Bounds on the minimum distance of additive quantum codes
Bounds on [[105,73]]2
lower bound: | 7 |
upper bound: | 10 |
Construction
Construction of a [[105,73,7]] quantum code:
[1]: [[105, 74, 7]] quantum code over GF(2^2)
cyclic code of length 105 with generating polynomial x^104 + w^2*x^102 + w*x^100 + w^2*x^99 + w*x^97 + x^96 + x^93 + w*x^91 + x^90 + w^2*x^89 + x^88 + w*x^87 + x^85 + w^2*x^83 + w*x^81 + w^2*x^79 + w^2*x^77 + w^2*x^76 + w^2*x^75 + w^2*x^74 + w^2*x^73 + w^2*x^71 + x^70 + w*x^69 + w*x^68 + x^67 + w*x^64 + x^63 + w*x^62 + w*x^61 + w*x^59 + w*x^58 + w*x^56 + w^2*x^55 + x^54 + w*x^53 + w^2*x^52 + w*x^51 + x^50 + w*x^49 + x^46 + x^45 + w*x^44 + w*x^43 + x^41 + w*x^40 + x^39 + x^38 + w^2*x^37 + w^2*x^35 + w*x^34 + w*x^33 + x^31 + w^2*x^28 + x^26 + w*x^25 + w*x^23 + w*x^22 + w^2*x^20 + w^2*x^18 + w*x^16 + w*x^15 + x^8
[2]: [[105, 73, 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 1 1 0 1 1 0 1 0 1 1 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 0 0 1 0 1 1 0 1 0 1 1 0 0 1 0 1 1 0 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0|1 0 0 1 0 1 0 0 1 0 0 1 1 0 0 1 1 1 1 1 1 1 1 1 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 1 1 1 0 1 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 1 0 1 0 1 0 0 1 1 0 1 0 1 0 1 0 0 1 0 0 0 1 1 1 0 1 1 1 1 0 0 1 1 0 0 0 1 0 1 1 0 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 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 0 0 1 0 1 1 0 1 0 1 1 0 0 1 0 1 1 0 1 1 1 0 0 1 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 1|0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 1 0 1 1 1 1 1 1 1 1 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 1 1 1 0 1 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 1 0 1 0 1 0 0 1 1 0 1 0 1 0 1 0 0 1 0 0 0 1 1 1 0 1 1 1 1 1 0 0 1 0 1 1 0 1 0 0 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 1 1 1 0 1 1 0 0 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 1 0 0 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 0|0 0 1 1 0 0 0 1 1 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 1 0 0 0 1 0 1 0 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 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 1 1 0 0 0 1 0 1 0 1 0 1 1 1 1]
last modified: 2022-08-02
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