Bounds on the minimum distance of additive quantum codes
Bounds on [[72,29]]2
lower bound: | 9 |
upper bound: | 15 |
Construction
Construction of a [[72,29,9]] quantum code:
[1]: [[65, 29, 9]] quantum code over GF(2^2)
cyclic code of length 65 with generating polynomial w*x^63 + x^62 + x^61 + w^2*x^59 + x^58 + x^57 + w^2*x^56 + x^55 + w*x^54 + w*x^53 + w^2*x^52 + w*x^51 + w*x^50 + x^48 + w^2*x^47 + w^2*x^46 + w*x^45 + w*x^44 + w^2*x^42 + w^2*x^41 + w*x^39 + w*x^38 + w^2*x^37 + w^2*x^36 + x^35 + w*x^33 + w*x^32 + w^2*x^31 + w*x^30 + w*x^29 + x^28 + w^2*x^27 + x^26 + x^25 + w^2*x^24 + x^22 + x^21 + w*x^20 + x^18 + 1
[2]: [[72, 29, 9]] quantum code over GF(2^2)
ExtendCode [1] by 7
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 1 0 1 0 1 1 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0|1 0 0 0 0 1 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 0 1 0 1 1 1 0 1 1 1 1 0 1 1 1 0 1 0 1 0 1 0 1 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 1 0 1 0 1 1 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0|1 1 0 0 0 0 1 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 0 1 0 1 1 1 0 1 1 1 1 0 1 1 1 0 1 0 1 0 1 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 1 0 0 0 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0|1 1 1 0 0 1 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 0 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 1 0 1 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 1 0 0 0 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 0 1 1 1 0 1 0 0 0 0 0 0 0 0|0 1 1 1 0 0 1 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 0 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 1 0 1 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 1 0 0 0 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 0 1 1 1 0 1 0 0 0 0 0 0 0|0 0 1 1 1 0 0 1 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 0 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 1 0 1 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 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
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