Bounds on the minimum distance of additive quantum codes

Bounds on [[28,12]]₂

Construction

Construction of a [[28,12,6]] quantum code:
[1]:  [[28, 12, 6]] quantum code over GF(2^2)
     V. D. Tonchev
Construction from a stored generator matrix

    stabilizer matrix:

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

last modified: 2010-07-16

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.