MinT - Best Known (237−117, 237, s)-Nets in Base 3

Best Known (237−117, 237, s)-Nets in Base 3

(237−117, 237, 77)-Net over F₃ — Constructive and digital

Digital (120, 237, 77)-net over F₃, using

net from sequence [i] based on digital (120, 76)-sequence over F₃, using
- base reduction for sequences [i] based on digital (22, 76)-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]

(237−117, 237, 120)-Net over F₃ — Digital

Digital (120, 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]

(237−117, 237, 924)-Net in Base 3 — Upper bound on s

There is no (120, 237, 925)-net in base 3, because

1 times m-reduction [i] would yield (120, 236, 925)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 40638 777510 422628 429003 360873 902227 715638 049178 505032 773772 481076 012370 895358 119017 048145 661888 177856 333111 287841 > 3²³⁶ [i]