Bounds on the minimum distance of additive quantum codes

Bounds on [[16,1]]₂

Construction

Construction of a [[16,1,6]] quantum code:
[1]:  [[16, 2, 6]] quantum code over GF(2^2)
     Construction from a stored generator matrix
[2]:  [[16, 1, 6]] quantum code over GF(2^2)
     Subcode of [1]

    stabilizer matrix:

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

last modified: 2005-06-24

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.