Bounds on the minimum distance of additive quantum codes
Bounds on [[81,18]]2
lower bound: | 12 |
upper bound: | 22 |
Construction
Construction of a [[81,18,12]] quantum code:
[1]: [[85, 13, 16]] quantum code over GF(2^2)
cyclic code of length 85 with generating polynomials [
w^2*x^84 + w^2*x^83 + x^82 + w^2*x^81 + w^2*x^80 + w^2*x^79 + w*x^78 + x^77 + w*x^76 + x^75 + w^2*x^74 + w*x^72 + w*x^71 + w*x^70 + w*x^69 + x^68 + w^2*x^67 + w^2*x^66 + x^63 + w*x^62 + w*x^61 + x^60 + w^2*x^58 + x^57 + w^2*x^55 + w*x^54 + x^53 + w*x^52 + x^51 + w^2*x^50 + w*x^49 + w^2*x^48 + w*x^47 + w*x^46 + x^45 + x^43 + x^40 + x^39 + x^38 + w*x^37 + x^36 + 1,
x^84 + x^83 + w*x^82 + x^81 + x^80 + x^79 + w^2*x^78 + w*x^77 + w^2*x^76 + w*x^75 + x^74 + w^2*x^72 + w^2*x^71 + w^2*x^70 + w^2*x^69 + w*x^68 + x^67 + x^66 + w*x^63 + w^2*x^62 + w^2*x^61 + w*x^60 + x^58 + w*x^57 + x^55 + w^2*x^54 + w*x^53 + w^2*x^52 + w*x^51 + x^50 + w^2*x^49 + x^48 + w^2*x^47 + w^2*x^46 + w*x^45 + w*x^43 + w*x^40 + w*x^39 + w*x^38 + w^2*x^37 + w*x^36 + w
]
[2]: [[80, 18, 12]] quantum code over GF(2^2)
Shortening of the stabilizer code of [1] at { 1, 2, 7, 18, 85 }
[3]: [[81, 18, 12]] quantum code over GF(2^2)
ExtendCode [2] by 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 0|0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 1 0 1 1 1 0 0 0 1 1 0 1 0 1 0 0 0 0 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 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 0 0 0 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 1 0 1 0 0 1 1 1 0 1 0 1 1 1 1 1 0 1 1 1 1 1 0|0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 1 1 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 0 0 1 1 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 1 1 0 0 1 0 0 1 0 0 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 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 1 1 1 1 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 1 0|0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 1 1 0 1 1 1 0 1 1 1 1 1 0 1 0 0 0 1 1 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 0 1 0 1 1 0 0 0 0]
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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.
This page is maintained by
Markus Grassl
(codes@codetables.de).
Last change: 23.10.2014