MinT - Best Known (166−103, 166, s)-Nets in Base 8

Best Known (166−103, 166, s)-Nets in Base 8

(166−103, 166, 98)-Net over F₈ — Constructive and digital

Digital (63, 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]

(166−103, 166, 144)-Net over F₈ — Digital

Digital (63, 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]

(166−103, 166, 2336)-Net in Base 8 — Upper bound on s

There is no (63, 166, 2337)-net in base 8, because

1 times m-reduction [i] would yield (63, 165, 2337)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 103181 662289 564935 439436 526330 391950 779464 213597 366680 287837 556398 878496 068930 196855 105699 946416 269351 230827 965172 337722 149102 718430 102826 678868 709120 > 8¹⁶⁵ [i]