Bounds on the minimum distance of additive quantum codes
Bounds on [[30,18]]2
lower bound: | 4 |
upper bound: | 4 |
Construction
Construction of a [[30,18,4]] quantum code:
[1]: [[40, 30, 4]] quantum code over GF(2^2)
QuasiCyclicCode of length 40 stacked to height 2 with generating polynomials: 1, w^2*x^4 + w*x^3 + w^2*x^2 + w*x + w^2, x^4 + w^2*x^2 + w^2*x + w^2, x^4 + w*x^3 + x^2, w*x^4 + x^3 + w^2*x, w*x^4 + x^3 + x^2 + x, w^2*x^4 + x^2 + w*x, x^4 + w^2*x^3 + w^2*x^2 + x + 1, w, x^4 + w^2*x^3 + x^2 + w^2*x + 1, w*x^4 + x^2 + x + 1, w*x^4 + w^2*x^3 + w*x^2, w^2*x^4 + w*x^3 + x, w^2*x^4 + w*x^3 + w*x^2 + w*x, x^4 + w*x^2 + w^2*x, w*x^4 + x^3 + x^2 + w*x + w
[2]: [[30, 20, 4]] quantum code over GF(2^2)
Shortening of [1] at { 2, 3, 4, 5, 11, 13, 27, 29, 37, 40 }
[3]: [[30, 18, 4]] quantum code over GF(2^2)
Subcode of [2]
stabilizer matrix:
[1 0 0 0 0 0 1 1 0 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 0 0 0 1|0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0]
[0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0|1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 1 1 1 0 0 1 1 0 1 1 1 0 0 0 0]
[0 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 1|0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 0 0 0 0 1 0 0 1 1 1]
[0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0|0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 1 1 1 0 0 1 1 1 1 0 1 1 1 1]
[0 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 0 1 1 0 1 0 1 0 0 1 0|0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0]
[0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 0 1|0 0 1 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 0 1 1 1 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1 0 0 0 1 0|0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 0 0 0 1 0 0 1 0 0 0 0 1 1 1 1]
[0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0|0 0 0 1 0 0 0 1 0 1 1 1 0 0 0 1 1 1 1 1 0 0 0 1 0 1 1 1 0 1]
[0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 1 0 0 1 1 1 1 0 1 0|0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 0 0 1 0 0 1 1 1 0 0 1 0 0 1]
[0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 0 0 1 0 0 1 1 1 0 0 0 0 1 0|0 0 0 0 1 0 1 0 0 0 1 0 1 1 1 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1]
[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 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 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 1 0 1 1 1 1|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 1 1 1 0 0]
last modified: 2005-06-24
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 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