Bounds on the minimum distance of additive quantum codes

Bounds on [[34,16]]₂

Construction

Construction of a [[34,16,6]] quantum code:
[1]:  [[34, 16, 6]] quantum code over GF(2^2)
     cyclic code of length 34 with generating polynomial x^32 + x^31 + w^2*x^30 + w*x^29 + w^2*x^28 + w^2*x^27 + w*x^25 + w^2*x^24 + w^2*x^23 + x^20 + w*x^19 + w^2*x^18 + x^16 + w*x^15 + w^2*x^14 + w*x^13 + w*x^12 + w^2*x^11 + w^2*x^9 + 1

    stabilizer matrix:

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

last modified: 2005-06-29

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.