Bounds on the minimum distance of additive quantum codes

Bounds on [[178,160]]₂

lower bound:	4
upper bound:	5
Construction

Construction of a [[178,160,4]] quantum code:
[1]:  [[252, 238, 4]] quantum code over GF(2^2)
     Construction from a stored generator matrix
[2]:  [[178, 164, 4]] quantum code over GF(2^2)
     Shortening of [1] at { 9, 11, 13, 15, 17, 19, 23, 24, 27, 30, 36, 38, 39, 41, 49, 56, 63, 65, 77, 82, 96, 105, 116, 119, 121, 123, 124, 125, 129, 130, 133, 135, 138, 143, 145, 149, 164, 166, 170, 171, 176, 177, 187, 189, 197, 199, 200, 201, 204, 208, 211, 217, 219, 221, 224, 226, 228, 229, 230, 231, 233, 234, 235, 239, 240, 241, 243, 244, 245, 246, 248, 249, 250, 252 }
[3]:  [[178, 160, 4]] quantum code over GF(2^2)
     Subcode of [2]

    stabilizer matrix:

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

last modified: 2009-02-08
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 the minimum distance of additive quantum codes

Bounds on [[178,160]]2

Construction

Notes

Bounds on [[178,160]]₂
