MinT - Best Known (166−99, 166, s)-Nets in Base 8

Best Known (166−99, 166, s)-Nets in Base 8

(166−99, 166, 98)-Net over F₈ — Constructive and digital

Digital (67, 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]

(166−99, 166, 144)-Net over F₈ — Digital

Digital (67, 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]

(166−99, 166, 2969)-Net in Base 8 — Upper bound on s

There is no (67, 166, 2970)-net in base 8, because

1 times m-reduction [i] would yield (67, 165, 2970)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 102576 028627 836647 837421 110198 455333 032816 466079 016493 732201 465159 663404 481779 911514 646554 458842 879900 143803 538301 601864 378696 295322 007079 559475 743692 > 8¹⁶⁵ [i]