Bounds on the minimum distance of quantum codes

Bounds on [[87,69]]₄

lower bound:	6
upper bound:	8
Construction

Construction of a [[87,69,6]] quantum code:
[1]:  [[87, 69, 6]] quantum code over GF(2^4)
     from Quantum Twisted code [[255,237,6]]_4
Construction from a stored generator matrix

    stabilizer matrix:

      [1 0 0 0 0 0 0 0 0 0 a^2 0 a a 1 1 1 0 1 0 a^2 1 1 a 0 a^2 1 0 a 1 1 0 0 1 a 0 0 1 1 a a^2 a^2 a^2 1 1 a 0 0 0 0 a 0 a a^2 a^2 0 a^2 a^2 a^2 1 a^2 1 1 a^2 1 1 a a^2 a^2 1 a 1 1 a^2 1 0 a^2 a 1 a^2 1 a^2 a^2 1 0 a 1|0 0 0 0 0 0 0 0 0 1 a^2 a 1 0 1 0 a^2 1 0 a^2 1 a^2 0 0 a^2 a 0 0 0 a^2 a^2 0 a^2 1 1 a^2 0 a 1 0 a^2 1 0 1 1 a^2 0 1 0 a 0 0 1 a a a 0 0 a 1 a a^2 0 0 a^2 a^2 a^2 a^2 a^2 0 1 a^2 a 0 a 0 a a a a a 0 a 1 0 1 a]
      [0 0 0 0 0 0 0 0 0 a^2 a 1 a^2 0 a^2 0 a a^2 0 a a^2 a 0 0 a 1 0 0 0 a a 0 a a^2 a^2 a 0 1 a^2 0 a a^2 0 a^2 a^2 a 0 a^2 0 1 0 0 a^2 1 1 1 0 0 1 a^2 1 a 0 0 a a a a a 0 a^2 a 1 0 1 0 1 1 1 1 1 0 1 a^2 0 a^2 1|1 0 0 0 0 0 0 0 0 a^2 1 1 1 a a 1 a^2 a^2 1 a 0 a^2 1 a a a 1 0 a a^2 a^2 0 a a 1 a 0 0 a a 1 0 a^2 a a 0 0 a^2 0 1 a 0 1 a a 1 a^2 a^2 a a a a^2 1 a^2 a^2 a^2 0 1 1 1 1 a^2 0 a^2 0 0 a a^2 0 a 0 a^2 a a 0 1 0]
      [a 0 0 0 0 0 0 0 0 a^2 a^2 1 0 a^2 1 a 0 a^2 a a a 0 a a^2 a 0 a 0 a^2 0 0 0 a 1 0 a 0 a^2 1 a^2 a^2 a 1 1 1 1 0 a^2 0 1 a^2 0 0 0 0 1 1 1 0 1 0 0 a 1 0 0 1 a^2 a^2 a 0 0 a^2 1 a^2 0 0 a a^2 0 a^2 1 0 1 0 0 a^2|1 0 0 0 0 0 0 0 0 1 0 a a^2 a 0 1 a 1 1 a^2 a a 1 a a^2 1 1 0 a a a 0 a^2 0 a^2 a^2 0 a^2 0 a 0 a a^2 0 0 1 0 1 0 a a 0 a^2 1 1 a a^2 a^2 1 0 1 a 1 a^2 a a 1 0 0 1 a^2 a a^2 a^2 a^2 0 1 0 a^2 1 a^2 a^2 1 0 0 a^2 a^2]
      [a 0 0 0 0 0 0 0 0 a 0 a^2 1 a^2 0 a a^2 a a 1 a^2 a^2 a a^2 1 a a 0 a^2 a^2 a^2 0 1 0 1 1 0 1 0 a^2 0 a^2 1 0 0 a 0 a 0 a^2 a^2 0 1 a a a^2 1 1 a 0 a a^2 a 1 a^2 a^2 a 0 0 a 1 a^2 1 1 1 0 a 0 1 a 1 1 a 0 0 1 1|a^2 0 0 0 0 0 0 0 0 0 a 0 1 1 a^2 a^2 a^2 0 a^2 0 a a^2 a^2 1 0 a a^2 0 1 a^2 a^2 0 0 a^2 1 0 0 a^2 a^2 1 a a a a^2 a^2 1 0 0 0 0 1 0 1 a a 0 a a a a^2 a a^2 a^2 a a^2 a^2 1 a a a^2 1 a^2 a^2 a a^2 0 a 1 a^2 a a^2 a a a^2 0 1 a^2]
