Bounds on the minimum distance of additive quantum codes

Bounds on [[45,1]]2

lower bound:13
upper bound:15

Construction

Construction of a [[45,1,13]] quantum code:
[1]:  [[45, 1, 13]] Quantum code over GF(2^2)
     cyclic code of length 45 with generating polynomials [
w^2*x^44 + w^2*x^43 + w*x^42 + x^41 + w^2*x^39 + w*x^38 + x^37 + w*x^36 + w^2*x^34 + x^33 + x^32 + w*x^31 + w^2*x^27 + w^2*x^26 + x^25 + w*x^24 + w*x^23 + w*x^18 + w^2*x^17 + w*x^16 + x^15 + x^14 + w^2*x^13 + w^2*x^12 + w*x^11 + w^2*x^10 + w^2*x^9 + x^8 + w^2*x^7 + w*x^6 + w^2*x^3 + w^2*x^2 + w^2*x + w^2,
w*x^42 + w^2*x^41 + x^40 + x^39 + w*x^36 + w^2*x^35 + w^2*x^34 + w^2*x^31 + x^29 + x^28 + w^2*x^26 + w*x^25 + w^2*x^24 + w*x^22 + w*x^21 + x^20 + w*x^19 + x^18 + w*x^17 + x^16 + x^15 + x^14 + w^2*x^13 + w*x^12 + w*x^10 + x^8 + w*x^7 + x^6 + x^4 + w^2*x^2 + w^2*x + w^2
]

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

last modified: 2006-04-07

Notes


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