Bounds on the minimum distance of additive quantum codes
Bounds on [[63,8]]2
lower bound: | 13 |
upper bound: | 20 |
Construction
Construction of a [[63,8,13]] quantum code:
[1]: [[63, 8, 13]] quantum code over GF(2^2)
cyclic code of length 63 with generating polynomial x^62 + w*x^61 + w*x^59 + w*x^58 + x^57 + w*x^55 + x^54 + x^53 + w*x^52 + w*x^51 + w^2*x^50 + x^48 + x^46 + x^44 + w^2*x^43 + w^2*x^42 + x^41 + w^2*x^40 + x^39 + w^2*x^37 + x^36 + w*x^35 + x^33 + w^2*x^31 + w*x^29 + w*x^27 + w
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 1 0 0 1 0 0 1 1|0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 1 0 1 1 0 1 0|0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 1 1 0 0 1 1 0 1 1 1 0 1 0 0 0 1 0 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 0 1|0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 1 1 0 0 1 1 0 1 1 1 0 1 0 0 0 1 0 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 1 0 1|0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 1 0 1 0 1 1 1 0 1 0 0 0 0 1 0 1 0 1 1 1 1 0 1 0 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 1|0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 0 0 0 1 0 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 1|0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 1 0 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0]
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[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 1 1 1 0|0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 0 1 1 0 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 1|0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
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