Bounds on the minimum distance of additive quantum codes
Bounds on [[117,92]]2
lower bound: | 5 |
upper bound: | 8 |
Construction
Construction of a [[117,92,5]] quantum code:
[1]: [[117, 93, 5]] quantum code over GF(2^2)
cyclic code of length 117 with generating polynomial w^2*x^116 + w^2*x^115 + w^2*x^113 + w*x^112 + x^111 + x^110 + w*x^109 + w*x^108 + w^2*x^106 + w*x^105 + w*x^104 + x^103 + x^100 + w^2*x^99 + w^2*x^98 + x^97 + x^95 + w*x^93 + w*x^92 + w*x^90 + x^89 + x^88 + w*x^87 + x^86 + w*x^85 + w^2*x^83 + w^2*x^82 + w*x^81 + w*x^80 + w^2*x^78 + w*x^77 + w*x^74 + w*x^71 + x^69 + w*x^68 + w^2*x^67 + w*x^66 + w*x^65 + x^64 + x^63 + x^62 + x^61 + x^60 + w^2*x^58 + w^2*x^56 + w*x^55 + x^53 + w*x^52 + w^2*x^51 + w^2*x^49 + w^2*x^48 + w^2*x^47 + w*x^46 + x^44 + x^41 + x^40 + w^2*x^39 + w^2*x^37 + x^36 + w*x^34 + w^2*x^33 + w^2*x^32 + w^2*x^31 + x^30 + w^2*x^29 + w^2*x^27 + w^2*x^26 + w*x^24 + w*x^23 + x^20 + w*x^19 + w^2*x^18 + w^2*x^17 + x^16 + x^15 + x^14 + w*x^11 + 1
[2]: [[117, 92, 5]] quantum code over GF(2^2)
Subcode of [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 1 0 1 0 0 1 0 0 1 1 1 0 0 1 0 1 0 1 0 0 1 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 1 1 0 1 1 0 1 1 1 1 0 0 1 0 0 0 1 1 1 0 1 1 1 1 1 0 0 0 1 1 1 1 1 0 0 0 1 0 0 1 0 1 1 1 1 1 1 1|0 1 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 0 1 1 0 1 1 1 0 1 0 0 0 0 0 1 0 1 0 1 0 1 1 0 1 1 1 0 1 0 1 0 0 0 1 0 0 1 0 0 1 0 1 1 1 0 0 1 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 1 1 0 1 0 0 0 1 0 0 0 0 1 0 0 0 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 1 1 1 1 0 1 1 0 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 1 0 0 0 0 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 1 1 0 1 1 1 0 1 1|1 1 0 1 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 0 0 1 1 1 1 1 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 1 1 1 1 1 0 1 1 0 0 1 0 1 1 1 0 0 0 0 0 1 0 1 1 0 0 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 0 1 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 1 1 1 1 0 1 1 0 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 1 0 0 0 0 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 1 1 0 1 1|1 1 1 0 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 1 1 0 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 0 0 1 1 1 1 1 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 1 1 1 1 1 0 1 1 0 0 1 0 1 1 1 0 0 0 0 0 1 0 1 1 0 0 1 1 1 0 0 1 0 0 1 0 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 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 1 0 0 0 0 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 1 0 1 1 1 0 0 0 0|0 1 1 1 0 1 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 0 0 1 1 1 1 1 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 1 1 1 1 1 0 1 1 0 0 1 0 1 1 1 0 0 0 0 0 1 0 1 1 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 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 0 0 0 1 1 1 1 0 1 1 0 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 1 0 0 0 0 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 1 0 1 1 1 0 0 0|0 0 1 1 1 0 1 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 0 0 1 1 1 1 1 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 1 1 1 1 1 0 1 1 0 0 1 0 1 1 1 0 0 0 0 0 1 0 1 1 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 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 0 0 0 1 1 1 1 0 1 1 0 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 1 0 0 0 0 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1|1 0 0 1 1 1 0 1 0 0 0 1 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 0 0 1 1 1 1 1 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 1 1 1 1 1 0 1 1 0 0 1 0 1 1 1 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0 0 1 0 0 0 1 1 0 1 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 1 1 1 1 0 1 1 0 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 1 0 0 0 0 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 1 0 1 1 1 0|0 1 0 0 1 1 1 0 1 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 0 0 1 1 1 1 1 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 1 1 1 1 1 0 1 1 0 0 1 0 1 1 1 0 0 0 0 0 1 0 1 1 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1]
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[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 0 1 1 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 1 1 1 1 0 0 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 1 0 0 1 0 0 0 0 1 1|1 0 0 0 0 1 1 0 1 1 1 1 1 1 0 0 1 0 1 0 0 1 0 1 1 1 1 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 1 1 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 1 1 1 1 1 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1]
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[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 1 0 1 1 0 1 0 1 0 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 1 1 0 1 0 0 1 0 1 0 0 0 1 0 0 0 1 1 0 0 1 0 1 1 0 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1 0 0 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 0 1 1 0 0 0 1 0|0 1 1 0 1 1 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 1 1 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0 0 1 0 0 0 0 1 0 1 1 1 1 1 0 1 1 1 0 0 0 0 1 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 0 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 1 0 1 0 0 1 0 0 1 1 1 0 0 1 0 1 0 1 0 0 1 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 1 1 0 1 1 0 1 1 1 1 0 0 1 0 0 0 1 1 1 0 1 1 1 1 1 0 0 0 1 1 1 1 1 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 1|1 0 0 0 0 0 1 0 0 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 1 0 0 0 0 0 1 0 1 0 1 0 1 1 0 1 1 1 0 1 0 1 0 0 0 1 0 0 1 0 0 1 0 1 1 1 0 0 1 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 1 1 0 1 0 0 0 1 0 0 0 1 1 1 0 0 1 1 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 1 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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Last change: 23.10.2014