Bounds on the minimum distance of additive quantum codes
Bounds on [[88,25]]2
lower bound: | 14 |
upper bound: | 22 |
Construction
Construction of a [[88,25,14]] quantum code:
[1]: [[85, 25, 14]] quantum code over GF(2^2)
cyclic code of length 85 with generating polynomials [
x^84 + x^83 + w^2*x^82 + w^2*x^81 + x^80 + x^79 + w^2*x^78 + w*x^77 + w^2*x^76 + w*x^75 + w*x^74 + x^73 + w^2*x^72 + w^2*x^69 + x^68 + x^66 + w^2*x^62 + w*x^59 + x^58 + x^57 + w^2*x^55 + w^2*x^54 + x^53 + w*x^52 + w*x^51 + x^50 + w^2*x^49 + w*x^48 + x^45 + x^44 + x^43 + w*x^42 + w*x^40 + w^2*x^38 + w*x^36 + w^2*x^35 + x^34 + x^33 + x^32 + x^31 + x^30 + 1,
w*x^84 + w*x^83 + x^82 + x^81 + w*x^80 + w*x^79 + x^78 + w^2*x^77 + x^76 + w^2*x^75 + w^2*x^74 + w*x^73 + x^72 + x^69 + w*x^68 + w*x^66 + x^62 + w^2*x^59 + w*x^58 + w*x^57 + x^55 + x^54 + w*x^53 + w^2*x^52 + w^2*x^51 + w*x^50 + x^49 + w^2*x^48 + w*x^45 + w*x^44 + w*x^43 + w^2*x^42 + w^2*x^40 + x^38 + w^2*x^36 + x^35 + w*x^34 + w*x^33 + w*x^32 + w*x^31 + w*x^30 + w
]
[2]: [[88, 25, 14]] quantum code over GF(2^2)
ExtendCode [1] 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 1 0 1 1 0 1 1 1 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 0 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 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 1 1 0 0 1 0 0 1 1 0 1 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 1 0 0 1 0 1 0 0 1 1 0 1 1 1 0 0 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 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 1 1 1 0 0 0 1 1 0 0 0 1 0 1 0 1 1 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 1 0 0 0|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 1 1 1 0 1 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 1 0 1 1 1 1 0 1 0 1 1 0 0 1 0 0 1 0 0 1 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