Bounds on the minimum distance of additive quantum codes

Bounds on [[15,0]]₂

Construction

Construction of a [[15,0,6]] quantum code:
[1]:  [[15, 0, 6]] self-dual quantum code over GF(2^2)
     cyclic code of length 15 with generating polynomial x^14 + w^2*x^13 + w^2*x^12 + x^11 + w^2*x^10 + w^2*x^9 + x^8 + w*x^7 + w

    stabilizer matrix:

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

last modified: 2006-04-17

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 even 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.