MinT - Best Known (109−87, 109, s)-Nets in Base 9

Best Known (109−87, 109, s)-Nets in Base 9

(109−87, 109, 78)-Net over F₉ — Constructive and digital

Digital (22, 109, 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]

(109−87, 109, 88)-Net over F₉ — Digital

Digital (22, 109, 88)-net over F₉, using

t-expansion [i] based on digital (21, 109, 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]

(109−87, 109, 500)-Net in Base 9 — Upper bound on s

There is no (22, 109, 501)-net in base 9, because

1 times m-reduction [i] would yield (22, 108, 501)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 12 286170 889438 105407 592915 736660 805599 988128 562635 935070 018952 576523 584987 179306 706956 207186 571770 548089 > 9¹⁰⁸ [i]