Bounds on the minimum distance of additive quantum codes

Bounds on [[54,10]]2

lower bound:11
upper bound:16

Construction

Construction of a [[54,10,11]] quantum code:
[1]:  [[52, 10, 10]] quantum code over GF(2^2)
     ConstructionX using cyclic codes [51,21], [51,20], and auxiliary code [1,1]
[2]:  [[54, 10, 11]] quantum code over GF(2^2)
     ExtendCode [1] by 2

    stabilizer matrix:

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

last modified: 2020-07-29

Notes


This page is maintained by Markus Grassl (grassl@ira.uka.de). Last change: 23.10.2014