Bounds on the minimum distance of additive quantum codes

Bounds on [[82,73]]₂

Construction

Construction of a [[82,73,3]] quantum code:
[1]:  [[128, 119, 3]] quantum code over GF(2^2)
     Construction from a stored generator matrix
[2]:  [[82, 73, 3]] quantum code over GF(2^2)
     Shortening of [1] at { 3, 5, 13, 16, 19, 21, 27, 28, 33, 35, 37, 38, 40, 41, 43, 44, 45, 46, 52, 53, 55, 57, 58, 59, 62, 65, 68, 72, 73, 75, 81, 85, 89, 90, 92, 93, 95, 97, 100, 103, 105, 107, 110, 112, 118, 125 }

    stabilizer matrix:

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

last modified: 2006-04-03

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 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.