Bounds on the minimum distance of additive quantum codes
Bounds on [[105,71]]2
lower bound: | 7 |
upper bound: | 10 |
Construction
Construction of a [[105,71,7]] quantum code:
[1]: [[105, 72, 7]] quantum code over GF(2^2)
cyclic code of length 105 with generating polynomial w^2*x^104 + x^102 + w^2*x^101 + w*x^99 + x^98 + w*x^97 + w^2*x^94 + x^93 + w^2*x^91 + x^90 + w^2*x^89 + w^2*x^88 + x^87 + x^86 + w^2*x^84 + w^2*x^83 + w^2*x^82 + x^80 + x^79 + w*x^78 + w*x^77 + x^76 + w^2*x^75 + w*x^74 + w*x^73 + w^2*x^72 + x^71 + w*x^68 + w*x^65 + w^2*x^64 + x^63 + x^61 + w*x^60 + w^2*x^59 + w^2*x^57 + w*x^56 + x^55 + w*x^54 + w^2*x^52 + x^50 + w^2*x^48 + x^47 + w^2*x^45 + w^2*x^42 + x^41 + w*x^40 + x^39 + w^2*x^37 + w*x^36 + x^35 + w*x^31 + x^29 + w^2*x^27 + w*x^26 + w*x^24 + x^23 + x^22 + w^2*x^21 + w^2*x^19 + w^2*x^18 + w*x^17 + w*x^16 + x^15 + x^13 + w*x^11 + w^2*x^10 + x^8 + x^6 + w^2*x^5 + w*x^4 + w*x^3 + x^2 + w^2*x + w
[2]: [[105, 71, 7]] 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 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0 1 0 1 1 0 1 0 1 0 0 0 1 0 0 1 1 0 1 1 0 1 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0|1 0 1 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 1 0 1 1 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 1 1 1 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 1 0 1 0 1 1 0 1 1 0 0 1 1 0 1 0 0 1 1 1 0 0 1 1 0 1 1 1 1 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 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0 1 0 1 1 0 1 0 1 0 0 0 1 0 0 1 1 0 1 1 0 1 1 1 1 0 1 0 1 0 1 1 0 1 1 1 0 0 1 1 1|1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 1 1 1 0 0 1 0 1 1 0 1 1 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 1 1 1 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 1 0 1 0 1 1 0 1 1 0 0 1 1 0 1 0 0 1 1 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 1 0 0 1 0 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 0 1 0 1 1 1 1 0 1 1 0 1 0 0 1 0 0 0 0 0 1 1 0 1 0 1 1 1 1 0 1|1 1 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 1 1 1 0 0 1 1 1 0 1 0 1 1 0 1 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 1 1 0 1 0 1 0 1 0 1 0 1 0 0 0 1 1 1 0 1 1 0 1 1 0 0 0 1 0 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 1 1 1 0 1 0 1 1 0 1 0|0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 0 0 1 0 1 1 1 1 0 1 1 1 0 0 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 0 1 0 1 0 0 0 1 1 0 0 1 0 0 0 1 1 1 0 1 1 1 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 1 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 1 1 0 1 0 1 0 1 0 1 0 1 0 0 0 1 1 1 0 1 1 0 1 1 0 0 0 1 0 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 0|0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 0 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 0 1 0 1 0 0 0 1 1 0 0 1 0 0 0 1 1 1 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 1 1 0 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 1 0 0]
last modified: 2020-09-17
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