Bounds on the minimum distance of additive quantum codes

Bounds on [[27,18]]₂

Construction

Construction of a [[27,18,3]] quantum code:
[1]:  [[40, 33, 3]] quantum code over GF(2^2)
     Construction from a stored generator matrix
[2]:  [[25, 18, 3]] quantum code over GF(2^2)
     Shortening of [1] at { 2, 5, 6, 13, 15, 16, 17, 19, 21, 26, 27, 30, 31, 32, 37 }
[3]:  [[26, 18, 3]] quantum code over GF(2^2)
     ExtendCode [2] by 1
[4]:  [[27, 18, 3]] quantum code over GF(2^2)
     ExtendCode [3] by 1

    stabilizer matrix:

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

last modified: 2005-06-27

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.