      [0 1 0 0 0 0 0 0 0 a 1 a^2 a 1 a a a 0 a 0 a^2 0 1 a a^2 a^2 1 0 a^2 a^2 0 0 0 a^2 a^2 a^2 1 a^2 a 1 a^2 a^2 a^2 1 0 a 0 a^2 a^2 a^2 a^2 a^2 a^2 a^2 0 0 0 0 a 0 0 0 a^2 0 a 0 1 1 a a^2 a 1 a 0 0 a 0 a a^2 0 a a a a 1 0 a^2|0 0 0 0 0 0 0 0 0 a^2 a^2 a^2 1 a^2 a 0 0 a a 1 a a 0 a a a^2 0 a^2 0 a 1 0 0 a 1 a 0 1 a a a^2 a^2 a a^2 1 0 0 1 a^2 0 a 0 a a 1 1 1 a^2 1 a^2 0 1 0 a 1 a^2 a 1 1 0 0 a^2 0 a 0 1 0 1 0 1 a^2 1 a^2 a^2 0 1 a]
      [0 0 0 0 0 0 0 0 0 a a a a^2 a 1 0 0 1 1 a^2 1 1 0 1 1 a 0 a 0 1 a^2 0 0 1 a^2 1 0 a^2 1 1 a a 1 a a^2 0 0 a^2 a 0 1 0 1 1 a^2 a^2 a^2 a a^2 a 0 a^2 0 1 a^2 a 1 a^2 a^2 0 0 a 0 1 0 a^2 0 a^2 0 a^2 a a^2 a a 0 a^2 1|0 1 0 0 0 0 0 0 0 0 a^2 1 1 a^2 a^2 a a 1 a^2 a^2 a 1 1 a^2 a 1 1 a a^2 a a^2 0 0 a 0 a 1 0 a^2 0 1 1 a a^2 a^2 a 0 0 1 a^2 a a^2 a a a^2 a^2 a^2 a 1 a 0 a^2 a^2 1 1 a 0 a 1 a^2 a a^2 a 1 0 1 0 1 a^2 a^2 0 1 0 0 1 a^2 a]
      [0 a 0 0 0 0 0 0 0 1 0 a^2 0 0 a a^2 a^2 1 a a^2 0 1 a a 0 a^2 a a 1 0 a^2 0 0 0 a 0 a a a a^2 a^2 a^2 0 0 a^2 a^2 0 a a^2 1 0 1 0 0 a^2 a^2 a^2 a 0 a 0 a^2 1 1 0 a a^2 1 0 1 a^2 0 a^2 1 0 0 0 0 1 a^2 1 0 1 1 a a^2 0|0 1 0 0 0 0 0 0 0 1 a 0 a^2 a 0 a a a 0 1 1 a 1 0 1 0 1 a^2 a^2 1 1 0 0 1 a 1 1 a 0 a^2 0 0 1 a 1 a 0 a 0 a^2 1 a^2 1 1 1 1 1 a^2 a^2 a^2 0 1 a^2 a a^2 a^2 a^2 0 a^2 a^2 a a a a 0 a^2 0 a^2 a^2 1 1 a^2 1 1 1 1 1]
      [0 a 0 0 0 0 0 0 0 a a^2 0 1 a^2 0 a^2 a^2 a^2 0 a a a^2 a 0 a 0 a 1 1 a a 0 0 a a^2 a a a^2 0 1 0 0 a a^2 a a^2 0 a^2 0 1 a 1 a a a a a 1 1 1 0 a 1 a^2 1 1 1 0 1 1 a^2 a^2 a^2 a^2 0 1 0 1 1 a a 1 a a a a a|0 a^2 0 0 0 0 0 0 0 1 a^2 a 1 a^2 1 1 1 0 1 0 a 0 a^2 1 a a a^2 0 a a 0 0 0 a a a a^2 a 1 a^2 a a a a^2 0 1 0 a a a a a a a 0 0 0 0 1 0 0 0 a 0 1 0 a^2 a^2 1 a 1 a^2 1 0 0 1 0 1 a 0 1 1 1 1 a^2 0 a]
      [0 0 1 0 0 0 0 0 0 a a^2 0 a^2 a^2 a^2 a^2 a^2 a a^2 a a a^2 a a^2 0 a a 0 a a^2 a a 0 1 0 0 0 a^2 1 a^2 a^2 0 1 1 0 a a a 0 a a a^2 0 a^2 1 a 0 a^2 a^2 1 a^2 a^2 1 a a^2 1 1 a 0 a 1 a 0 0 0 a^2 0 a^2 a^2 1 a^2 1 0 a^2 1 a^2 a^2|0 0 0 0 0 0 0 0 0 1 0 a a a^2 a^2 a 1 0 a 1 a 0 a 1 0 a^2 a 1 a^2 0 a^2 0 a 1 a 1 0 a 0 1 1 a 0 1 1 a^2 a^2 1 1 a a 0 a 0 1 1 0 1 a^2 a a a^2 0 a^2 1 a^2 1 0 0 1 0 a a^2 a a 0 0 1 0 0 a^2 a^2 a^2 a^2 1 0 a]
