MinT - Best Known (234−79, 234, s)-Nets in Base 3

Best Known (234−79, 234, s)-Nets in Base 3

(234−79, 234, 162)-Net over F₃ — Constructive and digital

Digital (155, 234, 162)-net over F₃, using

12 times m-reduction [i] based on digital (155, 246, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 123, 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]

(234−79, 234, 351)-Net over F₃ — Digital

Digital (155, 234, 351)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(234−79, 234, 5417)-Net in Base 3 — Upper bound on s

There is no (155, 234, 5418)-net in base 3, because

1 times m-reduction [i] would yield (155, 233, 5418)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1478 720863 719785 122029 677416 580570 324397 601119 424457 626784 155963 857199 921188 367538 014274 040363 983279 075661 067145 > 3²³³ [i]