MinT - Best Known (75−35, 75, s)-Nets in Base 9

Best Known (75−35, 75, s)-Nets in Base 9

(75−35, 75, 232)-Net over F₉ — Constructive and digital

Digital (40, 75, 232)-net over F₉, using

1 times m-reduction [i] based on digital (40, 76, 232)-net over F₉, using
- trace code for nets [i] based on digital (2, 38, 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]

(75−35, 75, 236)-Net over F₉ — Digital

Digital (40, 75, 236)-net over F₉, using

1 times m-reduction [i] based on digital (40, 76, 236)-net over F₉, using
- trace code for nets [i] based on digital (2, 38, 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]

(75−35, 75, 12772)-Net in Base 9 — Upper bound on s

There is no (40, 75, 12773)-net in base 9, because

1 times m-reduction [i] would yield (40, 74, 12773)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 41142 385077 283939 582262 759124 282822 766014 706678 110553 719871 033541 037993 > 9⁷⁴ [i]