Bounds on the minimum distance of additive quantum codes
Bounds on [[78,10]]2
lower bound: | 14 |
upper bound: | 24 |
Construction
Construction of a [[78,10,14]] quantum code:
[1]: [[75, 9, 15]] quantum code over GF(2^2)
cyclic code of length 75 with generating polynomial w^2*x^74 + w^2*x^73 + w^2*x^72 + w^2*x^71 + w*x^70 + x^69 + w*x^68 + w*x^67 + w^2*x^65 + w^2*x^64 + w*x^63 + w*x^62 + x^61 + w^2*x^59 + w^2*x^58 + x^57 + w*x^55 + w^2*x^54 + x^53 + w^2*x^52 + w*x^51 + x^50 + x^48 + x^47 + w*x^46 + x^43 + w^2*x^41 + x^39 + x^36 + w^2*x^35 + x^33 + 1
[2]: [[74, 10, 14]] quantum code over GF(2^2)
Shortening of the stabilizer code of [1] at { 75 }
[3]: [[78, 10, 14]] quantum code over GF(2^2)
ExtendCode [2] by 4
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 1 1 0 1 0 0 1 0 1 0 0 0 0|0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 0 0 0 1 0 0 1 1 0 0 1 1 0 1 0 0 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 0 0 0|0 1 1 1 0 1 1 1 0 0 1 1 1 1 0 0 0 1 1 1 0 1 1 1 1 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1 0 1 1 1 1 1 1 1 1 0 0 1 1 0 0 0 0 0 1 0 0 0 0]
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last modified: 2008-02-16
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