Bounds on the minimum distance of additive quantum codes
Bounds on [[88,43]]2
lower bound: | 10 |
upper bound: | 15 |
Construction
Construction of a [[88,43,10]] quantum code:
[1]: [[85, 45, 9]] quantum code over GF(2^2)
cyclic code of length 85 with generating polynomials [
x^84 + w*x^83 + x^82 + w*x^81 + w*x^80 + x^79 + w*x^78 + x^75 + w*x^74 + w*x^71 + w*x^69 + x^68 + w*x^67 + w^2*x^66 + w^2*x^65 + x^64 + w^2*x^62 + x^61 + x^57 + w^2*x^56 + w*x^55 + x^54 + w^2*x^53 + w^2*x^51 + w*x^50 + w*x^49 + w*x^48 + w^2*x^47 + w^2*x^46 + x^45 + w^2*x^44 + w*x^43 + w*x^41 + x^40 + w^2*x^39 + x^38 + w^2*x^35 + x^34 + x^33 + w*x^29 + w^2*x^28 + w*x^27 + w^2*x^26 + w^2*x^25 + x^24 + x^23 + x^20 + 1,
w*x^84 + w^2*x^83 + w*x^82 + w^2*x^81 + w^2*x^80 + w*x^79 + w^2*x^78 + w*x^75 + w^2*x^74 + w^2*x^71 + w^2*x^69 + w*x^68 + w^2*x^67 + x^66 + x^65 + w*x^64 + x^62 + w*x^61 + w*x^57 + x^56 + w^2*x^55 + w*x^54 + x^53 + x^51 + w^2*x^50 + w^2*x^49 + w^2*x^48 + x^47 + x^46 + w*x^45 + x^44 + w^2*x^43 + w^2*x^41 + w*x^40 + x^39 + w*x^38 + x^35 + w*x^34 + w*x^33 + w^2*x^29 + x^28 + w^2*x^27 + x^26 + x^25 + w*x^24 + w*x^23 + w*x^20 + w
]
[2]: [[86, 44, 10]] quantum code over GF(2^2)
standard lengthening of [1]
[3]: [[86, 43, 10]] quantum code over GF(2^2)
Subcode of [2]
[4]: [[88, 43, 10]] quantum code over GF(2^2)
ExtendCode [3] by 2
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 1 0 0 0 0 1 0 1 1 1 0 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 1 0 0 1 0 0 0 0 0 1 0 0|0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 0 0 1 1 0 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 1 0 1 0 0 1 1 0 0 1 0 0 1 1 1 0 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 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 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 0 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0|0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 1 0 0 0 1 1 1 1 1 0 0 1 0 1 1 0 1 0 1 1 1 1 0 1 1 1 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 1 1 0 0 0 0 0 0 0 1 1 1 0 0]
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[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 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 0 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 0 0|0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 1 0 0 0 1 1 0 0 1 1 1 0 0 0 0 1 0 1 0 1 1 1 0 1 1 0 1 1 0 1 0 1 1 1 1 1 1 1 0 1 0 0 1 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 0 0 0 0 1 0 1 1 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 0 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 0 1 1 0 1 1 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 1 1 0 1 1 1 0 1 1 0 1 0 0 1 1 1 1 1 0 0 1 1 1 0 1 0 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 1 1 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.
This page is maintained by
Markus Grassl
(codes@codetables.de).
Last change: 23.10.2014