Bounds on the minimum distance of additive quantum codes
Bounds on [[87,2]]2
lower bound: | 20 |
upper bound: | 29 |
Construction
Construction of a [[87,2,20]] quantum code:
[1]: [[85, 1, 21]] quantum code over GF(2^2)
cyclic code of length 85 with generating polynomials [
x^84 + w*x^83 + w^2*x^82 + x^81 + w^2*x^80 + w^2*x^79 + x^78 + w^2*x^77 + w^2*x^76 + w*x^75 + w*x^74 + x^73 + x^72 + w*x^71 + w^2*x^70 + x^69 + w*x^68 + w^2*x^67 + w^2*x^66 + x^65 + w*x^64 + w*x^63 + w^2*x^61 + w^2*x^59 + x^58 + x^55 + w^2*x^53 + w*x^52 + w*x^51 + w*x^50 + w^2*x^48 + w^2*x^47 + x^46 + x^44 + w*x^43 + x^40 + w*x^39 + w^2*x^36 + w^2*x^35 + w^2*x^32 + w*x^31 + x^30 + w^2*x^28 + x^23 + w^2*x^22 + x^21 + w^2*x^19 + x^16 + w*x^15 + w*x^14 + w^2*x^12 + w*x^11 + w^2*x^10 + w*x^8 + w*x^7 + x^6 + w^2*x^5 + w^2*x^3 + w^2*x^2 + x,
x^83 + w^2*x^82 + x^81 + w*x^80 + w*x^79 + x^78 + w^2*x^76 + w^2*x^75 + w*x^74 + x^73 + w^2*x^72 + w*x^71 + w*x^69 + x^68 + w*x^65 + x^64 + w*x^62 + x^61 + w*x^60 + x^59 + w*x^58 + x^57 + w^2*x^56 + x^55 + w*x^52 + w*x^51 + w*x^49 + x^48 + x^47 + w^2*x^46 + w^2*x^45 + x^44 + w^2*x^43 + w*x^42 + w^2*x^41 + w^2*x^40 + x^39 + w*x^37 + w^2*x^35 + w^2*x^34 + w^2*x^33 + w^2*x^32 + x^30 + w^2*x^29 + x^28 + x^27 + x^26 + w*x^25 + w*x^23 + x^22 + x^21 + w*x^20 + x^18 + w*x^17 + w^2*x^16 + w*x^15 + x^13 + w*x^12 + w*x^11 + x^9 + w^2*x^8 + w^2*x^7 + x^6 + x^5 + w*x^4 + w^2*x^3 + w^2*x^2 + x
]
[2]: [[84, 2, 20]] quantum code over GF(2^2)
Shortening of the stabilizer code of [1] at 85
[3]: [[87, 2, 20]] quantum code over GF(2^2)
ExtendCode [2] by 3
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 0 0 0 0 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 1 0 0 1 1 1 1 1 1 1 0 0 0|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 0 0 0 1 0 1 1 1 1 0 0 1 0 0 1 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 0 0 0 0 0 0 0 0 0 0 0 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 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 1 1 1 1 0 1 1 0 1 1 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 0 0 0 0 1 1 1 1 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 0 0 0 0 1 1 1 0 0 1 1 1 0 1 0 1 0 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 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 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 1 1 1 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 1 0 1 1 1 0 0 0 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 0 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 1 0 1 1 0 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0]
last modified: 2006-04-07
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.
This page is maintained by
Markus Grassl
(codes@codetables.de).
Last change: 23.10.2014