Bounds on the minimum distance of additive quantum codes

Bounds on [[37,5]]2

lower bound:8
upper bound:12

Construction

Construction of a [[37,5,8]] quantum code:
[1]:  [[31, 5, 8]] Quantum code over GF(2^2)
     cyclic code of length 31 with generating polynomial w^2*x^29 + x^28 + w^2*x^25 + w*x^24 + w*x^23 + w^2*x^20 + w*x^19 + w^2*x^17 + w^2*x^16 + w^2*x^14 + w*x^13 + w*x^12 + 1
[2]:  [[32, 5, 8]] Quantum code over GF(2^2)
     ExtendCode [1] by 1
[3]:  [[35, 5, 8]] Quantum code over GF(2^2)
     ExtendCode [2] by 3
[4]:  [[37, 5, 8]] Quantum code over GF(2^2)
     ExtendCode [3] by 2

    stabilizer matrix:

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

last modified: 2006-04-06

Notes


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