Bounds on the minimum distance of additive quantum codes

Bounds on [[52,9]]2

lower bound:10
upper bound:16

Construction

Construction of a [[52,9,10]] quantum code:
[1]:  [[51, 11, 10]] Quantum code over GF(2^2)
     cyclic code of length 51 with generating polynomials [
w*x^49 + x^47 + x^46 + x^45 + w^2*x^44 + x^43 + x^40 + w^2*x^38 + w^2*x^37 + x^34 + w*x^33 + x^32 + w^2*x^31 + w^2*x^29 + x^28 + w*x^26 + w*x^24 + w^2*x^22 + x^20 + 1,
w^2*x^49 + w*x^47 + w*x^46 + w*x^45 + x^44 + w*x^43 + w*x^40 + x^38 + x^37 + w*x^34 + w^2*x^33 + w*x^32 + x^31 + x^29 + w*x^28 + w^2*x^26 + w^2*x^24 + x^22 + w*x^20 + w
]
[2]:  [[51, 9, 10]] Quantum code over GF(2^2)
     Subcode of [1]
[3]:  [[52, 9, 10]] Quantum code over GF(2^2)
     ExtendCode [2] by 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 1 0 1 0 0 1 1 1 1 0 0 0 0 1 0|0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 1 0 1 0 0 0 1 1 1 0 1 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 1 0 0 1 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 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 1 1 1 0 1 0 1 0 0|0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 1 1 1 0 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 1 0 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 0|0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 1 1 1 0 0 1 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 0 0 0 1 1 1 0 0 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 1 1 0 1 1 0 1 1 0 0 1 1 1 1 0 1 0|0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 1 1 1 0 1 1 1 0 0 0 1 1 1 1 0 1 0 0 1 0 1 0 1 1 0 1 1 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 1 0 0 0 1 1 0 1 1 0 0 0 0 0 1 1 0|0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 1 1 0 1 1 1 1 1 1 0 1 1 0 1 1 0 1 1 0 1 0 0 1 1 1 1 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 1 0 1 1 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0|0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 0 1 1 0 1 1 0 0 1 0 0 1 1 0 1 0 0 1 0 1 1 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 1 0 1 0 0 1 0 0 0 1 0 1 1 1 0 0 1 1 0|0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 0 1 1 0 0 0 0 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 1 0 0 1 1 0 1 0 1 1 0 1 0 1 1 0 1 1 0|0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 1 1 0 1 0 1 0 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 1 0 0 0 0 1 0 1 1 0 0 1 0 0 1 1 1 1 0|0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 1 1 0 0 1 0 0 0 0 0 0 1 1 1 1 1 0 0 0 1 0 0 1 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 1 0 0 0 1 0 1 0 0 0 1 1 0 0 0 1 0 1 0|0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 0 1 1 0 1 0 0 1 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 0 1 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 1 0 0 0 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0|0 0 0 0 0 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 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 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 1 1 0|0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 1 1 0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 0 1 1 1 1 1 1 0 1 0 0 0 1 1 0 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 1 0 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0|0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 0 0 0 0 1 1 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 1 0 1 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 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 1 1 1 1 0 1 1 0|0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 1 0 0 0 1 1 0 1 1 1 1 0 1 0 0 0 0 1 1 1 0 0 1 0 1 1 1 0 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 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 1 1 1 0 1 1 0 1 1 1 0 1 1 1 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 0 0 0 0 1 1 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 1 0 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 0|0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 1 0 0 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 1 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 1 1 0 0 0 0 0|0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 1 1 1 0 1 1 1 1 1 1 0 1 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 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 0 1 1 1 0 0 1 1 0 0 0 1 1 1 1 1 0|0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 0 1 0 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 1 0 0 1 0 1 0 0 1 1 0 1 0 1 0 0|0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 0 1 0 1 1 1 1 1 0 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 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 0 0|0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 1 0 0 1 1 0 0 0 1 1 1 1 1 0 1 1 0 0 0 1 1 1 1 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 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 0 1 0 0 1 1 1 0 1 0|0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 1 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 1 0 1 0 0 1 1 0|0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 0 0 1 0 1 1 1 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 1 0 0 0 1 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 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 0 1 1 0|0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 0 1 0 1 1 0 0 1 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 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 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0|0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 0 0 1 1 0 1 1 1 0 1 1 0 1 0 0 0 1 0 1 1 0 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 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 0 0 0 1 0|0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 1 0 1 0 1 1 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 1 1 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 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 1 1 1 0 0 1 0 0 1 0 0|0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 1 0 0 0 1 0 1 0 0 1 1 1 0 1 0 0 0 1 0 0 1 0 1 1 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 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 1 0 0|0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 1 0 0 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 0|0 0 0 0 0 0 0 0 1 1 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 0 1 0 0 0 0 0 0 1 1 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 1 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 1 0|0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 1 1 1 0 0 1 0 1 0 0 1 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 1 0 0 0 1 0 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0|0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 1 1 0 1 1 1 1 0 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 0 0 0 0 1 0 1 0 1 1 1 0|0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 1 0 1 0 0 1 1 0 1 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 1 1 0 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 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 1 1 1 0 0 1 0 0 1 0|0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 1 0 0 1 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 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 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 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 0 0 0 0 1 1 1 0 1 1 1 1 1 1 1 0 1 0 1 0 1 1 0 0 1 1 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 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 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 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 1 1 0 0 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 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|1 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 1 0 0 1 0 1 0 1 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 0 0 0 0 0|0 1 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0 0 0 0 1 1 0 1 1 1 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|0 0 1 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0 0 0 0 1 1 0 1 1 1 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 0 0 1 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0 0 0 0 1 1 0 1 1 1 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 0 0 1 0 0 0 1 1 0 1 0 0 1 1 0 1 0 1 0 1 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 1 0 0 0 1 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 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 1 0 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 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 0 0 0 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0 0 0 0 1 1 0 1 1 1 1 1 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 1 1 0 0 0 1 1 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 0]

last modified: 2006-04-03

Notes


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