Bounds on the minimum distance of additive quantum codes
Bounds on [[56,25]]2
lower bound: | 8 |
upper bound: | 11 |
Construction
Construction of a [[56,25,8]] quantum code:
[1]: [[55, 25, 8]] quantum code over GF(2^2)
cyclic code of length 55 with generating polynomial w*x^53 + w*x^52 + w*x^51 + w^2*x^50 + w*x^49 + w^2*x^48 + x^47 + w^2*x^45 + w^2*x^43 + x^42 + w*x^40 + x^39 + w^2*x^38 + x^36 + x^35 + w*x^34 + w*x^33 + w^2*x^32 + w*x^31 + w^2*x^28 + w^2*x^24 + x^23 + x^22 + w^2*x^21 + w^2*x^20 + x^18 + w*x^16 + w*x^15 + w*x^14 + w
[2]: [[56, 25, 8]] quantum code over GF(2^2)
ExtendCode [1] by 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 1 0 1 1 0 0 1 1 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 0|0 1 1 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 1 1 0 1 1 0 0 1 1 1 0 1 0 1 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 1 1 1 0 1 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0|1 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 1 0 1 0 0 0 1 0 1 0 1 1 1 0 1 1 0 0 1 1 1 1 0 1 1 0 1 0 1 0 0 1 1 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 1 1 1 0 1 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0|1 1 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 1 0 1 0 0 0 1 0 1 0 1 1 1 0 1 1 0 0 1 1 1 1 0 1 1 0 1 0 1 0 0 1 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 1 0 0 0 1 0 0 1 1 0 0 0 1 1 1 1 0 0 0 1 0 0 0 1 1 0|1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 0 1 0 0 1 1 0 1 0 1 1 1 0 0 0 0 0 1 1 0 0 0 1 1 0 0 1 1 0 1 0 0 1 0 0 1 1 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 1 1 1 1 0 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0|0 0 1 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 1 1 1 1 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 0 0 1 1 1 0 0 1 1 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 1 1 1 1 0 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0|0 0 0 1 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 1 1 1 1 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 0 0 1 1 1 0 0 1 1 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 1 0 0 0 1 1 1 0 1 1 0 1 1 0 1 0 1 0 0 0 0 0 1 1 1 0|1 1 1 0 1 1 1 0 1 0 1 1 0 1 1 0 1 0 0 0 1 0 1 1 1 1 1 0 1 0 0 1 0 1 1 0 1 1 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 1 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 1 0 0 0 1 0 0 1 1 1 0 1 1 0 0 0 0 0 0 0 0|0 0 0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 0 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 0 1 0 1 0 0 1 1 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 1 1 1 1 0 1 0 0 0 1 0 0 1 1 1 0 1 1 0 0 0 0 0 0 0|1 0 0 0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 0 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 0 1 0 1 0 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 1 1 1 1 0 1 0 0 0 1 0 0 1 1 1 0 1 1 0 0 0 0 0 0|1 1 0 0 0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 0 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 0 1 0 1 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 1 1 1 1 0 1 0 0 0 1 0 0 1 1 1 0 1 1 0 0 0 0 0|0 1 1 0 0 0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 0 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 0 1 0 1 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 1 1 1 1 0 1 0 0 0 1 0 0 1 1 1 0 1 1 0 0 0 0|0 0 1 1 0 0 0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 0 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 0 1 0 1 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 1 1 1 1 0 1 0 0 0 1 0 0 1 1 1 0 1 1 0 0 0|1 0 0 1 1 0 0 0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 0 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 0 1 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 1 1 1 1 0 1 0 0 0 1 0 0 1 1 1 0 1 1 0 0|0 1 0 0 1 1 0 0 0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 0 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 0 1 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 1 1 1 1 0 1 0 0 0 1 0 0 1 1 1 0 1 1 0|1 0 1 0 0 1 1 0 0 0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 0 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 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 1 0 1 1 0 0 1 1 1 1 0 1 0 1 1 1 1 1 0 0 1 1 1 1 0 0|0 0 1 1 0 1 1 0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 1 0 0 0 0 0 1 1 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0 1 0 1 1 1 1 1 0 0 1 1 1 1 0|0 0 0 1 1 0 1 1 0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 1 0 0 0 0 0 1 1 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 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 0 0 0|1 1 1 0 1 0 0 0 0 1 1 1 0 0 1 1 1 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 1 0 1 0 1 1 0 1 0 0 0 0 1 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 1 0 0 1 1 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 0 0|1 1 1 1 0 1 0 0 0 0 1 1 1 0 0 1 1 1 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 1 0 1 0 1 1 0 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 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 0|0 1 1 1 1 0 1 0 0 0 0 1 1 1 0 0 1 1 1 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 1 0 1 0 1 1 0 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 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 1 1 0 0 1 0 1 0 0 1 1 1 1 1 0|1 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 0 1 1 0 1 1 1 0 0 1 1 1 0 0 1 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 1 0 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 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 0 1 0 0 1 1 0 0 0|1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 1 0 0 1 1 1 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 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 0 1 0 0 1 1 0 0|1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 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 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 0 1 0 0 1 1 0|1 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 1 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 1 0 0 0 0 0 1 0 1 0 1 1 1 1 0 1 0 0 1 0 1 0 0 1 1 0 0 1 0 1 0 0|1 0 0 1 0 1 0 0 1 1 1 1 0 0 1 0 1 0 0 1 1 1 1 0 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0 1 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 1 0 0 0 0 0 1 0 1 0 1 1 1 1 0 1 0 0 1 0 1 0 0 1 1 0 0 1 0 1 0|1 1 0 0 1 0 1 0 0 1 1 1 1 0 0 1 0 1 0 0 1 1 1 1 0 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0 1 1 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 1 0 0 0 1 0 0 1 1 0 0 0 1 1 1 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0|0 0 0 0 0 0 0 0 1 1 1 1 1 0 1 0 0 1 1 0 1 0 1 1 1 0 0 0 0 0 1 1 0 0 0 1 1 0 0 1 1 0 1 0 0 1 0 0 1 1 0 1 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 1 0 0 1 1 1 1 1 1 1 1 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 1 0|1 1 1 0 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1 0 0 1 1 0 1 0 1 0 0 0 1 1 0 1 1 1 1 1 0 1 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 1 0 1 1 0 0 1 1 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0|1 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 1 1 0 1 1 0 0 1 1 1 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 1 0 1 1 0 0 1 1 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0|1 1 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 1 1 0 1 1 0 0 1 1 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 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]
last modified: 2022-08-02
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
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Last change: 23.10.2014