MinT - Best Known (75−35, 75, s)-Nets in Base 8

Best Known (75−35, 75, s)-Nets in Base 8

(75−35, 75, 160)-Net over F₈ — Constructive and digital

Digital (40, 75, 160)-net over F₈, using

3 times m-reduction [i] based on digital (40, 78, 160)-net over F₈, using
- trace code for nets [i] based on digital (1, 39, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]

(75−35, 75, 194)-Net over F₈ — Digital

Digital (40, 75, 194)-net over F₈, using

1 times m-reduction [i] based on digital (40, 76, 194)-net over F₈, using
- trace code for nets [i] based on digital (2, 38, 97)-net over F₆₄, using
  - net from sequence [i] based on digital (2, 96)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 2 and N(F) ≥ 97, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

(75−35, 75, 8738)-Net in Base 8 — Upper bound on s

There is no (40, 75, 8739)-net in base 8, because

1 times m-reduction [i] would yield (40, 74, 8739)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6 749158 043173 365425 344411 853860 345933 693461 480394 128769 429744 882692 > 8⁷⁴ [i]