Bounds on the minimum distance of additive quantum codes

Bounds on [[30,16]]₂

Construction

Construction of a [[30,16,5]] quantum code:
[1]:  [[30, 16, 5]] quantum code over GF(2^2)
     QuasiCyclicCode of length 30 stacked to height 2 with generating polynomials: x^14 + w*x^13 + w*x^12 + x^10 + w*x^9 + w*x^8 + x^7 + w*x^6 + w^2*x^5 + w*x^4 + w*x^2 + x + 1,  w^2*x^14 + x^13 + w^2*x^10 + w^2*x^9 + x^8 + w*x^7 + w^2*x^6 + w^2*x^5 + x^4 + w^2*x^2 + w^2,  w*x^13 + w*x^11 + w^2*x^8 + x^7 + w*x^6 + x^4 + w^2*x^3 + x^2 + w*x + 1,  w^2*x^14 + w*x^13 + x^12 + w*x^11 + w^2*x^10 + w^2*x^9 + x^8 + w^2*x^7 + w^2*x^6 + w*x^5 + x^4 + x^3 + w*x^2 + w^2*x

    stabilizer matrix:

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

last modified: 2005-06-27

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.