      [0 0 0 0 0 0 0 0 0 a^2 0 1 1 a a 1 a^2 0 1 a^2 1 0 1 a^2 0 a 1 a^2 a 0 a 0 1 a^2 1 a^2 0 1 0 a^2 a^2 1 0 a^2 a^2 a a a^2 a^2 1 1 0 1 0 a^2 a^2 0 a^2 a 1 1 a 0 a a^2 a a^2 0 0 a^2 0 1 a 1 1 0 0 a^2 0 0 a a a a a^2 0 1|0 0 1 0 0 0 0 0 0 1 a^2 1 a 1 1 a 0 a a 1 a^2 a^2 a^2 0 0 0 a^2 a^2 0 a^2 0 a 1 a 1 a^2 0 a 1 0 0 1 1 a a^2 0 0 1 a^2 a^2 a^2 a^2 1 a^2 a 1 0 0 1 0 a 1 1 0 0 a^2 a a 0 1 1 a^2 a 1 1 a^2 0 0 a^2 1 1 a^2 a 1 a a^2 a]
      [0 0 a 0 0 0 0 0 0 0 1 1 0 a^2 a^2 0 a a^2 0 0 a 1 a a 0 1 a a^2 1 1 1 a^2 1 1 1 a^2 0 0 a a a 1 a 1 a^2 1 1 0 a^2 a a 1 1 1 1 0 0 a a^2 a^2 0 a^2 a 1 a 0 1 a^2 0 0 a a a 1 1 1 0 a 1 a a^2 0 a a^2 1 1 0|0 0 1 0 0 0 0 0 0 a^2 a^2 a 1 0 0 1 a a 1 a^2 0 a^2 0 a 0 1 0 1 1 a^2 1 a a 0 a 1 0 1 1 a a a 1 0 1 1 1 a^2 1 0 0 a^2 a a^2 0 a^2 0 a 0 a^2 1 0 1 1 a a 0 a 0 a^2 1 0 a^2 a a a^2 0 a a^2 1 0 a a^2 0 0 a^2 1]
      [0 0 a 0 0 0 0 0 0 1 1 a^2 a 0 0 a a^2 a^2 a 1 0 1 0 a^2 0 a 0 a a 1 a a^2 a^2 0 a^2 a 0 a a a^2 a^2 a^2 a 0 a a a 1 a 0 0 1 a^2 1 0 1 0 a^2 0 1 a 0 a a a^2 a^2 0 a^2 0 1 a 0 1 a^2 a^2 1 0 a^2 1 a 0 a^2 1 0 0 1 a|0 0 a^2 0 0 0 0 0 0 1 a 0 a a a a a 1 a 1 1 a 1 a 0 1 1 0 1 a 1 1 0 a^2 0 0 0 a a^2 a a 0 a^2 a^2 0 1 1 1 0 1 1 a 0 a a^2 1 0 a a a^2 a a a^2 1 a a^2 a^2 1 0 1 a^2 1 0 0 0 a 0 a a a^2 a a^2 0 a a^2 a a]
      [0 0 0 1 0 0 0 0 0 a a a^2 a^2 0 a 1 a 0 0 0 a 0 a^2 0 a^2 1 a^2 0 a 1 a^2 a 0 a^2 a^2 a^2 1 a^2 a a^2 a a 0 0 a a^2 a^2 1 1 0 0 a^2 0 a 0 a^2 0 1 1 1 a^2 1 1 1 1 a^2 0 0 1 1 1 1 a^2 a^2 1 a^2 1 a^2 0 a a 1 a^2 0 0 a^2 a|0 0 0 0 0 0 0 0 0 a^2 a 1 a^2 a^2 a 1 0 0 a 0 0 a^2 0 a a 0 0 a^2 a^2 1 1 1 a^2 0 0 0 a^2 a a^2 a 0 a a 1 a a^2 0 a^2 a^2 a a^2 a a 1 a a 1 a^2 a 1 a^2 1 0 a^2 1 1 1 a 0 a^2 a 0 a 1 a 1 1 1 a^2 0 a 1 a^2 1 a^2 0 1]
