Bounds on the minimum distance of additive quantum codes
Bounds on [[61,38]]2
lower bound: | 6 |
upper bound: | 8 |
Construction
Construction of a [[61,38,6]] quantum code:
[1]: [[60, 38, 6]] quantum code over GF(2^2)
cyclic code of length 60 with generating polynomial w*x^59 + w*x^55 + w*x^54 + w*x^51 + w*x^49 + w*x^48 + w*x^47 + w*x^46 + w*x^45 + w*x^44 + w*x^42 + w*x^40 + x^38 + x^37 + x^36 + x^35 + x^34 + w*x^31 + w*x^30 + w*x^29 + w^2*x^28 + w^2*x^27 + x^23 + x^21 + w^2*x^20 + w^2*x^19 + x^17 + x^14 + w*x^13 + w^2*x^12 + w*x^11 + w*x^10 + x^9 + x^7 + w^2*x^6 + x^5 + w^2*x^3 + x^2 + w^2*x + 1
[2]: [[61, 38, 6]] 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 1 1 1 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0|0 0 1 0 1 0 1 1 1 1 1 1 0 1 0 0 1 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 1 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 1 0 0 0 1 1 0 0 1 1 1 0 1 1 1 1 0 1 1 1 0 0 1 1 1 0 0 1 0 1 0 0 0 0 1 0 0 0 0|0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 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 1 0 0 0 1 1 0 0 1 1 1 0 1 1 1 1 0 1 1 1 0 0 1 1 1 0 0 1 0 1 0 0 0 0 1 0 0 0|0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 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 1 0 0 0 1 1 0 0 1 1 1 0 1 1 1 1 0 1 1 1 0 0 1 1 1 0 0 1 0 1 0 0 0 0 1 0 0|0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 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 1 0 0 0 1 1 0 0 1 1 1 0 1 1 1 1 0 1 1 1 0 0 1 1 1 0 0 1 0 1 0 0 0 0 1 0|0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 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 1 1 1 1 1 1 1 1 1 0 0 0 0 1 0 0 1 0 1 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1 1 1 1 1 0|0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 1 1 0 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 1 0 1 0 1 1 1 1 0 0 0 0 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 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0|0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 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 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0|1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 0 0 1 0 0 1 0 1 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 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0|0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 0 0 1 0 0 1 0 1 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 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0|0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 0 0 1 0 0 1 0 1 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 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0|0 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 0 0 1 0 0 1 0 1 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 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0|0 0 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 0 0 1 0 0 1 0 1 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 1 1 1 1 0 1 0 1 0 1 1 0 1 0 0 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0|0 0 1 0 1 1 1 1 0 0 0 0 1 0 1 1 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 1 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 1 0 0 0 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0|0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 1 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 1 1 1 1 0 0 0 1 0 0 0 0 1 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 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0|0 0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 1 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 1 1 1 1 0 0 0 1 0 0 0 0 1 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 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0|0 0 0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 1 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 1 1 1 1 0 0 0 1 0 0 0 0 1 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 1 0 0 0 0 0 1 1 1 0 0 1 1 0 1 1 1 1 0 1 0 1 1 0 0 1 0 0 0 1 0 1 1 0 1 0 0 0 1 0 1 1 1 0 0|0 0 1 0 1 1 0 0 0 1 1 1 1 0 1 0 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 1 0 0 1 0 0 0 1 0 1 1 1 0 0 0 0 1 1 1 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 1 1 1 0 0 1 1 0 1 1 1 1 0 1 0 1 1 0 0 1 0 0 0 1 0 1 1 0 1 0 0 0 1 0 1 1 1 0|1 0 0 1 0 1 1 0 0 0 1 1 1 1 0 1 0 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 1 0 0 1 0 0 0 1 0 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 1 0 0 0 1 1 0 0 1 1 1 0 1 1 1 1 0 1 1 1 0 0 1 1 1 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0|1 1 1 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 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 1 0 0 0 1 1 0 0 1 1 1 0 1 1 1 1 0 1 1 1 0 0 1 1 1 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0|1 1 1 1 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 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 0 0 0 1 0 0 0 1 1 0 0 1 1 1 0 1 1 1 1 0 1 1 1 0 0 1 1 1 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0|1 1 1 1 1 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 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 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 1 0|0 1 0 1 0 1 1 1 1 1 1 0 1 0 0 1 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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]
last modified: 2024-04-24
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