Bounds on the minimum distance of additive quantum codes

Bounds on [[22,12]]₂

Construction

Construction of a [[22,12,4]] quantum code:
[1]:  [[40, 30, 4]] quantum code over GF(2^2)
     quasicyclic code of length 40 stacked to height 2 with 16 generating polynomials
[2]:  [[22, 12, 4]] quantum code over GF(2^2)
     Shortening of [1] at { 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 20, 30, 31, 33, 40 }

    stabilizer matrix:

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

last modified: 2005-06-29

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.