Bounds on the minimum distance of additive quantum codes
Bounds on [[51,25]]2
lower bound: | 7 |
upper bound: | 9 |
Construction
Construction of a [[51,25,7]] quantum code:
[1]: [[51, 26, 7]] quantum code over GF(2^2)
cyclic code of length 51 with generating polynomial w^2*x^47 + w*x^43 + w*x^41 + x^40 + x^38 + w^2*x^37 + x^36 + w^2*x^35 + w*x^34 + w^2*x^33 + w*x^32 + w^2*x^30 + x^29 + w^2*x^28 + x^27 + w*x^24 + w*x^23 + w*x^22 + w^2*x^21 + x^20 + w^2*x^19 + x^18 + w^2*x^17 + w^2*x^16 + w*x^15 + w*x^14 + x^13 + w*x^12 + w
[2]: [[51, 25, 7]] quantum code over GF(2^2)
Subcode of [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 1 0 1 1 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 1|0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 1 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 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 1 1 0 1 0 1 1 1 0 0 0|0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 1 0 0 0 1 0 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 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 1 1 0|0 0 1 1 1 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0 1 1 0 0 1 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 0 1 0 0 0 0 0 1 1 1 1 0 1 0 1 1 1 1 1 1 0 0|0 1 1 0 0 0 1 1 1 1 0 1 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 1 0 1 0 0 0 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 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 1 0 0 0 0 1 0 1|0 0 1 1 0 0 0 1 1 1 1 0 0 1 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 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 1 0 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 0 1 0 1 1 0|0 1 1 0 1 1 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 1 0 1 1 1 1 1 1 0 1 1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 1 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 1 0 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 0 0 1 0|0 1 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 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 1 0 1 1 1 1 1 0 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 1 1|0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 1 1 1 1 0 1 1 1 0 0 0 1 0 0 1 1 1 1 0 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 1 0 1 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 0 0 0 1 0 1|0 1 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 1 1 1 1 0 1 0 0 0 1 0 0 1 0 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 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 1 1 1 0 0 0 1 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 0|0 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 1 1 1 0 1 1 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 1 0 1 0 0 1 1 0 1 0 1 1 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 1 1 0 0 0 1 1 1 1 0 1 0 1 1 1 0 0 1 0 1 0 0 0 0 0|0 1 1 0 1 1 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 0 1 1 0 0 1 1 0 1 0 0 0 0 1 0 1 0 0 1 1 1 0 1 1]
[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 1 1 0 0 0 1 1 1 1 0 1 0 1 1 1 0 1 1 1 1 1 0 0 1|0 0 1 1 0 1 1 0 0 1 1 0 1 1 0 1 1 1 0 0 0 1 0 0 0 0 0 1 0 1 1 0 0 1 1 0 1 0 0 0 1 1 1 0 1 1 0 1 1 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 1 0 1 0 1 1 0 0 0 1 0 1 0 0 1 1 1 0 0 0 1 0 1 0 0 1 1|0 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 1 1 1 0 0 1 0 0 1 1 0 0 0 1 0 1 1 0 0 1 1 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 1 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0|0 1 1 1 1 1 0 1 0 0 0 0 1 1 1 1 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 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 1 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1|0 1 0 0 0 0 0 1 0 1 1 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 1 0 1 0 0 1 0 1 0 0 1 1 0 1 1 1 0|0 1 0 1 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 1 1 0 1 1 1 0 1 1 0 0 0 0 1 1 0 1 1 1 1 1 0 1 1 0 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 1 0 0 0 1 0 0 1 1 0 1 0 1 0 0 1 0 0 0 1 1 0 0 1 0 1|0 0 1 0 1 0 1 0 1 0 0 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 0 1 1 1 0 1 1 0 0 0 0 1 1 1 1 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 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 0|0 1 1 0 0 0 0 0 1 1 1 1 0 1 0 1 1 1 1 1 1 1 0 1 0 0 0 1 0 1 0 1 0 0 1 0 1 1 1 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 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 1 0 0 1 1 0 1 1 0 1 0|0 1 0 0 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 0 1 0 0 1 1 1 1 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 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 1 0 0 1 1 0 1 1 0 1|0 0 1 0 0 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 0 1 0 0 1 1 1 1]
[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 1 0 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1|0 1 1 0 0 1 1 0 0 1 0 1 1 1 0 1 0 1 1 1 1 1 1 1 1 0 1 0 0 1 1 0 1 0 1 1 0 1 1 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 1 0 0 1 1 1 0 0 0 0 1 0 0 0 1 1 0 0 1 1 1 0 0 1 1 1 0 0 0 1|0 1 0 0 0 1 1 0 1 0 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 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 1 0 1 1 0 0 0 0 0 0 1 0 1 0 0 1 0 1 1 0 0 1 0 1 1 1 1 1 0|0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 1 1 0 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 1 0 1 1 0 0 0 0 0 0 1 0 1 0 0 1 0 1 1 0 0 1 0 1 1 1 1 1|0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 1 1 0 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 1 0 1 0 1 0 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 0 0 0 0 1 0 1 1 1 1 1 0 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 0 0 0 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]
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