Bounds on the minimum distance of additive quantum codes
Bounds on [[86,45]]2
lower bound: | 9 |
upper bound: | 14 |
Construction
Construction of a [[86,45,9]] quantum code:
[1]: [[85, 45, 9]] quantum code over GF(2^2)
cyclic code of length 85 with generating polynomial x^84 + x^83 + w^2*x^82 + w*x^81 + w^2*x^80 + w^2*x^79 + x^78 + w*x^75 + w^2*x^72 + w^2*x^71 + w*x^70 + w*x^69 + x^67 + w*x^66 + w^2*x^64 + w*x^63 + x^62 + w^2*x^61 + x^60 + x^59 + x^58 + w*x^57 + w^2*x^56 + w^2*x^55 + w^2*x^54 + x^53 + w*x^52 + w^2*x^51 + x^50 + x^49 + x^48 + x^46 + w*x^45 + w^2*x^44 + w*x^41 + w*x^40 + w^2*x^39 + x^38 + x^37 + w*x^35 + x^33 + w*x^30 + w*x^28 + x^27 + x^23 + w^2*x^21 + x^20 + 1
[2]: [[86, 45, 9]] quantum code over GF(2^2)
ExtendCode [1] 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 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 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 1 0 0 1 0 1 0 1 1 1 1 0 0|0 0 0 0 1 0 1 0 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 1 0 1 1 0 0 0 1 0 0 1 0 0 1 0 0 1 1 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 0 1 1 0 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 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 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 1 0 0 1 0 1 0 1 1 1 1 0|1 0 0 0 0 1 0 1 0 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 1 0 1 1 0 0 0 1 0 0 1 0 0 1 0 0 1 1 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 0 1 1 0 1 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 1 1 0 1 0 0 0 0 1 1 1 1 1 1 0 1 0 1 0 1 0 1 1 1 1 0 0 1 0 1 0 1 1 0 1 1 1 0 0 0 0 1 0 0 1 0|0 1 0 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 1 0 1 1 1 0 1 0 1 1 0 1 1 0 1 1 1 0 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 0 1 1 0 1 0 1 1 0 1 1 0 0 0 1 1 0 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0|1 0 1 0 1 1 1 0 1 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 1 1 1 1 0 0 1 0 0 1 0 0 0 0 1 1 0 1 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 0 0 1 1 0 1 1 0 0 1 1 1 0 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 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 1 0 0 0 0 1 1 0 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 0 1 0|1 1 0 1 0 1 1 1 0 1 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 1 1 1 1 0 0 1 0 0 1 0 0 0 0 1 1 0 1 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 0 0 1 1 0 1 1 0 0 1 1 1 0 0 0 1 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 1 1 0 0 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 1 0 0 0 0|1 1 1 0 0 0 0 1 0 0 0 1 0 1 0 0 1 1 1 0 1 1 0 0 0 1 1 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 0 0 0 0 0 1 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 1 0 0 0|1 1 1 1 0 0 0 0 1 0 0 0 1 0 1 0 0 1 1 1 0 1 1 0 0 0 1 1 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 0 0 0 0 0 1 1 1 1 0 1 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 1 1 0 0 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 1 0 0|0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 1 0 0 1 1 1 0 1 1 0 0 0 1 1 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 0 0 0 0 0 1 1 1 1 0 1 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 1 1 0 0 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 1 0|0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 1 0 0 1 1 1 0 1 1 0 0 0 1 1 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 0 0 0 0 0 1 1 1 1 0 1 0]
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[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 1 1 1 0 0 1 1 0 1 1 0 0 1 0 1 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 0|1 1 1 1 0 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 1 1 0 1 1 1 1 1 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1 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 0 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 0 1 1 0 1 1 0 0 1 0 1 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0|1 1 1 1 1 0 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 1 1 0 1 1 1 1 1 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1 0 1 1 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 1 1 1 0 0 1 1 0 1 1 0 0 1 0 1 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0|0 1 1 1 1 1 0 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 1 1 0 1 1 1 1 1 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1 0 1 1 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 1 1 1 0 0 1 1 0 1 1 0 0 1 0 1 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0|0 0 1 1 1 1 1 0 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 1 1 0 1 1 1 1 1 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1 0 1 1 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 1 1 1 0 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 0 0 1 0 1 1 0 0 0 0 1 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0|1 0 0 1 0 1 0 1 1 0 0 1 0 0 0 1 0 1 1 1 0 0 1 1 1 1 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0 0 1 1 0 1 0 0 1 1 0 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 1 1 1 1 1 1 0 0 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 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 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 1 0 0 1 0 1 0 1 1 1 1 0 1 0|0 0 0 1 0 1 0 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 1 0 1 1 0 0 0 1 0 0 1 0 0 1 0 0 1 1 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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]
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