MinT - Best Known (45, 45+69, s)-Nets in Base 8

Best Known (45, 45+69, s)-Nets in Base 8

(45, 45+69, 98)-Net over F₈ — Constructive and digital

Digital (45, 114, 98)-net over F₈, using

t-expansion [i] based on digital (37, 114, 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+69, 144)-Net over F₈ — Digital

Digital (45, 114, 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+69, 1918)-Net in Base 8 — Upper bound on s

There is no (45, 114, 1919)-net in base 8, because

1 times m-reduction [i] would yield (45, 113, 1919)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 123570 134070 289787 287169 843662 696938 837584 179342 991082 093706 238330 101540 323466 492906 399442 119653 221015 > 8¹¹³ [i]