MinT - Best Known (165−105, 165, s)-Nets in Base 8

Best Known (165−105, 165, s)-Nets in Base 8

(165−105, 165, 98)-Net over F₈ — Constructive and digital

Digital (60, 165, 98)-net over F₈, using

t-expansion [i] based on digital (37, 165, 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]

(165−105, 165, 144)-Net over F₈ — Digital

Digital (60, 165, 144)-net over F₈, using

t-expansion [i] based on digital (45, 165, 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]

(165−105, 165, 2004)-Net in Base 8 — Upper bound on s

There is no (60, 165, 2005)-net in base 8, because

1 times m-reduction [i] would yield (60, 164, 2005)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 13016 610115 977830 362411 610834 510464 900425 723775 255710 889191 397230 603244 467535 604360 781392 612806 302196 184526 822543 730251 396683 682627 905085 732515 804894 > 8¹⁶⁴ [i]