MinT - Best Known (168−59, 168, s)-Nets in Base 2

Best Known (168−59, 168, s)-Nets in Base 2

Digital (109, 168, 68)-net over F₂, using

8 times m-reduction [i] based on digital (109, 176, 68)-net over F₂, using
- trace code for nets [i] based on digital (21, 88, 34)-net over F₄, using
  - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
      - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(109, 168, 70)-net in base 2, using

2 times m-reduction [i] based on (109, 170, 70)-net in base 2, using
- trace code for nets [i] based on (24, 85, 35)-net in base 4, using
  - net from sequence [i] based on (24, 34)-sequence in base 4, using
    - base expansion [i] based on digital (48, 34)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 2 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

Digital (109, 168, 107)-net over F₂, using

There is no (109, 168, 591)-net in base 2, because

1 times m-reduction [i] would yield (109, 167, 591)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 194 289645 764398 566438 290430 506899 261440 288604 179440 > 2¹⁶⁷ [i]