Bounds on the minimum distance of additive quantum codes

Bounds on [[55,21]]2

lower bound:8
upper bound:12

Construction

Construction of a [[55,21,8]] quantum code:
[1]:  [[55, 21, 8]] Quantum code over GF(2^2)
     cyclic code of length 55 with generating polynomial w^2*x^49 + x^48 + x^47 + x^46 + w*x^44 + w^2*x^43 + w*x^41 + w*x^37 + x^35 + w*x^34 + w*x^32 + w^2*x^31 + x^30 + w^2*x^27 + w*x^25 + x^24 + x^22 + 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 1 0 0 1 0 1 0 0 1 1 1 0 1 1 1 0 1 1 0 1 1|0 1 1 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 0 0 0 1 0 0 1 1 0 0 0 1 1 1 1 0 1 0 0 1 0 1 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 0 1 1 0 0 1 1 0 1 1 0|0 1 0 1 0 0 0 1 0 1 1 1 1 0 1 0 1 0 1 1 0 1 0 0 0 0 1 1 0 1 0 1 0 0 1 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 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 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 0 1 1 0 0 1 1 0 1 1|0 0 1 0 1 0 0 0 1 0 1 1 1 1 0 1 0 1 0 1 1 0 1 0 0 0 0 1 1 0 1 0 1 0 0 1 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 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 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 1 0 0 1 0 0 0 1 0 1 1 0|0 1 1 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 1 1 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 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 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 1 0 0 1 0 0 0 1 0 1 1|1 0 1 1 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 1 1 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 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 1 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 1 1 1 0|1 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 1 1 1|0 1 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 1 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 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0|1 1 0 0 1 1 1 0 1 0 0 1 1 0 0 0 1 1 0 0 1 0 0 1 0 1 1 1 1 1 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 0 0 1 1 0 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 0 0 1 0 1 1 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0|0 1 1 0 0 1 1 1 0 1 0 0 1 1 0 0 0 1 1 0 0 1 0 0 1 0 1 1 1 1 1 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 0 0 1 1 0 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 0 0 1 0 1 1 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1|1 0 1 1 0 0 1 1 1 0 1 0 0 1 1 0 0 0 1 1 0 0 1 0 0 1 0 1 1 1 1 1 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 0 0 1 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 1 0 0 0 0 0 1 1 1 0 0 1 0 0 1 0 1 1 1 0 0|0 0 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 1 1 1 0 1 1 0 0 0 1 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 0 1 0 1 1 1 0|0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 1 1 1 0 1 1 0 0 0 1 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 0 1 0 1 1 1|1 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 1 1 1 0 1 1 0 0 0 1 1 1 1 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 1 0 0 0 0 1 0 0 1 0 0 1 1 1 0 0 1 0 0 0 0|1 0 1 0 0 1 1 0 1 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 1 0 0 1 1 0 0 1 1 1 1 0 1 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 1 0 0 0 0 1 0 0 1 0 0 1 1 1 0 0 1 0 0 0|1 1 0 1 0 0 1 1 0 1 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 1 0 0 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 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 0 1 0 0 1 1 1 0 0 1 0 0|0 1 1 0 1 0 0 1 1 0 1 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 1 0 0 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 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 0 1 0 0 1 1 1 0 0 1 0|1 0 1 1 0 1 0 0 1 1 0 1 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 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 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 0 1 0 0 1 1 1 0 0 1|0 1 0 1 1 0 1 0 0 1 1 0 1 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 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 1 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 1 0 1 0 0 0 1 1 1|1 1 0 0 1 1 0 0 1 0 0 1 1 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 0 1 0 1 0 1 0 1 1 1 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 1 0 0 0 1 0 1 1 1 1 0 0 0|1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 0 0 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 1 0 0 0 1 0 1 1 1 1 0 0|1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 0 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 1 1 0 1 1 1 0 0 1 0 0 0 1 0 1 1 1 1 0|1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 0 0 1 1 1 1 1 1 1 0 0 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 1 0 0 0 1 0 1 1 1 1|1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 0 0 1 1 1 1 1 1 1 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 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 0 1 1 0 0|1 0 0 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 1 1 1 0 1 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 1 0 0 0 0 0 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 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 0 1 1 0|1 1 0 0 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 1 1 1 0 1 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 1 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 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 0 1 1|1 1 1 0 0 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 1 1 1 0 1 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 1 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 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 0 1 0|0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 1 0 0 0 0 0 1 1 0 1 0 1 0 0 1 1 1 0 0 1 0 0 0 0 1 1 1 1 0 0 1 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 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 0 1|1 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 1 0 0 0 0 0 1 1 0 1 0 1 0 0 1 1 1 0 0 1 0 0 0 0 1 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 0 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1|1 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 1 1 1 0 0 0 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 1 0 0 0 1 0 1 0 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 1 0 0 0 0 1 1 0 0 1 1 1 0 0 1 1 0 0 0 0 0 1 1 1 1 0|1 0 1 1 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 1 0 1 1 1 1 1 0 1 0 1 0 0 0 1 1 0 1 0 0 0 1 0 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 1 0 0 0 0 1 1 0 0 1 1 1 0 0 1 1 0 0 0 0 0 1 1 1 1|1 1 0 1 1 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 1 0 1 1 1 1 1 0 1 0 1 0 0 0 1 1 0 1 0 0 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 1 0 0 1 0 1 0 0 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0 0|0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 0 0 0 1 0 0 1 1 0 0 0 1 1 1 1 0 1 0 0 1 0 1 0 0 0 0 0 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 1 0 1 0 0 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0|1 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 0 0 0 1 0 0 1 1 0 0 0 1 1 1 1 0 1 0 0 1 0 1 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 1 0 1 1 0 1 1 1|1 1 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 0 0 0 1 0 0 1 1 0 0 0 1 1 1 1 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0]

last modified: 2008-02-14

Notes


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