MinT - Best Known (20−9, 20, s)-Nets in Base 8

Best Known (20−9, 20, s)-Nets in Base 8

(20−9, 20, 160)-Net over F₈ — Constructive and digital

Digital (11, 20, 160)-net over F₈, using

trace code for nets [i] based on digital (1, 10, 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]

(20−9, 20, 162)-Net over F₈ — Digital

Digital (11, 20, 162)-net over F₈, using

trace code for nets [i] based on digital (1, 10, 81)-net over F₆₄, using
- net from sequence [i] based on digital (1, 80)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 81, using
    - sharp bound on the number of rational points on curves with genus 1 [i]

(20−9, 20, 6158)-Net in Base 8 — Upper bound on s

There is no (11, 20, 6159)-net in base 8, because

1 times m-reduction [i] would yield (11, 19, 6159)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 144147 070361 954633 > 8¹⁹ [i]