MinT - Best Known (157−71, 157, s)-Nets in Base 3

Best Known (157−71, 157, s)-Nets in Base 3

(157−71, 157, 72)-Net over F₃ — Constructive and digital

Digital (86, 157, 72)-net over F₃, using

1 times m-reduction [i] based on digital (86, 158, 72)-net over F₃, using
- trace code for nets [i] based on digital (7, 79, 36)-net over F₉, using
  - net from sequence [i] based on digital (7, 35)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 7 and N(F) ≥ 36, using
      - a function field by SÃ©mirat [i]

(157−71, 157, 108)-Net over F₃ — Digital

Digital (86, 157, 108)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(157−71, 157, 896)-Net in Base 3 — Upper bound on s

There is no (86, 157, 897)-net in base 3, because

1 times m-reduction [i] would yield (86, 156, 897)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 272 068036 788350 726835 509080 649266 428982 490180 792145 302133 016859 339390 011803 > 3¹⁵⁶ [i]