Bounds on the minimum distance of additive quantum codes
Bounds on [[99,1]]2
lower bound: | 21 |
upper bound: | 33 |
Construction
Construction of a [[99,1,21]] quantum code:
[1]: [[91, 1, 21]] quantum code over GF(2^2)
cyclic code of length 91 with generating polynomials [
w*x^89 + x^88 + w^2*x^87 + w^2*x^86 + w*x^85 + w^2*x^84 + w*x^83 + x^82 + x^80 + x^79 + w*x^78 + x^76 + w*x^75 + w^2*x^73 + x^72 + x^71 + w*x^70 + w*x^69 + x^66 + x^65 + x^64 + x^63 + w^2*x^61 + w*x^60 + w^2*x^59 + w^2*x^58 + w^2*x^57 + w^2*x^55 + w^2*x^53 + w^2*x^52 + x^51 + w*x^50 + x^49 + w*x^48 + w^2*x^47 + x^46 + x^45 + 1,
w^2*x^89 + w*x^88 + x^87 + x^86 + w^2*x^85 + x^84 + w^2*x^83 + w*x^82 + w*x^80 + w*x^79 + w^2*x^78 + w*x^76 + w^2*x^75 + x^73 + w*x^72 + w*x^71 + w^2*x^70 + w^2*x^69 + w*x^66 + w*x^65 + w*x^64 + w*x^63 + x^61 + w^2*x^60 + x^59 + x^58 + x^57 + x^55 + x^53 + x^52 + w*x^51 + w^2*x^50 + w*x^49 + w^2*x^48 + x^47 + w*x^46 + w*x^45 + w
]
[2]: [[99, 1, 21]] quantum code over GF(2^2)
ExtendCode [1] by 8
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 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 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 1 1 1 1 1 1 0 1 0 0 1 0 1 0 1 1 1 1 1 1 1 0 1 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 1 0 0 1 1 0 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 1 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
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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