MinT - Best Known (164−57, 164, s)-Nets in Base 2

Best Known (164−57, 164, s)-Nets in Base 2

Digital (107, 164, 68)-net over F₂, using

8 times m-reduction [i] based on digital (107, 172, 68)-net over F₂, using
- trace code for nets [i] based on digital (21, 86, 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]

(107, 164, 70)-net in base 2, using

2 times m-reduction [i] based on (107, 166, 70)-net in base 2, using
- trace code for nets [i] based on (24, 83, 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 (107, 164, 108)-net over F₂, using

There is no (107, 164, 599)-net in base 2, because

1 times m-reduction [i] would yield (107, 163, 599)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 11 907268 710305 773636 730983 135711 136592 532267 945800 > 2¹⁶³ [i]