MinT - Best Known (42, 42+37, s)-Nets in Base 9

Best Known (42, 42+37, s)-Nets in Base 9

(42, 42+37, 232)-Net over F₉ — Constructive and digital

Digital (42, 79, 232)-net over F₉, using

1 times m-reduction [i] based on digital (42, 80, 232)-net over F₉, using
- trace code for nets [i] based on digital (2, 40, 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]

(42, 42+37, 236)-Net over F₉ — Digital

Digital (42, 79, 236)-net over F₉, using

1 times m-reduction [i] based on digital (42, 80, 236)-net over F₉, using
- trace code for nets [i] based on digital (2, 40, 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]

(42, 42+37, 12874)-Net in Base 9 — Upper bound on s

There is no (42, 79, 12875)-net in base 9, because

1 times m-reduction [i] would yield (42, 78, 12875)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 269 930352 342576 966392 330079 905479 287828 231073 604359 555722 507316 941780 442801 > 9⁷⁸ [i]