      [0 0 0 0 0 0 0 0 0 a 1 a^2 a a 1 a^2 0 0 1 0 0 a 0 1 1 0 0 a a a^2 a^2 a^2 a 0 0 0 a 1 a 1 0 1 1 a^2 1 a 0 a a 1 a 1 1 a^2 1 1 a^2 a 1 a^2 a a^2 0 a a^2 a^2 a^2 1 0 a 1 0 1 a^2 1 a^2 a^2 a^2 a 0 1 a^2 a a^2 a 0 a^2|0 0 0 1 0 0 0 0 0 0 a^2 0 1 a a^2 a a 0 1 0 a a a^2 1 a 1 a^2 a 0 a 0 1 a a^2 a^2 a^2 a^2 a 0 a a a^2 1 a^2 a^2 1 a^2 a^2 a^2 1 a a 1 1 1 a a^2 a^2 0 a 1 a 1 a^2 a 0 a^2 1 1 a^2 0 1 a 0 0 0 a 0 a a a^2 a 1 a^2 a a^2 1]
      [0 0 0 a 0 0 0 0 0 1 a a a^2 a a 1 a^2 0 1 0 a^2 a 1 1 0 a 1 a 1 1 a 0 a 1 1 1 0 0 1 0 a^2 a 1 a^2 a a^2 1 0 0 1 a 0 1 0 1 0 a^2 0 a^2 1 a^2 1 a 0 1 a a^2 1 a 0 a^2 a 0 a a^2 a 1 a a a^2 a 1 a^2 a^2 a 1 0|0 0 0 1 0 0 0 0 0 1 0 a 0 a^2 0 0 a 0 a 0 a a^2 a^2 a 1 1 a^2 a^2 1 0 a a^2 a^2 a^2 a^2 a^2 a 1 1 1 a 0 a 1 0 0 a^2 a a a a^2 1 a a^2 a 1 1 a a^2 0 0 0 1 a 0 a 1 a 1 a a^2 1 1 a a^2 a 0 a a^2 a 0 0 0 1 a^2 a^2 a^2]
      [0 0 0 a 0 0 0 0 0 a 0 a^2 0 1 0 0 a^2 0 a^2 0 a^2 1 1 a^2 a a 1 1 a 0 a^2 1 1 1 1 1 a^2 a a a a^2 0 a^2 a 0 0 1 a^2 a^2 a^2 1 a a^2 1 a^2 a a a^2 1 0 0 0 a a^2 0 a^2 a a^2 a a^2 1 a a a^2 1 a^2 0 a^2 1 a^2 0 0 0 a 1 1 1|0 0 0 a^2 0 0 0 0 0 1 1 a a 0 1 a^2 1 0 0 0 1 0 a 0 a a^2 a 0 1 a^2 a 1 0 a a a a^2 a 1 a 1 1 0 0 1 a a a^2 a^2 0 0 a 0 1 0 a 0 a^2 a^2 a^2 a a^2 a^2 a^2 a^2 a 0 0 a^2 a^2 a^2 a^2 a a a^2 a a^2 a 0 1 1 a^2 a 0 0 a 1]
      [0 0 0 0 1 0 0 0 0 1 0 a 0 1 a 1 1 0 a 1 a 0 0 1 1 1 a^2 1 a^2 0 0 1 1 a^2 1 1 1 1 a^2 0 1 a a^2 a 1 0 1 0 a a^2 a^2 a a 0 a^2 0 a 1 0 a 0 a^2 1 a a^2 a a^2 0 1 a^2 0 0 a^2 1 a a^2 0 1 1 0 a a 0 a a^2 a^2 1|0 0 0 0 0 0 0 0 0 a a a a a 0 a^2 a 0 1 1 0 a^2 a a^2 1 a^2 a^2 a^2 1 0 0 0 a^2 a a 1 0 a 0 a a^2 a^2 a^2 1 a a a^2 a a 1 a^2 a a 0 a a^2 0 a 1 0 a^2 a^2 0 0 a a 1 a^2 a 1 a^2 a^2 a^2 a a^2 1 0 0 1 a a 0 a 0 a^2 a^2 1]
      [0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 a 1 0 a^2 a^2 0 a 1 a a^2 a a a a^2 0 0 0 a 1 1 a^2 0 1 0 1 a a a a^2 1 1 a 1 1 a^2 a 1 1 0 1 a 0 1 a^2 0 a a 0 0 1 1 a^2 a 1 a^2 a a a 1 a a^2 0 0 a^2 1 1 0 1 0 a a a^2|0 0 0 0 1 0 0 0 0 0 1 a^2 1 0 a a^2 0 0 1 a a a 1 a^2 a a^2 1 a^2 0 0 0 1 a^2 a 0 a 1 0 a^2 1 a^2 0 1 1 0 1 a^2 1 a^2 0 1 a^2 a^2 0 a a a 0 a^2 a a 1 1 a a a^2 0 a 0 0 a a 1 0 0 0 0 1 a 1 a^2 a 1 a 1 1 a]
