MinT - Best Known (109−67, 109, s)-Nets in Base 8

Best Known (109−67, 109, s)-Nets in Base 8

(109−67, 109, 98)-Net over F₈ — Constructive and digital

Digital (42, 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]

(109−67, 109, 129)-Net over F₈ — Digital

Digital (42, 109, 129)-net over F₈, using

t-expansion [i] based on digital (38, 109, 129)-net over F₈, using
- net from sequence [i] based on digital (38, 128)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 38 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(109−67, 109, 1676)-Net in Base 8 — Upper bound on s

There is no (42, 109, 1677)-net in base 8, because

1 times m-reduction [i] would yield (42, 108, 1677)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 34 231879 893453 905996 757929 554415 338401 480278 591173 042104 111223 186474 196842 050757 490798 956860 689296 > 8¹⁰⁸ [i]