Bounds on the minimum distance of additive quantum codes
Bounds on [[39,6]]2
lower bound: | 9 |
upper bound: | 12 |
Construction
Construction of a [[39,6,9]] quantum code:
[1]: [[40, 6, 10]] quantum code over GF(2^2)
QuasiCyclicCode of length 40 with generating polynomials: w^2*x^18 + w^2*x^17 + w^2*x^16 + w*x^15 + x^3, w^2*x^19 + w*x^18 + x^17 + w*x^15 + w*x^14 + w*x^12 + x^11 + w*x^10 + x^9 + w*x^7 + w*x^5
[2]: [[39, 6, 9]] quantum code over GF(2^2)
Puncturing of [1] at { 40 }
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 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 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 0 0 1 0 0 1 1 1 0 0 1 0 0 0 0 0 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 0 0 0 1 1 1|1 1 0 0 1 1 0 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 1 1 1 1 0 0 1 0 1 0 0 0 1 1 1 0 1]
[0 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 0 0 1 1 1|0 1 1 0 1 0 1 1 1 0 0 1 1 0 1 0 0 0 0 1 0 1 1 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 0]
[0 0 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 0 0 0 0|1 0 1 1 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 0 0 1 0 1 1 0 0 0]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 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 0 0 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 1 0 1 0 0 0 0 1 1 0 1 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 1 1 1|1 0 1 0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 0 0 1 1 0 0 0 1 1 1 0 0 0 1 0 1 1]
[0 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 0 0 0 0 0 0 0 0 0 0 1 1 1|0 1 0 1 1 1 0 1 1 0 1 1 1 1 1 1 0 0 1 1 0 1 1 0 1 1 1 1 0 1 1 1 0 0 0 0 1 1 1]
[0 0 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 0 0 0 0 0 0 0 0 0 1 1 1|1 0 1 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 0 0 0 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 0 1]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 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 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1]
[0 0 0 0 0 0 0 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 0 0 0 0 1 1 1|1 1 1 0 0 1 1 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 1 1 0 1 1 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|0 1 1 1 0 1 1 1 0 0 0 1 0 1 0 1 0 1 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 1 1 1 1]
[0 0 0 0 0 0 0 0 0 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 0 0 1 1 1|0 0 1 1 0 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 1 1 0 0 0 1 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 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1|0 1 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 1 0]
[0 0 0 0 0 0 0 0 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 1 0 0 1 0 1 0 0 1 0 0 0 1 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 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 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 1 0 1 0 0 1 0 0 0 1 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 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 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 1 0 1 0 0 1 0 0 0 1 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 0 0 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 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 0 0 0|1 1 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 0 0 1 0 0 1 1 1 0 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1|0 0 0 0 0 1 0 0 1 1 1 0 0 1 0 0 0 0 0 1 1 1 1 1 1 0 0 0 1 1 0 0 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 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0|1 0 0 0 0 1 1 0 1 0 0 1 0 1 1 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 1 0 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0|1 1 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 1 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 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 1 0 0 0 0 1 1 0 1 0 0 1 0 1 1 0 0 0 1 0 1 0 0 0 0 1 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 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 0 1 1 0 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 1 0 1 0 0 0 0 1 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 1 0 0 0 0 0 0 0 0 0 0 0 1 0|1 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 1 1 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 0 1 0 0 0 0 0 0 0 0 0 1 1 0|1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 1 1 0 0 1 1 0 1 0 0 1 1 1 1 1 1 1 1 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 1 0 0 0 0 0 0 0 0 1 1 1|1 1 1 0 1 1 1 0 0 1 1 1 0 1 1 1 1 1 1 1 0 1 0 1 1 0 1 0 1 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 1 0 0 0 0 0 0 0 0 1 1|1 1 1 1 0 1 1 1 0 0 1 1 1 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 0 1 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 1 0 0 0 0 0 0 0 1 0|1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 1 1 0 0 1 1 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 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1|0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 1 1 0 0 1 1 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 0 0 0 0 0 0 1 0 0 0 0 1 1 1|1 0 1 1 0 1 1 0 0 0 0 1 0 1 1 0 0 1 0 1 1 1 0 0 1 0 1 0 0 0 0 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0|1 1 0 1 0 0 1 0 1 1 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 1 0 0 1 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 0 0 0 0 0 0 1 0 0 0 1 0|0 1 1 0 1 0 0 1 0 1 1 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 1 0 0 1 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 0 0 0 0 0 0 1 0 0 0 1|0 0 1 1 0 1 0 0 1 0 1 1 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 1 0 0 1 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 0 0 0 0 0 0 1 0 0 0|0 0 0 1 1 0 1 0 0 1 0 1 1 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1]
last modified: 2023-01-04
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