MinT - Best Known (67−41, 67, s)-Nets in Base 9

Best Known (67−41, 67, s)-Nets in Base 9

(67−41, 67, 78)-Net over F₉ — Constructive and digital

Digital (26, 67, 78)-net over F₉, using

t-expansion [i] based on digital (22, 67, 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]

(67−41, 67, 110)-Net over F₉ — Digital

Digital (26, 67, 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]

(67−41, 67, 1450)-Net in Base 9 — Upper bound on s

There is no (26, 67, 1451)-net in base 9, because

1 times m-reduction [i] would yield (26, 66, 1451)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 955 842413 907198 066555 871334 576164 289339 533734 137605 068168 950241 > 9⁶⁶ [i]