MinT - Best Known (57−35, 57, s)-Nets in Base 9

Best Known (57−35, 57, s)-Nets in Base 9

(57−35, 57, 78)-Net over F₉ — Constructive and digital

Digital (22, 57, 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]

(57−35, 57, 88)-Net over F₉ — Digital

Digital (22, 57, 88)-net over F₉, using

t-expansion [i] based on digital (21, 57, 88)-net over F₉, using
- net from sequence [i] based on digital (21, 87)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 21 and N(F) ≥ 88, using
    - a curve from the manYPoints database [i]

(57−35, 57, 1237)-Net in Base 9 — Upper bound on s

There is no (22, 57, 1238)-net in base 9, because

1 times m-reduction [i] would yield (22, 56, 1238)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 273969 108219 842115 677079 404538 949491 155314 579376 303025 > 9⁵⁶ [i]