Bounds on the minimum distance of additive quantum codes

Bounds on [[44,2]]2

lower bound:12
upper bound:15

Construction

Construction of a [[44,2,12]] quantum code:
[1]:  [[43, 1, 13]] Quantum code over GF(2^2)
     cyclic code of length 43 with generating polynomial w*x^41 + w^2*x^40 + x^39 + w^2*x^38 + w^2*x^37 + w*x^36 + w*x^34 + x^33 + x^32 + x^31 + x^30 + w*x^29 + w*x^27 + w^2*x^26 + w^2*x^25 + x^24 + w^2*x^23 + w*x^22 + x^20 + 1
[2]:  [[42, 2, 12]] Quantum code over GF(2^2)
     Shortening of the stabilizer code of [1] at 43
[3]:  [[44, 2, 12]] Quantum code over GF(2^2)
     ExtendCode [2] by 2

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

last modified: 2006-04-07

Notes


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