Bounds on the minimum distance of additive quantum codes
Bounds on [[111,37]]2
lower bound: | 14 |
upper bound: | 25 |
Construction
Construction of a [[111,37,14]] quantum code:
[1]: [[109, 37, 14]] quantum code over GF(2^2)
cyclic code of length 109 with generating polynomial w^2*x^106 + w^2*x^104 + w^2*x^103 + w^2*x^102 + w^2*x^101 + w^2*x^99 + w^2*x^98 + w^2*x^97 + x^95 + w^2*x^93 + w^2*x^92 + x^91 + w*x^90 + x^89 + w*x^87 + w^2*x^86 + x^85 + w*x^84 + w^2*x^83 + x^82 + w^2*x^81 + w*x^80 + w^2*x^79 + w^2*x^78 + w^2*x^75 + x^74 + w^2*x^73 + w^2*x^72 + x^71 + w^2*x^70 + w^2*x^67 + w^2*x^66 + w*x^65 + w^2*x^64 + x^63 + w^2*x^62 + w*x^61 + x^60 + w^2*x^59 + w*x^58 + x^56 + w*x^55 + x^54 + w^2*x^53 + w^2*x^52 + x^50 + w^2*x^48 + w^2*x^47 + w^2*x^46 + w^2*x^44 + w^2*x^43 + w^2*x^42 + w^2*x^41 + w^2*x^39 + x^36 + 1
[2]: [[111, 37, 14]] quantum code over GF(2^2)
ExtendCode [1] by 2
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 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0|0 1 1 0 1 1 1 1 1 0 1 1 0 1 1 0 0 0 1 0 1 1 1 1 0 0 1 0 1 1 0 0 0 1 0 1 0 1 0 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 0 0 1 1 0 1 1 0 1 1 1 1 1 0 1 1 0 1 1 1 0 1 0 1 1 0 1 1 1 0 0 1 0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 1 1 1 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.
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Markus Grassl
(codes@codetables.de).
Last change: 23.10.2014