MinT - Best Known (237−93, 237, s)-Nets in Base 3

Best Known (237−93, 237, s)-Nets in Base 3

(237−93, 237, 156)-Net over F₃ — Constructive and digital

Digital (144, 237, 156)-net over F₃, using

7 times m-reduction [i] based on digital (144, 244, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 122, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(237−93, 237, 229)-Net over F₃ — Digital

Digital (144, 237, 229)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(237−93, 237, 2478)-Net in Base 3 — Upper bound on s

There is no (144, 237, 2479)-net in base 3, because

1 times m-reduction [i] would yield (144, 236, 2479)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 40197 132502 576368 798436 887508 264073 441314 422327 672824 889957 010490 209749 977652 066187 570063 309515 999232 580661 299589 > 3²³⁶ [i]