MinT - Best Known (237−119, 237, s)-Nets in Base 3

Best Known (237−119, 237, s)-Nets in Base 3

(237−119, 237, 75)-Net over F₃ — Constructive and digital

Digital (118, 237, 75)-net over F₃, using

net from sequence [i] based on digital (118, 74)-sequence over F₃, using
- base reduction for sequences [i] based on digital (22, 74)-sequence over F₉, using
  - s-reduction 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]

(237−119, 237, 120)-Net over F₃ — Digital

Digital (118, 237, 120)-net over F₃, using

t-expansion [i] based on digital (113, 237, 120)-net over F₃, using
- net from sequence [i] based on digital (113, 119)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 113 and N(F) ≥ 120, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(237−119, 237, 867)-Net in Base 3 — Upper bound on s

There is no (118, 237, 868)-net in base 3, because

1 times m-reduction [i] would yield (118, 236, 868)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 41819 066275 433095 711816 022499 776732 563637 376472 629245 882342 780451 066003 334094 518053 402556 817459 401591 340897 665521 > 3²³⁶ [i]