MinT - Best Known (168−107, 168, s)-Nets in Base 8

Best Known (168−107, 168, s)-Nets in Base 8

(168−107, 168, 98)-Net over F₈ — Constructive and digital

Digital (61, 168, 98)-net over F₈, using

t-expansion [i] based on digital (37, 168, 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]

(168−107, 168, 144)-Net over F₈ — Digital

Digital (61, 168, 144)-net over F₈, using

t-expansion [i] based on digital (45, 168, 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]

(168−107, 168, 2028)-Net in Base 8 — Upper bound on s

There is no (61, 168, 2029)-net in base 8, because

1 times m-reduction [i] would yield (61, 167, 2029)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6 623856 543499 557074 170457 125592 649703 908649 125774 142329 194119 148039 183496 795483 415291 258660 759609 443624 071937 231726 932229 115606 326763 565647 366758 475840 > 8¹⁶⁷ [i]