MinT - Best Known (50, 50+107, s)-Nets in Base 8

Best Known (50, 50+107, s)-Nets in Base 8

(50, 50+107, 98)-Net over F₈ — Constructive and digital

Digital (50, 157, 98)-net over F₈, using

t-expansion [i] based on digital (37, 157, 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]

(50, 50+107, 144)-Net over F₈ — Digital

Digital (50, 157, 144)-net over F₈, using

t-expansion [i] based on digital (45, 157, 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]

(50, 50+107, 1305)-Net in Base 8 — Upper bound on s

There is no (50, 157, 1306)-net in base 8, because

1 times m-reduction [i] would yield (50, 156, 1306)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 765 810845 753173 552688 478716 222166 795122 185524 002168 259451 602974 546415 600751 761563 396828 417146 822490 250208 626809 030878 400091 465593 724137 855344 > 8¹⁵⁶ [i]