Bounds on the minimum distance of additive quantum codes

Bounds on [[162,0]]₂

Construction

Construction of a [[162,0,20]] quantum code:
[1]:  [[82, 0, 20]] self-dual quantum code over GF(2^2)
     QuasiCyclicCode of length 82 with generating polynomials: x^27,  w^2*x^39 + w^2*x^38 + w^2*x^37 + w^2*x^36 + x^34 + x^33 + w*x^32 + x^30 + x^29 + w^2*x^27 + w^2*x^26 + x^25 + x^24 + w*x^23 + w^2*x^21 + w*x^20 + w*x^17 + w^2*x^16 + w*x^14 + w*x^13 + w^2*x^11 + w*x^10 + x^9 + w^2*x^7 + w*x^6 + w^2*x^5 + x^3 + x^2 + w^2*x
[2]:  [[80, 0, 20]] self-dual quantum code over GF(2^2)
     QuasiCyclicCode of length 80 with generating polynomials: 1,  w*x^36 + w^2*x^35 + x^34 + w*x^33 + x^32 + x^30 + w*x^29 + x^28 + w*x^27 + w^2*x^25 + w*x^24 + w*x^23 + w*x^21 + w*x^20 + x^19 + x^18 + w^2*x^17 + x^16 + x^15 + x^14 + w^2*x^13 + w*x^12 + w*x^11 + x^9 + x^5 + w*x^4 + x^3 + x + w
[3]:  [[162, 0, 20]] self-dual quantum code over GF(2^2)
     direct sum of [2] and  [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 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 1 0 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 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 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 1 1 0 1 1 1 0 1 0 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 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 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]
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last modified: 2019-05-16

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.