Bounds on the minimum distance of additive quantum codes
Bounds on [[79,37]]2
lower bound: | 9 |
upper bound: | 14 |
Construction
Construction of a [[79,37,9]] quantum code:
[1]: [[78, 38, 9]] quantum code over GF(2^2)
QuasiCyclicCode of length 78 stacked to height 2 with generating polynomials: w^2*x^38 + x^37 + w*x^36 + w*x^35 + w*x^34 + w*x^33 + w*x^32 + w^2*x^30 + w^2*x^29 + w^2*x^28 + w^2*x^27 + x^25 + w*x^24 + w*x^23 + w*x^21 + x^11, x^37 + w^2*x^36 + w^2*x^35 + w^2*x^34 + w^2*x^33 + w*x^32 + w^2*x^31 + x^30 + w*x^29 + w^2*x^28 + w*x^24 + x^23 + w*x^21 + w*x^20 + w*x^19 + w^2*x^18 + w*x^17 + x^16 + w^2*x^15 + x^13 + w^2*x^12 + w^2*x^11 + x^10 + w^2*x^9 + w^2*x^6 + x^5 + w^2*x^4 + x^3 + x^2 + w^2, x^38 + w*x^37 + w^2*x^36 + w^2*x^35 + w^2*x^34 + w^2*x^33 + w^2*x^32 + x^30 + x^29 + x^28 + x^27 + w*x^25 + w^2*x^24 + w^2*x^23 + w^2*x^21 + w*x^11, w*x^37 + x^36 + x^35 + x^34 + x^33 + w^2*x^32 + x^31 + w*x^30 + w^2*x^29 + x^28 + w^2*x^24 + w*x^23 + w^2*x^21 + w^2*x^20 + w^2*x^19 + x^18 + w^2*x^17 + w*x^16 + x^15 + w*x^13 + x^12 + x^11 + w*x^10 + x^9 + x^6 + w*x^5 + x^4 + w*x^3 + w*x^2 + 1
[2]: [[78, 37, 9]] quantum code over GF(2^2)
Subcode of [1]
[3]: [[79, 37, 9]] quantum code over GF(2^2)
ExtendCode [2] by 1
stabilizer matrix:
[1 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 1 1 1 1 0 1 1 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 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 1 1 1 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 0 0 1 0 1 0 1 1 0 1 1 1 0 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 1 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 1 0 0 0 0 0 0 0 1 0 0 1 1 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 0 1 0 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 1 0 1 1 0 0 0 0 1 1 1 1 0 1 1 1 0 0 1 0 0 0 0 1 0 0 1 1 0 0 1 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 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 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 0 0 0 0 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 0 0 0 0 1 1 0 0 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 0 0 0 0 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 0 1 1 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 0 0 1 1 0 0 1 1 1 0 0 1 1 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 0 0 0 1 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 1 1 1 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 1 0 1 0 1 1 0 0 0 0 1 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 1 1 0 0 1 1 1 0 0 1 1 1 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 1 0 1 0 1 0 0 1 0 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 0 1 1 0 0 1 1 1 1 0 0 1 0 0 0 1 1 0 1 1 1 1 0 1 0 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 1 0 1 0 0 1 0 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 0 1 1 0 0 1 1 1 1 0 0 1 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 1 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 1 1 1 0 0 1 1 1 0 0 1 0 1 0 0 1 1 1 0 1 1 1 1 1 0 1 1 1 0 1 1 1 1 0 0 0 0 1 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 1 1 0]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 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 0 1 0 1 1 0 0 0 0 1 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 1 1 0 0 0 0 1 0 1 0 0]
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[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 0 1 1 1 0 0 1 0 0 1 0 1 1 1 0 0 0 1 0 0 1 1 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 0 1 0 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 0 1 0 0 0 1 1 0 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0]
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[0 0 0 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 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 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 1 1 1 1 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 1 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 0 0 1 0 0 1 1 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 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 1 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 1 1 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 1 1 1 0 1 0 1 0 0 0 1 1 1 0 0 1 0 1 0 0 1 1 1 0 1 0 0 0 1 0 1 0 0 0 0 1 0 1 1 1 0 1 1 1 1 0 1 0 0 1 0 1 0 1 1 0 0 1 0]
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[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 1 0 0 0 1 0 1 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 0 1 0 0 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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]
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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(codes@codetables.de).
Last change: 23.10.2014