      [0 0 0 0 a 0 0 0 0 a^2 1 a 1 a^2 a^2 0 a^2 0 0 1 a^2 a 1 0 1 0 a^2 0 a 0 0 a 0 0 a^2 1 a a^2 1 1 0 1 a^2 0 a^2 1 0 1 a a a^2 a a 0 0 a a^2 a^2 a^2 a^2 a a^2 a a^2 0 a a a a^2 a a a a^2 a^2 1 a 0 a 1 1 a a^2 1 a^2 a^2 a^2 1|0 0 0 0 1 0 0 0 0 a^2 a 0 a a^2 a a a^2 0 a^2 0 a a^2 a a 0 a 0 a a 0 0 1 a 1 a^2 0 1 a^2 a^2 a a 1 0 a^2 a^2 a a a 0 a 0 0 0 0 1 a^2 a a^2 1 a a^2 0 1 a 1 0 a a^2 a^2 a a^2 a^2 0 a^2 1 a 0 1 0 a 0 a a a 0 0 0]
      [0 0 0 0 a 0 0 0 0 1 a^2 0 a^2 1 a^2 a^2 1 0 1 0 a^2 1 a^2 a^2 0 a^2 0 a^2 a^2 0 0 a a^2 a 1 0 a 1 1 a^2 a^2 a 0 1 1 a^2 a^2 a^2 0 a^2 0 0 0 0 a 1 a^2 1 a a^2 1 0 a a^2 a 0 a^2 1 1 a^2 1 1 0 1 a a^2 0 a 0 a^2 0 a^2 a^2 a^2 0 0 0|0 0 0 0 a^2 0 0 0 0 a^2 0 1 0 a^2 1 a^2 a^2 0 1 a^2 1 0 0 a^2 a^2 a^2 a a^2 a 0 0 a^2 a^2 a a^2 a^2 a^2 a^2 a 0 a^2 1 a 1 a^2 0 a^2 0 1 a a 1 1 0 a 0 1 a^2 0 1 0 a a^2 1 a 1 a 0 a^2 a 0 0 a a^2 1 a 0 a^2 a^2 0 1 1 0 1 a a a^2]
      [0 0 0 0 0 1 0 0 0 a 1 0 0 1 a a^2 a^2 a^2 a^2 a a^2 a^2 a a^2 1 0 a 0 a^2 a a^2 0 a 0 0 a a a a 1 0 1 a 1 a^2 a 1 a^2 a^2 1 a a a a^2 a a a a 1 a^2 0 1 1 a a^2 a a^2 a^2 0 a^2 a^2 0 1 a 1 a a 1 a 1 a^2 1 a^2 0 a 0 1|0 0 0 0 0 0 0 0 0 0 a 1 1 a^2 1 1 a^2 a a 1 1 1 a^2 a^2 0 a a^2 a^2 a^2 a^2 0 1 a^2 0 a a^2 a^2 a a 0 a 1 0 1 1 1 1 1 a 0 a^2 a^2 a a 0 a^2 a^2 0 0 a^2 1 a^2 a 0 a^2 a^2 a a^2 1 a a 1 a a^2 0 1 a a 1 0 0 1 a^2 0 1 a 0]
      [0 0 0 0 0 0 0 0 0 0 1 a^2 a^2 a a^2 a^2 a 1 1 a^2 a^2 a^2 a a 0 1 a a a a 0 a^2 a 0 1 a a 1 1 0 1 a^2 0 a^2 a^2 a^2 a^2 a^2 1 0 a a 1 1 0 a a 0 0 a a^2 a 1 0 a a 1 a a^2 1 1 a^2 1 a 0 a^2 1 1 a^2 0 0 a^2 a 0 a^2 1 0|0 0 0 0 0 1 0 0 0 a 0 a^2 a^2 a^2 1 0 1 a a 1 0 0 0 1 1 1 0 a 1 0 a^2 a^2 0 0 1 0 0 a^2 a^2 1 1 a a a 0 1 a 0 a 1 0 0 a^2 a a 0 0 a 1 1 a^2 a^2 0 a 1 0 a 1 a^2 a a a^2 0 0 1 1 a^2 0 1 1 a^2 a 1 0 1 1 1]
      [0 0 0 0 0 a 0 0 0 a^2 a^2 a^2 a^2 0 0 a a^2 0 0 0 a a 1 a^2 a 1 1 a a^2 1 1 a^2 1 0 1 1 1 a a a 1 1 a^2 1 a 0 1 a 0 a 1 1 a 0 a^2 1 1 a^2 a a^2 a^2 0 a^2 a^2 a^2 1 0 a^2 a^2 0 0 a^2 a^2 1 a 0 a a^2 0 a 1 1 a^2 0 0 1 a|0 0 0 0 0 1 0 0 0 a a^2 1 1 a a^2 a 0 1 1 a^2 a a 1 0 1 a 1 a^2 0 1 a^2 1 1 0 a 1 1 0 0 1 a 0 a 0 a a^2 0 a 1 1 1 1 0 1 a 1 1 a 1 0 1 a a^2 a 0 1 1 0 1 1 1 1 a^2 1 1 a^2 0 a^2 a^2 1 a^2 0 0 0 a^2 a 1]
