Bounds on the minimum distance of additive quantum codes
Bounds on [[54,34]]2
lower bound: | 5 |
upper bound: | 7 |
Construction
Construction of a [[54,34,5]] quantum code:
[1]: [[93, 73, 5]] quantum code over GF(2^2)
QuasiCyclicCode of length 93 stacked to height 2 with generating polynomials: x^30 + x^28 + x^27 + x^26 + x^25 + w*x^24 + x^23 + x^20 + x^19 + w^2*x^18 + x^17 + x^16 + w^2*x^14 + w*x^13 + w*x^11 + w^2*x^10 + w^2*x^9 + w^2*x^8 + w^2*x^7 + w*x^5 + w*x^3 + w*x^2, w*x^27 + x^26 + x^25 + w*x^24 + w*x^23 + w*x^21 + x^20 + x^19 + w*x^18 + w^2*x^16 + x^14 + w^2*x^13 + w*x^10 + w^2*x^8 + x^7 + x^6 + x^5 + w*x^4 + w^2*x^3 + w*x, w*x^30 + w^2*x^28 + x^27 + w^2*x^26 + x^24 + w^2*x^23 + w*x^22 + w^2*x^19 + x^18 + w*x^17 + w*x^15 + w^2*x^14 + w^2*x^13 + w^2*x^12 + w*x^10 + w^2*x^9 + w^2*x^8 + x^7 + x^6 + x^4 + w^2*x^3 + w*x, w*x^30 + w*x^28 + w*x^27 + w*x^26 + w*x^25 + w^2*x^24 + w*x^23 + w*x^20 + w*x^19 + x^18 + w*x^17 + w*x^16 + x^14 + w^2*x^13 + w^2*x^11 + x^10 + x^9 + x^8 + x^7 + w^2*x^5 + w^2*x^3 + w^2*x^2, w^2*x^27 + w*x^26 + w*x^25 + w^2*x^24 + w^2*x^23 + w^2*x^21 + w*x^20 + w*x^19 + w^2*x^18 + x^16 + w*x^14 + x^13 + w^2*x^10 + x^8 + w*x^7 + w*x^6 + w*x^5 + w^2*x^4 + x^3 + w^2*x, w^2*x^30 + x^28 + w*x^27 + x^26 + w*x^24 + x^23 + w^2*x^22 + x^19 + w*x^18 + w^2*x^17 + w^2*x^15 + x^14 + x^13 + x^12 + w^2*x^10 + x^9 + x^8 + w*x^7 + w*x^6 + w*x^4 + x^3 + w^2*x
[2]: [[54, 34, 5]] quantum code over GF(2^2)
Shortening of [1] at { 1, 3, 9, 14, 17, 20, 23, 24, 25, 28, 30, 31, 36, 38, 39, 40, 41, 43, 46, 47, 48, 49, 51, 52, 56, 58, 60, 63, 66, 67, 68, 71, 73, 74, 75, 77, 83, 88, 91 }
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 0 0 1 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 0 1 1 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 1 0 1 1 1 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 1 1 1 0 0 1 0 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 1 0 1 1 1 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 1 1 1 0 0 1 0 1 1 0 1|1 0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 1 1 1 1 0 0]
[0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 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 1 0 1 1 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 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 1 0 1 1 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 1 1 0 0 0|0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 1 0 0 0 1 1 1 0 1 0 0 1 0 1 0 1 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0]
[0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 0 0 1 0 0 1 1 0 1 1 1 0 0 1 0 1 0 0 1 1 1 0 1 1 1 1 0 1 1|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 1 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 0 1 1 0 0 1]
[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 1 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 0 1 1 0 0 1|0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 1 0 0 0 0 1 0 0 0 1 0 1 1 1 0 1 1 0 0 1 0 0 0 1 0]
[0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 1 0 0 1 1 0 0 0 0 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 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 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 1 0 1 1 1 1 1 0 0 1 1 0 0 1 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 1 0 0 0 0 1 1 0 0 1 1 0 1 1 1 1 1 0 0 1 1 0 0 1 1 1 1 0 1 0 0 1 1 1|0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 1 0 0 1 1 1 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 1 0 0 1 1 1 0 0 1 0 0 1 1 1 0 1 0 1]
[0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 0 0 1 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 1 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 1 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 1 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0|0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1 1 0 1 0 0 0 0 0 0 0]
[0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 1 1 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 1 0 1 1 0 1 0 0 0 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 0 1 1 0 1 1 0 1 0 0 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 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 1 1 0 1 1 0 1 0 0 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 1 1 0 1 0 0 0 0 0|0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 1 1 0 1 1 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 0 1 1 1 0 1 0 0 1 1 0 1 0 0 0 0]
[0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 1 0 0 1 0 0 0 0 1 0 0 1 1 1 1 0 1 1 0 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 1 0 1 0 0 1 0 1 1 1 0 1 0 0 1 0 0 0 1 1 0 1 1 0 1 0 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 1 0 1 0 0 1 0 1 1 1 0 1 0 0 1 0 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 1 1 1 1|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 1 0 1 0 0 1 1 0 1 0 0 1 1 1 1 0 0 1 1 1 0 1 0 0 1 0 1 0 0]
[0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 0 1 1 0 0 0 1 0 0 1 0 0 0 0 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 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 1 1 1 0 0 0 0 0 1 1 0 0 1 1 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 1 1 1 1 0 0 1 1 1 0 0 0 0 0 1 1 0 0 1 1 1 1 0 1 0 1 1 1 0 0 0|0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 1 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0]
[0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 0 1 1 0 1 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 1 1 1 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 1 1 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 0 1 1 1 1 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 1 1 0 0 1 1 1 1 1 1 0 0 0|0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 1 1 1 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 1 1 0 0 1 0 0 0 0 0 0 1 1 0 1 1 1 0 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 1 0 1 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 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 1 0 1 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 1 0 1 1|0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 0 1 0 1 0 1 0 1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 1 1 0 0 1 0 0]
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 even 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: 23.10.2014