MinT - Best Known (159−113, 159, s)-Nets in Base 8

Best Known (159−113, 159, s)-Nets in Base 8

(159−113, 159, 98)-Net over F₈ — Constructive and digital

Digital (46, 159, 98)-net over F₈, using

t-expansion [i] based on digital (37, 159, 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]

(159−113, 159, 144)-Net over F₈ — Digital

Digital (46, 159, 144)-net over F₈, using

t-expansion [i] based on digital (45, 159, 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]

(159−113, 159, 1060)-Net in Base 8 — Upper bound on s

There is no (46, 159, 1061)-net in base 8, because

1 times m-reduction [i] would yield (46, 158, 1061)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 50871 865033 527693 206017 964508 576024 850378 098441 198785 074109 107986 309463 204874 961699 952417 610718 457443 551600 941807 637840 230433 620954 561536 528512 > 8¹⁵⁸ [i]