Bounds on the minimum distance of additive quantum codes

Bounds on [[41,2]]2

lower bound:11
upper bound:14

Construction

Construction of a [[41,2,11]] quantum code:
[1]:  [[42, 1, 12]] Quantum code over GF(2^2)
     cyclic code of length 42 with generating polynomials [
w^2*x^41 + w*x^39 + w*x^38 + x^33 + x^31 + x^30 + w^2*x^29 + x^28 + w*x^27 + w*x^26 + x^18 + x^17 + x^15 + 1,
w^2*x^41 + w^2*x^40 + x^38 + w*x^37 + x^36 + w*x^35 + w^2*x^34 + w^2*x^33 + x^30 + w*x^29 + w*x^28 + x^27 + w*x^26 + w
]
[2]:  [[41, 2, 11]] Quantum code over GF(2^2)
     Shortening of the stabilizer code of [1] at 42

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

last modified: 2006-04-03

Notes


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