MinT - Best Known (109−65, 109, s)-Nets in Base 8

Best Known (109−65, 109, s)-Nets in Base 8

(109−65, 109, 98)-Net over F₈ — Constructive and digital

Digital (44, 109, 98)-net over F₈, using

t-expansion [i] based on digital (37, 109, 98)-net over F₈, using
- net from sequence [i] based on digital (37, 97)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 37 and N(F) ≥ 98, using
    - F₅ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]

(109−65, 109, 130)-Net over F₈ — Digital

Digital (44, 109, 130)-net over F₈, using

net from sequence [i] based on digital (44, 129)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 44 and N(F) ≥ 130, using
  - a curve from the manYPoints database [i]

(109−65, 109, 2020)-Net in Base 8 — Upper bound on s

There is no (44, 109, 2021)-net in base 8, because

1 times m-reduction [i] would yield (44, 108, 2021)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 34 515754 676617 347030 494420 003670 203400 165595 378247 736544 725638 939858 110735 993871 814950 233685 281707 > 8¹⁰⁸ [i]