Bounds on the minimum distance of additive quantum codes

Bounds on [[56,34]]2

lower bound:5
upper bound:8

Construction

Construction of a [[56,34,5]] quantum code:
[1]:  [[87, 59, 6]] Quantum code over GF(2^2)
     quasicyclic code of length 87 with 2 generating polynomials
[2]:  [[84, 62, 5]] Quantum code over GF(2^2)
     Shortening of the stabilizer code of [1] at { 3, 24, 30 }
[3]:  [[56, 34, 5]] Quantum code over GF(2^2)
     Shortening of [2] at { 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55 }

    stabilizer matrix:

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