Bounds on the minimum distance of additive quantum codes
Bounds on [[46,31]]2
| lower bound: | 4 |
| upper bound: | 5 |
Construction
Construction of a [[46,31,4]] quantum code:
[1]: [[51, 27, 6]] quantum code over GF(2^2)
cyclic code of length 51 with generating polynomial w*x^50 + w^2*x^48 + x^47 + w*x^46 + x^45 + x^43 + x^41 + w*x^40 + x^38 + w*x^36 + x^35 + w*x^34 + w^2*x^32 + w*x^31 + w^2*x^29 + w^2*x^28 + w*x^27 + w^2*x^25 + w^2*x^23 + w^2*x^22 + w*x^20 + x^18 + w^2*x^17 + w^2*x^16 + w^2*x^15 + x^13 + x^12 + 1
[2]: [[46, 32, 4]] quantum code over GF(2^2)
Shortening of the stabilizer code of [1] at { 47, 48, 49, 50, 51 }
[3]: [[46, 31, 4]] quantum code over GF(2^2)
Subcode of [2]
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 0 0 0 0 1 1 0 0 0 1 1 0 1 0 1 0 1 0 0 1 1 0 1 1 0 1 1 1 0 1|0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 1 0 0 0 0 1 0 1 1 1 0 0 1 1 1 1 1 0 0 1 0 1 1 0 1 0 0 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 1 0 1 1 1 0 0 1 1 1 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0|1 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 0 1 1 0 1 1 0 1 1 0 1 0 1 1 0 0]
[0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 1|0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 0 1 0 0 0 0 1 1 0 1 1 0 1 0 1]
[0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 0 1 0 0 0 0 1 1 0 1 1 0 0 0 0|0 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 0 1 1 0 1 1 0 1]
[0 0 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 0 1 1 1 1 1 0 0|0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 1 0 0 1 1 1 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 1 1 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 1 0 0 1 1 1 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1|0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 1 1 1]
[0 0 0 1 0 0 0 0 1 0 1 0 0 1 1 0 1 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0|0 0 0 0 0 0 0 1 1 0 1 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 1]
[0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 0 1 1 1 0 0 1 1 1 0 0 0 0 1|0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 1 1 0 1 0 0 0 1 1 1 1 0 1 1 0 1 0 0 0 0 1 1 1 1 0 1 1 1 0]
[0 0 0 0 1 0 0 0 0 1 1 0 1 0 1 0 0 0 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 0 0 1 1 1 1|0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 1 1 0]
[0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 0 0|0 0 0 0 1 0 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 1 0 1 1 1 1 0 1 0 0 1 0 0 0 0 1 0 0 1 0 1 1 1 0 0]
[0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 0 1 1 0 1 1 1 1 1 1 0 1 0 1 0 1|0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 1 1]
[0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 1 0 0 1 1 0 0 0 0 1 1 1 1 1 1 0|0 0 0 0 0 1 0 1 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 1 0 1 0 0 1 1 1 1 0 1 1 1 1 0 0 1 0 1 0 1]
[0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 1 1 0 1 0 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 0 0 1 1 1|0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1 1 1 1]
[0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0|0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 1 1 0 1 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 0 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 1 0 1]
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 necessarily 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: 10.06.2024