MinT - Best Known (34−15, 34, s)-Nets in Base 8

Best Known (34−15, 34, s)-Nets in Base 8

(34−15, 34, 160)-Net over F₈ — Constructive and digital

Digital (19, 34, 160)-net over F₈, using

2 times m-reduction [i] based on digital (19, 36, 160)-net over F₈, using
- trace code for nets [i] based on digital (1, 18, 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]

(34−15, 34, 194)-Net over F₈ — Digital

Digital (19, 34, 194)-net over F₈, using

trace code for nets [i] based on digital (2, 17, 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]

(34−15, 34, 8730)-Net in Base 8 — Upper bound on s

There is no (19, 34, 8731)-net in base 8, because

1 times m-reduction [i] would yield (19, 33, 8731)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 634009 685734 397118 788339 642120 > 8³³ [i]