MinT - Best Known (169−103, 169, s)-Nets in Base 8

Best Known (169−103, 169, s)-Nets in Base 8

(169−103, 169, 98)-Net over F₈ — Constructive and digital

Digital (66, 169, 98)-net over F₈, using

t-expansion [i] based on digital (37, 169, 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]

(169−103, 169, 144)-Net over F₈ — Digital

Digital (66, 169, 144)-net over F₈, using

t-expansion [i] based on digital (45, 169, 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]

(169−103, 169, 2644)-Net in Base 8 — Upper bound on s

There is no (66, 169, 2645)-net in base 8, because

1 times m-reduction [i] would yield (66, 168, 2645)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 52 579043 776583 054704 461867 592482 220690 730687 417600 556843 303016 829717 738329 224365 767103 775260 919830 945702 724478 558402 656966 101689 924304 629868 872480 528824 > 8¹⁶⁸ [i]