Bounds on the minimum distance of additive quantum codes
Bounds on [[70,3]]2
| lower bound: | 16 |
| upper bound: | 23 |
Construction
Construction of a [[70,3,16]] quantum code:
[1]: [[69, 3, 16]] quantum code over GF(2^2)
QuasiCyclicCode of length 69 stacked to height 2 with generating polynomials: x^68 + x^67 + x^63 + w^2*x^60 + w^2*x^59 + w^2*x^57 + w*x^56 + w*x^55 + w^2*x^54 + w^2*x^52 + w*x^51 + w*x^50 + x^49 + w^2*x^48 + x^47 + w*x^46 + x^45 + x^44 + w^2*x^41 + w*x^40 + x^39 + w^2*x^38 + w^2*x^37 + w^2*x^36 + w^2*x^35 + w^2*x^33 + w*x^32 + w*x^29 + w*x^27 + w^2*x^26 + x^25 + w*x^20 + w^2*x^18 + x^17 + w^2*x^16 + x^15 + x^14 + w^2*x^13 + w*x^12 + w*x^11 + w^2*x^10 + w^2*x^9 + w*x^8 + w*x^7 + w*x^6 + w*x^3 + w*x + w, x^68 + w*x^67 + w*x^65 + x^64 + w*x^63 + w^2*x^62 + w*x^61 + w*x^60 + w*x^59 + x^58 + x^57 + w^2*x^55 + w^2*x^54 + w*x^53 + w^2*x^52 + w*x^51 + w*x^50 + x^49 + w^2*x^47 + x^46 + w*x^45 + w^2*x^44 + w*x^43 + w^2*x^42 + x^41 + w^2*x^39 + x^37 + w^2*x^36 + w*x^35 + w*x^34 + x^33 + x^32 + w*x^31 + w*x^30 + w*x^29 + x^28 + w^2*x^27 + x^26 + w*x^24 + w*x^22 + x^21 + x^19 + w^2*x^17 + w^2*x^16 + w*x^15 + w^2*x^14 + w^2*x^13 + w^2*x^12 + w*x^11 + w^2*x^9 + w*x^8 + w^2*x^7 + w*x^5 + w^2*x^4 + w*x^3 + x^2 + x + w
[2]: [[70, 3, 16]] quantum code over GF(2^2)
ExtendCode [1] by 1
stabilizer matrix:
[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 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 1 0 0 1 1 1 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 0 1 1 0 0 1 0]
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last modified: 2020-10-24
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
(codes@codetables.de).
Last change: 10.06.2024