Bounds on the minimum distance of additive quantum codes

Bounds on [[136,116]]₂

lower bound:	4
upper bound:	6
Construction

Construction of a [[136,116,4]] quantum code:
[1]:  [[252, 238, 4]] quantum code over GF(2^2)
     Construction from a stored generator matrix
[2]:  [[136, 122, 4]] quantum code over GF(2^2)
     Shortening of [1] at { 1, 5, 7, 8, 9, 10, 11, 12, 19, 22, 24, 25, 28, 29, 30, 31, 32, 34, 36, 39, 41, 42, 44, 45, 46, 51, 53, 54, 57, 58, 59, 62, 69, 73, 76, 82, 87, 88, 92, 93, 94, 96, 97, 98, 99, 111, 115, 117, 119, 123, 124, 125, 127, 128, 130, 131, 133, 137, 139, 140, 141, 142, 147, 148, 149, 150, 151, 152, 157, 158, 159, 161, 162, 163, 165, 168, 170, 171, 173, 176, 177, 180, 181, 185, 189, 192, 194, 197, 199, 201, 202, 206, 207, 208, 212, 213, 214, 217, 218, 219, 220, 226, 228, 229, 230, 232, 233, 234, 236, 237, 239, 244, 245, 246, 249, 250 }
[3]:  [[136, 116, 4]] quantum code over GF(2^2)
     Subcode of [2]

    stabilizer matrix:

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

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 [[136,116]]2

Construction

Notes

Bounds on [[136,116]]₂
