MinT - Best Known (166−105, 166, s)-Nets in Base 8

Best Known (166−105, 166, s)-Nets in Base 8

(166−105, 166, 98)-Net over F₈ — Constructive and digital

Digital (61, 166, 98)-net over F₈, using

t-expansion [i] based on digital (37, 166, 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]

(166−105, 166, 144)-Net over F₈ — Digital

Digital (61, 166, 144)-net over F₈, using

t-expansion [i] based on digital (45, 166, 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]

(166−105, 166, 2087)-Net in Base 8 — Upper bound on s

There is no (61, 166, 2088)-net in base 8, because

1 times m-reduction [i] would yield (61, 165, 2088)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 103791 617115 141498 894906 943235 272352 499059 185732 577345 588082 085267 286716 556911 115471 106864 322998 992880 745623 304562 418492 067241 698615 891013 283339 666304 > 8¹⁶⁵ [i]