Bounds on the minimum distance of additive quantum codes

Bounds on [[137,121]]₂

lower bound:	4
upper bound:	4
Construction

Construction of a [[137,121,4]] quantum code:
[1]:  [[252, 238, 4]] quantum code over GF(2^2)
     Construction from a stored generator matrix
[2]:  [[137, 123, 4]] quantum code over GF(2^2)
     Shortening of [1] at { 3, 4, 5, 8, 10, 12, 15, 16, 21, 22, 23, 24, 27, 29, 30, 33, 35, 36, 37, 42, 44, 45, 49, 52, 54, 55, 57, 63, 66, 67, 73, 76, 77, 78, 79, 80, 82, 83, 85, 86, 88, 89, 90, 91, 92, 94, 101, 103, 105, 106, 107, 108, 110, 112, 114, 116, 117, 119, 122, 123, 127, 131, 134, 135, 140, 141, 144, 147, 150, 154, 159, 160, 163, 165, 169, 170, 179, 180, 182, 185, 186, 187, 189, 191, 192, 194, 195, 197, 200, 203, 204, 205, 206, 207, 208, 209, 210, 213, 215, 218, 220, 222, 223, 226, 228, 237, 240, 242, 244, 245, 246, 248, 249, 251, 252 }
[3]:  [[137, 121, 4]] quantum code over GF(2^2)
     Subcode of [2]

    stabilizer matrix:

      [1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 0 0 1 1 0 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 1 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 1 1 0 0 0 1 0 1 1 1 1 1 1 0 0 0 0 1 0 1 1 1 1 1 0 1 0 0 0 0 1 1|0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 0 0 1 1 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 1 0 0 1 0 0 0 0 1 1 0 1 1 1 1 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 1 1 1 0 0 1 1 1 1 0 1 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0]
      [0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 0 0 1 1 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 1 0 0 1 0 0 0 0 1 1 0 1 1 1 1 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 1 1 1 0 0 1 1 1 1 0 1 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0|1 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 1 0 1 0 0 0 1 1 1 0 1 0 1 1 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 0 1 1 0 0 1 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 1 1 1 0 1 0 1 1 0]
      [0 1 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 1 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 1 1 1 0 0 0 0 0 0 1 0 0 0 1 1 1 1 1 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 1|0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 0 0 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 1 0 1 1 0 0 0 0 0]
      [0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 0 0 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 1 0 1 1 0 1 1 0 1|0 1 0 0 0 0 0 0 1 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 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 1 0 1 1 1 1 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 1 1 0]
      [0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 0 1 0 0 1 1 0 1 1 1 1 0 1 1|0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0 1 0 1 1 0 1 1 1 0 1 1 0 0 1 0 1 0 0 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 1 1 0 1 1 1 0 1 1 1 1 0 1 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0]
      [0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0 1 0 1 1 0 1 1 1 0 1 1 0 0 1 0 1 0 0 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 1 1 0 1 1 1 0 1 1 1 1 0 1 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0|0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 1 1 1 0 0 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 1 1 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 0 0 1]
      [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 0 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 1 0 1 1 0 1 1 1 0 1|0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 0 0 1 0 0 0 1 0 1 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 0 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 1 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 0]
      [0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 0 0 1 0 0 0 1 0 1 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 0 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 1 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 0 1 0 0 1 0|0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 1 0 1 0 0 1 1 0 1 1 0 1 0 0 0 1 0 1 0 1 0 0 1 1 0 0 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 1 1 1 0 0 1 1 1 1 1 0 1 0 0 0 0 0 0 1 0 0 0 1 1 1 0 1 0 1 0 0 1 0 1 1 0 0 0 0 0 1 0 1 0 1 0 1 0 0 1 1 0 1 0 0 1 0 0 1 0 0 0]
      [0 0 0 0 1 0 0 1 1 0 0 1 1 1 0 1 0 1 0 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 1 0 0 1 0 1 0 1 0 1 1 1 0 0 0 0 1 0 0 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 1 0 0 0 1 0 0 0 0 0 0 1|0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 0 0 1 1 1 1 0 0 1 1 1 0 1 1 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 0 1 1 0 1 1]
      [0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 0 0 1 1 1 1 0 0 1 1 1 0 1 1 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 0 1 0 1 1 0|0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 1 0 1 1 1 1 1 0 0 0 1 1 0 0 0 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 1 1 1 0 1]
      [0 0 0 0 0 1 0 1 1 1 0 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 1 1 1 1 0 0 0 0 1 1 0 1 0 1 0 0 1 1 0 0 1 0 1 0 0 0 1 1 1 1 1 1 0 1 1 1 0 0 0 0 1 1 0 0 1 0 1 1 0 1 0 1 1 1 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 0 0 0 1 1 0 0 1 1 0 1 0 0 1 0 0 0 1 0 0 1 1|0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 1 1 0 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 1 0 0 1 0 1 1 1 0 1 0 1 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 1 0 0 0 0 1 0 0 1 1 0]
      [0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 1 1 0 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 1 0 0 1 0 1 1 1 0 1 0 1 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 1 0 0 0 0 1 0 1 0 1 1|0 0 0 0 0 1 0 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 1 1 0 1 1 1 0 1 1 1 0 1 0 1 1 0 1 1 0 0 1 0 1 0 1 0 0 1 1 0 0 0 1 0 1 1 0 0 0 1 0 0 0 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 1 1 0 0 1 1 0 1 0 0 1 0 1 0 0 1 1 1 0 1 0 1 1 1 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 0 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 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 0 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 1 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 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 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 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
      [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 1 0 0 0 0 0 0 0 0 0 0 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 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 1 1 1 1 1 1 1 1 0 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 [[137,121]]2

Construction

Notes

Bounds on [[137,121]]₂
