Bounds on the minimum distance of additive quantum codes
Bounds on [[93,62]]2
lower bound: | 7 |
upper bound: | 10 |
Construction
Construction of a [[93,62,7]] quantum code:
[1]: [[93, 63, 7]] quantum code over GF(2^2)
cyclic code of length 93 with generating polynomial x^92 + x^90 + x^88 + x^87 + w*x^86 + x^83 + w^2*x^81 + w^2*x^80 + w*x^79 + w^2*x^78 + w*x^77 + x^76 + x^75 + x^74 + x^72 + x^71 + w^2*x^70 + w*x^68 + x^67 + w*x^65 + x^64 + w*x^63 + w*x^61 + w*x^60 + x^59 + x^58 + w^2*x^57 + w*x^55 + w^2*x^54 + w^2*x^53 + w*x^52 + x^51 + w^2*x^50 + w*x^49 + x^48 + x^47 + x^45 + w^2*x^43 + w^2*x^41 + x^37 + x^36 + w^2*x^35 + x^34 + w^2*x^32 + w*x^29 + w^2*x^28 + w*x^27 + w*x^26 + w*x^25 + w*x^24 + w^2*x^22 + w*x^21 + w*x^19 + w*x^18 + w*x^17 + w*x^16 + w*x^14 + x
[2]: [[93, 62, 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 1 1 0 0 1 0 1 0 1 0 1 1 1 0 1 1 1 1 1 0 0 0 1 0 0 1 1 0 1 1 1 0 1 1 0 1 1 1 1 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0|0 1 0 1 0 1 0 0 1 1 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 1 0 1 1 1 1 1 0 0 1 0 1 0 0 0 1 0 1 1 0 1 1 1 1 1 1 0 0 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 1 1 1 0 1 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 1 1 0 0 1 0 1 0 1 0 1 1 1 0 1 1 1 1 1 0 0 0 1 0 0 1 1 0 1 1 1 0 1 1 0 1 1 1 1 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1|1 0 1 0 1 0 1 0 0 1 1 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 1 0 1 1 1 1 1 0 0 1 0 1 0 0 0 1 0 1 1 0 1 1 1 1 1 1 0 0 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 1 1 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 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0 1 1 0 1 0 1 1 1 1 0 1 0 1 0 1 1 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 0 1 1 1 1 1 1 0 0 1 1 0 0|1 0 0 0 0 0 0 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 0 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 0 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 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0 1 1 0 1 0 1 1 1 1 0 1 0 1 0 1 1 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 1|0 1 0 0 0 0 0 0 1 1 1 1 1 0 1 1 1 1 0 0 1 1 1 1 0 0 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 0 1 0 0 1 1 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 1 1 1 1 0 1 0 0 1 0 1 1 1 1 1 0 1 0 1 0 0 1 0 0 1 1 0 1 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 0 0 1 1 1 0|1 1 1 1 0 1 0 0 1 0 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 1 1 0 1 1 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 1 1 1 1 0 1 0 0 1 0 1 1 1 1 1 0 1 0 1 0 0 1 0 0 1 1 0 1 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 1 1 1 1 0 1 0 1 0 1 0 0|1 1 1 1 1 0 1 0 0 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 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 0 0 0 1 1 1 1 0 1 0 0 1 0 1 1 1 1 1 0 1 0 1 0 0 1 0 0 1 1 0 1 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 1 1 1 1 0 1 0 1 0 1 0|0 1 1 1 1 1 0 1 0 0 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 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 0 0 0 1 1 1 1 0 1 0 0 1 0 1 1 1 1 1 0 1 0 1 0 0 1 0 0 1 1 0 1 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 1 1 1 1 0 1 0 1 0 1|1 0 1 1 1 1 1 0 1 0 0 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 1]
[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 1 1 0 0 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 1 1 1 0 0 1 1 0 1 1 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 0|1 0 0 0 1 0 1 1 1 0 0 0 1 1 1 1 1 0 1 0 0 1 1 1 1 1 0 1 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 0 1 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 0]
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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 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 1 1 1 0 0 1 1 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 0 1|0 0 1 1 0 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 0 1 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 0 0 1 1 0 0 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 1 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 1 0 1 0 1 1 1 1 0 0 1 1 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 1 1 1 0 0 1 0 1 0 0 0 0 0 0 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