Bounds on the minimum distance of additive quantum codes
Bounds on [[76,3]]2
| lower bound: | 17 |
| upper bound: | 25 |
Construction
Construction of a [[76,3,17]] quantum code:
[1]: [[75, 3, 17]] quantum code over GF(2^2)
QuasiCyclicCode of length 75 stacked to height 2 with generating polynomials: x + 1, w^2*x^12 + x^11 + x^9 + x^8 + w*x^7 + x^6 + x^4 + x^3 + w*x^2 + w^2*x, w*x^24 + x^23 + x^22 + w*x^21 + x^19 + x^18 + w^2*x^17 + x^13 + w*x^12 + w*x^11 + x^10 + w^2*x^9 + x^8 + w*x^7 + x^6 + x^5 + w*x^4 + w^2*x^3 + x^2 + w^2*x, 0, x^13 + w^2*x^12 + w^2*x^11 + x^10 + w*x^8 + x^7 + x^6 + w*x^5 + x^3 + w^2*x^2 + w^2*x + 1, w*x^24 + w*x^23 + w*x^22 + w*x^21 + w*x^19 + w^2*x^18 + x^16 + w^2*x^15 + w*x^13 + w*x^11 + w*x^10 + x^9 + w*x^7 + w^2*x^6 + w*x^5 + w^2*x^2 + w^2*x + w^2
[2]: [[76, 3, 17]] 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 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 0 1 0 1 1 0 0 0 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 1 0 1 1 0 1 0 0 1 1 1 0 1 1 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0]
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last modified: 2024-05-23
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
(codes@codetables.de).
Last change: 10.06.2024