MinT - Best Known (166−107, 166, s)-Nets in Base 8

Best Known (166−107, 166, s)-Nets in Base 8

(166−107, 166, 98)-Net over F₈ — Constructive and digital

Digital (59, 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]

(166−107, 166, 144)-Net over F₈ — Digital

Digital (59, 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]

(166−107, 166, 1872)-Net in Base 8 — Upper bound on s

There is no (59, 166, 1873)-net in base 8, because

1 times m-reduction [i] would yield (59, 165, 1873)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 102455 534549 939272 942876 004286 157311 054625 293045 778183 995582 175713 058604 462912 952496 317728 530496 383134 472412 247705 357157 920398 841448 299539 830865 697856 > 8¹⁶⁵ [i]