      [0 0 0 0 0 a 0 0 0 a^2 1 a a a^2 1 a^2 0 a a 1 a^2 a^2 a 0 a a^2 a 1 0 a 1 a a 0 a^2 a a 0 0 a a^2 0 a^2 0 a^2 1 0 a^2 a a a a 0 a a^2 a a a^2 a 0 a a^2 1 a^2 0 a a 0 a a a a 1 a a 1 0 1 1 a 1 0 0 0 1 a^2 a|0 0 0 0 0 a^2 0 0 0 1 a^2 0 0 a^2 1 a a a a 1 a a 1 a a^2 0 1 0 a 1 a 0 1 0 0 1 1 1 1 a^2 0 a^2 1 a^2 a 1 a^2 a a a^2 1 1 1 a 1 1 1 1 a^2 a 0 a^2 a^2 1 a 1 a a 0 a a 0 a^2 1 a^2 1 1 a^2 1 a^2 a a^2 a 0 1 0 a^2]
      [0 0 0 0 0 0 1 0 0 a a a a^2 a a 0 1 a a^2 a^2 1 1 a 1 0 a 0 a^2 0 1 a^2 a^2 1 1 0 a^2 0 a 1 0 0 a a^2 1 1 0 1 a^2 a a^2 0 0 a a^2 a^2 0 1 1 a^2 a^2 0 a^2 0 a 0 a 0 a a^2 a^2 a 0 0 a^2 a^2 a 1 a 1 1 a a a^2 a 1 a^2 a|0 0 0 0 0 0 0 0 0 0 1 1 1 a^2 0 a 1 0 1 a^2 a a a^2 a^2 1 1 1 1 1 a a^2 0 0 a a a^2 0 a^2 a^2 0 a^2 1 a a 0 a 1 a 1 a^2 a^2 a^2 a a^2 a^2 0 1 a 0 0 a a a^2 a^2 0 0 1 0 a^2 1 a^2 a a 1 0 0 1 1 a^2 a a^2 a^2 1 0 1 1 a]
      [0 0 0 0 0 0 0 0 0 0 a^2 a^2 a^2 a 0 1 a^2 0 a^2 a 1 1 a a a^2 a^2 a^2 a^2 a^2 1 a 0 0 1 1 a 0 a a 0 a a^2 1 1 0 1 a^2 1 a^2 a a a 1 a a 0 a^2 1 0 0 1 1 a a 0 0 a^2 0 a a^2 a 1 1 a^2 0 0 a^2 a^2 a 1 a a a^2 0 a^2 a^2 1|0 0 0 0 0 0 1 0 0 a 1 1 0 0 a 1 a a 0 1 0 0 0 a^2 a^2 1 a^2 0 a^2 0 1 a^2 1 0 1 1 0 0 a^2 0 a 1 a 0 1 1 a a 1 1 a a a^2 1 1 0 a 0 a^2 a^2 1 a a 0 0 a a^2 a 1 0 0 1 1 0 a^2 a a 1 a^2 0 0 0 0 a a 0 a^2]
      [0 0 0 0 0 0 a 0 0 a^2 0 0 a 1 a^2 1 1 a^2 a a^2 a^2 a^2 1 0 a^2 0 a^2 a a^2 a^2 a^2 1 a a^2 1 a^2 0 1 0 0 a 0 0 a^2 a 1 1 0 0 a^2 a a a a^2 a^2 0 1 a^2 1 1 1 0 a 1 0 a^2 a^2 a^2 a^2 a 1 1 1 a 1 a^2 1 0 0 a^2 1 1 a a^2 1 a a|0 0 0 0 0 0 1 0 0 a a^2 a^2 a 1 a a 0 a a 0 a^2 a^2 1 a 1 a^2 1 a 1 a^2 0 a^2 1 a^2 a 0 0 1 a 0 a^2 a^2 1 a^2 1 a 0 1 a^2 0 a^2 a^2 0 0 0 0 0 a^2 a^2 a^2 a 1 a^2 1 0 a 1 a 0 a 1 a a a a^2 a 0 a^2 a a^2 1 1 a a 0 a 0]
