MinT - Best Known (99, 99+109, s)-Nets in Base 3

Best Known (99, 99+109, s)-Nets in Base 3

(99, 99+109, 66)-Net over F₃ — Constructive and digital

Digital (99, 208, 66)-net over F₃, using

net from sequence [i] based on digital (99, 65)-sequence over F₃, using
- base reduction for sequences [i] based on digital (17, 65)-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]

(99, 99+109, 96)-Net over F₃ — Digital

Digital (99, 208, 96)-net over F₃, using

t-expansion [i] based on digital (89, 208, 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]

(99, 99+109, 655)-Net in Base 3 — Upper bound on s

There is no (99, 208, 656)-net in base 3, because

1 times m-reduction [i] would yield (99, 207, 656)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 616 834345 539460 033610 360654 841338 949924 391381 523451 171544 185256 318525 708866 269542 472648 717796 218721 > 3²⁰⁷ [i]