Bounds on the minimum distance of additive quantum codes
Bounds on [[41,13]]2
| lower bound: | 8 |
| upper bound: | 10 |
Construction
Construction type: EzermanGrasslLingOzbudakOzkaya
Construction of a [[41,13,8]] quantum code:
[1]: [39, 14] Quasicyclic of degree 3 Linear Code over GF(2^2)
QuasiCyclicCode of length 39 stacked to height 2 with generating polynomials: 1, x^9 + w^2*x^8 + w^2*x^7 + x^6 + w*x^5 + x^3 + x^2 + w, x^12 + w*x^9 + w^2*x^7 + w^2*x^6 + w^2*x^5 + w*x^3 + w*x^2 + x + w, 0, x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
[2]: [[41, 13, 8]] quantum code over GF(2^2)
QuantumConstructionX applied to [1] with e = 2
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 1 0 1|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 1 1 0 1 0 0 1 0 0 0 1 0 1 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 1 1 0 1 0 0 1 0 0 0 1 0 1 1 1 0 0|1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 0 1 1 1 0 0 0 0 1 1 1 1 1 0 1 1 0 0 1]
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 1 0 1 0 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 1 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 1|0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 0 0 0 1 0 0 0 1]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 1 0 1 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 1 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 0 1 1 1 0 1 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 0 1 1 1 0 1 0 1 1|0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 0 0 0 1 0 0 1]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 1 0 1 0 0 0 1 1 1 0 0 1 0|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 0 1 1 1 0 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 0 1 1 1 0 1 1 1|0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 0 0 0 1 0 1]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 1 0 0 0 1 1 1 0 1 0 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 1 0 0 0 0 1 0 1 1 1 0 0 0 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 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 0 1 0 1 1 0 1 1 1 0 0 0|0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 1 1 1 0]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 1 0 0 1 1 1 1 0 1 0 1 1 1 0 0 0 0 1|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 0 1 1|0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 0 0 1 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 1 0 0 1 1 1 1 0 1 0 1 1 1 0 0 0 1|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 1 0 1 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 1 0 0 0 1 0 1 1 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 0 1 0]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 1 0 0 1 1 1 1 0 1 0 1 1 1 0 1 0|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 0 1 0 1 1 0 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 0 1 0 1 1 0 1 0 0|0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 1 0 1 0 0 1 0 1 1 1 0 1 1 0 0 0 0 0 1 1 1 0]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 0 0 0 0 1|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0|0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 0 0 1 0|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 0 1 0 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 0 1 0 1 1 1 1|0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 1 1 1 1 0 1 1 0 0 0 1 1 1 1 0 1 0 1 1 0|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 1 0 1 1 1 0 1 0 0 0 1 0 1 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 1 0 0 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0|0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 1 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 0 1 0]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 1 0 1 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 1 0 1 1 0 1 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 1 0 1 1 0 0 0 0 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0 0 0|0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 1 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 1 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 1 0 1 1 0 0 0 0 1 0 1 1 0 1 1 1 0 1 0 0 0 1 1 1]
[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 1 0 1 1 0 1 1 1 0 1 0 0 0 1 1 1|0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1]
[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 0 0|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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|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]
last modified: 2024-06-14
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