Bounds on the minimum distance of additive quantum codes
Bounds on [[46,23]]2
lower bound: | 6 |
upper bound: | 8 |
Construction
Construction of a [[46,23,6]] quantum code:
[1]: [[92, 70, 5]] quantum code over GF(2^2)
QuasiCyclicCode of length 92 stacked to height 2 with generating polynomials: w^2*x^22 + w^2*x^21 + w^2*x^20 + x^19 + w*x^18 + w^2*x^17 + w^2*x^16 + w*x^15 + x^13 + w^2*x^12 + w^2*x^11 + w*x^10 + w*x^9 + w^2*x^8 + w*x^7 + x^6 + w*x^3 + w*x^2 + x + w, w^2*x^22 + w*x^21 + x^20 + w^2*x^19 + x^18 + w^2*x^17 + w^2*x^16 + w^2*x^15 + w^2*x^13 + w*x^12 + x^11 + x^10 + w^2*x^9 + w*x^8 + x^7 + w*x^5 + w^2*x^4 + x^3 + x^2 + x, x^22 + x^21 + w*x^20 + w^2*x^18 + x^17 + x^14 + w*x^12 + x^11 + w*x^9 + w^2*x^7 + w*x^6 + x^4 + w*x^2 + w*x, w^2*x^21 + x^18 + x^17 + w*x^15 + x^14 + x^13 + w^2*x^12 + x^11 + w^2*x^10 + w^2*x^9 + x^8 + x^6 + w^2*x^5 + w^2*x^4 + x^3 + x^2 + x + w, x^22 + x^21 + x^20 + w*x^19 + w^2*x^18 + x^17 + x^16 + w^2*x^15 + w*x^13 + x^12 + x^11 + w^2*x^10 + w^2*x^9 + x^8 + w^2*x^7 + w*x^6 + w^2*x^3 + w^2*x^2 + w*x + w^2, x^22 + w^2*x^21 + w*x^20 + x^19 + w*x^18 + x^17 + x^16 + x^15 + x^13 + w^2*x^12 + w*x^11 + w*x^10 + x^9 + w^2*x^8 + w*x^7 + w^2*x^5 + x^4 + w*x^3 + w*x^2 + w*x, w*x^22 + w*x^21 + w^2*x^20 + x^18 + w*x^17 + w*x^14 + w^2*x^12 + w*x^11 + w^2*x^9 + x^7 + w^2*x^6 + w*x^4 + w^2*x^2 + w^2*x, x^21 + w*x^18 + w*x^17 + w^2*x^15 + w*x^14 + w*x^13 + x^12 + w*x^11 + x^10 + x^9 + w*x^8 + w*x^6 + x^5 + x^4 + w*x^3 + w*x^2 + w*x + w^2
[2]: [[46, 24, 6]] quantum code over GF(2^2)
Shortening of [1] at { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69 }
[3]: [[46, 23, 6]] quantum code over GF(2^2)
Subcode of [2]
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 1 0 1 1 0 1 1 0 0 1 1 0 1 1 1 1 1 0 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 1 0 0 1 0 1 0 1 0 1 0 0 1 1 1 1 0 1 1 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 1 0 0 1 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 1|1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1]
[0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 0 0 1 0 1 1 0 1 1 0 0 1 1 0 1 1 0 1 0 1 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 1 1 0 0 1 0 1 0 1 0 1 0 0 1 0 0 1 0 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 0 0 0 1 1 0 0 1 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1|0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 0 0 1 1 0 1 0 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1]
[0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 0 1 0 0 1 1 1 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 1 1 0 0 0 0 0 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 1 1 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 1 0 0|0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 1 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0]
[0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 0 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 1 1 1 0 0 0 0 0 0 0 1 1 0 1 1 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 1 1 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 1 0|0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 1 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 0 0 0 0 1 1]
[0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 1 1 0 0 0 1 0 0 1 0 0 1 0 0 0 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 1 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 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 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 1 1 1 0|0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 1 0 1 0 1 1 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 1 1 1 0 1 0 0 0 1 0 0 1 1 1 1 1 0 1 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 1 1 0 1 0 0 0 1 1 1 1 0 1 0 0 0 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 0 0 0 0 1 1 0 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 0|0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 0 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 1 0 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 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 0 0 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 0 0 0 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 0 0 0 0 1 0 1 1|0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 1 1 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 1 0 1]
[0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 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 1 1 1 1 1 1 1 0 1 1 0 1 1 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 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 0 0 0 0 1 0 1|0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 1 1 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 1 0]
[0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 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 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 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 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 0 0 0 0 1 0|0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 1 1 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 1]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 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 0 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 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 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 0 0 0 0 1|0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1]
[0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 1 0 0 1 1 0 1 1 0 0 0 0 1 0 0 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 1 0 1 0 1 0 1 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 1 1|0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 0 0 1 0 1 1 0 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 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 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 1 1 0 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
(codes@codetables.de).
Last change: 10.06.2024