MinT - Best Known (45, 45+105, s)-Nets in Base 8

Best Known (45, 45+105, s)-Nets in Base 8

(45, 45+105, 98)-Net over F₈ — Constructive and digital

Digital (45, 150, 98)-net over F₈, using

t-expansion [i] based on digital (37, 150, 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]

(45, 45+105, 144)-Net over F₈ — Digital

Digital (45, 150, 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]

(45, 45+105, 1085)-Net in Base 8 — Upper bound on s

There is no (45, 150, 1086)-net in base 8, because

1 times m-reduction [i] would yield (45, 149, 1086)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 371 797693 358733 849902 045044 009741 395639 957991 957551 079172 467196 675269 443069 565841 250795 476009 187601 579136 330960 396059 874797 069165 970673 > 8¹⁴⁹ [i]