Bounds on the minimum distance of additive quantum codes

Bounds on [[156,133]]₂

lower bound:	4
upper bound:	6
Construction

Construction of a [[156,133,4]] quantum code:
[1]:  [[252, 238, 4]] quantum code over GF(2^2)
     Construction from a stored generator matrix
[2]:  [[156, 142, 4]] quantum code over GF(2^2)
     Shortening of [1] at { 1, 2, 4, 5, 8, 12, 15, 19, 21, 23, 24, 25, 28, 34, 37, 46, 48, 49, 61, 62, 65, 69, 71, 72, 79, 81, 83, 88, 92, 94, 98, 100, 103, 104, 105, 109, 114, 116, 118, 119, 120, 123, 128, 129, 130, 135, 139, 140, 144, 146, 149, 156, 158, 159, 161, 165, 169, 170, 173, 178, 182, 185, 186, 187, 190, 191, 194, 196, 197, 207, 208, 209, 213, 217, 218, 219, 221, 222, 223, 225, 231, 232, 234, 235, 238, 239, 240, 241, 242, 245, 246, 247, 249, 250, 251, 252 }
[3]:  [[156, 133, 4]] quantum code over GF(2^2)
     Subcode of [2]

    stabilizer matrix:

      [1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 1 0 1 1 1 0 1 1 0 1 1 1 1 0 1 0 1 0 1 1 1 1 1 1 1 0 1 1 0 0 1 1 1 1 1 0 1 1 1 1 1 1 0 0 1 0 0 0 0|0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 0 1 0 1 0 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 1 1 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 1 1 1 0 1 0 0 1 1 0 1 0]
      [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 0 1 0 1 0 0 0 0 1 0 1 1 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 1 1 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 1 0 0 1 1 1 1 0 1 0 1|1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 0 0 1 0 1 1 1 1 1 1 0 1 1 1 1 1 0 0 0 1 1 0 0 0 1 1 0 1 1 1 1 0 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 1 1 0 0 1 1 1 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1 1 0 0 1 0 0 0 0 1 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 1]
      [0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 0 1 1 0 1 1 0 1 1 1 1 1 1 1 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 0 0 0 1 1 1 1 0 0 0 1 0 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 1 0 1 1 1 1 1 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 1 0 1 0 0 1 1 0 0 0 0 0 1|0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 0 0 0 1 1 1 0 0 1 0 1 1 0 1 0 0 0 0 1 0 0 0 1 1 1 0 0 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 1 1 0 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 1]
      [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 0 0 0 1 1 1 0 0 1 0 1 1 0 1 0 0 0 0 1 0 0 0 1 1 1 0 0 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 1 1 0 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 0 0 0 1 1 1 0|0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 1 1 1 0 1 1 0 1 1 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 1 0 1 1 1 0 0 1 1 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 1 1 1 0 0 1 0 0 0 1 0 1 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 0 1 1 0 0 0 0 0 1 1]
      [0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 1 1 1 0 1 0 0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 1 1 1 1 0 1 1 0 0 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1 0 0 1 0 1 1 0 1 0 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 0 0 0 1 0 0 0 1 1 0 0 1 1 0 0 1 1 1 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0|0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 1 1 0 1 1 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1 1 1 1 1 1 0 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 1 1 0 1 1 0 1 1 1 0 0 1 1 0 0 0 0 1 1 0 1 0 1 0 1 1 0 0 0]
      [0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 1 1 0 1 1 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 0 1 1 1 1 1 1 1 1 0 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 1 1 0 1 1 0 1 1 1 0 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 1 1 1 0|0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 0 0 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1 1 1 0 0 1 1 1 0 0 1 0 0 1 1 0 1 0 1 0 1 0 0 1 1 1 1 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 1 0 0]
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      [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 1 1 1 0 1 1 0 1 1 1 1 1 1 1 0 1 0 0 0 1 0 0 1 1 1 1 1 1 0 1 0|0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 1 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 0 0 1 1 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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]
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      [0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 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 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 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 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 0 1 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 0 0 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0]
      [0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 0 1 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 0 0 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 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 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 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 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1 1 0 0 0 1 0 0 1]
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      [0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 1 1 0 0 1 1 1 0 0 1 1 0 1 1 0 1 0 1 0 1 0 0 1 1 1 1 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 1 1 0 1 0 1 1 1|0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 0 0 1 1 1 1 0 0 1 0 1 0 0 0 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 1 0 1 0 1 1 1 1 1 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 1 0 0 0 1 0 1 0 1 1 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 0 0 1 0 1 0 1 0 0 0 1 1 0 1 0 0 1 0 1 0 0 0 1 0 1 1 0 0 0 0 1 1 0 1 1 0 0]
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      [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 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 1 1 0 0 0 1 1]
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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 1 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 1 0 0 1 0 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 0 0 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 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 [[156,133]]2

Construction

Notes

Bounds on [[156,133]]₂
