MinT - Best Known (51−23, 51, s)-Nets in Base 9

Best Known (51−23, 51, s)-Nets in Base 9

(51−23, 51, 232)-Net over F₉ — Constructive and digital

Digital (28, 51, 232)-net over F₉, using

1 times m-reduction [i] based on digital (28, 52, 232)-net over F₉, using
- trace code for nets [i] based on digital (2, 26, 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]

(51−23, 51, 236)-Net over F₉ — Digital

Digital (28, 51, 236)-net over F₉, using

1 times m-reduction [i] based on digital (28, 52, 236)-net over F₉, using
- trace code for nets [i] based on digital (2, 26, 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]

(51−23, 51, 13340)-Net in Base 9 — Upper bound on s

There is no (28, 51, 13341)-net in base 9, because

1 times m-reduction [i] would yield (28, 50, 13341)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 515452 064005 525936 416464 615713 660027 057704 738105 > 9⁵⁰ [i]