      [0 0 0 0 0 0 a 0 0 a^2 1 1 a^2 a a^2 a^2 0 a^2 a^2 0 1 1 a a^2 a 1 a a^2 a 1 0 1 a 1 a^2 0 0 a a^2 0 1 1 a 1 a a^2 0 a 1 0 1 1 0 0 0 0 0 1 1 1 a^2 a 1 a 0 a^2 a a^2 0 a^2 a a^2 a^2 a^2 1 a^2 0 1 a^2 1 a a a^2 a^2 0 a^2 0|0 0 0 0 0 0 a^2 0 0 1 1 1 a 1 1 0 a^2 1 a a a^2 a^2 1 a^2 0 1 0 a 0 a^2 a a a^2 a^2 0 a 0 1 a^2 0 0 1 a a^2 a^2 0 a^2 a 1 a 0 0 1 a a 0 a^2 a^2 a a 0 a 0 1 0 1 0 1 a a 1 0 0 a a 1 a^2 1 a^2 a^2 1 1 a 1 a^2 a 1]
      [0 0 0 0 0 0 0 1 0 a 0 1 a^2 a a^2 1 a^2 a^2 a^2 0 1 a a^2 0 0 a 1 0 a 0 0 0 a a^2 1 0 a 1 0 a 0 0 1 a 0 a^2 a a^2 a^2 a a^2 1 0 a a 0 0 a^2 a^2 1 a a^2 a^2 a^2 a a 0 a^2 0 0 a^2 1 a 1 a^2 0 a^2 a 1 a 1 a 1 0 a^2 0 a^2|0 0 0 0 0 0 0 0 0 a a^2 a a^2 0 a a^2 a^2 0 1 a 1 a^2 a 0 1 1 a^2 a^2 0 a^2 0 1 0 a^2 a^2 a^2 1 a a^2 a 1 0 a^2 0 1 1 a 1 a^2 0 a^2 a^2 a a^2 a^2 a^2 a 1 a 0 0 1 a^2 a a^2 a 1 a a a a^2 a^2 a^2 1 1 1 0 a 0 a a a 1 a a 0 a^2]
      [0 0 0 0 0 0 0 0 0 1 a 1 a 0 1 a a 0 a^2 1 a^2 a 1 0 a^2 a^2 a a 0 a 0 a^2 0 a a a a^2 1 a 1 a^2 0 a 0 a^2 a^2 1 a^2 a 0 a a 1 a a a 1 a^2 1 0 0 a^2 a 1 a 1 a^2 1 1 1 a a a a^2 a^2 a^2 0 1 0 1 1 1 a^2 1 1 0 a|0 0 0 0 0 0 0 1 0 a^2 a 0 1 a a a^2 1 a^2 0 1 a 0 a 0 a^2 1 a^2 a a a 0 a^2 a 1 a^2 a 1 0 a a^2 a^2 0 a^2 a a^2 0 a^2 0 1 a 1 a^2 1 0 0 a 1 0 a 1 a 0 1 a 0 a^2 a^2 a 1 1 1 a^2 0 a 0 a^2 a^2 a^2 1 a^2 0 a^2 a 1 a 0 1]
      [0 0 0 0 0 0 0 a 0 a a a^2 a^2 a^2 0 0 a^2 1 a 1 1 1 0 0 a^2 0 0 a a^2 a 0 a^2 a^2 a^2 0 a 0 a^2 a a a^2 0 0 a^2 a^2 a a a a^2 a^2 a^2 0 1 1 1 a 1 a 0 a a^2 a a^2 0 1 a a^2 0 1 1 a^2 0 1 1 a a^2 1 a a a a^2 a 1 1 0 0 a^2|0 0 0 0 0 0 0 1 0 0 a^2 a^2 0 a 1 a 0 a^2 a a 0 1 1 0 1 a^2 a a^2 a a^2 0 1 a 0 a a^2 a^2 a^2 a^2 0 1 0 a a 1 a 0 a 0 a 0 a a 1 1 a^2 a a 1 1 a a 0 1 1 0 1 1 a a 0 a 1 0 a 1 a^2 0 1 0 a^2 0 0 a 1 0 0]
      [0 0 0 0 0 0 0 a 0 0 1 1 0 a^2 a a^2 0 1 a^2 a^2 0 a a 0 a 1 a^2 1 a^2 1 0 a a^2 0 a^2 1 1 1 1 0 a 0 a^2 a^2 a a^2 0 a^2 0 a^2 0 a^2 a^2 a a 1 a^2 a^2 a a a^2 a^2 0 a a 0 a a a^2 a^2 0 a^2 a 0 a^2 a 1 0 a 0 1 0 0 a^2 a 0 0|0 0 0 0 0 0 0 a^2 0 1 0 a^2 a 1 a a^2 a a a 0 a^2 1 a 0 0 1 a^2 0 1 0 0 0 1 a a^2 0 1 a^2 0 1 0 0 a^2 1 0 a 1 a a 1 a a^2 0 1 1 0 0 a a a^2 1 a a a 1 1 0 a 0 0 a a^2 1 a^2 a 0 a 1 a^2 1 a^2 1 a^2 0 a 0 a]
