MinT - Best Known (165−107, 165, s)-Nets in Base 8

Best Known (165−107, 165, s)-Nets in Base 8

(165−107, 165, 98)-Net over F₈ — Constructive and digital

Digital (58, 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]

(165−107, 165, 144)-Net over F₈ — Digital

Digital (58, 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]

(165−107, 165, 1799)-Net in Base 8 — Upper bound on s

There is no (58, 165, 1800)-net in base 8, because

1 times m-reduction [i] would yield (58, 164, 1800)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 12934 015613 602431 627677 297933 477220 471444 186039 000861 712381 112043 192797 382427 851821 408299 026908 083245 327271 631898 852407 558472 266763 971391 843027 637456 > 8¹⁶⁴ [i]