Bounds on the minimum distance of additive quantum codes

Bounds on [[45,29]]₂

Construction

Construction of a [[45,29,5]] quantum code:
[1]:  [[45, 29, 5]] quantum code over GF(2^2)
     cyclic code of length 45 with generating polynomial w*x^44 + x^43 + w^2*x^42 + w*x^41 + w*x^40 + w^2*x^39 + w*x^36 + x^35 + w^2*x^33 + w*x^31 + w*x^30 + x^28 + w^2*x^27 + w^2*x^25 + w*x^24 + w*x^22 + w*x^21 + x^19 + w^2*x^17 + w^2*x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^7 + w^2*x^6 + w^2*x^5 + w^2*x^4 + w*x^3 + x^2 + x + 1

    stabilizer matrix:

      [1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 1 0 1 0|0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 1 1 0 1 1 1 1 1 1]
      [0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 0 0 1|1 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 0 1 1 0 1 1 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 1 1]
      [0 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0 1 1 0 1 1 0 1 1 1 1 1 1 1 0 0 0 1 1 0 1 1 0 0 1 0 0|0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 0 0 1 0 0 0 1 0 1 1 1 0 0]
      [0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 0 1 1 0 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 1 1|0 1 0 0 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 1 0 0 1 1 0 1 0 1 0 0 0 0 1 0 1 1 1 0 1 1 1 1 1 0 1]
      [0 0 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0 1 1 0 1 1 0 1 1 1 1 1 1 1 0 0 0 1 1 0 1 1 0 0 1 0|0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 0 0 1 0 0 0 1 0 1 1 1 0]
      [0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 1 1 1 1 0 1 0 1 0 1 0|0 0 1 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 1 0 0 0 0 0 1 0]
      [0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0 1 1 0 1 1 0 1 1 1 1 1 1 1 0 0 0 1 1 0 1 1 0 0 1|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 0 0 1 0 0 0 1 0 1 1 1]
      [0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 1 1 1 1 0 1 0 1 0 1|0 0 0 1 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 1 0 0 0 0 0 1]
      [0 0 0 0 1 0 0 0 1 1 1 1 0 1 0 0 1 1 1 1 0 1 0 1 0 0 1 1 0 1 0 1 1 1 0 0 1 0 0 1 1 0 1 1 1|0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1]
      [0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 0 0 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 0 1 0 0 1 0 1 1 0 0 0 1|0 0 0 0 1 0 0 0 1 1 0 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 0 1 1 1 0 0]
      [0 0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 1 0 0 0 0 0 0|0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 0 1 1 1 0 1 0 1 1 1 1 1 0 0 1 1 1]
      [0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 1 0 0 1 1 0 0 1 1 1 0 0 1 0|0 0 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 0 0 0 1]
      [0 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0 1 1 0 1 1 0 1 1 1 1 1 1 1 0 0 0 1 1 0 1 1 0 0 1 0 0 0 1|0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 0 0 1 0 0 0 1 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 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0|0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 1 0 1 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 1 1]
      [0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0|0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 0 1 1 0 1 1 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 1 1 1]
      [0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 1 0 1 0 1|0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 1 1 0 1 1 1 1 1 1 0]

last modified: 2020-09-17

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.