MinT - Best Known (237−81, 237, s)-Nets in Base 3

Best Known (237−81, 237, s)-Nets in Base 3

(237−81, 237, 162)-Net over F₃ — Constructive and digital

Digital (156, 237, 162)-net over F₃, using

11 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]

(237−81, 237, 342)-Net over F₃ — Digital

Digital (156, 237, 342)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(237−81, 237, 5110)-Net in Base 3 — Upper bound on s

There is no (156, 237, 5111)-net in base 3, because

1 times m-reduction [i] would yield (156, 236, 5111)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 40096 415806 892959 674903 569409 014446 639466 906204 916255 482345 732623 838339 447423 436564 627006 416998 850825 785983 429681 > 3²³⁶ [i]