Bounds on the minimum distance of additive quantum codes
Bounds on [[69,0]]2
lower bound: | 17 |
upper bound: | 24 |
Construction
Construction of a [[69,0,17]] quantum code:
[1]: [[69, 0, 17]] self-dual quantum code over GF(2^2)
cyclic code of length 69 with generating polynomial w^2*x^67 + w^2*x^66 + w*x^65 + x^64 + w*x^62 + w^2*x^61 + w^2*x^60 + x^58 + w*x^56 + x^55 + w*x^54 + w^2*x^53 + w^2*x^52 + x^51 + x^49 + w^2*x^48 + x^47 + x^45 + w^2*x^44 + x^43 + w^2*x^42 + w*x^41 + w^2*x^40 + x^38 + w*x^35 + w*x^32 + 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0|0 1 0 1 0 0 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 1 0 1 0 1 1 1 0 1 1 0 1 1 0 0 1 1 0 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 0 0 1 0 0 1 1 1 0 0 0 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 0 0 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 1 0 1 0 1 1 1 0 1 1 0 1 1 0 0 1 1 0 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 0 0 1 0 0 1 1 1 0 0 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0|0 1 0 0 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 1 0 1 0 1 1 1 0 1 1 0 1 1 0 0 1 1 0 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 0 0 1 0 0 1 1 1 0 0 0 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 1 0 1 0 1 1 1 0 1 1 0 1 1 0 0 1 1 0 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 0 0 1 0 0 1 1 1 0 0 0 1 0 1 0]
last modified: 2008-02-09
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