0 1 0 1 0 1 0 0 1 1 1 1 0 0 1 0 0 1 1 1 0 1 1 0 0 0 0 1 0 0 0 1 1 1 0 1 1 1 0 1 0 0 0 0|0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 1 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 1 0 1 0 1 0 0 0 0 0 0 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 0 0 1 0 0 1 1 0 0 1 0 0 1 1 0 0 1 0 0 0 0 1 1 1 1 1 0 1 0 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 0|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 0 0 1 1 0 0 1 0 1 1 1 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 1 0 0 1 1 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 0 0 1 1 0 0 1 0 1 1 1 0 1 0 0 0 0|0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 0 0 0 1 0 1 0 1 1 1 1 1 1 1 0 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 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 0 0 0 0 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]
last modified: 2006-04-03
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
(codes@codetables.de).
Last change: 23.10.2014