Bounds on the minimum distance of additive quantum codes

Bounds on [[39,27]]₂

Construction

Construction of a [[39,27,4]] quantum code:
[1]:  [[39, 27, 4]] quantum code over GF(2^2)
     QuasiCyclicCode of length 39 with generating polynomials: w*x^11 + w*x^10 + x^9 + x^7 + w^2*x^5 + x^4 + w*x^3 + w^2*x^2 + x + w,  w^2*x^12 + w^2*x^11 + w*x^9 + w^2*x^8 + x^7 + x^6 + x^5 + w*x^3 + w^2*x^2 + 1,  x^12 + x^11 + w^2*x^10 + w^2*x^9 + w^2*x^8 + x^7 + w*x^5 + w*x^4 + w^2*x^2 + 1

    stabilizer matrix:

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

last modified: 2006-04-07

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.