MinT - Best Known (72−45, 72, s)-Nets in Base 9

Best Known (72−45, 72, s)-Nets in Base 9

(72−45, 72, 78)-Net over F₉ — Constructive and digital

Digital (27, 72, 78)-net over F₉, using

t-expansion [i] based on digital (22, 72, 78)-net over F₉, using
- net from sequence [i] based on digital (22, 77)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
    - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(72−45, 72, 110)-Net over F₉ — Digital

Digital (27, 72, 110)-net over F₉, using

t-expansion [i] based on digital (26, 72, 110)-net over F₉, using
- net from sequence [i] based on digital (26, 109)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 26 and N(F) ≥ 110, using
    - a curve from the manYPoints database [i]

(72−45, 72, 1346)-Net in Base 9 — Upper bound on s

There is no (27, 72, 1347)-net in base 9, because

1 times m-reduction [i] would yield (27, 71, 1347)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 57 089287 833255 297446 838954 053969 038331 523164 464522 215903 103692 390545 > 9⁷¹ [i]