Bounds on the minimum distance of additive quantum codes
Bounds on [[94,84]]2
lower bound: | 3 |
upper bound: | 3 |
Construction
Construction of a [[94,84,3]] quantum code:
[1]: [[99, 89, 3]] quantum code over GF(2^2)
QuasiCyclicCode of length 99 with generating polynomials: w^2*x^10 + w*x^9 + x^7 + x^6 + w*x^5 + w^2*x^4 + w^2*x^3 + w*x + 1, x^8 + x^5 + x^4 + w^2*x^3 + w*x^2 + x + 1, w*x^10 + x^9 + x^8 + x^6 + w*x^5 + w^2*x^4 + w^2*x^2 + w^2*x + w, w^2*x^10 + w^2*x^9 + w^2*x^7 + w^2*x^6 + w^2*x^5 + x^4 + w*x^3 + x^2 + w*x + w^2, w^2*x^10 + w*x^8 + w*x^6 + x^4 + w^2*x^3 + 1, x^10 + x^6 + w*x^5 + w^2*x^3 + w*x^2 + w^2, x^10 + x^9 + x^8 + x^7 + w^2*x^5 + w*x^2 + 1, x^10 + w*x^8 + w*x^7 + w^2*x^6 + w^2*x^5 + x^3 + w^2*x^2 + x + w, w*x^10 + x^9 + w^2*x^8 + w*x^7 + w^2*x^6 + w^2*x^5 + w^2*x^4 + w*x^3 + w*x^2 + w*x + w^2
[2]: [[94, 84, 3]] quantum code over GF(2^2)
Shortening of [1] at { 9, 55, 78, 86, 90 }
stabilizer matrix:
[1 0 0 0 0 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 0 0 1 1 1 0 1 0 1 0 0 0 1 1 0 1 1 0 0 1 0 0 0 0 0 1 1 1 1 0 0 1 0 0 1 1 0 1 0 1 0 1 0 0 1 1 0|0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 1 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 1 1 0 1 1 0 0 0 1 0 1 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 1 1 0 0 1 0 1]
[0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 1 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 1 1 0 1 1 0 0 0 1 0 1 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 1 1 0 0 1 0 1|1 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 0 0 1 0 1 1 0 0 0 1 1 1 0 0 1 0 1 0 0 1 0 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1]
[0 1 0 0 0 1 0 1 0 1 1 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 0 1 1 0 1 1 0 1 1 0 1 1 0 0 1 1 1 1 0 1 0 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0 0|0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 1 1 0 1 1 0 1 0 0 1 0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 0 1 1 0 1 1 1 0 1 1 1 0 1 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0]
[0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 1 1 0 1 1 0 1 0 0 1 0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 0 1 1 0 1 1 1 0 1 1 1 0 1 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0|0 1 0 0 0 0 0 0 1 1 0 1 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 1 1 0 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 0 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 1 0 0 1 1 1 0 0 1 0 0 1 0 0 1 1 1 0 0 0 0 1 0 0]
[0 0 1 0 0 1 1 0 1 1 1 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 0 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 1 1 0 0 1 1 1 1 1 0 1 1 1 0|0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 0 1 0 0 1 1 1 0 1 1 1 1]
[0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 0 1 0 0 1 1 1 0 1 1 1 1|0 0 1 0 0 1 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 1 1 1 0 0 1 0 1 1 0 1 1 0 0 1 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 0 1 0 1 0 1 1 0 0 0 0 1 1 1 0 1 1 1 0 0 1 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 1]
[0 0 0 1 0 1 1 1 0 0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 0 1 1 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 1|0 0 0 0 0 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 1 1 0 0 1 0 1 1 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 0 0 0 0 1 0 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 1 0 0 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 0 1 0]
[0 0 0 0 0 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 1 1 0 0 1 0 1 1 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 0 0 0 0 1 0 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 1 0 0 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 0 1 0|0 0 0 1 0 1 0 0 0 1 0 0 1 0 1 1 1 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 1 1 0 1 1 0 1 1 0 1 1 1 0 1 1 1 1 0 1 0 0 1 0 1 0 1 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1 1]
[0 0 0 0 1 0 0 0 1 1 1 0 1 1 0 1 1 1 1 1 0 0 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 1 1 0 1 0 1 0 1 0 0 1 1 1 0 1 0 1 0 0 0 1 1 0 1 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 1 1 0 1 1 1 0 1 0 0 1 1 0 1|0 0 0 0 0 1 1 0 1 0 1 1 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 1 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 1 0 0 0 1 0 1 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 1 1 1 1 1 0 1 1 0 0 1 0 1 0]
[0 0 0 0 0 1 1 0 1 0 1 1 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 1 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 1 0 0 0 1 0 1 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 1 1 1 1 1 0 1 1 0 0 1 0 1 0|0 0 0 0 1 1 1 0 0 1 0 1 0 0 1 1 0 0 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 0 0 0 1 0 1 1 0 0 0 1 1 1 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 1 1 1]
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
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Last change: 23.10.2014