MinT - Best Known (140−55, 140, s)-Nets in Base 3

Best Known (140−55, 140, s)-Nets in Base 3

(140−55, 140, 128)-Net over F₃ — Constructive and digital

Digital (85, 140, 128)-net over F₃, using

4 times m-reduction [i] based on digital (85, 144, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 72, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(140−55, 140, 145)-Net over F₃ — Digital

Digital (85, 140, 145)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(140−55, 140, 1535)-Net in Base 3 — Upper bound on s

There is no (85, 140, 1536)-net in base 3, because

1 times m-reduction [i] would yield (85, 139, 1536)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 101700 129413 775370 986383 734201 601521 709141 917447 441963 071872 702465 > 3¹³⁹ [i]