      [0 0 0 0 0 0 0 0 1 1 a^2 a^2 0 a 1 0 0 a^2 0 a 0 a^2 0 a 0 a a^2 a^2 a a^2 1 0 a a^2 a 0 0 a^2 a^2 1 0 1 1 0 a^2 a a^2 a a^2 a a 1 a a^2 a a^2 1 0 0 a^2 1 1 a^2 0 0 a^2 a^2 0 1 a^2 1 a^2 a^2 1 a^2 1 0 a a a^2 0 a a 0 a^2 1 a^2|0 0 0 0 0 0 0 0 0 a 1 a^2 0 0 1 a 1 0 1 0 a^2 0 0 0 1 1 0 0 a^2 a 1 a a a^2 a^2 0 a^2 a 1 a a 1 0 0 a 1 0 1 a^2 0 a^2 a^2 a a^2 a 0 0 0 a 0 1 a^2 1 0 a^2 a^2 1 a 1 a^2 a^2 1 a^2 0 1 0 0 a^2 a^2 0 1 1 a 0 a 0 a]
      [0 0 0 0 0 0 0 0 0 1 a^2 a 0 0 a^2 1 a^2 0 a^2 0 a 0 0 0 a^2 a^2 0 0 a 1 a^2 1 1 a a 0 a 1 a^2 1 1 a^2 0 0 1 a^2 0 a^2 a 0 a a 1 a 1 0 0 0 1 0 a^2 a a^2 0 a a a^2 1 a^2 a a a^2 a 0 a^2 0 0 a a 0 a^2 a^2 1 0 1 0 1|0 0 0 0 0 0 0 0 1 0 0 1 0 a a 1 a^2 a^2 a^2 a a a^2 0 a a^2 1 a^2 a^2 0 a a 1 a^2 1 0 0 a a 0 0 1 a 1 0 a 1 a^2 1 1 a 0 a^2 a^2 1 a^2 a^2 1 0 1 a^2 a a^2 0 0 a 1 0 1 a 1 a^2 0 1 1 0 1 0 0 0 a^2 a^2 1 a^2 0 a 1 a]
      [0 0 0 0 0 0 0 0 a a^2 a a^2 0 a^2 1 1 a^2 1 a^2 a^2 a 1 0 a^2 a^2 0 1 1 1 0 1 1 a a^2 1 0 a 0 a a^2 1 1 a 0 0 0 1 0 a^2 a^2 1 0 a a^2 a 1 a 0 1 1 1 0 a 0 a a^2 a 1 1 a^2 0 a a^2 a a a 0 1 1 1 a^2 0 a 0 0 a 0|0 0 0 0 0 0 0 0 1 a^2 a 0 0 a 0 a 1 a^2 1 a a^2 a^2 0 a 1 a^2 a^2 a^2 1 1 0 a 0 0 1 0 a^2 1 a a^2 a 0 1 0 1 a^2 a^2 a^2 0 a 1 a 0 0 0 a^2 1 0 a a^2 0 a a 0 a^2 0 a a 0 0 a a 0 1 a 1 0 1 1 a^2 1 a^2 0 0 1 1 1]
      [0 0 0 0 0 0 0 0 a 1 a^2 0 0 a^2 0 a^2 a 1 a a^2 1 1 0 a^2 a 1 1 1 a a 0 a^2 0 0 a 0 1 a a^2 1 a^2 0 a 0 a 1 1 1 0 a^2 a a^2 0 0 0 1 a 0 a^2 1 0 a^2 a^2 0 1 0 a^2 a^2 0 0 a^2 a^2 0 a a^2 a 0 a a 1 a 1 0 0 a a a|0 0 0 0 0 0 0 0 a^2 a^2 a a 0 1 a^2 0 0 a 0 1 0 a 0 1 0 1 a a 1 a a^2 0 1 a 1 0 0 a a a^2 0 a^2 a^2 0 a 1 a 1 a 1 1 a^2 1 a 1 a a^2 0 0 a a^2 a^2 a 0 0 a a 0 a^2 a a^2 a a a^2 a a^2 0 1 1 a 0 1 1 0 a a^2 a]

last modified: 2024-05-13
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.
This page is maintained by Markus Grassl (codes@codetables.de). Last change: 10.06.2024
Bounds on [[87,69]]4

Construction

Notes

Bounds on [[87,69]]₄
