Bounds on the minimum distance of additive quantum codes
Bounds on [[78,16]]2
lower bound: | 12 |
upper bound: | 22 |
Construction
Construction of a [[78,16,12]] quantum code:
[1]: [[85, 9, 18]] quantum code over GF(2^2)
cyclic code of length 85 with generating polynomials [
x^38 + w*x^37 + w*x^36 + x^35 + w*x^34 + w^2*x^32 + x^31 + w*x^30 + w^2*x^29 + x^28 + w^2*x^27 + w^2*x^26 + w*x^25 + x^24 + w^2*x^22 + w*x^21 + x^19 + x^16 + w^2*x^15 + w*x^14 + w^2*x^12 + x^11 + w*x^10 + x^9 + w^2*x^8 + x^6 + x^5 + w*x^4 + x^3 + x^2 + x + 1,
w*x^38 + w^2*x^37 + w^2*x^36 + w*x^35 + w^2*x^34 + x^32 + w*x^31 + w^2*x^30 + x^29 + w*x^28 + x^27 + x^26 + w^2*x^25 + w*x^24 + x^22 + w^2*x^21 + w*x^19 + w*x^16 + x^15 + w^2*x^14 + x^12 + w*x^11 + w^2*x^10 + w*x^9 + x^8 + w*x^6 + w*x^5 + w^2*x^4 + w*x^3 + w*x^2 + w*x + w
]
[2]: [[80, 14, 14]] quantum code over GF(2^2)
Shortening of the stabilizer code of [1] at { 1, 15, 67, 69, 70 }
[3]: [[78, 16, 12]] quantum code over GF(2^2)
Shortening of the stabilizer code of [2] at { 79, 80 }
stabilizer matrix:
[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 1 1 1 0 1 0 1 0 0 1 0 1 0 0 0 0 0 1 1 0 1 1|0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 0 1 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 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 0 0 0 0 0 0 0 0 0 0 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 1 1 0 1 1 0 0 0 1 0 0 0 1 1 0 1 1 0 0 0 1 0|0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 0 1 0 1 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1]
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[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 0 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 0 1 0 0 0 1 1 1 0 1 0 1 1 1 1 1 0 0 1 0 1 1 0]
last modified: 2006-04-11
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.
This page is maintained by
Markus Grassl
(codes@codetables.de).
Last change: 23.10.2014