MinT - Best Known (234−83, 234, s)-Nets in Base 3

Best Known (234−83, 234, s)-Nets in Base 3

(234−83, 234, 162)-Net over F₃ — Constructive and digital

Digital (151, 234, 162)-net over F₃, using

4 times m-reduction [i] based on digital (151, 238, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 119, 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]

(234−83, 234, 303)-Net over F₃ — Digital

Digital (151, 234, 303)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(234−83, 234, 4112)-Net in Base 3 — Upper bound on s

There is no (151, 234, 4113)-net in base 3, because

1 times m-reduction [i] would yield (151, 233, 4113)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1486 211374 009095 893570 111794 981829 630865 183890 896457 145731 711778 652435 557862 050232 902311 580717 547231 835763 548547 > 3²³³ [i]