Bounds on the minimum distance of additive quantum codes
Bounds on [[126,87]]2
lower bound: | 7 |
upper bound: | 12 |
Construction
Construction of a [[126,87,7]] quantum code:
[1]: [[126, 90, 7]] quantum code over GF(2^2)
QuasiCyclicCode of length 126 stacked to height 2 with generating polynomials: w*x^61 + w^2*x^60 + x^59 + x^58 + w^2*x^57 + x^56 + w^2*x^55 + w*x^53 + w*x^52 + w^2*x^51 + x^50 + w^2*x^49 + w*x^48 + w^2*x^46 + x^45 + w*x^43 + x^39 + w^2*x^35 + w^2*x^34 + x^33 + w*x^32 + w^2*x^28 + x^27 + x^26 + x^25 + x^24 + w*x^23 + w*x^22 + x^20 + w^2*x^19 + x^18 + 1, w*x^62 + w^2*x^59 + w^2*x^58 + x^57 + x^56 + w^2*x^55 + w*x^54 + x^53 + x^51 + x^50 + x^49 + w*x^47 + w*x^46 + x^45 + x^42 + w*x^41 + w^2*x^40 + w^2*x^39 + w*x^38 + w^2*x^35 + w*x^34 + x^33 + w*x^32 + w*x^30 + x^29 + w*x^28 + x^26 + w^2*x^25 + x^23 + x^22 + x^21 + w^2*x^18 + w*x^16 + w*x^13 + w*x^12 + x^11 + w^2*x^10 + x^9 + x^8 + w^2*x^7 + x^6 + w^2*x^5 + w*x^4 + w^2*x^3 + x^2 + 1, w^2*x^61 + x^60 + w*x^59 + w*x^58 + x^57 + w*x^56 + x^55 + w^2*x^53 + w^2*x^52 + x^51 + w*x^50 + x^49 + w^2*x^48 + x^46 + w*x^45 + w^2*x^43 + w*x^39 + x^35 + x^34 + w*x^33 + w^2*x^32 + x^28 + w*x^27 + w*x^26 + w*x^25 + w*x^24 + w^2*x^23 + w^2*x^22 + w*x^20 + x^19 + w*x^18 + w, w^2*x^62 + x^59 + x^58 + w*x^57 + w*x^56 + x^55 + w^2*x^54 + w*x^53 + w*x^51 + w*x^50 + w*x^49 + w^2*x^47 + w^2*x^46 + w*x^45 + w*x^42 + w^2*x^41 + x^40 + x^39 + w^2*x^38 + x^35 + w^2*x^34 + w*x^33 + w^2*x^32 + w^2*x^30 + w*x^29 + w^2*x^28 + w*x^26 + x^25 + w*x^23 + w*x^22 + w*x^21 + x^18 + w^2*x^16 + w^2*x^13 + w^2*x^12 + w*x^11 + x^10 + w*x^9 + w*x^8 + x^7 + w*x^6 + x^5 + w^2*x^4 + x^3 + w*x^2 + w
[2]: [[126, 87, 7]] quantum code over GF(2^2)
Subcode of [1]
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 0 1 0 0 0 1 0 1 0 0 0 1 1 0 1 0 0 1 1 0 0 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 1 0 0 1 1 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 1 0 0 1 0 1 1 0 0 1 0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 1 1 1 0 0 0 0 0 0]
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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