MinT - Best Known (55−25, 55, s)-Nets in Base 9

Best Known (55−25, 55, s)-Nets in Base 9

(55−25, 55, 232)-Net over F₉ — Constructive and digital

Digital (30, 55, 232)-net over F₉, using

1 times m-reduction [i] based on digital (30, 56, 232)-net over F₉, using
- trace code for nets [i] based on digital (2, 28, 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]

(55−25, 55, 236)-Net over F₉ — Digital

Digital (30, 55, 236)-net over F₉, using

1 times m-reduction [i] based on digital (30, 56, 236)-net over F₉, using
- trace code for nets [i] based on digital (2, 28, 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]

(55−25, 55, 13005)-Net in Base 9 — Upper bound on s

There is no (30, 55, 13006)-net in base 9, because

1 times m-reduction [i] would yield (30, 54, 13006)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 3382 762973 649194 013491 979879 281203 074946 700978 323905 > 9⁵⁴ [i]