Bounds on the minimum distance of additive quantum codes

Bounds on [[45,25]]2

lower bound:5
upper bound:7

Construction

Construction of a [[45,25,5]] quantum code:
[1]:  [[51, 19, 9]] Quantum code over GF(2^2)
     cyclic code of length 51 with generating polynomials [
w*x^50 + w*x^47 + w^2*x^46 + x^45 + w*x^44 + x^42 + x^41 + x^38 + w^2*x^36 + w^2*x^34 + x^33 + x^31 + x^27 + w^2*x^26 + w*x^25 + w^2*x^24 + w*x^23 + w^2*x^22 + w*x^21 + w^2*x^20 + w*x^19 + w^2*x^18 + w*x^16 + 1,
w^2*x^50 + w^2*x^47 + x^46 + w*x^45 + w^2*x^44 + w*x^42 + w*x^41 + w*x^38 + x^36 + x^34 + w*x^33 + w*x^31 + w*x^27 + x^26 + w^2*x^25 + x^24 + w^2*x^23 + x^22 + w^2*x^21 + x^20 + w^2*x^19 + x^18 + w^2*x^16 + w
]
[2]:  [[45, 25, 5]] Quantum code over GF(2^2)
     Shortening of the stabilizer code of [1] at { 1, 3, 4, 18, 23, 35 }

    stabilizer matrix:

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