Bounds on the minimum distance of additive quantum codes
Bounds on [[75,4]]2
| lower bound: | 16 |
| upper bound: | 25 |
Construction
Construction of a [[75,4,16]] quantum code:
[1]: [[75, 4, 16]] quantum code over GF(2^2)
cyclic code of length 75 with generating polynomial w*x^74 + w^2*x^72 + x^70 + x^69 + x^67 + w^2*x^65 + w^2*x^64 + x^62 + x^58 + w*x^56 + w^2*x^55 + w*x^54 + w^2*x^53 + w*x^52 + w^2*x^51 + w^2*x^48 + x^47 + w^2*x^46 + w^2*x^45 + w^2*x^42 + x^40 + w^2*x^39 + x^37 + w^2*x^36 + 1
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 1 1 0 0|0 0 0 0 0 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 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 1 1 0|0 0 0 0 0 0 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 0 1 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 0 0 0 1 1 0 0 1|0 0 0 0 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0]
last modified: 2008-02-14
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