MinT - Best Known (90−43, 90, s)-Nets in Base 9

Best Known (90−43, 90, s)-Nets in Base 9

(90−43, 90, 232)-Net over F₉ — Constructive and digital

Digital (47, 90, 232)-net over F₉, using

trace code for nets [i] based on digital (2, 45, 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]

(90−43, 90, 236)-Net over F₉ — Digital

Digital (47, 90, 236)-net over F₉, using

trace code for nets [i] based on digital (2, 45, 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]

(90−43, 90, 11998)-Net in Base 9 — Upper bound on s

There is no (47, 90, 11999)-net in base 9, because

1 times m-reduction [i] would yield (47, 89, 11999)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 8 475783 589309 420118 550702 364580 855114 388238 062852 854194 163060 518539 981575 687889 324633 > 9⁸⁹ [i]