MinT - Best Known (72−31, 72, s)-Nets in Base 9

Best Known (72−31, 72, s)-Nets in Base 9

(72−31, 72, 320)-Net over F₉ — Constructive and digital

Digital (41, 72, 320)-net over F₉, using

trace code for nets [i] based on digital (5, 36, 160)-net over F₈₁, using
- net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
    - a function field by SÃ©mirat [i]

(72−31, 72, 334)-Net over F₉ — Digital

Digital (41, 72, 334)-net over F₉, using

trace code for nets [i] based on digital (5, 36, 167)-net over F₈₁, using
- net from sequence [i] based on digital (5, 166)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 167, using
    - a curve from the manYPoints database [i]

(72−31, 72, 26380)-Net in Base 9 — Upper bound on s

There is no (41, 72, 26381)-net in base 9, because

1 times m-reduction [i] would yield (41, 71, 26381)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 56 416996 946939 650521 525807 884863 123004 997307 370814 323670 942459 585753 > 9⁷¹ [i]