Bounds on the minimum distance of additive quantum codes

Bounds on [[49,3]]2

lower bound:12
upper bound:16

Construction

Construction of a [[49,3,12]] quantum code:
[1]:  [[49, 3, 12]] Quantum code over GF(2^2)
     cyclic code of length 49 with generating polynomial w*x^47 + x^46 + w*x^45 + w^2*x^44 + w^2*x^43 + x^42 + w^2*x^41 + w^2*x^39 + w^2*x^38 + w^2*x^37 + w*x^36 + w*x^34 + w*x^33 + x^32 + w^2*x^31 + w*x^29 + w^2*x^28 + w^2*x^27 + w*x^26 + w^2*x^25 + x^24 + w*x^22 + 1

    stabilizer matrix:

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

last modified: 2008-02-14

Notes


This page is maintained by Markus Grassl (grassl@ira.uka.de). Last change: 23.10.2014