Bounds on the minimum distance of additive quantum codes
Bounds on [[42,14]]2
| lower bound: | 8 |
| upper bound: | 10 |
Construction
Construction type: EzermanGrasslLingOzbudakOzkaya
Construction of a [[42,14,8]] quantum code:
[1]: [[42, 14, 8]] quantum code over GF(2^2)
QuasiCyclicCode of length 42 stacked to height 2 with generating polynomials: x^10 + x^9 + x^8 + w^2*x^6 + w^2*x^5 + w^2*x^4 + w^2*x^3 + w^2*x^2 + w*x + w, x^17 + w^2*x^15 + x^14 + w*x^13 + x^12 + w^2*x^8 + x^7 + w*x^6 + w^2*x^5 + w^2*x^4 + x^3 + w^2*x^2 + x + 1, 0, x^18 + x^16 + x^15 + x^14 + x^11 + x^9 + x^8 + x^7 + x^4 + x^2 + x + 1
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 1 1 0 1 1 0 1 0 1 1 0 1 0 1 1|0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 0]
[0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 0|1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1]
[0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 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 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 1 1 0 0 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 1 1 0 0 1 1 1|0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 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 1 0 0 0 0 1 0 0 1 0 0 0 1 1 1 0 0 1 1 1 1 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 1 1 0 0 1 1 1 1 0 1|0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 1 0 1 1 1 0 1 0 1]
[0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 0 0 0 1 0 1 1 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 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 1 0]
[0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 1 0|0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 0 0 1 1 0 0 0 1 0 0 1 0 1]
[0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 1 0 1 1 0 1 1 0 1 0 0 1 1|0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1 1 0 0 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1 1 0 0 1 0 0|0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 1 1 0 1 1 1]
[0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 1 1 0|0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 1 0 1 1 0 1 1 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 0 0 1 0 0 0 0 0 1 0 1 1 0 1 1 0 1 0 1 0 0 1 1 0 0 1|0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 0 1 1 0 1 0 0 0 0 1 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 1]
[0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 1 1 1 1 0 0 0 1 1 0 0 0 1 1 1 0 1 1 0 0 1 0 0 0 0|0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 1 1 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 1 1 1 0 0 0|0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 1 1 0 1 0 1 0 0 0 1 1 0 1 0 1 0 1 1 1 0 0 1 0 1 0 0 0]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 0 1 0 0 0 1 1|0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 0 1 0 0 0 1 0 1 0 1 0 0 1 0 1 1 1 1 0 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 0 1 0 0 0 1 0 1 0 1 0 0 1 0 1 1 1 1 0 0 1 1 0|0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 1 0 1]
[0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1 0 0 1 0 1|0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 1 0 0 0 0 1 0 0 1 1 1 1 1 1 0 0 1 0 0 0 1 0]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 1 0 0 0 0 1 0 0 1 1 1 1 1 1 0 0 1 0 0 0 1 0|0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 1 0 1 1 1 0 0 0 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 1 1]
[0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 1 1 0 0 1 1 1 0 0 1 1 0|0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 0 1 1 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 1 0 0 0 0 1 1 0 0 1 0 0 0 0 0 1 1 1 1 0 0 1 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 1 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 1 0 1 0 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 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 1 1 1 1 1 0 1 1 1 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 1 1 1 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 1 1|0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 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 1 0 0 1 1 1 0 1 0 0 1 1 1 0 1 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 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 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 1 0 0 1 1 1 0 1 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 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 1 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 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 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 1 0 0 1 1 1 0 1 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 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 1 1 1 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 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 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 1]
last modified: 2024-06-15
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