MinT - Best Known (140−87, 140, s)-Nets in Base 8

Best Known (140−87, 140, s)-Nets in Base 8

(140−87, 140, 98)-Net over F₈ — Constructive and digital

Digital (53, 140, 98)-net over F₈, using

t-expansion [i] based on digital (37, 140, 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]

(140−87, 140, 144)-Net over F₈ — Digital

Digital (53, 140, 144)-net over F₈, using

t-expansion [i] based on digital (45, 140, 144)-net over F₈, using
- net from sequence [i] based on digital (45, 143)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 45 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(140−87, 140, 1975)-Net in Base 8 — Upper bound on s

There is no (53, 140, 1976)-net in base 8, because

1 times m-reduction [i] would yield (53, 139, 1976)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 339346 279714 807632 399997 314144 999338 568437 869638 726009 439950 817591 284370 248272 621804 815274 884385 784055 868834 948818 626464 744094 > 8¹³⁹ [i]