Bounds on the minimum distance of additive quantum codes

Bounds on [[81,53]]2

lower bound:6
upper bound:9

Construction

Construction of a [[81,53,6]] quantum code:
[1]:  [[85, 49, 8]] quantum code over GF(2^2)
     cyclic code of length 85 with generating polynomial w^2*x^83 + x^82 + x^79 + w^2*x^76 + w*x^75 + w^2*x^74 + w^2*x^73 + w*x^71 + w*x^70 + x^69 + w^2*x^68 + w*x^67 + x^66 + w^2*x^64 + w^2*x^63 + w*x^62 + w*x^61 + w^2*x^59 + x^58 + x^57 + w^2*x^56 + w*x^55 + x^54 + x^52 + w^2*x^50 + w^2*x^49 + x^48 + x^45 + x^44 + w*x^42 + w^2*x^41 + w*x^35 + w^2*x^34 + w^2*x^33 + w*x^31 + x^30 + w^2*x^29 + x^27 + x^26 + x^24 + x^23 + w^2*x^22 + x^20 + x^19 + x^18 + 1
[2]:  [[81, 53, 6]] quantum code over GF(2^2)
     Shortening of the stabilizer code of [1] at { 5, 7, 39, 80 }

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

last modified: 2006-04-03

Notes


This page is maintained by Markus Grassl (codes@codetables.de). Last change: 23.10.2014