MinT - Best Known (156, 156+83, s)-Nets in Base 3

Best Known (156, 156+83, s)-Nets in Base 3

(156, 156+83, 162)-Net over F₃ — Constructive and digital

Digital (156, 239, 162)-net over F₃, using

9 times m-reduction [i] based on digital (156, 248, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 124, 81)-net over F₉, using
  - net from sequence [i] based on digital (32, 80)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
      - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(156, 156+83, 328)-Net over F₃ — Digital

Digital (156, 239, 328)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(156, 156+83, 4707)-Net in Base 3 — Upper bound on s

There is no (156, 239, 4708)-net in base 3, because

1 times m-reduction [i] would yield (156, 238, 4708)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 359812 883752 346568 570043 903125 005847 950516 195853 623878 774550 521432 700018 426855 444927 416844 564066 690509 062038 107593 > 3²³⁸ [i]