Bounds on the minimum distance of additive quantum codes

Bounds on [[45,8]]2

lower bound:9
upper bound:14

Construction

Construction of a [[45,8,9]] quantum code:
[1]:  [[42, 8, 9]] Quantum code over GF(2^2)
     quasicyclic code of length 42 stacked to height 2 with 4 generating polynomials
[2]:  [[45, 8, 9]] Quantum code over GF(2^2)
     ExtendCode [1] by 3

    stabilizer matrix:

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

last modified: 2006-04-03

Notes


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