MinT - Best Known (86−41, 86, s)-Nets in Base 9

Best Known (86−41, 86, s)-Nets in Base 9

(86−41, 86, 232)-Net over F₉ — Constructive and digital

Digital (45, 86, 232)-net over F₉, using

trace code for nets [i] based on digital (2, 43, 116)-net over F₈₁, using
- net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
    - a function field by SÃ©mirat [i]

(86−41, 86, 236)-Net over F₉ — Digital

Digital (45, 86, 236)-net over F₉, using

trace code for nets [i] based on digital (2, 43, 118)-net over F₈₁, using
- net from sequence [i] based on digital (2, 117)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 118, using
    - sharp bound on the number of rational points on curves with genus 2 [i]

(86−41, 86, 11784)-Net in Base 9 — Upper bound on s

There is no (45, 86, 11785)-net in base 9, because

1 times m-reduction [i] would yield (45, 85, 11785)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1291 528538 741417 149719 136593 789605 822645 612582 748151 999745 768973 397845 210536 630049 > 9⁸⁵ [i]