Bounds on the minimum distance of additive quantum codes

Bounds on [[51,25]]2

lower bound:6
upper bound:9

Construction

Construction of a [[51,25,6]] quantum code:
[1]:  [[51, 27, 6]] Quantum code over GF(2^2)
     cyclic code of length 51 with generating polynomial w*x^50 + w^2*x^48 + x^47 + w*x^46 + x^45 + x^43 + x^41 + w*x^40 + x^38 + w*x^36 + x^35 + w*x^34 + w^2*x^32 + w*x^31 + w^2*x^29 + w^2*x^28 + w*x^27 + w^2*x^25 + w^2*x^23 + w^2*x^22 + w*x^20 + x^18 + w^2*x^17 + w^2*x^16 + w^2*x^15 + x^13 + x^12 + 1
[2]:  [[51, 25, 6]] Quantum code over GF(2^2)
     Subcode of [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 1 1 1 0 0 0 1 1 1 1 0 1 0 1 0 1 1 1 1 1 1 0 0 0|0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 0 1 1 0 1 0 1 1 0 0 1 0 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 1 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 1 1|0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 0 1 0 0 0 1 1 0 1 1 0 1 1 1 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 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0|1 1 1 0 0 1 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 0 1 1 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 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 1 1 1 1 1 0 0 1 0|0 1 1 1 0 0 1 0 0 1 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 0 0 1 0 0 1 1 0 1 0 0 0 0 0 1 0 1 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 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 1 1 0 1 0 1 1 1|0 0 1 1 1 0 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 0 0 1 0 0 0 1 0 1 0 1 1 1 0 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 0 1 0 0 1 0 1 1 1 1 1 0 0 1 0 0 1 0 1 0 0 1 1 1 1|1 0 0 1 0 0 1 1 1 0 0 0 1 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1 1 0 0 1 0 1 1 1 1 0 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 1 0 1 1 1 1 1 0 1 0 1 0 1 0 1 0 0 0 1 0 1 1 1 1 1 0|1 0 0 0 1 0 0 1 0 0 0 1 0 1 0 1 1 0 1 1 1 0 0 1 1 0 0 1 0 0 1 0 1 1 0 1 1 0 0 1 1 1 0 0 1 0 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 1 0 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 0 0 1|1 1 0 0 0 1 0 0 1 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 1 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 0 0 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 1 0 1 1 1 0 1 0 1 1 1 1 1 0 1 0 0 0 1 1 1 0 0|0 1 1 0 1 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 1 0 1 1 1 1 1 1 1 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 1 1 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 1 0 0 1 1 1 1 1 0 0 0 0 1 0 0 1 0 1 0 0 1 1 0 0 1 1|1 1 1 1 1 0 0 1 0 1 1 0 1 0 1 1 0 0 0 0 1 0 1 1 1 1 1 0 0 0 1 0 1 1 1 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 1 0 1 0 1 1 1 0 1 1 1 1 1 0 0 1 0|0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 0 1 0 1 0 1 0 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 1 1 1 0 0 0 1 1 1 1 0 1 0 1 1 0 0 1 0 1 0 0 1 0 1|0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 1 0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 1 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 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 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 1 0 0 1 1 0 1 1 0 1 0 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 1 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1|1 1 0 0 1 1 1 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 0 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 1 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1|1 0 1 0 0 1 1 1 0 0 1 1 0 1 0 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 1 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 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 1 0 0 1|1 1 0 1 1 1 0 0 1 0 0 0 0 1 0 0 0 1 1 0 0 1 0 1 1 1 0 1 1 0 1 0 0 1 1 0 0 0 1 1 0 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 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 1 0 1 0 0 1 0 1 0 0 1 1 0 1 1 1|0 0 1 0 1 1 1 0 1 0 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 1 0 1 1 1 1 0 0 1 0 0 1 0 1 0 1 0 1 1 0 0 1 0 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 1 1 1 1 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 1 1 1 1 1 1|1 0 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 0 0 0 1 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 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 1 0 0 1 1 0 1 1 0 1 0|0 0 0 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 0 0 0 0 1 1 1 1 0 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 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1|1 0 0 0 0 1 1 0 0 1 1 1 0 0 1 0 1 0 1 0 1 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 0 0 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 1 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0|0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 0 0 1 1 0 0 1 0 1 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 1 0 0 0 1 0 0 1 1 0 1 0 1 0 0 1 0 1 0 0 0 1 1 1 1 1 0 1 0 0|1 1 1 0 1 0 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 1 0 1 1 1 1 0 0 1 0 0 1 0 1 0 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 1 0 0 1 1 0 1 1 0 0 1 1 1 1 0 1 1 0 0 0 0 0 0 1 1 0 1 0 1|1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 0 1 0 0 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 1 1 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1|0 0 0 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 0 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 0 0 0 0 0 1 0 1 0 1 1 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 0 0 1 0 1 1|1 0 0 0 0 0 0 1 1 0 1 1 0 1 1 0 1 0 1 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 1 0 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 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 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 1 0 0 0 0 1 1 1 0 0 0 0 0 1]

last modified: 2006-04-03

Notes


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