MinT - Best Known (227−129, 227, s)-Nets in Base 3

Best Known (227−129, 227, s)-Nets in Base 3

(227−129, 227, 65)-Net over F₃ — Constructive and digital

Digital (98, 227, 65)-net over F₃, using

net from sequence [i] based on digital (98, 64)-sequence over F₃, using
- base reduction for sequences [i] based on digital (17, 64)-sequence over F₉, using
  - s-reduction 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]

(227−129, 227, 96)-Net over F₃ — Digital

Digital (98, 227, 96)-net over F₃, using

t-expansion [i] based on digital (89, 227, 96)-net over F₃, using
- net from sequence [i] based on digital (89, 95)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 89 and N(F) ≥ 96, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(227−129, 227, 536)-Net in Base 3 — Upper bound on s

There is no (98, 227, 537)-net in base 3, because

1 times m-reduction [i] would yield (98, 226, 537)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 731341 166123 481163 899349 540438 427934 326409 058251 784392 422365 963557 672588 855182 170733 035761 697014 603731 049729 > 3²²⁶ [i]