MinT - Best Known (112−81, 112, s)-Nets in Base 9

Best Known (112−81, 112, s)-Nets in Base 9

(112−81, 112, 78)-Net over F₉ — Constructive and digital

Digital (31, 112, 78)-net over F₉, using

t-expansion [i] based on digital (22, 112, 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]

(112−81, 112, 120)-Net over F₉ — Digital

Digital (31, 112, 120)-net over F₉, using

net from sequence [i] based on digital (31, 119)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 31 and N(F) ≥ 120, using
  - a curve from the manYPoints database [i]

(112−81, 112, 852)-Net in Base 9 — Upper bound on s

There is no (31, 112, 853)-net in base 9, because

1 times m-reduction [i] would yield (31, 111, 853)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 8667 701673 518354 271527 826254 767127 750864 862190 319970 318749 557793 843507 149844 104485 132603 454982 607434 977089 > 9¹¹¹ [i]