Bounds on the minimum distance of additive quantum codes

Bounds on [[27,13]]₂

Construction

Construction of a [[27,13,5]] quantum code:
[1]:  [[27, 13, 5]] quantum code over GF(2^2)
     V. D. Tonchev, Quantum codes from caps, Disc. Math.(2008)
Construction from a stored generator matrix

    stabilizer matrix:

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

last modified: 2008-10-03

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.