MinT - Best Known (237−128, 237, s)-Nets in Base 3

Best Known (237−128, 237, s)-Nets in Base 3

(237−128, 237, 74)-Net over F₃ — Constructive and digital

Digital (109, 237, 74)-net over F₃, using

t-expansion [i] based on digital (107, 237, 74)-net over F₃, using
- net from sequence [i] based on digital (107, 73)-sequence over F₃, using
  - base reduction for sequences [i] based on digital (17, 73)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
      - a function field by GarcÃa and Quoos [i]

(237−128, 237, 104)-Net over F₃ — Digital

Digital (109, 237, 104)-net over F₃, using

t-expansion [i] based on digital (102, 237, 104)-net over F₃, using
- net from sequence [i] based on digital (102, 103)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 102 and N(F) ≥ 104, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(237−128, 237, 659)-Net in Base 3 — Upper bound on s

There is no (109, 237, 660)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 120521 127355 162321 076762 613251 644982 460212 983156 719302 978633 628697 269924 292900 618530 892670 168837 804345 414663 129601 > 3²³⁷ [i]